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A stochastic analysis of steady-state groundwater flow in a bounded domain Smith, James Leslie 1978

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A STOCHASTIC ANALYSIS OF STEADY-STATE GROUNTMATER FLOW IN A BOUNDED DOMAIN by JAMES LESLIE SMITH B.Sc., University of Alberta, 1974 A THESIS SUBMITTED IN PARTIAL IUIPILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES tepartment of Geological Sciences University of British Columbia Vfe accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1978 © James Leslie Smith In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my wri tten pe rm i ss ion. Department of G e o l o g i c a l Sciences The University of Bri t ish Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date September 22. 1978 i i ABSTRACT A stochastic analysis of groundwater flow leads to probability distributions on the predicted hydraulic head values. This variability reflects our uncertainty in the system being modeled due to the spatial heterogeneity of hydraulic conductivity. Monte Carlo techniques can be used to estimate the head distributions. This approach relies on the repetitive generation of discrete-block con-ductivity realizations. In this study, steady state flow through one and twcr^limensional flow domains is investigated. A space law based on a f i r s t order, nearest neighbour stochastic process model is used to generate the multilateral spatial dependence in the hydraulic conductivity values within the block structure. This allows consideration of both statistically isotropic and anisotropic autocorrelation functions. It is shown that the probability distribution of hydraulic head and the head gradient or the flux across the boundaries of the flow domain, must be interpreted in terms of: 1) The spatial variation of expected head gradients. 2) The standard deviation in the conductivity distribution. 3) The ratio of the integral scale of the autocorrelation function for conductivity to the distance between boundaries on the flow domain. 4) The arrangement of stationary units within the flow i i i domain. The standard deviations in hydraulic head increase with an increase in either the conductivity standard deviation or the strength of the correlation between neighbouring conductivity values. Provided the integral scales of the medium are preserved, the standard deviations in head show only a minor dependence on the discretization interval. The head standard deviations are approximately halved in a two-dimensional model from those in a one-dimensional model with an equivalent space law. Spatial trends in the mean conductivity can considerably alter the magnitude and spatial variation in the hydraulic head standard deviations. The geometric mean has been suggested by others as a suitable effective conductivity in a heterogeneous twcr-dimensional flow domain. This study shows that only in the case of uniform flow through a single stationary unit is this concept valid. If the mean gradient field is nonuniform, or i f the mean conductivity has a spatial trend, predictions based on the geometric mean do not satisfy the necessary equivalence criteria. Direct comparisons cannot be made, but the Monte Carlo and spectral approaches to the solution of the stochastic flow equations predict a similar behavior. A f i r s t order, nearest neighbour model i s matched to a data i set collected from a relatively uniform but stratified, unconsolidated sand deposit. The data show statistically anisotropic autocorrelation functions, both in the integral scale and in the functional form of the correlation. A broader class of spatial models may need to be considered to describe the cyclic behavior of sedimentary sequences. V CHAPTER ONE: CHAPTER TWO: CONTENTS INTRODUCTION DEFINITIONS, REVIEW OF PREVIOUS WORK 1) Hydraulic Conductivity as a Random Variable 2) Previous Work on Stochastic Flow Jfodels 3) Analyses on the Sp a t i a l V ariation i n Hydraulic Conductivity CHAPTER THREE: GENERATION OF COSIDUCTrvTTY REALIZATIONS Page 1 5 18 24 1) Introduction 27 2) Generation Technique 27 3) Nearest Neighbour Stochastic Process Model 42 4) Analysis of Conductivity Realizations 50 5) Examples of Model Realizations 65 6) Conclusions 69 CHAPTER FOUR: C^-DIMENSIONAL SIMULATION RESULTS 1) introduction 2) Steady State Flow Problem and Solution Technique 71 71 3) Influence of Conductivity Space Law Parameters 76 4) Variation i n Integrated Measures of Flow 90 5) Ergodicity i n Bounded Systems 96 6) Degenerate Autoregressive Models 7) Convergence of the Monte Carlo Technique 8) Conclusions 99 101 102 vi CHAPTER FIVE: CHAPTER SIX: TWO-DIMENSIONAL SIMULATION RESULTS 1) Introduction j 2) Steady State Flow Problem 3) Unidirectional Flow, Uniform Gradient Field 4) Nonuniform Gradient Field 5) Nonstationary Conductivity Field, Nonuniform Gradient Field 6) Additional Examples 7) Conclusions THE QUESTION: OF SCALE: A DISCUSSION 1) Introduction 2) Scale Classification 3) Block Structure 4) Scale of the Flow Analysis Page 107 109 115 136 150 156 161 166 167 169 172 CHAPTER SEVEN: COMPARISON AMONG MODELS 1) Introduction 2) Comparison with Spectral Techniques 3) Effective Conductivity 4) Dimension Dependence CHAPTER EIGHT: PARAMETER ANALYSIS OF QUADRA SANDS DATA 1) Introduction 2) Sample Site, Sample Plan, Measurement Technique 3) Assumptions Required 180 181 188 194 196 197 203 Page 4) Moment and Histogram Analysis 204 5) Conductivity Variability within a Sample 212 6) Trend Surface Analysis 213 7) Correlograms 217 8) Power Spectra For the Line Samples 228 9) Fitted Nearest Neighbour Models 234 10) Variograms 241 11) Correlation Between Flow Parameters 244 12) Conclusions 248 CHAPTER NINE: CONCLUSICNS, RECCMMENDATICNS FOR FUTURE RESEARCH 251 REFERENCES 265 APPENDICES 274 v i i i TABLES Table 8-7 Estimated correlation coefficients between flow parameters Page Table 2-1 Sample realization from a f i r s t order autoregressive model 15 Table 3-1 Summary of estimated integral scale moments, one-dimensional model 59 Table 3-2 Summary of estimated integral scale moments, two-dimensional model 64 Table 5-1 Flux variability in two-dimensional, nonuniform gradient field 145 Table 5-2 Gradient variability in two-dimensional, nonuniform gradient field 147 Table 5-3 Flux variability, trend in mean conductivity 155 Table 5-4 Flux variability for inflow to the tunnel 158 Table 5-5 Flux variability on outflow surface below dam 161 Table 7-1 Comparison pf flux estimate based on the geometric mean conductivity with the mean flux in a stochastic solution 193 Table 8-1 Estimated moments of flow parameters 206 Table 8-2 Estimated moments of percent finer grain size fractions 207 r Table 8-3 Percent explained sum of squares for trend analysis, line samples 218 Table 8-4 Percent explained sum of squares for trend analysis, square grid 219 Table 8-5 Twcr-dimensional sample autocorrelation matrices 226 Table 8-6 Fitted autoregressive parameters-first order nearest neighbour model 238 247 Table 8-8 Estimated correlation coefficients between flow parameters and grain size fraction, random sample 247 ix Page Table 1-1 Relations reguired in deriving equation 1-10 276 Table 1-2 Non-zero terms in row 25 of product matrix {( I - W ) • ( I - W )T• } 278 X ILLUSTRATIONS Page Figure 2-1 Hypothetical conductivity sequence 10 Figure 2-2 First order, autoregressive time series model 16 Figure 3-1 Schematic illustration of nearest neighbour grid, f i r s t order model 31 Figure 3-2 Two-dimensional system of conductivity blocks, uniform flow field 32 Figure 3-3 Vectors for the two-dimensional correlation function 37 Figure 3-4 Theoretical autocorrelation functions, one-dimensional model 54 Figure 3-5 Sample autocorrelation function averaged over a set of conductivity realizations, one-comensional model 54 Figure 3-6 Decay in autocorrelation through the exponential transform from the {Y} to {K} sequence 56 Figure 3-7 Autocorrelation function in a discrete block model 56 Figure 3-8 Theoretical autocorrelation functions, two-dimensional model 61 Figure 3-9 Sample autocorrelation function averaged over a set of conductivity realizations, two-dimensional model 63 Figure 3-10 Examples of realizations, one-dimensional nearest neighbour model 66 Figure 4-1 Che-dimensional stochastic flow field 73 Figure 4-2 Dependence of the standard deviations in hydraulic head on the integral scale 79 Figure 4-3 Dependence of the standard deviations in hydraulic head on the standard deviation in conductivity 81 Figure 4-4. Discretization, length of flow system 84 Figure 4-5 Influence of the integral scale on the hydraulic head solution 89 xi Page Figure 4-6 Coefficient of variation in flux, one-dimensional model 93 Figure 4-7 Coefficient of variation in head gradient, one-dimensional model 95 Figure 4-8 Degenerate autoregressive models 100 Figure 4-9 Convergence of the Monte Carlo technique 103 Figure 5-1 Surface of the standard deviations in hydraulic head on a two-dimensional flow field 113 Figure 5-2 Dependence of the standard deviations in hydraulic head on the integral scale 118 Figure 5-3 Dependence of the standard deviations in hydraulic head on the standard deviation in conductivity 119 Figure 5-4 Comparison of one and two-ciLmensional solutions of the stochastic flow equation 122 Figure 5-5 Coefficient of variation in head gradient, uniform gradient field 124 Figure 5-6 Coefficient of variation in flux, uniform gradient field 127 Figure 5-7 Influence of a statistically anisotropic covarianoe function in conductivity on the standard deviations in hydraulic head 129 Figure 5-8 Influence of two stationary units within the flow domain on the standard deviations in hydraulic head, different standard deviations in conductivity 133 Figure 5-9 Influence of two stationary units within the flow domain on the standard deviations in hydraulic head, different mean conductivities 135 Figure 5-10 Standard deviations in hydraulic head, two-dimensional nonuniform gradient field 138 Figure 5-11 Finite element grid for evaluation of discretization error 144 Figure 5-12 Influence of two stationary units within the flow domain on the standard deviations in hydraulic head, 148 x i i Page nonuniform gradient field Figure 5-13 Influence of a linearly trending mean 'conductivity on the standard deviations in hydraulic head 153 Figure 5-14 Inflow to a tunnel 157 Figure 5-15 Flow beneath an impermeable dam 160 Figure 6-1 Schematic diagram of geologic considerations for the interaction between scale and heterogeneity 176 Figure 7-1 Dependence of the head variance on the mean flux, one-dimensional model 185 Figure 7-2 Normalized error in flux estimate, network design problem 189 Figure 8-1 Sample site, Quadra Sands data 198 Figure 8-2 Exposure at sample locations 5 and 6 200 Figure 8-3 More detailed view of Quadra Sand, sample location 5 200 Figure 8-4 Frequency distributions for conductivity, porosity, and compressibility 209 Figure 8-5 Spatial variation in conductivity and porosity along the horizontal and vertical line transects 214 Figure 8-6 One-dimensional correlograms for conductivity and porosity 221 Figure 8-7 One-dimensional correlograms for the D^Q grain size fraction 224 Figure 8-8 Power spectra for conductivity porosity, and the D 5 Q grain size fraction 232 Figure 8-9 Comparison of sample correlograms for the Quadra Sands data with the theoretical correlograms for a fitted f i r s t order, nearest neighbour model 239 Figure 8-10 Grie-dimensional variograms for conductivity and porosity243 Figure 8-11 Variograms for conductivity, twc»-dimensional grid 245 x i i i L I S T OF SYMBOLS K hydraulic conductivity L / T Y log transformed conductivity X p o s i t i o n vector L mean conductivity L / T standard deviation i n conductivity L / T V v mean of log conductivity 0 ^ standard deviation i n log conductivity E expectation operator e uncorrelated random variable a standard deviation of e p (h) autocorrelation at l a g h C(h) autocovariance at l a g h P ( f ) power spectrum, frequency f Y (h) variogram, l a g h a x nearest neighbour autoregressive parameter, x-direction az nearest neighbour autoregressive parameter, ^ d i r e c t i o n Ax block length, x-direction L Az block length, e-direction L { e } vector of uncorrelated random variables { Y } vector of correlated random variables {K} vector of generated conductivity values L / T [w] nearest neighbour weight matrix xiv w^j scaled weight of blockj on blocki r number of contiguous blocks surrounding blockj s block length, square block L [*] identity matrix P total number of blocks in flow domain covariance^matrix (Rj correlation matrix [ M ] {( i - w ) • ( I - W )T} P ^ j correlation coefficient between blocks i and j n variance expansion factor NJjir'aJ normal distribution, mean U , standard deviation P L N [ V / ° ] lognormal distribution, meany , Standard deviation Q A y estimated mean A a estimated standard deviation A a estimated autoregressive parameter r(h) estimated autocorrelation at lag h P(_fL estimated power spectrum A integral scale L Y estimator, mean of I Y ) S estimator, standard deviation of ( Y > X estimated mean integral scale L estimated standard deviation in the integral scale L r(a,b) estimated two-dimensional autocorrelation function <|>' hydraulic head L L length of flow system, one-dimensional model L length of flow system, x-direction, two-dimensional model L length of flow system, e-direction, two-dimensional model L dimensionless distance, x-direction dimensionless distance, «-direction number of rows of blocks number of columns of blocks boundary head, X=0 L boundary head, X - L L estimated mean head L estimated standard deviation in head L dimensionless estimated mean head dimensionless estimated standard deviation in head distance over which average a flow estimate L estimator for mean flux L J / T 3 estimated standard deviation in flux L / T arithmetic mean harmonic mean estimated mean head gradient estimated standard deviation in head gradient number of conductivity realizations in Monte Carlo simulation flux estimated mean flux L / T geometric mean xvi K x x conductivity, x-direction L / T K_ conductivity, g-direction L / T A X integral scale, x-direction L X , integral scale, 2-direction L conditional mean of Y, given * regression coefficient in trending mean conductivity y z location of center of flow domain, z-coordinate L a,, „ conditional standard deviation of Y, given « y. z standard deviation in e, the grid cxxjrdinate for a linear trend L x^ ^y.z 6 Y a z St, microscale L L * nonhomogeneity scale L a decay parameter in exponential autocorrelation function e normalized error term q T estimated mean flux, average taken over simulation L / T estimated flux based on conductivities and head 3 gradients in center of flowlirie L / T * a 2 compressibility LT /M n porosity percent finer grain size fractions L w(h) weighting function in power spectral estimator N length of sample series for spectral estimation 9 confidence level X ^ eigenvalues of [WJ S x x f estimator of the power spectrum r (x, y) correlation coefficient between two random variables X and Y x v i i Acknowledgments Thanks are extended to Dr. R.A. Freeze for his guidance and encouragement during my stay at the University of British Columbia. The author is indebted to Dr. Freeze for his i n i t i a l suggestion of the usefulness of a stochastic approach to modeling flow in a heterogeneous medium. Thanks go to Drs. J. deVries, W.H. Mathews, and A.J. Sinclair; University of British Columbia, for their helpful comments on an early draft of this thesis. Dr. F. Schwartz; University of Alberta, provided valuable suggestions on the presentation of the results. I also benefitted from discussions with Dir. Schwartz on the probability model used to characterize spatial variations in hydraulic conductivity. Gordon Hodge drafted a l l of the figures in the thesis. This research was supported by a grant from the National Research Council of Canada. I would also like to thank the National Research Council for making available a postgraduate scholarship from 1974 to 1978. 1 CHAPTER 1 introduction Steady state flow models are routinely applied to a wide variety of subsurface flow problems. The traditional approach to modeling has been deterministic. Solutions are presented as single-valued predictions. They do not take into account the uncertainty that must exist in any prediction due to the spatial heterogeneity of geologic deposits and our uncertainty as to the exact spatial distribution of this heterogeneity. Natural depositional processes do not create homogeneous units; the cartitionly observed spread of values measured within a single aquifer provides evidence of significant variability. This nonuniformity i s often hidden in the deterministic approach by assuming that single equivalent values can be defined that account for the hydraulic behavior of the: spatial variations. Calibration techniques are used to bring model output in line with the known field response. In the stochastic approach to porous media flow, an attempt i s made to quantify the uncertainty in the model predictions when the heterogeneity of the medium is recognized but the spatial variations cannot be replicated in the numerical model. In this study, the deterministic formulation of the flow equations i s assumed to be physically sound. However, the equation parameters are taken to be random variables in space, with a known 2 probability distribution. Rather than obtaining single-valued predictions, probability distributions on the output variables are formed. These variabilities then reflect our uncertainty in the system being modeled. A discrete block representation of a conductivity field i s adopted. Each block i s assigned a single conductivity value. A probability model of the spatial dependence between neighbouring values of conductivity will be described. The flow domain is bounded in a l l directions. The objectives of this study can be stated as follows: 1) To characterize the uncertainty in the output variables in terms of the correlation between conductivity values in the block system. 2) To determine quantitative values of the standard deviations in the predicted head values for realistic values of input standard deviations in conductivity, for some two-dimensional steady flow fields. 3) To consider the differences in the prediction variabilities arising from one and twcr-dimensional flow models. 3 4) To consider the influence of spatial variations in the expected head gradients (uniform versus nonuniform flow fields) on the prediction variabilities. 5) To consider the influence of various models of spatial heterogeneity for conductivity on the prediction uncertainties. 6) To consider the effects of scale. From the consideration of these objectives, a physically based explanation for the dependence of the output variabilities on the number of blocks included inr^he bounded system can be given. A theoretical model is described to represent the spatial heterogeneity of a conductivity field. To complement this analysis, the model is matched to a data set collected from an unconsolidated sand deposit. By way of introduction, an outline is given of the directions taken in the succeeding chapters. In Chapter 2, the probability model used to characterize the heterogeneity of a porous medium is described. As well, a review of previous work and the alternate approaches in stochastic modeling is given. A technique by which synthetic conductivity fields with a known spatial structure can be randomly generated on the computer is detailed in Chapter 3. Following this, the results of a series of one and two-dimensional simulations are presented in Chapters 4 and 5. The interaction of the correlation between con-ductivity values, the size of the flow domain, the conductivity distribution, and the nature of the flow system in determining the prediction variabilities is demonstrated. A Monte Carlo technique is used to estimate the distributions of the output variables. In Chapter 6, a qualitative discussion of scale considerations is given. Chapter 7 follows with a comparison of the model output with other solution techniques. The study concludes with the analysis of the field data in Chapter 8. Following most chapters, a summary of the important conclusions arising from the particular problem considered in that chapter is presented. Then, in Chapter 9, the conclusions are redrawn in the context of the entire study. In addition, recommendations for future : research or extensions to the existing method are summarized. CHAPTER 2 Definitions, Review of Previous Work 1) Hydraulic Conductivity as a Random Variable In this section, the mathematical model used to characterize the spatial heterogeneity of a porous medium is detailed. Definitions and concepts wil l be reviewed as they are introduced. Throughout this thesis, a fairly rigorous terminology is employed, in keeping with that found in the statistical literature. For those readers not familiar with the terms or concepts, a descriptive glossary has been included in Appendix VI. a) complete structure Hadley (1969) defines a random variable as any numerical quantity whose value is determined by the outcome of a random experiment. In.this study, hydraulic conductivity is considered to be a random variable everywhere within the flow domain. A random function is defined by an ordered set of random variables. Here, the set of random variables- are the conductivity values at every point within the flow domain. Because the spatial variations in conductivity cannot be predicted with absolute certainty, i t can be considered a stochastic process. Matheron (1970) defines the space law of a random function to be the simultaneous probability distribution of random values taken 6 on by the random function. At any point in the flow domain, coiductiyity has its own probability density function. There is a set of joint probability density functions between this point and a l l other points within the flow domain. The spatial variations in conductivity at the field site are considered the single, unique realization of the space law. It forms just one of the infinite number of possible realizations of the random function. The infinite number of possible realizations is termed the ensemble. b) stationarity The above probability model depends upon an infinite number of unknown parameters; the moments of each of the point probability density functions and a l l of the joint moments between point values of conductivity. A simplifying hypothesis must be assumed that will reduce the number of parameters needed to define the space law. At the same time i t must be sufficiently general to be applicable to real media. The usual assumption made is second order stationarity (or equivalently, statistical homogeneity). Two constraints are imposed on the random function. First, conductivity must have the same expected value at every point. Secondly, the covariance between conductivity values at any two points must depend only upon the vector separating those points and not their absolute position. Second order stationarity can be expressed mathematically as follows. Let the random variable be K in a coordinate system X 7 (x^, x^, x^). The constant mean condition requires: E(K(X)] = constant V X 2-1 where E denotes the expectation. The covariance condition can be stated: E[|K(X1)K(X2)} ] = E^KfX^KfXj-}^)} .'] VX^X, 2-2 The covariance function defines the relation between the covariance and the vector separation (X^^^). If i t depends only upon the magnitude of the separation and not the orientation, the medium is statistically isotropic. If not, i t is statistically anisotropic. A stationary covariance implies that the variance of the conductivity distributions is the same at every point. The above probability model is cast in terms of hydraulic conductivity. The model also applies to other hydrogeologic parameters, such as porosity or compressibility. Jenkins and Watts (1968), Box and Jenkins (1976), ard'Lumley and Panofslcy (1964) discuss the stationarity assumption in more detail. c) ergodicity * The porous medium at a field site is the only realization of the space law that ever exists. Whenever statistical estimates are 8 calculated from a single realization, an ergodic theorem must be assumed. In our case, this states that averages taken across space are equivalent to averages formed across the ensemble. A general requirement for the ergodic theorem to hold i s that the covariance between point values of conductivity decays to zero over a finite distance. Then, different parts of the same realization can be regarded as independent events. Lumley and Panofsky (1964, pg. 35-39) provide a further discussion of ergodicity as i t applies to a spatial domain. d) autocorrelation function The correlation function is obtained by normalizing the covariance function by the variance of the process. In considering the correlation between the same variable but at different separations (lags), the prefix auto is attached. The autocorrelation function is defined for a stationary process as: (h) =E |jK(Xj+h)-E[K(X)] },{ K(Xx)-E[K(X)] }] ^E "{K(Xj+h)-E[K(X)] } 2 • E { K(X][) -E [K(X)] 2-3 = Cov^J^+h), KO^)] ^VarfKfXj+h)] .VarfKt^j] 2-4 9 for different lags h, h=l, 2,.... The mean of the stationary process is denoted E{K(X)J for a l l X in the domain of interest. The graphical representation of the autocorrelation function is the correlogram. It can be interpreted as a measure of the strength of the spatial dependence in the random variable as the separation between the two points increases. If conductivity is strongly dependent on its neighbouring values, we can expect its correlogram to decay slowly toward zero. If the spatial dependence i s weak, we expect to see a steep correlogram which decays to zero for small values of h. The autocorrelation p (h) takes the range | £ (h) | 31 and is dimensionless. Agterberg (1974), Davis (1973) and Schwarzacher (1975) analyze correlation functions in a geologic context. e) power spectra The autocorrelation function is a space domain representation of the strength of the dependence between neighbouring values of conductivity. A power spectrum allows us to consider the same information in the frequency domain. A hypothetical conductivity sequence is shown in Figure 2-1. It is assumed to be a realization from a second order stationary space law. The fluctuations in the conductivity values can be described by their amplitude and frequency. The power spectrum identifies those frequencies which are significant in contributing to the total variance within the sequence. Figure 2-1 Hypothetical conductivity sequence 1 1 The autocovariance f u n c t i o n and the power spectrum form a Fourier cosine transform pair. That is (Kisiel, 1969, pg. 40): P ( f ) = /C(h) cos2Trfh dh C(h) = /P ( f) cos27rhf df 2-5 where P ( f ) i s the pcwar spectrum, C(h) the autocovariance f u n c t i o n , h the s p a t i a l l a g FL], and f the frequency [ 1 / L 1 . The power spectrum d e f i n e s the d i s t r i b u t i o n o f the t o t a l v a r i a n c e o f the s t o c h a s t i c process over frequency. A process w i t h a s l o w l y decaying, p o s i t i v e a u t o c o r r e l a t i o n f u n c t i o n w i l l have a power spectrum w i t h a peak a t low frequencies. Any r e a l i z a t i o n w i l l appear f a i r l y smooth. I f the s p a t i a l dependence i s weaker, more power i s s h i f t e d i n t o the h i g h e r frequency end o f the spectrum. A r e a l i z a t i o n w i l l tend t o f l u c t u a t e more r a p i d l y about i t s mean va l u e . I n the l i m i t o f a s p a t i a l l y u n c o r r e l a t e d s e r i e s (white n o i s e ), the spectrum i s h o r i z o n t a l w i t h equal power a t a l l frequencies. I n t e g r a t i o n o f the area beneath the power spectrum y i e l d s the t o t a l v a r i a n c e o f the process. I n c r e a s i n g the v a r i a n c e w i l l s h i f t the e n t i r e spectrum upwards but w i l l not change the r e l a t i v e d i s t r i b u t i o n o f v a r i a n c e over frequency. Jenkins and Watts ( 1 9 6 8 ) p r o v i d e a b a s i c treatment o f s p e c t r a l a n a l y s i s . K i s i e l ( 1 9 6 9 ) and J u l i a n ( 1 9 6 7 ) ' 12 provide good summary treatments. Agterberg (1974) and Schwarzacher (1975) i n c l u d e examples o f the a p p l i c a t i o n o f s p e c t r a l a n a l y s i s t o g e o l o g i c data. f) variogram Second order s t a t i o n a r i t y i s not the o n l y assumption we can invoke t o reduce the complexity o f the space law. An a l t e r n a t e s e t o f weaker assumptions, the s o - c a l l e d i n t r i n s i c h y p othesis, i s o f t e n a p p l i e d i n s p a t i a l e s t i m a t i o n o r i n t e r p o l a t i o n (Matheron, 1970; Delhomme, 1976). Here, the d i f f e r e n c e s between p o i n t s are assumed s t a t i o n a r y r a t h e r than the p o i n t s themselves. That i s : E | K ( X x ) - K ( X 2 ) j = O ¥ X±,X2 V a r | K ( X 1 + h ) - K ( X 1 ) | = 2y(h) YX-^ where Var i s the v a r i a n c e o f the random v a r i a b l e d e f i n e d w i t h i n the brackets and Y(h) i s the variogram. The parameter h i s g e n e r a l l y a vec t o r s e p a r a t i o n . Equation 2-7 can be mo d i f i e d by a p p l y i n g the d e f i n i t i o n o f v a r i a n c e t o give -: ¥ X 1 2-8 Thus, the variogram measures the s p a t i a l s t r u c t u r e as the expected 2-6 2-7 Y(h) = J§E (K(X 1+h) - K(X 1)} 13 square of the difference between the random variable at two points as the separation between them increases. If the space law is at least second order stationary, then the variogram is the mirror image of the correlogram. g) stochastic process model A stochastic process model can be viewed as an equation which describes the probability structure of a sequence of observations. For discrete data, the stochastic process model is often a linear equation expressing the dependence of a random variable at a point on the same random variable at surrounding points. The process model can be used to generate synthetic realizations of a random function. Alternately, the process model can be fitted to a given realization using appropriate statistical estimators for the equation parameters. Box and Jenkins (1976) provide the basic treatment in the time domain. Process models in a spatial domain have only recently been considered in detail. Bartlett (1975) provides the most complete review to date. A further discussion of process models used to describe the spatial heterogeneity of hydraulic conductivity follows in Chapter 3. h) example To illustrate the probability model that has been introduced, consider the example of a first order, autoregressive time series model. The s t o c h a s t i c process equation i s d e f i n e d : 2-9 K ( t ) - y k = a ( K ( t - l ) - p k ) + E ( t ) where y T r i s the mean o f the s t a t i o n a r y process a i s the a u t o r e g r e s s i v e parameter (non-randcm) ex p r e s s i n g the s t r e n g t h o f the l i n e a r dependence between the r e a l i z a t i o n s a t t and t - 1 ( |a | -1 ) e i s an u n c o r r e l a t e d random v a r i a b l e w i t h a mean o f zero and v a r i a n c e o e A 20 p o i n t r e a l i z a t i o n generated from equation 2-9 i s l i s t e d i n Table 2-1. The mean and au t o r e g r e s s i v e parameter were s e t a t 0 and 0.8, r e s p e c t i v e l y . Each member o f the sequence ( e ( t ) } was generated from a normal d i s t r i b u t i o n w i t h a mean zero and a standard d e v i a t i o n o f 0.43 (NJb.O, 0.43J). The r e a l i z a t i o n i s p l o t t e d i n Fi g u r e 2-2a. A t each p o i n t t ^ , K(tj.) i s a normally d i s t r i b u t e d random v a r i a b l e . A l l 20 p o i n t s considered together d e f i n e the random f u n c t i o n on the time l i n e . The sequence i n Table 2-1 i s j u s t one o f an i n f i n i t e number o f p o s s i b l e r e a l i z a t i o n s t h a t c o u l d have been generated by the process model (eqn. 2-9). K. K. -0.36 -0.36 11 0.17 0.99 0.22 -0.07 12 -0.76 0.03 0.62 0.56 13 -0.36 -0.33 -0.21 0.24 14 0.27 0.00 -0.10 0.09 15 0.31 0.31 0.42 0.49 16 0.80 1.05 0.65 1.04 17 0.54 1.38 0.41 1.25 18 -0.87 0.23 10 0.22 0.05 1.22 1.02 19 20 0.37 0.55 0.51 0.95 Table 2-1 Sample R e a l i z a t i o n From a F i r s t Order A u t o r e g r e s s i v e Model 16 F i g u r e 2-2 F i r s t order a u t o r e g r e s s i v e time s e r i e s model 17 For the s t o c h a s t i c process model (2-9), the v a r i a n c e o f K(t) i s r e l a t e d t o the v a r i a n c e o f the random process e ( t ) by: °l ^ e ' 1 " " 2 2-10 The t h e o r e t i c a l a u t o c o r r e l a t i o n f u n c t i o n f o r K(t) i s given as: p (h) = a h 2-11 where p(h) i s the c o r r e l a t i o n c o e f f i c i e n t f o r a l a g number h between d i s c r e t e p o i n t s on the time l i n e . Eguation 2-11 r e s u l t s from the e v a l u a t i o n o f the expectations i n equation 2-3 i n terms o f the process model, equation 2-9. Because the process model i s s t a t i o n a r y , p(h) i s independent o f t i , the p o s i t i o n a l ong the time l i n e . The power spectrum f o r the process model (2-9) i s given as: 2 a 2 Ptf» " -,+ 2 ' , . °<-f-1'2 2-12 1 + tX - i d C O S Z T T f For d e t a i l s on how these r e s u l t s are d e r i v e d , see Box and Jenkins (1976, pg. 46-63). The a u t o c o r r e l a t i o n f u n c t i o n and the power spectrum f o r the process model (2-9) are p l o t t e d i n F i g u r e s 2-2b and 2-2c. The parameters are the same as those used t o generate the sample r e a l i z a t i o n . The 18 r e l a t i o n s expressed i n equations 2-10, 2-11 and 2-12 apply t o the t h e o r e t i c a l (or ensemble) s t r u c t u r e o f the process. I f o n l y the s i n g l e r e a l i z a t i o n i s a v a i l a b l e , the parameters o f the process equation must be estimated by averaging over the e f f e c t o f the random component i n t h a t r e a l i z a t i o n . For example, one c o u l d o b t a i n the estimates y , a , a k , r(h) , P (f) f o r the mean, aut o r e g r e s s i v e parameter, v a r i a n c e , a u t o c o r r e l a t i o n f u n c t i o n and power spectrum o f the process model from which the r e a l i z a t i o n was generated. Our confidence i n these estimates would depend, i n p a r t , upon the l e n g t h o f the sample sequence. Rather than p r e s e n t i n g the e s t i m a t i o n s t a t i s t i c s here, they w i l l be d e t a i l e d as they are r e q u i r e d through the t e x t . 2) Previous Work on S t o c h a s t i c Flow Models S t o c h a s t i c concepts a p p l i e d t o groundwater fl o w problems can be grouped acc o r d i n g t o the form o f the boundary valu e problem, the c h a r a c t e r i z a t i o n o f the equation parameters as random v a r i a b l e s , the s o l u t i o n methodology, and the assumptions which must be made i n d e r i v i n g t h a t s o l u t i o n . The p i o n e e r i n g work on the e f f e c t s o f porous-medium heterogeneity on model p r e d i c t i o n was c a r r i e d out by Warren and P r i c e (1961). They n u m e r i c a l l y s o l v e d both a steady s t a t e and t r a n s i e n t flow equation i n a 'homogeneously heterogeneous' medium. The medium was formed by p e r t u r b a t i n g a three-dimensional a r r a y o f randomly s e l e c t e d homogeneous p e r m e a b i l i t y b l o c k s chosen from a given p r o b a b i l i t y d e n s i t y f u n c t i o n . They were p r i m a r i l y t r y i n g t o determine whether a s i n g l e 19 permeability value could characterize the behavior of a heterogeneous medium. They concluded the most probable behavior of an equivalent homogeneous medium is given by the geometric mean of the permeabilities in the constituent blocks. The heterogeneity of a porous medium has been considered in terms of both permeability and conductivity. If conductivity is used, then the influence of the variability in the fluid properties is implicitly included in the conductivity variation. These terms are not interchangeable. In suitmarizing studies that utilized the permeability of the medium, reference is made to permeability. This study uses hydraulic conductivity as a basis flow parameter. McMillan (1966) combined a numerical solution of a steady state flow equation with repetitive Monte Carlo simulation to analyze the output variability in the potential field. He considered the effects of the standard deviation in a lognormally distributed con-ductivity distribution, the (dimension of the analysis, the mean potential gradient, and the basin size and shape. The homogeneous conductivity blocks were assigned values randomly chosen from their probability density function. Vandenberg (1977) assessed the adequacy of simple analytic models in the analysis of complex aquifers. He used the Theis method to evaluate simulated drawdown curves from a finite difference aquifer model. Medium heterogeneity was generated from randomly selected transmissivity values with a uniform distribution. He found that 20 c a l c u l a t e d t r a n s n d s s i v i t i e s were d i s t r i b u t e d i n a narrow range about the average o f the values assigned t o the nodes. Various workers have commented on sources o f p r e d i c t i o n u n c e r t a i n t y besides the porous medium heterogeneity. Sagar and K i s i e l (1972) i n v e s t i g a t e d the i n f l u e n c e o f u n i f o r m l y d i s t r i b u t e d e r r o r s i n the a q u i f e r parameters. Bibby and Sunada (1971) s o l v e d the w e l l f l o w equation n u m e r i c a l l y t o i n v e s t i g a t e the e f f e c t s on the s o l u t i o n o f normally d i s t r i b u t e d measurement e r r o r s a n the equation parameters. They a l s o analyzed u n c e r t a i n t y i n t r o d u c e d i n t o the s o l u t i o n due t o s p a t i a l averaging o f a l i n e a r l y t r e n d i n g p e r m e a b i l i t y f u n c t i o n i n o r d er t o o b t a i n nodal p e r m e a b i l i t y values. The p e r m e a b i l i t i e s were chosen from a normal d i s t r i b u t i o n . Sagar (1978) approached the s t o c h a s t i c problem i n i t s most complete form, as a s t o c h a s t i c boundary val u e problem i n a continuous medium. He presents an a n a l y t i c a l s o l u t i o n f o r one-dimensional t r a n s i e n t flow i n a nonrandom medium but w i t h a random f o r c i n g f u n c t i o n . I n general however, a n a l y t i c a l techniques f o r s t o c h a s t i c boundary value problems o f the complexity encountered i n groundwater hydrology are not y e t a v a i l a b l e . I n a r e c e n t study, Dagan ( i n press) reconsidered the problem ,of" r e p l a c i n g a heterogeneous medium by a s i n g l e - v a l u e d e f f e c t i v e p e r m e a b i l i t y . Both the steady s t a t e and t r a n s i e n t f l o w equations were i n v e s t i g a t e d . The r e s u l t s are presented as a s e r i e s o f boundary 21 values for the effective permeability, together with its best estimate. From this model, he was able to derive relations for the variances in head gradient, specific discharge, and head for one, two and three-dimensional models. An alternate approach to characterizing porous media heterogeneity is to work from a set of lithologies rather than with the conductivity distribution. After generating a realization of discrete blocks or layers, conductivity values are associated with each member of the lithology set. The prediction variability can then be determined via Monte Carlo simulation techniques. Wu et al. (1973) used this approach in constructing seepage models for an excavation. Lippman (1973) used a Markovian stratigraphic model to analyze heterogeneity effects in a regional flow model. Schwartz (1977) considered a dispersion problem by distributing various sets of low conductivity inclusions within a higher conductivity matrix. A one-dimensional stochastic analysis of both a steady state groundwater flow equation and a transient consolidation problem was considered by Freeze (1975, 1977). Monte Carlo techniques were used to generate the output distributions of hydraulic head and compaction. Conductivity values were inserted into a discrete set of blocks by randomly choosing values from the probability density function for conductivity. Multivariate generation of conductivity, porosity and compressibility values was required for the transient problem. 22 Gelhar et al (1977) and Bakr (1976) quite rightly point out that this analysis was limited by the nature of the spatial covariance between conductivity values along the flowline. In effect, the sequence was perfectly correlated within each block and uncorrelated between any two adjacent blocks. M l of the discrete block models previously discussed follow this same assumption. Many of the available solutions to stochastic groundwater flow equations have relied on Monte Carlo simulation, probably because of its simplicity rather than i t being a preferred solution technique. There are no apparent limitations on the magnitude of the moments of the conductivity distribution. Assumed conditions on the probabilistic structure of the heterogeneity and on the flow system are very flexible. The obvious drawback to the method is its requirement for repetitive solution of the flow equation. Two other approximate solution techniques have been investigated as alternatives to Monte Carlo simulation. Bakr (1976) utilized a continuum approach to solve a similar one-dimensional flow problem to that of Freeze (1975). His method is based on spectral analysis and perturbation theory. The flow equation is solved in a stationary medium. The spatial covariance between point values of conductivity is described by a decaying exponential function. This function leads to the definition of the integral scale for a random medium. The integral scale can be considered as the average distance over which 23 point values of conductivity are positively correlated. This concept was f i r s t introduced in stocliastic turbulence theory (Lumley and Panofsky, 1964). Bakr (1976) and Gelhar (1977) were able to show that the output uncertainties are strongly controlled by the nature of the spatial dependence whenever distances over which flow calculations are made are low order multiples of the integral scale. Commenting on his own solution, Freeze (1975) and later Dagan (1976) suggested a one-dimensional flow model may exert a strong control on the extent of the predicted variance in hydraulic head. Bakr (1976) and Gelhar (1977) have compared one, two and three-dimensional spectral solutions of the steady state flow equation. They found the variability in hydraulic head was reduced by an order of magnitude in the multi-dimensional analysis. A more detailed comparison of spectral and Monte Carlo techniques follows in Chapter 7. It suffices here to state that the weakness of spectral techniques l i e in their requirement for small standard deviations in the conductivity distribution and unbounded systems in the direction of flow. Their strength lies in the continuum approach and the functional form of the solution. A second alternative to Monte Carlo simulation was introduced into the groundwater literature by Tang and Pinder (1977). They propose a numerical solution of the stochastic differential equation based on perturbation theory. The random coefficient is separated into a mean and a perturbed term at each point in the flow domain. They apply 24 the method to both a convective-dispersion equation and the transient groundwater flow equation. The magnitude of the perturbed component must be small compared to the mean to get an accurate solution. The stochastic basis of the spatial distribution of the flow parameters seems more closely allied with a discrete block Monte Carlo solution, although the mathematics are more rigorous in solving the flow equation. 3) Analyses on the Spatial Variation of Hydraulic Conductivity Analyses on the spatial distribution of hydrogeologic parameters are now becoming available in the literature, although samples collected over a series of scales and/or on a regular grid are s t i l l scarce. Davis (1969) provides a general discussion on the hydraulic conductivity and porosity of porous media. Bakr (1976) presents a good summary treatment of the geologic factors controlling variations in conductivity on both micro and macro scales. Natural depositional processes do not create homogeneous units. Significant variability in the flow parameters will result from spatial and temporal variability in the depositional process, the available material and sedimentation rates, and in any secondary process such as cementation. However, spatial continuity in these processes suggest the parameter values at neighbouring points should be correlated. Attempts to f i t permeability or conductivity data to a pro-25 bability density function have generally found the lognormal distribution TCO be the most satisfactory (McMillan, 1966; Bakr, 1976; among others). Law (1944) was the f i r s t to propose this distribution on the basis of core data from a carbonate o i l field reservoir. Bennion and Guffiths (1966), working with 60,000 samples from a sandstone reservoir, found the permeability distribution to be too skewed to the right and too leptokurtic to be lognormal. Freeze (1975) provides a table that summarizes some of the earlier data collected. If we define Y=log K, where K is lognormally distributed, then Y is normally distributed. For conductivity values log transformed in base 10, the standard deviation of Y generally lies between 0.2 and 2.0. Mote the standard deviation provides a measure of the nonurdformity of the porous medium over the sampled domain but not its spatial continuity. The spectral and perturbation techniques can only be applied to media at the very low end of the reported range in conductivity standard deviations. Several studies have considered the variability of hydraulic conductivity in the soil zone. Among those of note are Nielson et al (1973), Biggar and Nielson (1976) and Baker(1978). Using data from the Illinois State Water Survey, Bakr (1976) provides the most complete analysis to date of gridded permeability measurements. He computed correlation functions and spectra for eight sets of permeability and porosity values taken along a vertically 26 cored sequence sampled at 0.30 m intervals. The samples were taken from the Mount Simon aquifer in northeastern Illinois at depths of over 460 m. The length of the sample sequences varied from 45 to 92 m. The unit is a fine to coarse grained orthoquartzite. On each core, both horizontal and vertical permeabilities were measured. Standard deviations on the log of permeability ranged from 0.67 to 0.95. The integral scale, estimated by assuming an exponential correlation function, ranged from 0.33 to 1.02 m. Data collected in other deposits and over different sample scales suggests much larger integral scales may be possible. This problem will be considered in Chapter 6. Oil reservoir heterogeneity has been a particular concern of petroleum engineers. Several studies have concentrated on providing a deterministic description of gross reservoir heterogeneity from a set of well log data. Techniques have included analyzing empirical continuity ratios based on lithologic stratification (Alpay, 1972) and statistical zonation techniques based on analysis of variance, multiple range tests and trend surface analysis (Testerman, 1962; Bennion and Guffiths, 1966). Bennion and Hope (1974) suggested more sophisticated correlation and spectral techniques. Polasek and Hutchinson (1967) analyze the nonurdformiti.es within a sandstone outcrop in terms of its spatial permeability variation and i t s probable behavior as a fluid reservoir. Harris and Hewitt (1977) review geolo-gical controls that aid in the description of reservoir continuity and thickness patterns. 27 CHAPTER 3 Generation of Conductivity Realizations 1) Introduction In this chapter, a model is presented from which conductivity realizations obeying a known space law can be generated. The model is based on subdividing the flow domain into p eguisized blocks with a single conductivity value assigned to each block. Conductivity values in neighbouring blocks are autocorrelated. Following a brief introduction to the generation of realizations from a known space law, the details of the above technique are discussed. The presentation is carried out for a twcr-dimensional network of blocks. Simplifications which occur in reducing the analysis to one dimension will be noted. The probabilistic structure of the model is then discussed as i t relates to the conductivity field and the flow domain. The chapter concludes with an analysis of the conductivity realizations generated by the model. 2) Generation Technique a) available techniques The method used in this study to generate conductivity fields with a known oovariance structure was suggested from Martin (1974). He generated two-dimensional realizations of a random function to study correlation problems in trend surface analysis. Spatial 28 dependence was b u i l t i n t o a network o f b l o c k s by s o l v i n g a system o f l i n e a r equations which r e l a t e d the r e a l i z a t i o n w i t h i n one b l o c k t o the r e a l i z a t i o n s i n neighbouring b l o c k s , coupled w i t h the a d d i t i o n o f a random component. Th i s approach t o generating i n t e r n a l l y c o r r e l a t e d r e a l i z a t i o n s depends upon the correspondence between a covariance f u n c t i o n and an a s s o c i a t e d s t o c h a s t i c process model. These techniques are w e l l known i n time s e r i e s modeling. As a p a r t i c u l a r example i n hydrology, s y n t h e t i c sequences o f streamflow can be generated t o extend the h i s t o r i c a l r e c o r d (eg. Matalas, 1967). Various a u t o r e g r e s s i v e and/or moving average models have been used. For c e r t a i n s e t s o f model parameters, they generate sequences w i t h e x p o n e n t i a l l y decaying covariance f u n c t i o n s . The p r o b a b i l i s t i c d i f f e r e n c e between a space and a time dependent s e r i e s must be recognized. I n a time s e r i e s , the s t r u c t u r e can extend i n o n l y d i r e c t i o n ; w i t h dependence on p r e v i o u s l y r e a l i z e d v a lues. I n space, the dependence i s m u l t i d i r e c t i o n a l . The e x t e n s i o n from u n i d i r e c t i o n a l t o m u l t i d i r e c t i o n a l dependence i n t r o d u c e s a d d i t i o n a l c o m p l e x i t i e s i n t o t he a n a l y s i s o f the s t o c h a s t i c process models. The s p a t i a l model u t i l i z e d i n t h i s study f a l l s i n the g e n e r a l category o f n e a r e s t neighbour processes. I t i s most e a s i l y d e s c r i b e d as an a u t o r e g r e s s i v e scheme extended i n t o the s p a t i a l domain. I n 29 discussing the mechanics of the generating technique, numerous properties of the nearest neighbour model will be introduced. Therefore, a more detailed discussion of the model is postponed until after the generating technique is outlined. For completeness, two alternate approaches should be mentioned. Mejia and Rodriquez-Iturbe (1974) describe a method for generating statistically isotropic random fields based on repeated sampling from the spectrum. Journel (1974), Delhomme (1976) and others describe the turning bands method. It allows the realization to be generated in one dimension with then a simple extension to two or three dimensions as a summation process. b) generation via the nearest neighbour stochastic process model In this section, the model from which conductivity realizations are generated is detailed. The simultaneous, nearest neighbour auto-regressive relation in two dimensions can be written as: Y. . = a (Y. . . + Y. •) + a (Y- . , + Y. . , J + e . . 3-1 ax l - l , j 1+1,1 2 i/D-1 l/l+l l l is the random variable satisfying the nearest neighbour relation is an uncorrelated random variable associated with each realization of Y.. i l where Y.. i l e i j 30 a x is an autoregressive parameter expressing the degree of spatial dependence of Y^_. on its two neighbouring values in the x-direction, Y. , . and Y. , . i — i , j x+x,j similarily expresses the dependence in the 2-direction on Y. . and Y. . , n x,3-l i,D+l Figure 3-1 schematically illustrates this process equation for the block model. Let x and « be the space coordinates. The coordinate systems of the stochastic process model and the flow domain are aligned. The space scale is defined on the discrete interval Ax = X. . - X. , . --rj 1 —-L/3 and A 2 = a- . ,. It will generally be required that AX and A « ifD --r3~ I be equal. In a stationary nedium, equation 3-1 holds for every block within that medium. The autocovariance function wil l be statistically anisotropic whenever the autoregressive parameters are a function of direction. If ax equals a a, the model is statistically isotropic. Following Martin (1974), consider now the twc>-dimerisional system of conductivity blocks shown in Figure 3-2. The process model for the entire set of p blocks (m rows, n columns) can be written as a system of p linear equations: {Y} = [w] { Y} + { e } 3-2 where jVj is a matrix spatial lag operator of scaled weights w^... Y U+1 Y i-1.i Y u -1 AZ AX gure 3-1 Schematic i l l u s t r a t i o n of the nearest neighbour grid, f i r s t order model / / / f (1.D (1,n) (2,1) <P =0o (m,1) (m,n) 0 = 0 L ; Figure 3-2 Two-dimensional system of conductivity blocks, uniform flow f i e l d UJ These s c a l e d weights are defin e d : w.. i = 1, ... ,p w. . = -i^- , j = 1, ...,p 3-3 i l i ? j * w i t h w. . = cs i f b l o c k s i and j are contiguous i n the i l x x - d i r e c t i o n = i f b l o c k s i and j are contiguous i n the 2 - d i r e c t i o n - 0 otherwise r = t o t a l number o f contiguous b l o c k s surrounding b l o c k The [w] m a t r i x i n d i c a t e s which blo c k s are l i n e a r l y r e l a t e d t o each other. The s c a l i n g i s r e q u i r e d t o preserve s t a t i o n a r i t y i n the generated sequence. The random f u n c t i o n { e } forms an i n t e r n a l l y u n c o r r e l a t e d , p member v e c t o r o f random v a r i a b l e s . Each term has a mean o f zero and a v a r i a n c e , a2 . These c o n d i t i o n s are expressed mathematically e as: E [{ E }] = 0 3-4 C o v [ { e } - { e } T J = E { e } • { e } T J = a * • 1 3-5 where [ i ] i s the i d e n t i t y m a t r i x and T denotes a transpose o f the 2 v e c t o r . The v a r i a n c e a g w i l l c o n t r o l the magnitude o f the c o n d u c t i v i t y f l u c t u a t i o n s but not the dependence between the random v a r i a b l e a t neighbouring p o i n t s . Equation 3-2 can be r e w r i t t e n as {Y} - [w].{Y} = { e } 3-6 and then put i n the form ([l] - JwJMY} = { e } 3-7 S o l v i n g f o r the v e c t o r {Y} y i e l d s : {Y} = ( [l] - [w] r 1 • {e} 3-8 provided the i n v e r s e e x i s t s . For a p bl o c k system, the (p x p) m a t r i x i n v e r s e ( [lj - [w]) ^ can be regarded as a f i l t e r o p e r a t i n g on a random (p) v e c t o r { e } t o y i e l d an output (p) v e c t o r {Y} w i t h an i n t e r n a l c o r r e l a t i o n depending upon the form o f the s p a t i a l l a g o p e r a t o r [w] and the parameters a and a . A s i m i l a r s e t o f equations w i l l apply i n one dimension; o n l y the s t r u c t u r e o f the s p a t i a l l a g operator [w] changes. The mean o f the Y process i s given as: [{Y}] = E [( [I] - fT-T]) 1 • {e} ( [ i ] - [W]) 1 -E {£} 3-9 s i n c e the w^j are constants f o r a s t a t i o n a r y medium. Using equation 3-4, we f i n d {Y} = {0} 3-10 The covariance o f the Y process can be shown t o be: Gov M - E = E = E ({Y}- E [{Y}]) • ( {Y} - E ({Y}{Y} T)J {Y} ([I]- [W]) 1 { £}{ E} T [([I] - [w])"1] T = " ( P J - fWj )([I] - [WJ) T _ 1 E J{e}{e} T 3-11 L e t [M] equal the r e s u l t o f the above m a t r i x i n v e r s e o p e r a t i o n o u t s i d e the expected value operator. L e t the covariance m a t r i x be denoted [c] Then, usi n g equation 3-5, the covariance m a t r i x can be w r i t t e n as: [Cj = [Mj f l] o2e = [M] a2e 3-12 The c o r r e l a t i o n m a t r i x , denoted by [R] , i s r e l a t e d t o [C] by: fcj = o2 [RJ 3-13 36 where a i s the v a r i a n c e o f the Y process. T h i s v a r i a n c e i s found y along the di a g o n a l o f jc]. S u b s t i t u t i n g equation 3-13 i n t o 3-12, the c o r r e l a t i o n m a t r i x can be w r i t t e n as: 2 M - —A— M 3-14 y The two-dimensional c o r r e l a t i o n m a t r i x takes t he gen e r a l form: b l o c k 1 2 3 p P12 P 13 ... P i p P 2 1 1 P 2 3 . . . P 2 P P P p i Pp2 where, f o r example, P ^ 2 i s the c o r r e l a t i o n c o e f f i c i e n t between b l o c k s 1 and 2. The exact numerical form o f t h i s m a t r i x depends upon the numbering g r i d d e f i n i n g the c o n d u c t i v i t y b l o c k s . Because the process model i s s t a t i o n a r y , must equal p 23/ P34/ ••• P p _ i p« I - t f o l l o w s P 1 3 must equal P 2 ^ , . . . P ^ 2 p' e t c * ^ d i t i o n a l l y , "the c o r r e l a t i o n m a t r i x i s symmetric; t h a t i s , p ^  equals P 2 ^ . By d e f i n i n g one b l o c k as the o r i g i n o f the c o r r e l a t i o n m a t r i x , we can s p e c i f y a s e r i e s o f one-dimensional c o r r e l a t i o n f u n c t i o n s f o r vec t o r s moving away from t h a t block. E i g h t such v e c t o r s are shown 37 in Figure 3-3. If the conductivity field i s statistically isotropic (a = a ), a l l the one—(dimensional correlation functions w i l l be equal. X 2 If not, the vector orientated in the direction of the autoregressive parameter with the greater magnitude wil l have the slowest decaying correlation. The correlation along the perpendicular vector w i l l decay the most rapidly. Vectors such as that labeled B w i l l have a correlation function intermediate between these two. 1 I _ E / 3 > / \ 1 Figure 3-3 Vectors for the two-dimensional co r r e l a t i o n function In one dimension, the autocorrelation function for the Y process can be plotted from the f i r s t row of [RJ. In the above analysis, we ignore for the moment boundary effects in the stochastic process 38 model. The f i l t e r operating on the random sequence { e} wil l expand 2 2 its variance from a to a in {Y}. The variance expansion will e y be a function of the spatial lag operator [w] and the autoregressive parameters. In generating synthetic realizations, we wish to 2 simulate a predetermined variance a . Starting from a random sequence { e } with a standard deviation of one, this vector can be premultiplied by an appropriate factor to yield the desired value 2 for a . This factor, denoted n , can be calculated by modifying the original system of equations (3-2) by: { Y } = jw] { Y } + n • {e } 3-15 Following the same procedure as Martin (1974) used to derive equation 3-12, the covariance matrix is now given as: [C] = [M]T,2 o2& 3-16 Applying equation 3-13 gives: % [R] = M n 2 o2e 3-17 Now let c?e equal one. Then: o2 [R] = [M]n 2 3-18 M u l t i p l y both s i d e s o f t h i s equation by [MJ 1 t o g i v e : .2 r„i r»,l-l !VI 2 °i M M - W« 3 - 1 9 This m a t r i x equation can then be s o l v e d f o r n . For a twci-dimensional system, i t can be shown each d i a g o n a l element o f the product m a t r i x [R] [M] ^ i s computed from the r e l a t i o n : 4 p z ( 2 ) + a z a x p ( l t l ) + a z p z ( 1 > - a x P x C 1 ) + a l +"1 8 2 8 8 3-20 2 ,-, ,a p (2) + 1+ x x 8 where # (2), p (1) l a g 2 and l a g 1 ^ - d i r e c t i o n c o r r e l a t i o n c o e f f i c i e n t s p (2), p (1) l a g 1 and l a g 1 x - d i r e c t i o n c o r r e l a t i o n c o e f f i c i e n t s P (1,1) l a g 1 i n x - d i r e c t i o n , l a g 1 i n g - d i r e c t i o n c o r r e l a t i o n c o e f f i c i e n t a , a 2 X a - d i r e c t i o n , x - d i r e c t i o n a u t o r e g r e s s i v e parameters L e t t h i s r e l a t i o n be denoted by Q . Then n i s g i v e n as: 40 n = a y N[Q~ 3-21 I f the medium i s s t a t i s t i c a l l y i s o t r o p i c , t h i s reduces t o : n = a y a 2 ( 2 ) + a 2 p ( l , l ) - 2 a p ( l ) + a 2 + l 3-22 \|4 2 4 D e t a i l s o f d e r i v i n g r e l a t i o n 3-20 from equation 3-19 are gi v e n i n Appendix I . For a one-dimensional system, r e l a t i o n 3-20 reduces t o : g 2 + l - 2 p ( l ) a + g 2 p ( 2 ) 3-23 Therefore, n i s g i v e n as: |g2+l-2p ( l ) a + o t 2 p (2) 3-24 n = a S o l u t i o n f o r the {Y } sequence then f o l l o w s from the m o d i f i e d form o f equation 3-8. That i s : { ¥ } = ( [i] - [w] ) _ 1 ( iv { e } ) 3-25 The nearest neighbour f i l t e r r e q u i r e s t h a t both the { e} and (Y) sequences be zero-meaned. By adding the constant, y , t o each element o f {Y} and assuming the mean t o be independent o f the covariance o f the process; any mean f o r Y can be s p e c i f i e d w i t h o u t 41 a f f e c t i n g the second order moments. The system o f equations f o r the n e a r e s t neighbour process can then be w r i t t e n as: {Y} = y y + [w] • {Y} + {e} 3-26 However, y i s not added u n t i l a f t e r equation 3-25 i s s o l v e d . The advantage o f t h i s technique i n r e p e t i t i v e l y generating s p a t i a l l y c o r r e l a t e d c o n d u c t i v i t y f i e l d s can be seen i n equation 3-25. A t the s t a r t o f any s i m u l a t i o n , the m a t r i x ( [ i j - [w]) must be i n v e r t e d o n l y once. For each subsequent r e a l i z a t i o n , the i n v e r t e d m a t r i x i s simply m u l t i p l i e d by the e a s i l y generated random v e c t o r {n*e} The obvious drawback i s i n computing the i n v e r s e m a t r i x i t s e l f . F i r s t , note t h a t the m a t r i x ( [ i j - jwj) w i l l be sparse f o r a f i r s t o r d e r , nearest neighbour model. I n the one-dimensional problem, i t i s t r i d i a g o n a l l y banded. For a p b l o c k system, we must i n v e r t a (p x p) m a t r i x . This w i l l p l a c e a p r a c t i c a l l i m i t on the number o f b l o c k s which can be simultaneously generated f o r a s i n g l e s t a t i o n a r y porous medium. \ I n summary, the parameters needed t o generate a c o n d u c t i v i t y r e a l i z a t i o n can be l i s t e d : 1) the number o f rows and columns o f b l o c k s forming the c o n d u c t i v i t y f i e l d 42 2) the m a t r i x o f s c a l e d weights s p e c i f y i n g the neighbour dependence i n the b l o c k system 3) the a u t o r e g r e s s i v e parameters i n the two coordinate d i r e c t i o n s 4) the mean c o n d u c t i v i t y 5) the v a r i a n c e f a c t o r t o y i e l d a s p e c i f i e d c o n d u c t i v i t y v a r i a n c e The m a t r i x o f s c a l e d weights i s s e t up by u s i n g equation 3-3. The v a r i a n c e f a c t o r i s determined from equation 3-21. The p r o b a b i l i t y d e n s i t y f u n c t i o n from which the random sequence {e} i s generated i s c a r r i e d through i n t o the ( Y ) sequence. 3) Nearest Neighbour S t o c h a s t i c Process Model The nearest neighbour models were developed t o d e s c r i b e random media i n which the s p a t i a l dependence i s l o c a l . The medium i s u s u a l l y d i v i d e d i n t o a l a t t i c e o f p o i n t s o r r e g i o n s , w i t h which ar e a s s o c i a t e d e i t h e r continuous o r d i s c r e t e random v a r i a b l e s . I r r e g u l a r i l y spaced p o i n t s can be handled by an a p p r o p r i a t e l y d e f i n e d w e i g h t i n g f u n c t i o n . Thus, the nearest neighbour models cover a f a i r l y g e n e r a l c l a s s o f random processes. I n t h i s study, we apply the model t o a d i s c r e t i z e d r e p r e s e n t a t i o n o f a continuous random f u n c t i o n . The l a t t i c e p o i n t s 43 apply at the centers of the stochastic blocks. The random variable is then taken to be constant within each block. a) probabilistic structure There are two approaches to defining the spatial dependence in the nearest neighbour model. The dependence can be formulated in terms of a Markovian conditional probability or as a set of simultaneous joint probabilities. The differences will be briefly considered since the simultaneous autoregressive model; which is used in this study, is not Markovian. This departs from the autoregressive schemes in unilateral time series models. Whittle (1963), Besag (1974) and Bartlett (1975) discuss the consequences of these differences. The two formulations can be stated as follows: 1) Whittles' (1954, 1963) joint probability definition of the nearest neighbour model has been called the simultaneous, autoregressive model. In one dimension, i t can be written as: Y i = ^ \ - l + \ + l ) + H 3"27 where the terms are equivalent to those defined earlier in equation 3-1. 2) The spatial dependence can be defined in terms of conditional probabilities and 44 the Markov property. Here: p{Y. i a l l other values} = P { Y. |'Y. , Y. } 3-28 1 l i - i l+l where P represents a conditional probability. Brook (1964) shows that i f the dependence is unilateral (ie. only between Y^  and > then the conditional and joint probabilities are easily related. However, in a bilateral scheme, casting the conditional probabilities in terms of a set of joint probabilities leads to a unilateral structure. Bartlett (1975) shows further that the spectrums of the two processes are not equal. The difference in the two formulations will influence parameter estimation techniques, as discussed later in Chapter 8. Whittle (1954) and Bartlett (1975) show i t is possible to convert multilateral simultaneous models to a unilateral autoregression with an equivalent correlation structure, but such a transformation only appears useful in one dimension. Whittle (1954) questions the use of the exponential as an autocorrelation function in two dimensions because i t does not seem possible to construct simple two-dimensional stochastic process models that yield the exponential. Agterberg (1970) discusses this problem further. Rodriquez-Iturbe and Mejia (1974) used both an exponential function and a Bessel function representation of the two-dimensional spatial autocorrelation function to describe the distribution of precipitation in space. They found the form of the correlation function 45 d i d not seem t o matter provided the s c a l e o f the f u n c t i o n was adequately s p e c i f i e d . I n Chapter 4, the dependence o f the output standard d e v i a t i o n s i n head on ' d i f f e r e n t forms o f the c o n d u c t i v i t y a u t o c o r r e l a t i o n f u n c t i o n w i l l be i n v e s t i g a t e d . b) i n t e r p r e t a t i o n o f the model • T h e ^ i n t e r p r e t a t i o n o f the n e a r e s t neighbour model i n d e s c r i b i n g a c o n d u c t i v i t y f i e l d can e i t h e r be completely s p a t i a l o r l i n k e d t o a model d e s c r i b i n g the time sequence o f g e o l o g i c a l d e p o s i t i o n . The h o r i z o n t a l , m u l t i l a t e r a l s p a t i a l model c o u l d be separated from a time dependent v e r t i c a l model governing the t h i c k n e s s o f c o n d u c t i v i t y b l o c k s o r groups o f b l o c k s . The s p a t i a l and time models would be coupled through a c o r r e l a t i o n r e l a t i o n . This approach i s a k i n t o t h a t used i n the s y n t h e t i c generation o f sedimentary sequences (Schwarzacher, 1975) . Our use o f the model i s mechanistic. The time element i s ignored so t h a t the s p a t i a l model operates i n a l l d i r e c t i o n s . This approach i s a l s o adopted i n the s p a t i a l e s t i m a t i o n o f c o n d u c t i v i t y values u s i n g c o r r e l a t i o n a n a l y s i s o r k r i g i n g techniques (Delhomme, 1976) . The time element may be seen as b e i n g i n d i r e c t l y i n c l u d e d through an a n i s o t r o p i c s p a t i a l dependence. V e r t i c a l c o r r e l a t i o n lengths are expected t o be s h o r t e r s i n c e a g e o l o g i c process i s u s u a l l y thought t o be more continuous i n space than time. 46 The b l o c k s i z e s p e c i f i e d i n the model p r o v i d e s a good example o f the d i f f e r e n c e i n the two approaches. With the u n d e r l y i n g g e o l o g i c process i n mind, i t i s reasonable t o c o n s i d e r the b l o c k s as l a y e r s o f d i f f e r i n g c o n d u c t i v i t y o f random t h i c k n e s s o r l e n g t h . The b l o c k dLmensions would have an a s s o c i a t e d p r o b a b i l i t y d i s t r i b u t i o n . I n a s t a t i o n a r y medium, such an assumption i s incompatible w i t h a mechanistic nearest neighbour model and i t s s p a t i a l w e i g h t i n g m a t r i x . c) boundary e f f e c t s Consider a bounded, one-dimensional f l o w l i n e . I n generating a c o n d u c t i v i t y r e a l i z a t i o n , an i n i t i a l and t e r m i n a l b l o c k d e f i n i n g the {e} and {Y} v e c t o r s must always be s p e c i f i e d . Except f o r the two b l o c k s adjacent t o the upper and lower p o i n t boundaries, each b l o c k i s s u b j e c t t o the i d e n t i c a l s t o c h a s t i c process equation. However, a t the boundaries the dependence i s u n i d i r e c t i o n a l . Hence the process equations are n o n s t a t i o n a r y w i t h r e s p e c t t o the i n t e r n a l b l o c k s . The e f f e c t o f t h i s t r u n c a t i o n i s t o modify the covariance s t r u c t u r e o f the process f o r b l o c k s i n the immediate v i c i n i t y o f each o f the boundaries. As one moves f u r t h e r i n t o the system, the boundary e f f e c t s decay and the i n t e r n a l s t a t i o n a r i t y o f the process model r e e s t a b l i s h e s the s t a t i o n a r i t y i n the s p a t i a l covariance f u n c t i o n . T h i s e f f e c t occurs because the boundary b l o c k s are i n c l u d e d as unknowns i n the system o f equations. I f the c o n d u c t i v i t y values i n 4 7 these b l o c k s were s p e c i f i e d as boundary c o n d i t i o n s i n the model, the truncated equations disappear and the system i s everywhere s t a t i o n a r y . The e x t e n t o f the d e v i a t i o n s o f each o f the l a g covariances from the s t a t i o n a r y covariance f u n c t i o n can be c a l c u l a t e d from equation 3-12. I n the two-dimensional model, boundary e f f e c t s are more pronounced because o f the g r e a t e r number o f t r u n c a t e d nearest-neighbour equations i n c l u d e d i n the system. The e f f e c t s are g r e a t e s t i n the four corner regions o f the two-dimensional g r i d . F o r a f i x e d g r i d , the n o n s t a t i o n a r i t y i n the covariance f u n c t i o n i n c r e a s e s f o r l a r g e r values o f the a u t o r e g r e s s i v e parameters. Truncation has the e f f e c t o f i n c r e a s i n g the v a r i a n c e o f the process over t h a t o f the c e n t r a l s t a t i o n a r y r e g i o n as w e l l as i n c r e a s i n g the s t r e n g t h o f the c o r r e l a t i o n between neighbouring b l o c k s near the boundaries. I t f o l l o w s we w i l l always be r e p r e s e n t i n g a s t a t i o n a r y u n i t by r e a l i z a t i o n s which are nonstationary w i t h i n p a r t o f t h e i r domain. However, the magnitude o f the departure i s such t h a t i t w i l l n ot c o n s i d e r a b l y a l t e r the r e s u l t s i n t e r p r e t e d under the assumption o f s t a t i o n a r i t y . d) i n t e r p r e t a t i o n o f the boundaries The system boundaries must be i n t e r p r e t e d i n a s t o c h a s t i c framework. There are two c o i n c i d e n t boundaries t o c o n s i d e r ; those on 48 the flew equation and those on the boundary o f the c o n d u c t i v i t y f i e l d . Consider f i r s t those p h y s i c a l s i t u a t i o n s which can be adequately represented by a one-dimensional flow a n a l y s i s . I n t h i s study, constant head boundaries are de f i n e d f o r the flow equation. I t i s assumed they have no random component. A common i n t e r p r e t a t i o n f o r a steady s t a t e a n a l y s i s would be two head measurements i n piezometers a d i s t a n c e L apart. I n t h i s case, i t nay be app r o p r i a t e t o assume the c o n d u c t i v i t y f i e l d i s p a r t o f some l a r g e r s t a t i o n a r y u n i t . Then the imposed boundary on the c o n d u c t i v i t y f i e l d r e f l e c t s o n l y the dimensions o f the flow model and not an i d e n t i f i a b l e boundary i n the porous medium. An a l t e r n a t e i n t e r p r e t a t i o n a r i s e s i f we i n t r o d u c e g e o l o g i c boundaries i n the c o n d u c t i v i t y f i e l d . Consider a common two l a y e r c o n f i g u r a t i o n o f , say, a g l a c i a l l a k e c l a y o v e r l y i n g a f l u v i a l sand. L e t the two l a y e r s be o f approximately equal t h i c k n e s s and the s c a l e of the flow problem be such t h a t we are on l y i n t e r e s t e d i n flow from the top o f the c l a y t o the base o f the sand. I t i s important t o r e -cognize the c o n d u c t i v i t y f i e l d has a marked d e t e r m i n i s t i c component; t h a t o f the c l a y l a y e r o v e r l y i n g the sand. In a s t o c h a s t i c a n a l y s i s , t h i s cannot be a s c r i b e d t o a chance alignment i n one r e a l i z a t i o n from a space law t h a t d e s c r i b e s both u n i t s . The c l a y and the sand l a y e r s must be considered as two separate s t a t i o n a r y u n i t s w i t h t h e i r own c h a r a c t e r -i s t i c space laws. The model boundaries then r e f l e c t e x i s t i n g boundaries on s t a t i o n a r y u n i t s i n the f i e l d . 49 By defining each layer as a stationary unit, i t is implicitly assumed that the two layers are independent of each other. That is, there is no continuity in the space law across the interlayer boundary. It is also assumed that the units can be considered stationary everywhere up to their boundaries. The truncation of the direction dependence at the boundary is strictly a facet of the stochastic model and its boundary condition. In the twcr-dimensional model, the geometrical configuration of the boundaries and the flow conditions on those boundaries are assumed known without uncertainty. Unlike the one-<±Lmensional problem, no attempt is made to match the known boundary conditions with, say, the number of head measurements from a series of piezometers located along the boundary. If the domain under consideration is part of a larger stationary unit, then a l l boundaries are artificially imposed on the conductivity field. In a vertical cross-section, horizontal boundaries may or may not be coincident with geologic boundaries defining stationary units. Lateral boundaries on the conductivity field are less likely to be physically realizable as they require abrupt vertical boundaries between stationary units. The known boundary condition on the flow domain must also be associated with these vertical boundaries. Regardless of their physical interpretation, the boundaries are real in the sense that they are the conditions on which the model 50 is constructed and predictions made from its solution. It will be shown in the later flow analyses that the hydraulic head distributions must be interpreted in light of the imposed length parameters in our model of the flow system. 4) Analysis of Conductivity Realizations In this section, correlograms estimated from conductivity realizations are compared with theoretical correlograms calculated from equation 3-14. One and two-dimensional models are considered separately. First however, the necessary transformation required to obtain a lognormally distributed conductivity sequence is discussed. a) logarithmic transformation Hydraulic conductivity is assumed to be lognormally distributed for the model analysis. Base ten logarithms are used. A conductivity realization is formed by first generating the random sequence { E } from a normal distribution. The nearest neighbour f i l t e r (equation 3-25) is then applied to yield a normally distributed, correlated sequence {Y}. A non-zero mean ) is added to each member of {Y}. Finally, the exponential transform: $ ± = exp (2.3026 Y±) i = 1, p 3-29 is applied to {Y}, yielding a lognormally distributed sequence {K} of 51 conductivity values. The distributions of these sequences can be summarized: N N K : LN [0 , 0 [v%] yk'°k The general relation between the mean and variance of the normal and lognormal distributions is given as (Krumbein and Graybill, 1965): / ,1 2 2\ 3-30 y = exp Uv+hai a ) J K. y y a}= exp (2ay +a 2a 2) • (exp ( a 2 a 2 ) - l ) 3-31 k y y y where a = log ^ e and Y = log j_K Note that the mean and standard deviation of the conductivity series {K} each depend upon the mean and variance of the {Y} sequence. In working with conductivity data, only the mean of the log transformed (Y) sequence will depend upon the units in which conductivity is measured. The standard deviation is independent of the units of measurement. This can easily be seen by computing the expectations 52 on the relation Y = log (gK) where g is a scale parameter for different units of measurement. Matalas (1967) and Mejia and Rodriquez-Iturbe (1974) discuss the problem of preserving the correlation structure through the exponential transform. There will be a loss in the strength of the correlation in the (K } sequence compared to that in the {Y} sequence. 2 This loss is a function of both the variance a and the autoregressive y parameters a and a . The dependence will be examined in a succeeding subsection. Further complications arise in estimating the moments of the lognormal distribution from sample data. The mean and variance of the {Y} sequences are estimated by the usual statistics: Y = - £ Y 3-32 * p i=l i P y p-1 1=1 S 2 = - \ ? n (Y. - Y) 2 3-33 — 2 2 • • where Y and S estimate y and a , respectively. These statistics y y Y do not estimate the moments of the lognormal distribution. See Aitchison 2 and Brown (1957) for details on the estimators for y k and afc . 53 b) one-cltmensional models i) theoretical autocorrelation function The theoretical autocorrelation functions for the {Y } sequence for several values of the autoregressive parameter a are illustrated in Figure 3-4. The autocovariances, calculated from equation 3-12, were normalized by the variance associated with a central block in the grid. The block size was set to 10.0 x 10.0 units for plotting, although the autocorrelation-lag relation is independent of this block size in its computation. i i ) sample autocorrelation function Two sets of comparisons are made in considering the correlation structure in the realizations generated from the nearest neighbour model. In the first, the theoretical autocorrelation is compared with the estimated mean autocorrelation in the generated {Y} sequences. Secondly, the estimated correlations in the { K} sequences can be compared with their respective {Y} sequences to observe the decay in the correlation due to the exponential transform. At this stage, we are not reproducing a specific field situation, so no consideration is given to preserving the correlation function through the transformation. The estimator for the one-dimensional autocorrelation function is given as: 54 PM 1.0 0.8 0.6 A 0.4 0.2 \ \ \ Y l N h i y . a y ] m = 100 s = 10.0 \ \ \ \ a = 0.90 a = 0.50\ \ V a = 0.80^ 10 20 30 40 50 X (distance) 60 70 80 90 100 F i g u r e 3-4 T h e o r e t x c a l a u t o c o r r e l a t i o n f u n c t i o n s , o n e - d i m e n s i o n a l m o d e l F i g u r e 3-5 S a m p l e a u t o c o r r e l a t i o n f u n c t i o n a v e r a g e d o v e r a s e t o f c o n d u c t i v i t y r e a l i z a t i o n s , o n e - d i m e n s i o n a l m o d e l 55 r(h) = p-h p-h E i=l ( Y i + h " Y ) ( Y i " Y ) 3-34 1 P-l 77,2 . E. (Y. " Y) 1=1 1 where h is the lag number and p the number of conductivity blocks in the sequence. For any one realization, r(h) need not correspond to its theoretical equivalent because the random component built into each realization may not average out over the finite sample length. Because the space averages may not equal the ensemble parameters from which the realization is generated, the sequence is nonergodic. A detailed discussion of this divergence follows in Chapter 4. The estimated mean autocorrelation function for the Y process with an autoregressive parameter of 0.80 is plotted in Figure 3-5. The average was taken over 400 realizations. Also plotted is the theoretical autocorrelation function. To indicate the range over which each of the correlation coefficients varied in the simulation, a set of vertical crossbars are plotted to identify one standard deviation on either side of the estimated mean coefficient. An almost identical plot is obtained when the variability in the Y process is doubled or the number of realizations over which the average is formed is reduced to 200. The extent to which the spatial correlation is reduced in transforming the {Y} sequence to the conductivity sequence in {K} is illustrated in Figure 3-6. The autoregressive parameter is fixed 56 T 1 1 1 1 10 20 30 40 50 60 X (distance) F i g u r e 3-6 Decay i n a u t o c o r r e l a t i o n t h r o u g h t h e e x p o n e n t i a l t r a n s f o r m f r o m t h e {Y} t o {K} s e q u e n c e 1.0 0.8 0.6 0.4 0.2 H m = 100 s =10.0 a =0.80 1 I 10 20 30 40 50 X (distance) 60 70 80 90 100 F i g u r e 3-7 A u t o c o r r e l a t i o n f u n c t i o n i n a d i s c r e t e b l o c k m o d e l 57 w h i l e the standard d e v i a t i o n i n {Y} i s i n c r e a s e d . I t i s t h i s c o r r e l a t i o n s t r u c t u r e ( l a b e l e d by the curves T) t h a t i s i n s e r t e d i n t o the con-d u c t i v i t y b l o c k s . i i i ) i n t e g r a l s c a l e The i n t e g r a l s c a l e , as d e f i n e d by Lumley and Panofsky (1964), i s given f o r a continuous medium as: 7 o (5K) d x = A 3-35 0 where p(x) i s the one-dimensional a u t o c o r r e l a t i o n f u n c t i o n i n the x d i r e c t i o n . Thus, the i n t e g r a l s c a l e i s equal t o the area beneath the a u t o c o r r e l a t i o n f u n c t i o n . I t can be i n t e r p r e t e d as the average d i s t a n c e over which c o n d u c t i v i t y values are c o r r e l a t e d . I t i s used as a s c a l e parameter t o c h a r a c t e r i z e the extent o f the s p a t i a l dependence w i t h i n a heterogeneous medium. In a d i s c r e t e block system, the a u t o c o r r e l a t i o n f u n c t i o n takes the form o f a decreasing step f u n c t i o n . The same c o r r e l a t i o n i s a p p l i e d everywhere w i t h i n the given b l o c k l e n g t h . T h i s form i s i l l u s t r a t e d i n F i g u r e 3-7. The bl o c k l e n g t h i s s e t t o 10.0 u n i t s . I t f o l l o w s t h a t t h e mijiimum i n t e g r a l s c a l e ( i e . f o r an u n c o r r e l a t e d sequence o f blocks) i s determined by the b l o c k s i z e i n the system. 58 An approximate i n t e g r a l s c a l e can be computed by summing the area beneath the sample a u t o c o r r e l a t i o n f u n c t i o n u n t i l i t f i r s t becomes negative. The t h e o r e t i c a l a u t o c o r r e l a t i o n f u n c t i o n remains p o s i t i v e but v a n i s h i n g l y s m a l l a t l a r g e lags f o r f i r s t o r d e r , b i l a t e r a l auto-r e g r e s s i o n s . Therefore, o n l y a sm a l l e r r o r i s i n t r o d u c e d by t r u n c a t i n g the summation a f t e r a s u f f i c i e n t number o f l a g s have been i n c l u d e d . The sample a u t o c o r r e l a t i o n f u n c t i o n w i l l decay towards zero and then f l u c t u a t e about t h a t l e v e l f o r a l l l a r g e r l a g s . Because neighbouring l a g values o f the sample a u t o c o r r e l a t i o n f u n c t i o n are c o r r e l a t e d , i t may f a i l t o damp out as r a p i d l y as the t h e o r e t i c a l a u t o c o r r e l a t i o n f u n c t i o n . T h i s w i l l l e a d t o an overestimate o f the i n t e g r a l s c a l e . However, the e r r o r w i l l be s m a l l s i n c e the exte n s i o n i s dominantly r e f l e c t e d i n the t a i l o f the sample a u t o c o r r e l a t i o n f u n c t i o n , where the c o r r e l a t i o n c o e f f i c i e n t s are s m a l l e r . I n Table 3-1, the estimated mean and standard d e v i a t i o n o f the c o n d u c t i v i t y i n t e g r a l s c a l e s f o r s e v e r a l s e t s o f space law parameters are summarized. The estimates are formed from 400 r e a l i z a t i o n s o f a 100 block system. The bl o c k s i z e i s 10.0 u n i t s . Because the sample a u t o c o r r e l a t i o n f u n c t i o n v a r i e s f o r each o f the r e a l i z a t i o n s so too does the i n t e g r a l s c a l e . This again r e f l e c t s the nonergodic behavior o f each o f the bounded r e a l i z a t i o n s . 59 ct=0.00 a=0.80 ct=0.95 I 10.61 30.19 53.87 0.20 1.26 8.53 15.12 I 10.68 26.62 47.39 1 0 1 V 0.42 ! y s x 1.33 7.78 13.79 I 10.48 19.91 35.74 0.84 1.18 5.17 10.65 Note: block length 10.0 units Table 3-1 Summary of Estimated Integral Scale Moments, One-Dimensional Model Note the integral scale for an uncorrelated sequence (<*=0.0) is slightly greater than 10.0 units. It attains a value of 10.0 because of the perfect correlation within each block. The fraction greater than 10.0 is due to the random alignment of like conductivity values. Inspection of Table 3-1 also indicates the estimated mean integral scale decreases as the standard deviation (cr ) increases, 60 given a fixed autoregressive parameter. This reflects the loss in correlation in transforming from the {Y} to the { K } sequence. c) two-dimensional models i) theoretical autocorrelation function The theoretical autocorrelation function for the Y process for several sets of autoregressive parameters is illustrated in Figure 3-8. These correlation functions are calculated for the (10 x 20) grid shown in Figure 3-2. The autocovariances, calculated from equation 3-12, were normalized by the variance associated with a central block in the grid. Figure 3-8a shows a statistically isotropic correlogram with a and a equal to 0.80. The autocorrelation for the long axis of the system is plotted. First note that including a second dimension causes the autocorrelation function to decay more quickly than for the one-dimensional system with an equal autoregressive parameter. This is due to the interlocking of the dependence along the coordinate axes through the nearest neighbour relation. Figure 3-8b illustrates the correlogram for a statistically anisotropic medium. Here, a equals 0.95 and a equals 0.50. Y: N[My a Y] m = 10 n = 20 p s= 10.0 - 1 1 r 10 20 30 40 50 60 X (distance) Figure 3-8 Theoretical autocorrelation functions two-dimensional model 62 i i ) sample autocorrelation function The estimated mean autocorrelation function in the ( K ) sequence, calculated over 300 realizations, is plotted in Figure 3-9. The estimator for the tridimensional autocorrelation function i s given as (Agterberg, 1974): m-a n-b i 1 1 U - (Y. . - Y) r(a,b, = V n = 1 1 = 1 ^ 3-36 m n 1=1:=! < Y i j - ? ) 2 where a and b are lag numbers in the i and j coordinate directions, respectively. The number of data points in the i and j coordinate directions are m and n, respectively. Correlograms along the coordinate axes are computed by r(a,o) and r(o,b). For our plots, the origin of the function is set in the upper left-hand corner of the grid. Only positive lags in a and b are calculated. In Figure 3-9, both autoregressive parameters were set to 0.80. The standard deviation for each of the estimated autocorrelation coefficients was about 0.10 for a l l lags up to six. The variability calculated over the short dimension of the block system was slightly higher. 63 1.0 0.8 \ \ PM 0.6 4 \ 0.4 0.2 \ \ \ \ \ m = 10 n = 20 p = 200 s = 10.0 MC = 300 ax= a z = o.80 Y: N[-2.00, 0.43] — I 1 1 T — 10 20 30 40 50 60 X (distance) Figure 3-9 Sample autocorrelation function averaged over a set of conductivity r e a l i z a t i o n s , two-dimensional model 64 i i i ) i n t e g r a l s c a l e The i n t e g r a l s c a l e along each o f the two co o r d i n a t e axes can be estimated i n the same manner as f o r the one-dimensional r e a l i z a t i o n s . The estimated mean i n t e g r a l s c a l e and i t s standard d e v i a t i o n c a l c u l a t e d f o r a (10 x 20) block system from 300 r e a l i z a t i o n s are summarized i n Table 3-2 f o r s e v e r a l s e t s o f parameter v a l u e s . The bl o c k s i z e i s s e t t o 10.0 u n i t s f o r computation. a y a , a x z Xx Xx X z Xz 0.21 (.75,.75) 17.9 3.2 18.0 3.9 0.43 (.80,.80) 17.4 3.4 17.5 4.0 0.43 (0.0,0.0) 10.4 0.7 10.4 0.9 0.42 (.95,.95) 25.6 6.1 26.8 7.8 0.46 (.95,.50) 18.4 3.8 13.9 2.3 0.46 (.84,.84) 18.3 3.5 18.7 4.3 Note: b l o c k l e n g t h 10.0 u n i t s Table 3-2 Summary o f Estimated I n t e g r a l S c a l e Moments, Two-Dimensional Model 65 Boundary effects can create an apparently statistically anisotropic relation in the mean integral scale for an isotropic autoregressive scheme. This is seen for and a g set to 0.95. The integral scale i s greater along the short axis because the higher correlations between blocks near the boundary extends relatively further into the stationary region compared to the correlation along the long axis. 5) Examples of Model Realizations A method of generating realizations of a discrete conductivity field from a known space law has now been outlined. It is worthwhile to illustrate several one-dimensional realizations in order to get a frame of reference for the integral scale, the autoregressive parameter, and the form of the sequence itself. In the next set of Figures, one half of a 100 block sequence is plotted. The f i r s t plot, Figure 3-10a, is an internally uncorrelated realization in the {e} vector, generated from N|O.O, 0.2OJ. Figure 3-10b illustrates how this series is modified by application of the nearest neighbour f i l t e r with an autoregressive parameter of 0.80. This sequence is distributed as N 0.0, 0.4lj. For comparative purposes, a random series from an {e} vector generated from N^O.0, 0.4lj is plotted in Figure 3-10c. The integral scale in the second sequence is estimated as 31.7 units, assuming 10.0 unit blocks. Note the different i g u r e 3- 10 E x a m p l e s o f r e a l i z a t i o n s , o n e -d i m e n s i o n a l , n e a r e s t n e i g h b o u r m o d e l F i g u r e 3-10 Cont. h o r i z o n t a l s c a l e s on these p l o t s . 6) Conclusions 1) The simultaneous, nearest neighbour, a u t o r e g r e s s i v e model provides an adequate generation technique f o r c o n d u c t i v i t y r e a l i z a t i o n s s a t i s f y i n g a known space law i n a s t a t i o n a r y medium. Both s t a t i s t i c a l l y i s o t r o p i c and a n i s o t r o p i c c o r r e l a t i o n f u n c t i o n s are e a s i l y handled. The estimated sample a u t o c o r r e l a t i o n f u n c t i o n , averaged over a s e r i e s o f r e a l i z a t i o n s , c l o s e l y matches i t s t h e o r e t i c a l form. The method can be a p p l i e d t o systems represented by a moderately l a r g e number o f d i s c r e t e b l o c k s . 2) The nature o f the system boundaries must be i n t e r p r e t e d i n terms o f the s t o c h a s t i c framework o f the problem. There are two s e t s o f boundary c o n d i t i o n s t o c o n s i d e r ; those on the flow equation and those on the c o n d u c t i v i t y f i e l d . The l a t t e r can e i t h e r be c o i n c i d e n t w i t h g e o l o g i c boundaries d e f i n i n g s t a t i o n a r y u n i t s o r imposed on t h e s o l u t i o n by the flow model i t s e l f . 3) The t r u n c a t i o n o f the b i l a t e r a l dependence o f the n e a r e s t neighbour equations a t the boundary b l o c k s leads t o n o n s t a t i o n a r i t y i n the covariance f u n c t i o n w i t h i n p a r t o f the conductivity realization. The covariance matrix allows us to specify the deviation from stationarity and so have a measure of its importance. An approximate integral scale for each conduct!vity realization can be computed by summing the area beneath the sample autocorrelation function until i t first becomes negative. The calculation takes into account the step function form of the auto-correlation in a discrete medium. CHAPTER 4 One-Dimensional Simulation Results 1) Introduction The stochastic analysis of the one-dimensional, steady state groundwater flow eguation has recently been considered by several authors using different mathematical techniques. Although this study is chiefly directed towards the two-dimensional analysis, i t is useful to return to the one-dimensional model to reconsider several problems that arise in using a discrete block conductivity model. In addition, several concepts in the model analysis are more easily demonstrated in one dimension. In this chapter, we consider the effects of introducing a spatial covariance between the conductivity blocks on the output variabilities in our model predictions. The analysis will be extended to a two-dimensional flow equation in Chapter 5. The one-dLmensional analysis is an ex-tension of earlier work by Freeze (1975, 1977). 2) Steady State Flow Problem and Solution Technique The stochastic equation for one-dimensional, steady state, saturated flow in the x direction is given as: 3_ 9 x K ( x ) 3<j> ' 3x = 0 4-1 72 where <l>1, the h y d r a u l i c head, and K ( x ) , the h y d r a u l i c c o n d u c t i v i t y , are both random f u n c t i o n s i n space. The boundary c o n d i t i o n s on <f> w i l l be : *'(0) = * J 4-2 <t>* ( L ) = <fL f o r a f l o w l i n e o f l e n g t h L. Because the head values are f i x e d a t the two boundaries, the va r i a n c e i n the head d i s t r i b u t i o n w i l l be nonstationary along the f l o w l i n e ( i e . a f u n c t i o n o f p o s i t i o n ) . Even i f the boundary heads a r e allowed t o be random, say due t o u n c e r t a i n t y because o f measurement e r r o r , the head v a r i a n c e w i l l s t i l l be non-s t a t i o n a r y . T h i s i s because the random component i n the boundary heads i s chosen independently o f the v a r i a b i l i t y i n head, values w i t h i n the flo w domain. The flow domain i s d i v i d e d i n t o m e q u i s i z e d segments w i t h a s i n g l e c o n d u c t i v i t y v a l u e assigned t o each segment, i = l , 2, ...m. The segments can be thought o f as bl o c k s i n one dimension i f no v a r i a t i o n i n c o n d u c t i v i t y occurs p e r p e n d i c u l a r t o the f l o w l i n e . Then the flow remains one-dimensional. I n the d i s c r e t e approach, the conceptual choice must be made o f e i t h e r t o con s i d e r the c o n d u c t i v i t y values as constant w i t h i n the block o r t o con s i d e r them as r e p r e s e n t i n g some h y d r a u l i c a l l y e q u i v a l e n t value f o r the s m a l l e r s c a l e v a r i a t i o n s w i t h i n the blo c k . The l a t t e r i n t e r p r e t a t i o n i s adopted here. S o l u t i o n o f the flo w equation i s independent o f the c h o i c e made. 73 Figure 4-1 diagranmatically illustrates this boundary value problem. The analytical solution for cj>' (x) is developed by re-cognizing the interlayer continuity conditions. The solution is detailed in Appendix II. = 0 * i X2 X3 / / i = 2 i = 3 f / / > Xm-2 Xm-1 X = L i = m-1 i = m ^ vm-1 K m Figure 4-1 One-dimensional stochastic flow f i e l d A Monte Carlo technique is used to solve the stochastic boundary value problem (equations 4-1, 4-2). The basis of Monte Carlo simulation is the repetitive generation of conductivity realizations from a known space law and subsequent solution of a deterministic flow problem. By repeating the analysis over a series of runs, the distribution of hydraulic heads $>' at any point x can be analyzed to obtain estimates of the mean and variance in the output variable. 74 The variance can then be used as a measure of the uncertainty in our prediction. This uncertainty reflects the lack of knowledge in head values due to our inability to replicate in our model the pattern of spatial variation in conductivity as i t exists in the porous medium. The Monte Carlo technique is founded on two basic assumptions. These can be stated: 1) As the number of runs in a simulation increases, the moments of the output distribution converge towards unique values. That i s , we can approximate the variability from the infinite ensemble of conductivity realizations by a finite number of samples from that ensemble. 2) The estimated moments from a finite number of realizations converge to the moments that would be determined from an analytical solution to the equivalent problem posed as a continuous stochastic differential equation. Sagar (1978) refers to the Monte Carlo technique as the sample function approach and briefly comments on its applicability. Although the exact conditions for the validity of this approach are not established, we proceed under the assumption that the Monte Carlo 75 technique does y i e l d approximate s o l u t i o n s t o the s t o c h a s t i c boundary value problems considered. The porous medium i n Freeze's (1975) a n a l y s i s was d e s c r i b e d as nonuniform homogeneous. This terminology i s based on d e f i n i n g the flow domain as a s i n g l e p o p u l a t i o n r a t h e r than a s s o c i a t i n g a p r o b a b i l i t y d e n s i t y f u n c t i o n w i t h each p o i n t i n the flo w domain. I n order t o con s i d e r s p a t i a l covariance r e l a t i o n s w i t h i n the c o n d u c t i v i t y f i e l d , the l a t t e r d e f i n i t i o n must be adopted. A nonuniform homogeneous medium i s conceDtually e q u i v a l e n t t o a s t a t i o n a r y medium, w i t h o u t c o n s i d e r a t i o n being g i v e n t o the s p a t i a l covariance. The problem Freeze (1975) s o l v e d forms a subset o f the s o l u t i o n s t o be presented i n succeeding s e c t i o n s . H i s two major steady s t a t e r e s u l t s can be summarized: 1) As the degree o f nonuniformity i n c o n d u c t i v i t y i n c r e a s e s , so too does the standard d e v i a t i o n i n the h y d r a u l i c head d i s t r i b u t i o n . 2) As the number o f c o n d u c t i v i t y b l o c k s m i s in c r e a s e d f o r a f i x e d f l o w l i n e l e n g t h , the standard d e v i a t i o n i n h y d r a u l i c head decreases. For an unc o r r e l a t e d system o f b l o c k s , t h i s corresponds t o d e c r e a s i n g the i n t e g r a l s c a l e t o t h e s i z e o f the s m a l l e r b l o c k . 7 6 3) I n f l u e n c e o f C o n d u c t i v i t y Space Law Parameters a) i n t r o d u c t i o n The i n t e n t o f t h i s s e c t i o n i s t o s y s t e m a t i c a l l y i n v e s t i g a t e the i n f l u e n c e o f the c o n d u c t i v i t y space law parameters i n determining the output standard d e v i a t i o n s i n h y d r a u l i c head. The e f f e c t s t o be considered are: 1) dependence on the i n t e g r a l s c a l e 2) dependence on the standard d e v i a t i o n i n c o n d u c t i v i t y 3) d i s c r e t i z a t i o n i n t e r v a l 4) l e n g t h o f the flow system Before proceeding, some general comments on the one-dimensional s i m u l a t i o n s are presented. In p l o t t i n g the r e s u l t s , d i s t a n c e s and output standard d e v i a t i o n s are expressed as dimensionless q u a n t i t i e s , d e f i n e d by the r e l a t i o n s : D__ = x/L x <f> = (j.'/ * L - + 4-3 0 77 Here, L is the length of the flowline and <f> and <f> Q the two boundary heads. The estimated mean of the head distribution at a position x is denoted by <j>' ; the estimated standard deviation by S + I . Unless otherwise indicated, a l l the simulation results were calculated over 400 conductivity realizations on a 100 block system. Each block was 10 cm on the side. The boundaries have head values set at <j> = 100 cm and <j> L = 0 cm. Units are chosen here only for convenience; they do not represent any attempt to define or restrict the scale at which the model is applied. The standard deviation in head curves are specified by the parameters of {Y} (equation 3-25). The mean of the {Y} sequence was set to -2.00; the corresponding transformed mean conductivity (in cm/s) is a function of a , the standard deviation in {Y} (equation 3-30). The simulations were carried out at the University of British Columbia Computing Center on an IBM 370/168. For a 100 block system averaged over 400 runs, the computing time was about 40 s. b) estimated mean hydraulic head The dimensionaless estimated mean head f e l l linearly between the two boundary heads. It can be shown that the hydraulically equivalent single conductivity value for one-dimensional flow perpendicular to a layered system is equal to the harmonic mean conductivity of the layers (Bear, 1972). Using the harmonic mean conductivity, a single deterministic solution for the head distribution can be calculated. Any conductivity value will give the head solution, but only the harmonic mean will preserve the flux. Although the estimated mean head did not equal the single deterministic solution, deviations from i t were small with no apparent bias to overestimate or underestimate its value. For those simulations where the head distribution can be approximated by a normal distribution, i t i s possible to calculate confidence intervals. For these cases, the deterministic head was well within a 95% confidence interval on the estimated mean head. c) dependence on the integral scale Figures 4-2a, 4-2b and 4-2c show the effect of a larger integral scale. In each of the plots, the conductivity standard deviation is fixed. Three different conductivity standard deviations are considered. The curves labeled a = 0.0 refer to simulations in which uncorrelated conductivity blocks were used. They correspond to the curves presented by Freeze (1975). On each of the plots, we see that as the integral scale in-creases, so too do the standard deviations in the head values. In other words, as the extent of the spatial dependence increases in a conduct!-79 Figure 4-2 Dependence of the hydraulic head on standard deviations i n the i n t e g r a l scale 80 v i t y f i e l d , our u n c e r t a i n t y as t o the head d i s t r i b u t i o n between the two p o i n t boundaries w i l l i n c r e a s e . T h i s r e s u l t i s c o n d i t i o n a l on being given no a d d i t i o n a l head measurements i n t e r n a l t o the boundaries. The same r e s u l t was found by Bakr (1976) and Gelhar (1977) u s i n g s p e c t r a l a n a l y s i s techniques. An e x p l a n a t i o n o f t h i s behavior f o l l o w s a f t e r we con s i d e r the e f f e c t s o f the other space law parameters on the head v a r i a b i l i t y . d) dependence on the c o n d u c t i v i t y standard d e v i a t i o n Another way o f l o o k i n g a t the same i n f o r m a t i o n i s t o f i x the i n t e g r a l s c a l e and compare the standard d e v i a t i o n s i n head f o r d i f f e r e n t i n p u t standard d e v i a t i o n s i n c o n d u c t i v i t y . The mean i n t e g r a l s c a l e c o u l d o n l y be approximately s p e c i f i e d a p r i o r i by choosing ap p r o p r i a t e values o f the aut o r e g r e s s i v e parameter a and the v a r i a n c e f a c t o r n . Fi g u r e 4-3 i l l u s t r a t e s the dependence o f the head v a r i a b i l i t y on the c o n d u c t i v i t y standard d e v i a t i o n . As the heterogeneity o f the medium in c r e a s e s ( i n c r e a s i n g a y ), so too does the degree o f u n c e r t a i n t y a s s o c i a t e d w i t h the p r e d i c t e d h y d r a u l i c head values. The standard d e v i a t i o n s i n head a s s o c i a t e d w i t h d i f f e r e n t media have been compared by f i x i n g a l l the space law parameters but one, which was then v a r i e d . I n f i x i n g the i n t e g r a l s c a l e ( X ) and v a r y i n g the l o g c o n d u c t i v i t v standard d e v i a t i o n (a ) , we must assume y each c r ^ i s e q u a l l y l i k e l y a s s o c i a t e d w i t h t h a t X . I f there i s a 81 Figure 4-3 Dependence of the standard d e v i a t i o n s i n h y d r a u l i c head on the standard d e v i a t i o n i n h y d r a u l i c c o n d u c t i v i t y 82 tendency f o r s p a t i a l c o r r e l a t i o n s t o be s m a l l e r the l a r g e r the c o n d u c t i v i t y v a r i a b i l i t y ; then t o compare head v a r i a b i l i t i e s , a s m a l l e r c o n d u c t i v i t y standard d e v i a t i o n should be matched w i t h a l a r g e r i n t e g r a l s c a l e . T h i s w i l l tend t o reduce the d i f f e r e n c e s between the head v a r i a b i l i t i e s . Our comparisons can then be i n t e r p r e t e d s t r i c t l y o n l y i n terms o f the parameters which are f i x e d and which are v a r i e d . e) d i s c r e t i z a t i o n One o f the most d i f f i c u l t conceptual problems i n a p p l y i n g the nearest neighbour model and, more g e n e r a l l y , any d i s c r e t e r e p r e s e n t a t i o n of a continuous medium; i s the s i z e o f the b l o c k used t o c h a r a c t e r i z e c o n d u c t i v i t y v a r i a t i o n s i n space. I n t u i t i v e l y , one might expect a need f o r s m a l l e r b l o c k s i n media w i t h s m a l l e r i n t e g r a l s c a l e s and/or l a r g e r standard d e v i a t i o n s i n c o n d u c t i v i t y . T h i s assumes t h a t f o r these c o n d i t i o n s , the consequent accuracy o f s i n g l e values r e p r e s e n t i n g the h y d r a u l i c behavior o f the b l o c k decreases. An important f i e l d o f f u t u r e research would be the d e t a i l e d c o n s i d e r a t i o n o f the r e l a t i o n s h i p between a d i s c r e t e s t o c h a s t i c r e p r e s e n t a t i o n o f a porous medium and i t s h y d r a u l i c behavior i n terms o f macroscopic d e t e r m i n i s t i c f l o w theory. The fundamental assumption o f our modeling approach r e s t s on the nearest neighbour model generating an adequate r e p r e s e n t a t i o n o f the h y d r a u l i c behavior o f r e a l media. 83 What we l a c k i s a s e t o f o b j e c t i v e c r i t e r i a which c o u l d be used t o evaluate the d i s c r e t i z a t i o n i n t e r v a l w i t h i n a s t a t i o n a r y medium. A r a t h e r simple one can be developed from the nearest neighbour a u t o c o r r e l a t i o n f u n c t i o n when i t i s considered as an approximation t o a known continuous a u t o c o r r e l a t i o n f u n c t i o n . The sampling problem o f o b t a i n i n g the a u t o c o r r e l a t i o n f u n c t i o n from f i e l d d a t a i s not considered here. A s m a l l e r b l o c k s i z e w i l l always g i v e a more accurate r e p r e s e n t a t i o n o f the i n t e g r a l s c a l e . For example, we would not represent a 10 metre i n t e g r a l s c a l e by a s e t o f 10 metre b l o c k s . The i n t e g r a l s c a l e can be viewed as an upper l i m i t f o r the d i s c r e t i z a t i o n . Here the area under the a u t o c o r r e l a t i o n f u n c t i o n would be r e p l i c a t e d but not the decay i n the c o r r e l a t i o n f o r i n c r e a s i n g l a g s . The bl o c k s should be chosen i n such a way as t o preserve the i n t e g r a l s c a l e and approximate the s p a t i a l decay i n c o r r e l a t i o n . I t w i l l a l s o be r e l a t e d t o the s c a l e o f the f l o w a n a l y s i s and the d e f i n i t i o n o f s t a t i o n a r y u n i t s w i t h i n the flo w domain. An obvious problem t o co n s i d e r i s the e f f e c t on the head v a r i a b i l i t i e s o f u s i n g v a r y i n g b l o c k s i z e s t o represent a known s p a t i a l covariance s t r u c t u r e . To answer t h i s q u e s t i o n , a system o f 200 b l o c k s , each 5 u n i t s l o n g , i s generated w i t h an au t o r e g r e s s i v e parameter chosen t o pr o v i d e the same i n t e g r a l s c a l e as i n a 100 b l o c k system based on b l o c k s 10 u n i t s l o n g . A comparison o f the head standard d e v i a t i o n s i s shown i n F i g u r e 4-4a. The mean and v a r i a n c e o f the o Sty Y: N[-2.00, 0.42] 0O= 100.0 0 L = 0.0 MC = 400 A m = 200 s = 5.0 X=24.6 m= 100 s= 10.0 X= 26.6 Sty Y: N [-2.00, 0.43] .ty0= 100 0L= 0 m= 100 s= 10.0 X = 47.4 Y: N [-2.00, 0.42] 0O = 200 0 L = 0 m= 200 s= 10.0 X =49.3 Y: N[-2.00, 0.43] 4>Q = 200 <t>i = 0 m= 100 s= 10.0 X=26.6' Effects of d i s c r e t i z a t i o n , length of the flow system 85 c o n d u c t i v i t y d i s t r i b u t i o n are equal. The p l o t suggests t h a t the head v a r i a b i l i t i e s , aside from a minor s t a t i s t i c a l v a r i a t i o n , w i l l not depend upon the bl o c k s i z e used; provided a s u f f i c i e n t l y accurate r e p r e s e n t a t i o n o f the i n t e g r a l s c a l e i s made. For a system o f s m a l l e r b l o c k s , the l a r g e r a u t o r e g r e s s i v e parameter f o r c e s neighbouring c o n d u c t i v i t y values t o be more h i g h l y c o r r e l a t e d . S i n g l e c o n d u c t i v i t y values are a p p l i e d over s h o r t e r d i s t a n c e s f o r the s m a l l e r b l o c k s . However, the s p a t i a l c o n t i n u i t y o f the e n t i r e sequence i n terms o f i t s d e v i a t i o n s from the mean c o n d u c t i v i t y should remain about the same. Both Warren and P r i c e (1961) and Freeze (1975) found the output v a r i a b i l i t i e s would decrease as a bounded domain was d i s c r e t i z e d i n t o a l a r g e r number o f s m a l l e r b l o c k s . I n both s t u d i e s , u n c o r r e l a t e d sequences o f bl o c k s were used. We see now t h a t t h i s r e s u l t should not be i n t e r p r e t e d s o l e l y i n terms o f the s c a l e o f the nonuniformity, but r a t h e r i n l i g h t o f the s m a l l e r i n t e g r a l s c a l e s represented by the s m a l l e r u n c o r r e l a t e d b l o c k s . D i s c r e t i z i n g the flow domain i n t o s m a l l e r b l o c k s w h i l e m a i n t a i n i n g the i n t e g r a l s c a l e w i l l l e a d t o the same standard d e v i a t i o n s i n head a t any f i x e d p o i n t . f) l e n g t h o f flow system I n a l l o f the s o l u t i o n s considered so f a r there has been a f i x e d l e n g t h L between the two p o i n t boundaries. I f one were t o double 86 both the l e n g t h o f the f l o w l i n e and the head drop across the system, there would be a decrease i n the h y d r a u l i c head standard d e v i a t i o n a t any f i x e d The head drop i s doubled t o maintain the same expected g r a d i e n t across each bl o c k . A l l o t h e r space law parameters remain equal. T h i s r e s u l t i s shown i n F i g u r e 4-4b, f o r the curves l a b e l e d A and B. Equal head standard d e v i a t i o n s w i l l o n l y r e s u l t when the r a t i o X/L, the estimated mean i n t e g r a l s c a l e t o the l e n g t h o f the f l o w l i n e , i s equal. Again, a l l o t h e r parameters are equal. T h i s r e s u l t i s i n d i c a t e d by the curve l a b e l e d C i n F i g u r e 4-4b. Here both the l e n g t h o f the f l o w l i n e and the i n t e g r a l s c a l e have been halved. For the curves B and C, the r a t i o X/L i s approximately equal. T h i s i m p l i e s t h a t .for* bounded, one-dimensional systems, the r a t i o : 1 X/L i s as e q u a l l y important a s t o c h a s t i c parameter as i s the c o n d u c t i v i t y v a r i a b i l i t y o r the magnitude o f the i n t e g r a l s c a l e i t s e l f . An e x p l a n a t i o n f o r the dependence o f the standard d e v i a t i o n s i n head on the r a t i o X/L can be approached i n s e v e r a l ways. Consider the case where a s e r i e s o f h i g h o r low c o n d u c t i v i t y b l o c k s are adjacent t o each o t h e r . Then the head s o l u t i o n can move f u r t h e r away from i t s expected l i n e a r decrease along x than i f there were no such tendency f o r l i k e c o n d u c t i v i t y values t o f o l l o w each o t h e r . For s m a l l e r i n t e g r a l s c a l e s , i t i s more probable t h a t the head s o l u t i o n f o r a given r e a l i z a t i o n w i l l approach i t s mean s o l u t i o n because there 87 i s l e s s o f a tendency f o r the head drop t o be c o n s i s t e n t l y above o r below i t s expected value. Therefore, on averaging over a s e r i e s o f r e a l i z a t i o n s , alignment o f l i k e c o n d u c t i v i t y values w i t h i n each r e a l i z a t i o n tends t o i n c r e a s e the standard d e v i a t i o n i n the head d i s t r i b u t i o n f o r any p o i n t along the f l o w l i n e . This behavior can a l s o be e x p l a i n e d i n terms o f the c r o s s -c o r r e l a t i o n between c o n d u c t i v i t y and head va l u e s . With a more h i g h l y c o r r e l a t e d c o n d u c t i v i t y sequence ( i e . l a r g e r i n t e g r a l s c a l e s ) , d e v i a t i o n s i n the head values from the mean s o l u t i o n over some d i s t a n c e C w i l l be more h i g h l y c o r r e l a t e d . I f the head val u e a t a f i x e d p o i n t i s above i t s mean; then i t i s more probable t h a t neighbouring head values w i l l a l s o be higher than t h e i r means. On averaging over a s e r i e s o f r e a l i z a t i o n s t h i s c o r r e l a t i o n w i l l tend t o expand the standard d e v i a t i o n of the head ( d i s t r i b u t i o n a t any p o i n t . The l e n g t h o f the flo w system c o n t r o l s the number o f a s y m p t o t i c a l l y independent samples from the c o n d u c t i v i t y d i s t r i b u t i o n . For the fou r curves i n F i g u r e 4-2b, t h i s number i s 93, 63, 37, 21; r e s p e c t i v e l y , as one moves from a s m a l l t o a l a r g e r i n t e g r a l s c a l e . T h i s number was c a l c u l a t e d from the estimated mean i n t e g r a l s c a l e and not the t h e o r e t i c a l s t r u c t u r e o f the model. For t h i s reason, the u n c o r r e l a t e d sequence y i e l d s on estimate o f 93, r a t h e r than i t s t h e o r e t i c a l v a l u e o f 100. With fewer independent c o n d u c t i v i t y values i n each r e a l i z a t i o n , the head values along the f l o w l i n e are more s e n s i t i v e t o the t a i l s o f the 88 c o n d u c t i v i t y d i s t r i b u t i o n . I n c r e a s i n g the l e n g t h o f the system counteracts the e f f e c t s o f l i k e c o n d u c t i v i t y values f o l l o w i n g each o t h e r by reducing the r e l a t i v e importance o f any such alignment i n determining the head values along the e n t i r e f l o w l i n e . Thus, the i n v e r s e r e l a t i o n between the i n t e g r a l s c a l e and the f l o w l i n e l e n g t h f o l l o w s . I n F i g u r e 4-5, the o u t l i e r s o f the head d i s t r i b u t i o n along x from one s i m u l a t i o n are p l o t t e d . Three s o l u t i o n s f o r d i f f e r e n t i n t e g r a l s c a l e s but the same c o n d u c t i v i t y standard d e v i a t i o n are in c l u d e d . These p a r t i c u l a r r e a l i z a t i o n s have been s p e c i f i c a l l y chosen t o i l l u s t r a t e the concept and are not randomly s e l e c t e d . There are two s e t s o f standard d e v i a t i o n s and i n t e g r a l s c a l e s given on the diagram. One i s a s s o c i a t e d w i t h averages taken over the s i m u l a t i o n . The o t h e r , i d e n t i f i e d by the l a b e l s A and B, are the parameters a s s o c i a t e d w i t h the head s o l u t i o n p l o t t e d . As the i n t e g r a l s c a l e i n c r e a s e s , the head s o l u t i o n can wander f u r t h e r from i t s mean s o l u t i o n . The curves above the mean r e s u l t from e i t h e r a g r e a t e r alignment o f low c o n d u c t i v i t y b l o c k s i n the upper h a l f o f the sequence o r o f hi g h e r c o n d u c t i v i t y b l o c k s i n the lower h a l f . The curves below the mean r e s u l t from the opposite combination. g) output d i s t r i b u t i o n o f head values The p a r a b o l i c shape o f the standard d e v i a t i o n curves r e f l e c t s the n o n s t a t i o n a r i t y o f the head d i s t r i b u t i o n along the f l o w l i n e . This Y: N[-2.00, 0.43] m= 100 s= 10.0 0Q = 100.0 0L = 0.0 MC = 400 Figure 4-5 Influence of the i n t e g r a l scale on the hydraulic head solution occurs because the f i x e d head values t r u n c a t e the range o f p e r m i s s i b l e values t h a t occur near the boundaries. The t r u n c a t i o n leads t o h i g h l y skewed head d i s t r i b u t i o n s near the boundaries. The h o r i z o n t a l crossbars on F i g u r e 4-2 i n d i c a t e those p o s i t i o n s a t which the head d i s t r i b u t i o n s c o u l d be approximated by a normal d i s t r i b u t i o n a t the 95% s i g n i f i c a n c e l e v e l . A chi-square goodness o f f i t t e s t was used. Note t h a t as e i t h e r the i n t e g r a l s c a l e o r the c o n d u c t i v i t y standard d e v i a t i o n i n c r e a s e s , the range over which the head values along the f l o w l i n e are approximated by a normal d i s t r i b u t i o n decreases. 4) V a r i a t i o n i n I n tegrated Measures o f Flow S o l u t i o n o f the f l o w equation using these s t o c h a s t i c techniques y i e l d s the v a r i a b i l i t y i n the h y d r a u l i c head values as a f u n c t i o n o f p o s i t i o n . I n t e g r a t e d measures o f the u n c e r t a i n t y i n the model p r e d i c t i o n can a l s o be considered. Two such measures f o r the steady s t a t e problem are the volume f l u x through the system and the head gr a d i e n t between two p o i n t s along the f l o w l i n e . Each i s considered i n t u r n . a) f l u x v a r i a b i l i t y The f l u x through the one-dimensional system o f b l o c k s can be c a l c u l a t e d by s e l e c t i n g any b l o c k i n the sequence and s u b s t i t u t i n g t he g r a d i e n t across t h a t b l o c k together w i t h i t s c o n d u c t i v i t y value i n Darcy's Law. By c o n t i n u i t y , t h i s i s the f l u x through the e n t i r e 91 system. On averaging over a s e r i e s o f r e a l i z a t i o n s , a f l u x d i s t r i b u t i o n can be formed. I n one dimension, the u n i t s on the f l u x are cm/s per u n i t area p e r p e n d i c u l a r t o the flow. The f l u x can a l s o be c a l c u l a t e d from a h y d r a u l i c a l l y e q u i v a l e n t 'homogeneous' system. I n one dimension, the b l o c k system can be replaced by a system w i t h a s i n g l e c o n d u c t i v i t y value equal t o the harmonic mean o f the b l o c k c o n d u c t i v i t y v a l u e s . Thus: q = *H dL 4-4 where P z i = l '4-5 K. The f l u x i s denoted q, the harmonic mean K^, and H i s the head drop between the two boundaries a d i s t a n c e L apart. The moments o f the f l u x d i s t r i b u t i o n can be computed d i r e c t l y from those on the harmonic mean c o n d u c t i v i t y . I t f o l l o w s from equation 4-4 t h a t q dL h 4-6 Var Var 4-7 92 On each run, we need o n l y c a l c u l a t e the harmonic mean o f the c o n d u c t i v i t y sequence {K}; and then c a l c u l a t e i t s v a r i a b i l i t y over the s i m u l a t i o n t o p r e d i c t the f l u x v a r i a b i l i t y . The assumption o f a random f l u x i n a one-dimensional system w i t h f i x e d head boundaries and a known space law w i l l be j u s t i f i e d i n the next s e c t i o n . Since we are now c o n s i d e r i n g the v a r i a b i l i t y o f a space average, i t can be expected t h a t the f l u x v a r i a b i l i t y w i l l be i n t e r p r e t e d i n terms o f the d e v i a t i o n o f the space average on each r e a l i z a t i o n from the ensemble average. Computing the f l u x moments by e i t h e r a p p l y i n g Darcy's Law o r equations 4-6 and 4-7 y i e l d the same estimates. The f l u x v a r i a b i l i t y i s p l o t t e d as a c o e f f i c i e n t o f v a r i a t i o n ( S f / f ) versus the estimated mean i n t e g r a l s c a l e d i v i d e d by the l e n g t h o f the system ( x/L). The r e s u l t s o f three s e t s o f s i m u l a t i o n s f o r d i f f e r e n t standard d e v i a t i o n s i n c o n d u c t i v i t y are shown i n F i g u r e 4-6a. The v a r i a b i l i t y i n the f l u x estimate i n c r e a s e s w i t h an i n c r e a s e i n the i n t e g r a l s c a l e and/or t h e c o n d u c t i v i t y standard d e v i a t i o n . Note the l a r g e u n c e r t a i n t i e s i n the f l u x f o r the commonly reported range o f v a r i a t i o n i n c o n d u c t i v i t y . I n F i g u r e 4-6b, a frequency histogram o f the f l u x estimates c a l c u l a t e d from 500 r e a l i z a t i o n s i s p l o t t e d . The standard d e v i a t i o n i n the {Y } sequence was s e t t o 0.42 and the \\/L r a t i o t o 0.023. A chi-square t e s t i n d i c a t e s the logarithms o f the f l u x can be accepted as normally d i s t r i b u t e d a t the 95% s i g n i f i c a n c e l e v e l . However, on 70 60 50 -40 -30 20 -10 -ft 5 9 13 17 class interval 21 Y: N[-2.00, 0.42] m= 100 s= 10.0 <P0 = 100.0 </>L =0.0 MC = 500 min. 2.53 x 10~4 max. 1.32 x 10 del. 5.09x10" ! -3 Figure 4-6 C o e f f i c i e n t of va r i a t i o n i n flux, one-dimensional model 94 those s i m u l a t i o n s where a was reduced t o 0.20, the f l u x i t s e l f was y more n e a r l y symmetrically d i s t r i b u t e d about i t s mode and c o u l d be accepted as normal. b) g r a d i e n t v a r i a b i l i t y A second i n t e g r a t e d measure o f the u n c e r t a i n t y i n the model p r e d i c t i o n i s d e f i n e d by the g r a d i e n t v a r i a b i l i t y between 2 p o i n t s along the f l o w l i n e . I n one dimension, the g r a d i e n t v e c t o r i s simply { J }. A d d i t i o n a l l y , the expected head drop across each b l o c k i s constant. That i s , the flo w i s uniform. The g r a d i e n t v a r i a b i l i t y i s estimated by averaging over a s e r i e s o f r e a l i z a t i o n s . The r e s u l t s are p l o t t e d as a c o e f f i c i e n t o f v a r i a t i o n ( S j ^ / J^) versus the estimated mean i n t e g r a l s c a l e , d i v i d e d by the d i s t a n c e C over which the g r a d i e n t i s c a l c u l a t e d ( X/c). These curves are given i n F i g u r e 4-7. The g r a d i e n t was c a l c u l a t e d over 10 b l o c k s i n the center o f the f l o w l i n e . C was f i x e d and X v a r i e d f o r these p l o t s . The d i f f e r e n c e s i n the g r a d i e n t v a r i a b i l i t y c a l c u l a t e d over the same d i s t a n c e but f o r d i f f e r e n t regions o f the f l o w l i n e were s m a l l compared t o the magnitude o f the g r a d i e n t v a r i a b i l i t y i t s e l f . The g r a d i e n t v a r i a b i l i t y i n c r e a s e s w i t h an i n c r e a s e i n e i t h e r the i n t e g r a l s c a l e o f the c o n d u c t i v i t y sequence o r i t s standard d e v i a t i o n . The c o n d u c t i v i t y standard d e v i a t i o n c o n t r o l s the ex t e n t o f the v a r i a t i o n i n the head values a t e i t h e r end o f C. The dependence 95 Figure 4-7 C o e f f i c i e n t of v a r i a t i o n i n the head gradient, one-dimensional model on t i i e i n t e g r a l s c a l e i s due t o i t s c o n t r o l o f the expected departure o f the mean c o n d u c t i v i t y over C from the mean o f the s t a t i o n a r y medium. " F o r a l a r g e r i n t e g r a l s c a l e , the g r a d i e n t across C can d e v i a t e f u r t h e r from the expected g r a d i e n t . Then, on averaging over the s i m u l a t i o n , the g r a d i e n t v a r i a b i l i t y w i l l i n c r e a s e . The g r a d i e n t v a r i a b i l i t y i s dependent not o n l y on the con-d u c t i v i t y sequence along C but a l s o on the sequence everywhere along the f l o w l i n e . Thus, the curves are a l s o dependent upon L, the leng t h o f the f l o w l i n e . (Or, more p r e c i s e l y , on \V L ). Reducing t h i s r a t i o w i l l l e a d t o s m a l l e r g r a d i e n t v a r i a b i l i t i e s . 5) E r g o d i c i t y i n Bounded Systems I n t h i s s e c t i o n , the i m p l i c a t i o n s o f a d i s c r e t e b l o c k r e -p r e s e n t a t i o n o f a s t a t i o n a r y c o n d u c t i v i t y f i e l d i n a bounded r e g i o n are d i s c u s s e d . I n generating s y n t h e t i c sequences, the process equation i s e rgodic i f , from any of the r e a l i z a t i o n s formed, we can get back the ensemble parameters by averaging over t h a t r e a l i z a t i o n . Any s t o c h a s t i c process model t h a t e x h i b i t s s p a t i a l a u t o c o r r e l a t i o n w i l l o n l y be a s y m p t o t i c a l l y e r g o d i c . Lumley and Panofsky (1964) show t h a t the v a r i a b i l i t y o f a r e a l i z a t i o n mean about i t s ensemble mean i s a f u n c t i o n o f the ensemble v a r i a n c e and the r a t i o o f the ensemble i n t e g r a l s c a l e t o the le n g t h o f the averaging sequence. This concept c a r r i e s over i n t o the s p a t i a l averages formed 97 from the c o n d u c t i v i t y r e a l i z a t i o n s . Because the f l o w l i n e s are o f f i n i t e l e n g t h , s p a t i a l averages w i l l d e v i a t e from t h e i r ensemble values as a f u n c t i o n o f the r a t i o o f the i n t e g r a l s c a l e o f the c o n d u c t i v i t y sequence t o the l e n g t h o f the f l o w l i n e . We must then be concerned w i t h the e x t e n t t o which the standard d e v i a t i o n s i n head o r any o t h e r i n t e g r a t e d response depends upon the n o n e r g o d i c i t y o f each o f the c o n d u c t i v i t y r e a l i z a t i o n s . The a r i t h m e t i c mean (equation 3-32) o f each c o n d u c t i v i t y r e a l i z a t i o n i s an estimate o f the ensemble a r i t h m e t i c mean. I f a l a r g e enough number o f a s y m p t o t i c a l l y independent c o n d u c t i v i t y v a l u e s were present i n each r e a l i z a t i o n , then the v a r i a b i l i t y i n the a r i t h m e t i c mean would be n e g l i g i b l e . The number o f a s y m p t o t i c a l l y independent c o n d u c t i v i t y values can be approximated by the r a t i o L/X„ the l e n g t h o f the f l o w l i n e d i v i d e d by the i n t e g r a l s c a l e . Most g e n e r a l l y , ^ p j f ^ - p ^ " ? ^ K A t ' n o n e °^ w n i c h are e x a c t l y equal t o y ^ . Here, t i s the t o t a l number o f r e a l i z a t i o n s . The standard d e v i a t i o n o f the a r i t h m e t i c mean across the s i m u l a t i o n can be used as a measure o f the average departure from e r g o d i c i t y o f each o f the c o n d u c t i v i t y r e a l i z a t i o n s . S i m i l a r i l y , the standard d e v i a t i o n i n the harmonic mean conduct-i v i t y i s a l s o a measure o f the departure from e r g o d i c i t y . I t was shown e a r l i e r t h a t the v a r i a b i l i t y i n the harmonic mean was d i r e c t l y r e l a t e d t o the f l u x v a r i a b i l i t y i n the one-dimensional problem. I t 98 f o l l o w s t h a t the f l u x v a r i a b i l i t y i s r e l a t e d t o the c o n t r o l o f ( X/L) on the number o f a s y m p t o t i c a l l y independent observations w i t h i n any r e a l i z a t i o n . As t h i s r a t i o decreases, a space average over a r e a l i z a t i o n w i l l approach i t s ensemble average and the v a r i a b i l i t y across the s i m u l a t i o n approaches zero. Therefore, i n c r e a s i n g the number o f Monte C a r l o runs o r decreasing the b l o c k s i z e w h i l e m a i n t a i n i n g the same i n t e g r a l s c a l e w i l l not change the d e v i a t i o n from e r g o d i c i t y , nor the f l u x v a r i a b i l i t y . T h i s d i s c u s s i o n leads t o -a b a s i c q u e s t i o n i n the s t o c h a s t i c s o l u t i o n o f the one-dimensional fl o w equation o f whether o r not the f l u x should be considered a random v a r i a b l e . The f l u x on any r e a l i z a t i o n depends o n l y upon the harmonic mean o f the c o n d u c t i v i t y sequence, i r r e s p e c t i v e o f i t s s p a t i a l o r d e r i n g . I n the l i m i t o f an erg o d i c r e a l i z a t i o n on each r e a l i z a t i o n , the f l u x v a r i a b i l i t y w i l l go t o zero. How does t h i s r e l a t e c o n c e p t u a l l y t o the a p p l i c a t i o n o f s t o c h a s t i c models t o f i e l d problems? Consider an a q u i t a r d o f t h i c k n e s s L w i t h known h y d r a u l i c heads on i t s upper and lower boundaries. Assume the a q u i t a r d can be modeled as a s i n g l e s t a t i o n a r y u n i t w i t h a known space law. The r a t i o o f the medium's i n t e g r a l s c a l e t o the t h i c k n e s s o f the a q u i t a r d i s a f i x e d parameter i n the p h y s i c a l problem. The u n c e r t a i n t y i n the f l u x w i l l then represent a r e a l source o f v a r i a b i l i t y i n bounded systems i f the leng t h o f the system i s a low order m u l t i p l e o f the medium's i n t e g r a l 99 s c a l e . The f a c t t h a t our generating technique y i e l d s r e a l i z a t i o n s t h a t are nonergodic may then be a r e f l e c t i o n o f c o n d i t i o n s as they e x i s t i n the f i e l d . I f the model boundaries are c o i n c i d e n t w i t h g e o l o g i c boundaries d e f i n i n g s t a t i o n a r y u n i t s , the parameter X/L a r i s e s n a t u r a l l y i n the a n a l y s i s . When the boundaries are d e f i n e d by head measurements w i t h i n a s t a t i o n a r y medium; the imposed r a t i o o f X/L does not r e f l e c t a medium parameter. However, the v a r i a b i l i t y i n t r o d u c e d by the modeled r a t i o X/L i s none the l e s s an e s s e n t i a l component o f the s t o c h a s t i c problem s i n c e i t d e f i n e s the l e n g t h s c a l e i n the model f o r which the output v a r i a b i l i t i e s are generated. 6) Degenerate Au t o r e g r e s s i v e Models The dependence o f the h y d r a u l i c head standard d e v i a t i o n s on the form o f the a u t o c o r r e l a t i o n f u n c t i o n i s i n v e s t i g a t e d i n t h i s s e c t i o n . W h i t t l e (1954) r e f e r s t o a u n i l a t e r a l scheme extended i n t o a s p a t i a l domain as a degenerate autoregression. Generation o f r e a l i z a t i o n s from t h i s model are much e a s i e r s i n c e the dependence o n l y extends onto p r e v i o u s l y generated values w i t h i n the sequence. The auto-c o r r e l a t i o n f u n c t i o n s f o r the nearest neighbour and degenerate models are compared i n F i g u r e 4-8a, f o r equal values o f the a u t o r e g r e s s i v e parameter a . Note t h a t the c o r r e l a t i o n decays more q u i c k l y i n t h e nearest neighbour model w i t h i t s b i l a t e r a l dependence. Y : N [ M y a y ] m=. 100 s=.10.0 a = 0.80 T 1 1 1 1 1 1 i r 10 30 50 70 90 110 X (distance) A Y: N[-2.00, 0.37] m= 100 s= 10.0 ct = 0.935 X= 43.2 B Y: N[-2.00, 0.39] m = 100 s= 10.0 a =0.86 X=41.2 X (distance) O Q degenerate Y: N[-2.00, 0.37] m= 100 s= 10.0 a = 0.935 X=43.2 Y: N[-2.00, 0.37] m= 100 s_= 10.0 a = 0.86 X= 42.3 F i g u r e 4 - 8 D e g e n e r a t e a u t o r e g r e s s i v e m o d e l s 101 A comparison can be made between the standard d e v i a t i o n s i n head generated from the two models when the c o n d u c t i v i t y sequences have the same estimated mean i n t e g r a l s c a l e . From an o p e r a t i o n a l p o i n t o f view, i f the d i f f e r e n c e s are minor, we may want t o c o n s i d e r a p p l y i n g degenerate models t o those problems where the number o f nodal b l o c k s becomes e x c e s s i v e l y l a r g e f o r the m a t r i x i n v e r s i o n problem. The a u t o c o r r e l a t i o n f u n c t i o n i n the c o n d u c t i v i t y sequences and the parameters used t o y i e l d approximately equal i n t e g r a l s c a l e s are given i n F i g u r e 4-8b. The r e s u l t i n g standard d e v i a t i o n s i n head f o r a f i x e d c o n d u c t i v i t y standard d e v i a t i o n are p l o t t e d i n F i g u r e 4-8c. Curve A i d e n t i f i e s the nearest neighbour model; curve B the degenerate model. The d i f f e r e n c e s between the two curves a r e s m a l l . They l i e w i t h i n each others 95% confidence i n t e r v a l on the estimated standard d e v i a t i o n . T h i s i m p l i e s i t i s not the exact form o f the c o r r e l a t i o n f u n c t i o n but i t s i n t e g r a l s c a l e t h a t i s most important i n determining the head standard d e v i a t i o n s . Although the d i f f e r e n c e s may be s m a l l , the nearest neighbour model c o n c e p t u a l l y i s more r e a l i s t i c than the degenerate autor e g r e s s i o n . 7) Convergence o f the Monte C a r l o Technique To determine whether a Monte C a r l o s i m u l a t i o n based on 400 r e a l i z a t i o n s converges toward a unique s o l u t i o n , the head standard d e v i a t i o n s formed over v a r i o u s numbers o f runs can be compared. I f 102 the d i f f e r e n c e s are s n a i l , i t can be concluded t h a t averaging was c a r r i e d o ut over a s u f f i c i e n t number o f r e a l i z a t i o n s , a t l e a s t i n comparing the head v a r i a b i l i t y . F i g u r e 4-9 i s such a p l o t f o r a equal t o 0.42. Two s e t s o f runs are compared f o r c o n d u c t i v i t y r e a l i z a t i o n s w i t h two d i f f e r e n t i n t e g r a l s c a l e s . I n both cases the d i f f e r e n c e s a r e minor, so^averaging over 400 r e a l i z a t i o n s i s taken t o be s u f f i c i e n t . Because the standard d e v i a t i o n i s a h i g h e r order moment than the mean, s a t i s f a c t o r y con-vergence i n the mean f o l l o w s d i r e c t l y . By i n c r e a s i n g the number of runs, the w i d t h o f any confidence i n t e r v a l about the estimates can be reduced. From a s i m i l a r p l o t f o r a c o n d u c t i v i t y sequence generated w i t h a =0.84, the same c o n c l u s i o n can be reached. y 8) Conclusions 1) The main c o n c l u s i o n t o be drawn from the one-dimensional a n a l y s i s i s t h a t the r a t i o o f the i n t e g r a l s c a l e ( X ) t o the d i s t a n c e between the boundary p o i n t s (L) i s a fundamental, p h y s i c a l l y - b a s e d s t o c h a s t i c parameter i n modeling a s t a t i o n a r y medium. The output d i s t r i b u t i o n s formed must always be i n t e r p r e t e d i n l i g h t o f the r a t i o X/L. For systems w i t h a d i f f e r e n t L o r a d i f f e r e n t X , the standard d e v i a t i o n s i n head (S ) are equal o n l y when the Y : N[-2.00, 0.42] Dx .10 .20 S<f> Figure 4-9 Convergence of the Monte Carlo technique 104 r a t i o X/L i s equal; ot h e r parameters b e i n g f i x e d and equal. 2) For a f l o w l i n e o f l e n g t h L, i t was observed t h a t : a) S , i n c r e a s e s w i t h l a r g e r values o f a . <f> y the standard d e v i a t i o n i n the logarithms o f c o n d u c t i v i t y b) For a f i x e d a^,S^ a t a given p o i n t i n c r e a s e s as X i n c r e a s e s . 3) For a s e t t o 0.84 and X/L r a t i o s ranging from 0.01 y t o 0.036, S (expressed as a percent o f the t o t a l head drop) a t the midpoint o f the f l o w l i n e ranged from 15 t o 29%. 4) The output v a r i a b i l i t i e s , a s i d e from a minor v a r i a t i o n , w i l l not depend upon .the b l o c k s i z e used; provided a s u f f i c i e n t l y accurate r e p r e s e n t a t i o n o f X i s made. The i n t e g r a l s c a l e i s the upper l i m i t on d i s c r e t i z a t i o n i n a s t a t i o n a r y medium. 5) The dependence on X/L can be e x p l a i n e d as f o l l o w s : a) For l a r g e r values o f X , the tendency f o r l i k e c o n d u c t i v i t y values t o f o l l o w each o t h e r a l l o w s the head s o l u t i o n t o move > 105 further away from its expected linear decrease along x. b) Increasing L counteracts the effects of like conductivity values following each other by reducing the relative importance of any such alignment in determining the head values along the entire flowline. 6) Each of the bounded conductivity realizations is nonergodic because i t contains only a finite number of asymptotically independent conductivity values. The extent of the departure from ergodicity depends upon both X/L and a . . y 7) The coefficient of variation in flux increases with an increase in the ratio X/L and/or a y . For a X/L ratio of 0.03, the coefficient of variation in flux is estimated as 0.10, 0.29 and 0.86 for a values y of 0.20, 0.42 and 0.84, respectively. In the one-dimensional model, the flux variability is due entirely to the nonergodic behavior of each of the conductivity realizations. Aside from the confidence intervals on its estimate, the flux variability does not depend upon the number of Monte Carlo realizations used to form the estimate. 106 8) The g r a d i e n t v a r i a b i l i t y over a d i s t a n c e C inc r e a s e s w i t h an i n c r e a s e i n X and/or a y I t i s a l s o l a r g e r f o r a s m a l l e r L. For L=1000 cm. and a X/C r a t i o o f 0.3, the c o e f f i c i e n t o f v a r i a t i o n i n g r a d i e n t i s estimated as 0.28, 0.68 and 1.39 f o r a values o f 0.20, 0.42 y and 0.84, r e s p e c t i v e l y . The g r a d i e n t v a r i a b i l i t y can be e x p l a i n e d by the p r o b a b i l i t y o f l i k e c o n d u c t i v i t y values f o l l o w i n g each o t h e r over the g r a d i e n t d i s t a n c e C and the departure o f t h a t alignment from the mean c o n d u c t i v i t y . 9) The standard d e v i a t i o n s i n head are more s e n s i t i v e t o the i n t e g r a l s c a l e o f the a u t o c o r r e l a t i o n f u n c t i o n than i t s p r e c i s e form o f decay w i t h i n c r e a s i n g l a g . 107 CHAPTER 5 Two-Dimensional S i m u l a t i o n R e s u l t s 1) I n t r o d u c t i o n I n t h i s chapter, a two-dimensional s t o c h a s t i c a n a l y s i s o f f l o w through a random medium i n a bounded domain i s considered. By c a r r y i n g o ut the a n a l y s i s i n two dimensions, more f l e x i b i l i t y i s allowed i n the d e s c r i p t i o n o f the mediums heterogeneity and i n the c o n d i t i o n s on the flo w f i e l d . The dependence o f the p r e d i c t i o n v a r i a b i l i t i e s on the dimension o f the flow equation w i l l be i n v e s t i g a t e d . A d d i t i o n a l l y , two-dimensional models f i n d g r e a t e r a p p l i c a t i o n t o f i e l d - o r i e n t a t e d problems than do the one-dimensional r e p r e s e n t a t i o n s . Three steady s t a t e f l o w f i e l d s w i l l be i n v e s t i g a t e d t o determine the i n f l u e n c e o f the s p a t i a l v a r i a t i o n i n expected g r a d i e n t s on the h y d r a u l i c head v a r i a b i l i t i e s . The cases t o be considered a r e : 1) U n i d i r e c t i o n a l f l o w i n a uniform g r a d i e n t f i e l d 2) U n i d i r e c t i o n a l flow i n a nonuniform g r a d i e n t f i e l d 3) Nonuniform fl o w i n a nonuniform g r a d i e n t f i e l d The d e s c r i p t i o n o f the flo w f i e l d i s based on the s p a t i a l v a r i a t i o n o f the g r a d i e n t f i e l d as determined by the mean s o l u t i o n o f the s t o c h a s t i c problem. The g r a d i e n t v e c t o r over a f i x e d d i s t a n c e i s constant everywhere w i t h i n a uniform flow f i e l d . A nonuniform flow i m p l i e s the f l o w l i n e s are curved. S e v e r a l d i f f e r e n t c o n f i g u r a t i o n s o f s p a t i a l heterogeneity w i l l be considered. These are: 1) A s t a t i s t i c a l l y i s o t r o p i c , s i n g l e s t a t i o n a r y u n i t i n a bounded domain. 2) A s t a t i s t i c a l l y a n i s o t r o p i c * s i n g l e s t a t i o n a r y u n i t i n a bounded domain. 3) Two s t a t i s t i c a l l y i s o t r o p i c , s t a t i o n a r y u n i t s w i t h i n a bounded domain. 4) A no n s t a t i o n a r y u n i t due t o a l i n e a r t r e n d i n the mean c o n d u c t i v i t y . These c o n d u c t i v i t y f i e l d s w i l l be combined w i t h the d i f f e r e n t g r a d i e n t f i e l d s mentioned above t o show t h a t the magnitude and s p a t i a l d i s t r i b u t i o n o f the h y d r a u l i c head standard d e v i a t i o n s depend upon the i n t e r a c t i o n o f both the medium heterogeneity and the flo w f i e l d . I n o t h e r words, a p r e d i c t i o n u n c e r t a i n t y cannot be a s s o c i a t e d a p r i o r i w i t h a given degree o f s p a t i a l heterogeneity. I n the model a n a l y s i s t h a t f o l l o w s , i d e a l i z e d r e p r e s e n t a t i o n s o f flow systems are used t o c r e a t e the d e s i r e d g r a d i e n t f i e l d s . To complement t h a t a n a l y s i s , two more r e a l i s t i c flow problems are presented a t the c o n c l u s i o n o f t h i s chapter. 109 2) Steady St a t e Flow Problem a) boundary val u e problem The s t o c h a s t i c equation f o r steady s t a t e , s a t u r a t e d flow i n two dimensions i s given as: — — — — K 3d) ' xx-r1-3 x + 3 3z K 3d)' zz T 3 z = o 5-1 The h y d r a u l i c head (<f>' ) , the c o n d u c t i v i t y i n the x d i r e c t i o n (K ) , and the c o n d u c t i v i t y i n the z d i r e c t i o n ( K g g ) are a l l random f u n c t i o n s i n space. A d i s t i n c t i o n must be made between s t a t i s t i c a l a n i s o t r o p y i n the covariance f u n c t i o n and h y d r a u l i c a n i s o t r o p y i n the flow equation. A s t a t i o n a r y , s t a t i s t i c a l l y a n i s o t r o p i c random f i e l d has the same mean c o n d u c t i v i t y a t every p o i n t but an autocovariance f u n c t i o n dependent upon o r i e n t a t i o n . The assumption o f s t a t i o n a r i t y precludes h y d r a u l i c a n i s o t r o p y f o r a p o i n t v a l u e o f c o n d u c t i v i t y s i n c e t h i s would imply t h a t the mean c o n d u c t i v i t y i s d i r e c t i o n dependent. The s t a t i o n a r i t y assumption i n the c o n d u c t i v i t y t e n s o r , assuming cross terms t o be zero, can be w r i t t e n : K XX 0 K 0 0 0 K 5-2 110 A complete c h a r a c t e r i z a t i o n o f c o n d u c t i v i t y as a random f u n c t i o n would be d e r i v e d from i t s tensor r e p r e s e n t a t i o n . I n t h i s study, p o i n t values o f c o n d u c t i v i t y are assumed t o be h y d r a u l i c a l l y i s o t r o p i c . Therefore, the s c a l a r r e p r e s e n t a t i o n o f c o n d u c t i v i t y i s adopted. The f l o w equation i n a s t a t i o n a r y medium can be w r i t t e n : 3 K3<j> ' + 3 K9<|) ' 3x 3x 3z 9z Both no-flow and constant head c o n d i t i o n s w i l l be used on the boundaries o f the flow domain. That i s : M l = o , | i i - o 9 x . 9 z . l l 5-4 o r <j)' ( x i , z i ) = <j>^  f o r v a r i o u s boundary p o s i t i o n s i . The f l o w domain i s d i v i d e d i n t o p e q u i s i z e d b l o c k s (m rows, n columns) w i t h a s i n g l e c o n d u c t i v i t y value assigned t o each b l o c k i = l , p. R e p e t i t i v e generation o f c o n d u c t i v i t y r e a l i z a t i o n s from a twc>-dimensional nearest neighbour model and s o l u t i o n o f the d e t e r m i n i s t i c problem i s again used t o o b t a i n an approximation o f the output d i s -t r i b u t i o n i n h y d r a u l i c head. The boundary value problem was i l l u s t r a t e d I l l in Figure 3-2 for a specific set of boundary conditions. b) solution technique An analytical solution for the head ,(f> ' (x,z) in this blocked domain does not exist. Accordingly, we must resort to numerical techniques to determine <(> 1 (x,<z). & finite element technique was used to solve for the head values on a set of nodal points within the flow domain. Triangular elements were used in conjunction with the Raleigh-Ritz variational formulation of the system of linear algebraic equations (Remson et al., 1971). See Appendix II for a more detailed account of the procedure. Two triangular elements were included in each conductivity block. Because the order of the system of equations is determined by the number of nodal points and not the number of elements, there was l i t t l e advantage to using quadralateral elements that matched the conductivity blocks in size and shape. The numerical solution is not exact, unlike the one-dimensional problem presented earlier. The source of possible errors is twofold. First, a discretization (or truncation) error occurs with the super-position of the finite element grid on the continuously varying hydraulic head field. A linear basis function was used to specify the variation in head over each element. This forces the head gradient within each element to be constant. It can, of course, vary from element to element. 112 The accuracy o f the d i s c r e t i z a t i o n depends upon the t o t a l head drop across the flow domain, the r a t e o f change i n the head g r a d i e n t i n space and the s i z e o f the elements over which the l i n e a r approximation i s made. The e f f e c t s o f t h i s e r r o r w i l l be analyzed i n a l a t e r s e c t i o n . Secondly, roundoff e r r o r s are i n t r o d u c e d i n t o the head f i e l d ty * (x,z> d u r i n g s o l u t i o n o f the system o f equations. I t i s considered t h a t numerical e r r o r s are s u f f i c i e n t l y minimi zed-for the boundary value problems analyzed i n t h i s study. U n l i k e the a n a l y t i c a l s o l u t i o n o f the one-dimensional equation, the s o l u t i o n i s obtained a t a l l nodal p o i n t s f o r each c o n d u c t i v i t y r e a l i z a t i o n , i n s t e a d o f a p r e s e l e c t e d subset o f p o i n t s . The numerical s o l u t i o n i s t h e r e f o r e l e s s e f f i c i e n t and r e q u i r e s r e l a t i v e l y g r e a t e r OTtputing time than the s o l u t i o n o f the one-dimensional equation. c) p r e l i m i n a r y d e t a i l s on s o l u t i o n In' two dimensions, the estimated standard d e v i a t i o n i n head d i s t r i b u t e d over the flow domain i s a s u r f a c e . F i g u r e 5-1 d i a g r a i t m a t i c a l l y i l l u s t r a t e s t h i s v a r i a t i o n f o r the boundary c o n d i t i o n s on the conduct-i v i t y f i e l d i n F i g u r e 3-2. Our p l o t s o f the standard d e v i a t i o n i n head w i l l be presented e i t h e r f o r v e c t o r s i n the x z plane o r as a contoured s u r f a c e over the f l o w domain. 113 Figure 5-1 Surface of the standard deviations i n hydraulic head on a two-dimensional flow domain Conductivity realizations for the block model are generated from equation 3-25. After applying the exponential transform, the conductivity values are inserted into their appropriate locations in the finite element grid. In using lognormal distributions, care must be taken in com-paring head standard deviations arising from different space law parameters. The relations between the mean and standard deviation of the {Y } and {K } sequences were given in equations 3-30 and 3-31. The standard deviation in head curves to be presented are specified by the parameters of {Y} . Fixing the mean ( y ) of {Y} and varying 114 i t s standard d e v i a t i o n (a ) w i l l l e a d t o d i f f e r e n t mean c o n d u c t i v i t i e s y (y . ) i n {K} (equation3-30). However, f o r the steady s t a t e problem, the head v a r i a b i l i t y i s independent of y^, provided the higher order moments i n {K} are f i x e d . Therefore, i t i s s u f f i c i e n t t o c h a r a c t e r i z e the head v a r i a b i l i t y curves by a . I f both y and a • are d i f f e r e n t , y y y the output curves are no longer s u f f i c i e n t l y c h a r a c t e r i z e d by a y alone; s i n c e the two r e a l i z a t i o n s would have d i f f e r e n t standard d e v i a t i o n s i n {K} (equation 3-31) . This occurs even i f cry i s f i x e d . Due t o the behavior o f the numerical s o l u t i o n o f the flow equation, there was an upper bound on the s i z e o f the c o n d u c t i v i t y standard d e v i a t i o n t h a t c o u l d be considered. As the e x t e n t of the c o n d u c t i v i t y c o n t r a s t s between neighbouring b l o c k s was i n c r e a s e d , the s o l u t i o n o f the system o f equations f a i l e d t o converge. As an example from a p a r t i c u l a r s i m u l a t i o n , f o r a a y v a l u e o f 1.0, 10 o f 300 conduct-i v i t y r e a l i z a t i o n s d i d not y i e l d convergent s o l u t i o n s . I f t h i s o ccurs, the e n t i r e s i m u l a t i o n must be discounted. The l a r g e s t value o f a y used i n the two-dimensional a n a l y s i s i s 0.91 I t may be p o s s i b l e t o extend the range o f a y values t h a t can be considered by f u r t h e r i n v e s t i -g a t i o n o f the s o l u t i o n a l g o r i t h m . Unless otherwise i n d i c a t e d , a l l the r e s u l t s presented were c a l c u l a t e d over 300 r e a l i z a t i o n s on a 200 b l o c k (10 x 20) system'.'. Each b l o c k i s s e t t o 10 cm on the s i d e . An upper boundary head o f 100 cm and a lower boundary head o f 0 cm are s p e c i f i e d . These u n i t s do not represent 115 any attempt to define the scale at which the stochastic problem is applied to field situations. The mean of the {Y} sequence was set to -2.00 for conductivities measured in cm/s. For the above conductivity grid and a general anisotropic nearest neighbour relation, simulations over 300 realizations required about 9 minutes of computer time. A flowchart for the computer program written to carry out the simulation is given in Appendix III. 3) Unidirectional Flow, Uniform Gradient Field a) introduction The f i r s t set of plots to be presented consider unidirectional flow in a uniform gradient field. The flow domain and its boundary conditions are shown in Figure 3-2. Although the mean flow is uni-directional, on any realization, the flowlines are no longer constrained to one dimension. Media with statistically isotropic covariance functions are discussed first. The influence of the space law parameters on the head variabilities w i l l be demonstrated. A comparison is made between the one and two-cLLmensional problem. A statistically aniso-tropic covariance function is then introduced. Finally, the effect on the prediction variabilities of more than one stationary unit being present within the flow domain is considered. b) s t a t i s t i c a l l y isotropic covariance function i) mean solution The distance along the two coordinate axes i s expressed i n dimensionless quantities: D = x/L 5-5 x .. x D = z/L where L i s the distance between the two constant head boundaries and x L g i s the distance between the impermeable boundaries. The head solution i s set i n the dimensionless quantities defined earl i e r i n equation 4-3. The estimated mean head at a fixed D was not equal to the analytical solution for a uniform medium; nor was the mean head constant along D , given D . However, i n both instances, deviations were small with no apparent bias. For those cases i n which the head distribution could be approximated by a normal distribution, the deterministic solution was within a 95% confidence interval on the estimated mean head. 117 i i ) dependence on the i n t e g r a l s c a l e For the u n i d i r e c t i o n a l problem, the head v a r i a b i l i t y can be c h a r a c t e r i z e d by a p l o t o f S ^  versus D , the climensionless d i s t a n c e <p X along the x a x i s . The standard d e v i a t i o n s are recorded along a v e c t o r midway between the impermeable boundaries. The i n f l u e n c e o f l a r g e r i n t e g r a l s c a l e s i n the two coordinate d i r e c t i o n s on the standard d e v i a t i o n s i n head i s i l l u s t r a t e d i n Fi g u r e 5-2. The curve l a b e l e d A was generated from an u n c o r r e l a t e d sequence o f b l o c k s . As the i n t e g r a l s c a l e s i n c r e a s e , so too do the standard d e v i a t i o n s i n h y d r a u l i c head. The e x p l a n a t i o n f o l l o w s d i r e c t l y frcm t h a t i n one dimension; alignment o f l i k e c o n d u c t i v i t y values a l l o w s the head s o l u t i o n on any r e a l i z a t i o n t o move f u r t h e r away frcm i t s expected s o l u t i o n f o r t h a t r e g i o n w i t h i n the flow domain. i i i ) dependence on the c o n d u c t i v i t y standard d e v i a t i o n The dependence o f the standard d e v i a t i o n s i n head on the conduct-i v i t y standard d e v i a t i o n i s most c l e a r l y shewn f o r f i x e d i n t e g r a l s c a l e s i n the coordinate d i r e c t i o n s . Such a p l o t i s given i n F i g u r e 5-3. The mean i n t e g r a l s c a l e c o u l d o n l y be approximately s p e c i f i e d a p r i o r i by choosing a p p r o p r i a t e values o f the au t o r e g r e s s i v e parameters and the v a r i a n c e f a c t o r n . - Note the expected e f f e c t o f l a r g e r v a r i a -b i l i t i e s i n head f o r g r e a t e r standard d e v i a t i o n s i n c o n d u c t i v i t y . 118 Y: N[-2.00, 0.43] 0.02 0.06 0.10 0.14 0.18 S0 Figure 5-2 Dependence of the standard deviations i n hydraulic head on the i n t e g r a l scale 1 1 9 X 0.5 -\ Y: N[-2.00, a y] m=10 n = 20 p = 200 s= 10.0 MC = 300 0fO) = 1OO 0(Lx) = O A a y = 0.21 X x = 17.9 "^=18.0 C B O y = 0.43 X x= 17.4 Xz=17.5 C a y = 0.91 X x =16.6 Xz= 16.6 Sc6 F i g u r e 5-3 Dependence o f t h e s t a n d a r d d e v i a t i o n s i n h y d r a u l i c h e a d on t h e s t a n d a r d d e v i a t i o n i n h y d r a u l i c c o n d u c t i v i t y 120 iv) discretization A conductivity field discretized into 200 blocks was found to be near the practical computer limit in terms of inverting the nearest neighbour weight matrix and utilizing a finite element solution of the flow equation. Further efficiencies may be achieved by exploiting the form of the matrices, allowing larger numbers of blocks to be modeled. It was demonstrated in the one-dimensional analysis that the ratio of the integral scale to the length of the flow domain controls the output variabilities and not the discretization interval or the length of the system considered separately. This holds in the two-dimensional problem. By changing the integral scales along the coordinate axes, the influence of a larger system can be effectively considered. v) comparison of one and two-dimensional solutions In this section, one and two-dimensional models are compared to determine the extent to which the output variability is reduced in the higher dimension analysis. Constraining the flow to one dimension exaggerates the effect of the tails of the conductivity distribution in determining the standard deviation in the head values. Because the flow cannot be routed around low conductivity inclusions, their positioning in the sequence will to a large extent control the head distribution everywhere along the flowline. In two dimensions, the 121 effect of low conductivity inclusions is localized and their influence on the entire hydraulic .head field is reduced. To make the comparison, the systems must be of equal length. Therefore, another set of one-dimensional simulations were carried out on a 20 block sequence. The equivalent twc>-dimensional problem is unidirectional mean flow in, say, a (10 x 20) block system. The length perpendicular to the mean flow direction is arbitrarily chosen. The comparison must be interpreted in light of the ratio of the integral scale to the length between the impermeable boundaries ( x /L ) • This ratio would be 1 in the one-dimensional problem. A l l other parameters of the space law will be set equal. It should be obvious from Figures 5-2 and 5-3 that the head standard deviations are smaller than those which we faced in the one-dimensional analysis. Figures 5-4a, 5-4b, and 5-4c show the more direct comparison for three different a values for systems of equal y length. The head standard deviations are approximately halved for the two-dimensional problem. In Chapter 8 the dimension dependence pre-dicted by this model is compared with that suggested from other solution techniques to the stochastic flow problem. vi) output distribution of head values As a y decreases, the head distribution can be f i t to a normal distribution over a greater percentage of the flow domain. The same 122 Figure 5-4 Comparison of one and two-dimensional solutions of the stochastic flow equation 123 holds t r u e f o r s m a l l e r i n t e g r a l s c a l e s w i t h f i x e d values o f a . I n y both i n s t a n c e s , the e f f e c t s o f boundary t r u n c a t i o n i n c r e a t i n g skewed head d i s t r i b u t i o n s do not extend as f a r i n t o the flow domain. The h o r i z o n t a l crossbars on Figure s 5-4 i n d i c a t e a t which nodes the head d i s t r i b u t i o n c o u l d be approximated by a normal d i s t r i b u t i o n a t the 95% s i g n i f i c a n c e l e v e l . v i i ) g r a d i e n t v a r i a b i l i t y For u n i d i r e c t i o n a l mean f l o w i n the :x coordinate d i r e c t i o n , the g r a d i e n t v e c t o r { J , J } can be w r i t t e n { J , 0 } . On any one r e a l i z a t i o n , both J and J are nonzero b u t J i s the dominating term. ' x « x 3 The r e s u l t s are p l o t t e d as a c o e f f i c i e n t o f v a r i a t i o n i n g r a d i e n t (S / J ) versus the estimated mean i n t e g r a l s c a l e i n x , normalized by J A X the d i s t a n c e C over which the g r a d i e n t i s c a l c u l a t e d ( X /C). I n F i g u r e 5-5, three such curves are presented f o r d i f f e r e n t values o f ° y • T h e g r a d i e n t v a r i a b i l i t y i s l a r g e r f o r g r e a t e r standard d e v i a t i o n s i n c o n d u c t i v i t y o r l a r g e r i n t e g r a l s c a l e s . The dependence on the i n t e g r a l s c a l e a r i s e s from i t s c o n t r o l on the expected departure o f the mean c o n d u c t i v i t y over C from the mean o f the s t a t i o n a r y medium. For l a r g e r standard d e v i a t i o n s i n c o n d u c t i v i t y , the v a r i a b i l i t y i n the head values a t e i t h e r end o f C i n c r e a s e s , l e a d i n g t o a g r e a t e r g r a d i e n t v a r i a b i l i t y . 124 Y: N[-2.00, ay ] m = 10 n = 20 p = 200 s=10.0 MC = 300 0(0) = 100.0 0(Lx) = 0.0 1 1 0.4 0.6 C Ax/X(02) -X'f^) Figure 5-5 C o e f f i c i e n t of v a r i a t i o n i n head gradient, uniform gradient f i e l d 125 v i i i ) f l u x v a r i a b i l i t y The v a r i a b i l i t y i n the f l u x through the f l o w domain provides another i n t e g r a t e d measure o f the u n c e r t a i n t y i n the model p r e d i c t i o n . As d e t a i l e d i n Appendix I I , the f l u x e s a t nodal p o i n t s along the constant head boundaries are e a s i l y c a l c u l a t e d from the f i n i t e element model. The f l u x c a l c u l a t i o n a t a node i s d i s t r i b u t e d over t h a t l e n g t h (C) o f i t s a s s o c i a t e d elements' boundaries c o i n c i d e n t w i t h the constant head boundary. The f l u x i s then presented i n u n i t s o f (cm/sMcm) per u n i t t h i c k n e s s . For a bounded system, the f l u x v a r i a b i l i t y i s due t o the combined e f f e c t s o f : 1) the arrangement o f c o n d u c t i v i t y values w i t h i n the f l o w domain 2) the n o n e r g o d i c i t y o f each r e a l i z a t i o n . Because each r e a l i z a t i o n w i l l have a unique s e t o f c o n d u c t i v i t i e s i n the v i c i n i t y o f C, the f l u x w i l l vary on averaging over a s e r i e s o f r e a l i z a t i o n s . Therefore, u n l i k e the one-dimensional s i m u l a t i o n s , the f l u x v a r i a b i l i t y w i l l not go t o zero i n the l i m i t o f an e r g o d i c r e a l i -z a t i o n on each run. Only the t o t a l f l u x through the flow domain must e x h i b i t such a behavior. The f l u x v a r i a b i l i t y is p l o t t e d as a c o e f f i c i e n t o f v a r i a t i o n 126 (S^/f) versus the» estimated mean i n t e g r a l s c a l e d i v i d e d by the d i s t a n c e between the constant head boundaries ( X^/Lx) • Such a p l o t i s u s e f u l i n two dimensions o n l y f o r the u n i d i r e c t i o n a l mean flow problem. The v a r i a b i l i t y i s p l o t t e d f o r one nodal p o i n t on the out f l o w s u r f a c e . As shown i n F i g u r e 5-6, the c o e f f i c i e n t o f v a r i a t i o n i n c r e a s e s w i t h the mean i n t e g r a l s c a l e f o r a f i x e d a y , and i s g r e a t e r f o r l a r g e r values o f a y . An ex p l a n a t i o n f o r the l a t t e r i n c r e a s e f o l l o w s simply from the l a r g e r c r y a l l o w i n g the c o n d u c t i v i t y values near C t o have a h i g h e r p r o b a b i l i t y o f d e v i a t i n g f u r t h e r from the mean c o n d u c t i v i t y . The dependence on the i n t e g r a l s c a l e i s twof o l d . F or s m a l l e r i n t e g r a l s c a l e s , the v a r i a b i l i t y due t o d i f f e r e n c e s i n the t o t a l f l u x w i l l decrease as the number o f a s y m p t o t i c a l l y independent observations from the c o n d u c t i v i t y d i s t r i b u t i o n i n c r e a s e s . Secondly, f o r s m a l l e r i n t e g r a l s c a l e s , the p r o b a b i l i t y o f the head g r a d i e n t b e i n g c l o s e r t o i t s expected mean i s i n c r e a s e d . The c o n d u c t i v i t y sequence a l s o has a h i g h e r p r o b a b i l i t y o f b e i n g c l o s e r t o i t s expected mean. The estimated f l u x d i s t r i b u t i o n depends upon the C o n d u c t i v i t y d i s t r i b u t i o n , the g r a d i e n t d i s t r i b u t i o n i n the v i c i n i t y o f C and the d i s t r i b u t i o n o f the mean f l u x averaged over the s i m u l a t i o n . As X o r 0 ^ were i n c r e a s e d , the r i g h t skew o f the f l u x d i s t r i b u t i o n i n c r e a s e d . For equal t o 0.43, the logarithms o f the absolute value o f t h e . f l u x f i t a normal d i s t r i b u t i o n a t the 95% s i g n i f i c a n c e l e v e l . I f cr were reduced t o 0.21 and an u n c o r r e l a t e d c o n d u c t i v i t y sequence i n s e r t e d i n t o the f l o w domain, the f l u x i t s e l f c o u l d be F i g u r e 5 - 6 C o e f f i c i e n t o f v a r i a t i o n i n f l u x , u n i f o r m g r a d i e n t f i e l d 128 accepted as f i t t i n g a normal d i s t r i b u t i o n . The apparent d i s t r i b u t i o n dependence on o y probably a r i s e s from the i n a b i l i t y o f the c h i -square t e s t t o d i s t i n g u i s h between a normal and lognormal d i s t r i b u t i o n as the skew on the sample d i s t r i b u t i o n decreases (Agterberg, 1974). c) s t a t i s t i c a l l y ajxLsotropic covariance f u n c t i o n i ) a n i s o t r o p y v i a the space law Geologic c o n s i d e r a t i o n s suggest s t a t i s t i c a l l y a n i s o t r o p i c covariance f u n c t i o n s are probably the r u l e r a t h e r than the exception i n two-dimensional v e r t i c a l s e c t i o n s . I n t r o d u c i n g an a n i s o t r o p i c covariance f u n c t i o n f o r r e a l i z a t i o n s s u b j e c t t o u n i d i r e c t i o n a l mean, flow should not have dramatic e f f e c t s on the standard d e v i a t i o n s i n head. Here, the i n t e g r a l s c a l e i n the d i r e c t i o n o f mean f l o w dominates the f u n c t i o n a l dependence o f head v a r i a b i l i t y on the covariance f u n c t i o n because t h i s i s the d i r e c t i o n i n which the head gr a d i e n t s are the g r e a t e s t . By decreasing the c o r r e l a t i o n l e n g t h along the a x i s perpen-d i c u l a r t o the mean flow d i r e c t i o n , the p r o b a b i l i t y o f an elongate low c o n d u c t i v i t y zone o r i e n t a t e d a t r i g h t angles t o the mean flo w i s reduced. T h i s w i l l reduce the dependence o f the head f i e l d on the t a i l s o f the c o n d u c t i v i t y d i s t r i b u t i o n , l e a d i n g t o s m a l l e r head standard d e v i a t i o n s . The maxiittum p o s s i b l e r e d u c t i o n occurs f o r a s e t o f b l o c k s u n c o r r e l a t e d along the a x i s p e r p e n d i c u l a r t o the mean flo w d i r e c t i o n . The output standard d e v i a t i o n s i n head are compared f o r i s o t r o p i c and a n i s o t r o p i c covariance f u n c t i o n s i n F i g u r e 5-7a. The i n t e g r a l 129 s-<t> Figure 5-7 Influence of a s t a t i s t i c a l l y anisotropic covariance function i n hydraulic conductivity on the standard deviations i n hydraulic head 130 s c a l e i n the d i r e c t i o n o f mean flow and a are f i x e d . Because o f y the interdependence o f the auto r e g r e s s i v e parameters i n the covariance f u n c t i o n , d i f f e r e n t parameters are r e q u i r e d t o achieve the same mean i n t e g r a l s c a l e (X ). Th i s dependence a l s o r e s t r i c t s t h e range over which the r a t i o o f the two i n t e g r a l s c a l e s can vary. I f the two s c a l e parameters ( A , A ) are e q u a l , then i t would be imp o s s i b l e t o X 2 achieve X ^ / X g r a t i o s o f say, f i v e , u s i n g a s e t o f a u t o r e g r e s s i v e parameters ( ct , a ) t h a t are p h y s i c a l l y r e a l i z a b l e . X 2 i i ) a n isotropy v i a b l o c k expansion A n i s o t r o p i c covariance f u n c t i o n s can be generated from a s t a t i s t i c a l l y i s o t r o p i c space law by s c a l e expansion along one o f the coordinate axes. S i n g l e c o n d u c t i v i t y values are then i n s e r t e d i n t o a s e t o f r e c t a n g u l a r b l o c k s . U n l i k e square b l o c k s , t h i s i n troduces a p r e f e r r e d o r i e n t a t i o n i n t o the ensemble s t r u c t u r e o f the medium i n re p r e s e n t i n g the s p a t i a l heterogeneity by e q u i v a l e n t s i n g l e v a l u e s . S c a l e expansion i s not a t r u e a n i s o t r o p y i n the covariance f u n c t i o n i n the sense o f an an i s o t r o p y i n the au t o r e g r e s s i v e parameters. I t occurs because o f a m o d i f i c a t i o n i n the l e n g t h s c a l e o f the conduct-i v i t y b l o c k r a t h e r than i n the l e n g t h s c a l e o f the covariance f u n c t i o n . Upon expansion, the c o n d u c t i v i t y sequence i s a s t r e t c h e d v e r s i o n o f the i s o t r o p i c process. A more d e t a i l e d d i s c u s s i o n o f the b l o c k s c a l e i s d e f e r r e d u n t i l Chapter 6. 131 I n a n a l y z i n g the e f f e c t s o f coordinate expansion, the same expected head drop i s maintained across each b l o c k . Compared t o a s e t o f square b l o c k s , the head drop w i l l be g r e a t e r through the r e l a t i v e l y low c o n d u c t i v i t y b l o c k s and lower through the h i g h e r c o n d u c t i v i t y b l o c k s . For the boundary valu e problems s o l v e d , the d i f f e r e n c e s were not t h a t great. The r e s u l t o f such a coordinate expansion i s shown i n F i g u r e 5-7b. The head v a r i a b i l i t y i n c r e a s e s as the t a i l s o f the c o n d u c t i v i t y d i s t r i b u t i o n r e c e i v e s l i g h t l y more weight i n i n f l u e n c i n g the d i s t r i b u t i o n o f head w i t h i n any one r e a l i z a t i o n . These curves do not r e f l e c t an a n i s o t r o p y i n the covariance f u n c t i o n as they d i d e a r l i e r i n F i g u r e 5-7a f o r a n i s o t r o p y generated from the space law. I n the e a r l i e r problem, the Xg/L^ r a t i o s are d i f f e r e n t i n comparing the i s o t r o p i c and a n i s o t r o p i c cases (note X^/L was f i x e d ) . On expanding the b l o c k s c a l e t o achieve a n i s o t r o p y , not o n l y i s the i n t e g r a l s c a l e e f f e c t i v e l y i n c r e a s e d but so too i s the l e n g t h o f the f l o w system. Both curves shown i n F i g u r e 5-7b have i d e n t i c a l x y/^-^r X g / L g r a t i o s . The s i m i l a r i t y i n the curves f o l l o w from the equal number o f a s y m p t o t i c a l l y independent c o n d u c t i v i t y values w i t h i n the f l o w domain. Care must be taken i n i n t e r p r e t i n g the e f f e c t s o f c o o r d i n a t e expansion on the p r e d i c t i o n u n c e r t a i n t y . 132 d) e f f e c t o f two s t a t i o n a r y u n i t s w i t h i n the f l o w domain More than one s t a t i o n a r y u n i t may o ccur w i t h i n the f l o w domain. The p o s s i b i l i t y a r i s e s i n a l a y e r e d medium whenever the dimensions o f the i n d i v i d u a l l a y e r s are not s i g n i f i c a n t l y s m a l l e r than the dimensions o f the f l o w domain. A determinism i s i n t r o d u c e d i n t o the c o n d u c t i v i t y f i e l d t h a t cannot be a s c r i b e d t o the arrangement o f c o n d u c t i v i t i e s under a s t a t i o n a r y space law d e s c r i b i n g both u n i t s . The model should then take' account o f d i f f e r e n c e s i n the l a y e r means, standard d e v i a t i o n s o r s p a t i a l covariance r e l a t i o n s ; o r some combination o f these moments. Each o f the l a y e r s i s a r e a l i z a t i o n o f i t s own s t a t i o n a r y space law. The c o n d u c t i v i t y f i e l d i s generated by s o l v i n g the nearest neighbour equation (3-25) f o r each o f the l a y e r s and then crarfoiriing the sequences i n t h e i r a p p r o p r i a t e l o c a t i o n s w i t h i n the f i n i t e element g r i d . I t i s assumed the l a y e r boundaries are d i s t i n c t and t h e i r l o c a t i o n i s known. I n t h i s study, a two-layer problem w i t h each l a y e r o f the same dimension (5 x 20 blocks) i s presented. I n the f i r s t example, the i n t e g r a l s c a l e i s f i x e d i n each l a y e r b u t the c o n d u c t i v i t y standard d e v i a t i o n i n l a y e r 1 i s approximately one h a l f o f t h a t i n l a y e r 2. The head standard d e v i a t i o n s are p l o t t e d i n F i g u r e 5-8. A l s o p l o t t e d a r e the head v a r i a b i l i t i e s f o r the e q u i v a l e n t parameter one l a y e r problems. The r e s u l t s are c a l c u l a t e d along h o r i z o n t a l v e c t o r s o r i g i n a t i n g a t dimensionless v e r t i c a l d i s t a n c e s o f 0.3 and 0.7 along the constant head boundaries. 1 3 3 Figure 5-8 Influence of two s t a t i o n a r y u n i t s w i t h i n the flow domain on the standard d e v i a t i o n s i n h y d r a u l i c head, d i f f e r e n t standard d e v i a t i o n s i n h y d r a u l i c c o n d u c t i v i t y , 134 Two conc l u s i o n s can be reached. F i r s t , the standard d e v i a t i o n i n head i s g r e a t e r i n the l a y e r w i t h the l a r g e r standard d e v i a t i o n i n i t s c o n d u c t i v i t y d i s t r i b u t i o n . Secondly, the head v a r i a b i l i t y a t a p o i n t i s dependent upon the v a r i a b i l i t y a t a l l p o i n t s w i t h i n the flow domain. The presence o f l a y e r 1 a f f e c t s the v a r i a b i l i t y i n l a y e r 2 (and v i c e versa) so as t o decrease (or increase) the v a r i a b i l i t y compared t o a s i n g l e l a y e r problem w i t h a space law e q u i v a l e n t t o one of the two l a y e r s . Thus, the two-layer curves are bounded by those o f the one l a y e r problem. Note t h a t a t an equal dimensionless d i s t a n c e , the expected mean s o l u t i o n i s the same f o r a l l 4 curves i n F i g u r e 5-8. The same conc l u s i o n s are reached when a i s the same i n both y l a y e r s and the i n t e g r a l s c a l e i s in c r e a s e d i n the d i r e c t i o n o f mean flow i n one o f the two l a y e r s . A low c o n d u c t i v i t y c l a y o v e r l y i n g a h i g h e r c o n d u c t i v i t y sand i s a commonly encountered hydrogeologic c o n f i g u r a t i o n . A frequency histogram formed from a sample randomly d i s t r i b u t e d w i t h i n the f l o w domain would r e v e a l a bimodal c o n d u c t i v i t y d i s t r i b u t i o n . The e f f e c t o f the d i f f e r i n g mean c o n d u c t i v i t i e s on the head standard d e v i a t i o n s i s shown i n F i g u r e 5-9. The u n i d i r e c t i o n a l mean flow problem i s not w e l l s u i t e d t o such an a n a l y s i s b u t i t serves t o i l l u s t r a t e two p o i n t s . A p p l y i n g equations 3-30 and 3-31, the parameters i n the two c o n d u c t i v i t y sequences are given as: 1 135 Figure 5-9 Influence of two stationary units within the flow domain on the standard deviations i n hydraulic head, d i f f e r e n t mean hydraulic conductivities 2.75 x 10 -1 cm/s 6.60 x 10 cm/s v k l a y e r 1 5.30 x 10 -2 cm/s l a y e r 2 1.27 x 10 cm/s k The g r e a t e r head standard d e v i a t i o n s i n l a y e r 2 r e s u l t s from i t s g r e a t e r standard d e v i a t i o n i n the c o n d u c t i v i t y d i s t r i b u t i o n . I f the { Y}" parameters had been chosen so t h a t 0^ was equal i n both l a y e r s , then the head standard d e v i a t i o n s would a l s o be equal f o r the u n i -d i r e c t i o n a l mean flow problem. Secondly, i f the l a y e r i n g were not recognized and the conduct-i v i t y f i e l d assumed s t a t i o n a r y everywhere w i t h i n the flow domain, our estimate o f the standard d e v i a t i o n i n c o n d u c t i v i t y would be i n f l a t e d . In t h i s example, the standard d e v i a t i o n o f the combined {Y} sequence i s 1.09. The standard d e v i a t i o n s i n l a y e r 1 and 2 are 0.43 and 0.44, r e s p e c t i v e l y . I n s e r t i n g a space law t h a t combines both l a y e r s i n t o a s t a t i o n a r y u n i t would l e a d t o head v a r i a b i l i t i e s g r e a t l y exceeding those o f the a c t u a l two-layer problem. 4) Nonuniform Gradient F i e l d a) i n t r o d u c t i o n The s t o c h a s t i c problem i s now g e n e r a l i z e d t o co n s i d e r a non-uniform g r a d i e n t f i e l d . Here, the mean g r a d i e n t v e c t o r v a r i e s w i t h i n the flow domain. The f l o w l i n e s are no longer u n i d i r e c t i o n a l . I t w i l l be shown t h a t the magnitude and s p a t i a l v a r i a t i o n o f the head standard d e v i a t i o n s are determined by the s p a t i a l v a r i a t i o n i n the head g r a d i e n t s . 137 The i n f l u e n c e o f the space law parameters i n determining the f l u x and g r a d i e n t v a r i a b i l i t y i s a l s o i n v e s t i g a t e d . The boundary value problem which w i l l be i n v e s t i g a t e d i s shown i n F i g u r e 5-10a. /Also p l o t t e d are the d e t e r m i n i s t i c contours o f h y d r a u l i c head based on a s o l u t i o n u s i n g a s i n g l e c o n d u c t i v i t y everywhere w i t h i n the f l o w domain. There no longer e x i s t s any obvious v e c t o r along which t o p l o t the v a r i a t i o n i n the standard d e v i a t i o n i n head. Instead, the standard d e v i a t i o n s u r f a c e i s contoured. These p l o t s o n l y approximately d e t a i l i t s s p a t i a l v a r i a t i o n as the contouring i s performed manually on the s m a l l s e t o f nodal p o i n t s a t which the head d i s t r i b u t i o n was formed. b) comparison w i t h uniform g r a d i e n t f i e l d The f i r s t c o n d u c t i v i t y f i e l d t o be analyzed i s a s t a t i s t i c a l l y i s o t r o p i c medium. The head standard d e v i a t i o n s are contoured i n F i g u r e 5-10b. The c o n t o u r i n g was based on 20 p o i n t s . Two c o n c l u s i o n s can be drawn: 1) The head standard d e v i a t i o n s are g r e a t e s t i n t h a t r e g i o n o f the f l o w domain where the expected g r a d i e n t s are h i g h e r than average; y e t the r e g i o n i s f a r enough away from the constant head boundaries so t h a t t r u n c a t i o n o f the p e r m i s s i b l e range o f head values 138 m = 10 n = 20 p = 200 deterministic solution for hydraulic head Y: N[-2.00, 0.43] \=M.6 \ = 16.9 s= 10.0 MC = 300 Figure 5-10 Standard deviations i n hydraulic head, nonuniform gradient f i e l d 139 Y: N[-2.00, 0.43] \ = 25.6 ^ = 26.3 s= 10.0 MC = 300 Y: N[-2.00, 0.42] ^ = 1 0 . 4 X^=10.4 s= 10.0 MC = 300 Figure 5-10 Cont. 140 e) f y / / y / / y y y / / S y / y / y / / / y y s y y s y y y y y , y / y y y y y P.05 0.10 0.10 0.05 y / y / y / / y y y y y y y y y y y y y y y y y / y y y y y y / y / / y ) > y y j Y: N[-2.00, 0.43] m= 10 n= 20 p = 200 s= 10.0 MC = 300 0(0) = 1.0 0(Lx) =0 ^ = 17.4 "^=17.5 f ) 0.052 x 0.090 x 0.111 0.103 X 0.136 x 0.113 x 0.087 / / / / / / / y / y y / / j / y / y ) / / / / y / / / / / y / / / / / / / / £0.011 0.133 C0.143 Y: N[-2.00, 0.43] \ = 18.1 \ = 13.3 s=10.0 MC = 300 Figure 5-10 Cont. dees not a c t t o s i g n i f i c a n t l y reduce the v a r i a b i l i t y . 2) The v a r i a b i l i t i e s are g r e a t e r f o r t h i s boundary c o n f i g u r a t i o n than i n the u n i d i r e c t i o n a l mean flow problem i n a uniform g r a d i e n t f i e l d (Figure 5-10e). Both the c o n d u c t i v i t y space law and the t o t a l head drop were equal f o r the two g r a d i e n t f i e l d s . The u n c e r t a i n t y i n a flo w a n a l y s i s depends upon both the heterogeneity o f the c o n d u c t i v i t y f i e l d and the nature o f the flo w system o p e r a t i n g w i t h i n t h a t medium. We cannot a s s o c i a t e a p r i o r i a l e v e l o f s o l u t i o n v a r i a b i l i t y w i t h a porous medium o f known heterogeneity. c) v a r i a b l e i n t e g r a l s c a l e The i n t e g r a l s c a l e was inc r e a s e d along both coordinate axes w h i l e 0 ^ was kept constant. The r e s u l t i n g standard d e v i a t i o n s i n head are contoured i n F i g u r e 5-10c. Note the expansion o f the r e g i o n enclosed by the 0.15 contour. The standard d e v i a t i o n s i n head f o r an u n c o r r e l a t e d s e t o f c o n d u c t i v i t y b l o c k s are contoured i n F i g u r e 5-10d. The v a r i a b i l i t y here i s much lower than t h a t i n the two previous p l o t s . 142 d) s t a t i s t i c a l l y a n i s o t r o p i c covariance f u n c t i o n I n the uniform f l o w f i e l d , the expected head g r a d i e n t s were a l i g n e d w i t h the coordinate d i r e c t i o n o f the maximum i n t e g r a l s c a l e . For a nonuniform g r a d i e n t f i e l d i n a s t a t i s t i c a l l y a n i s o t r o p i c medium, the expected g r a d i e n t s have a v a r i a b l e o r i e n t a t i o n w i t h r e s p e c t t o the coordinate axes o f the covariance f u n c t i o n . The i n t e g r a l s c a l e a l i g n e d w i t h the coordinate a x i s i n which the expected g r a d i e n t s are g r e a t e s t (when averaged, over- the e n t i r e f l o w domain) w i l l have the most marked e f f e c t on the head standard d e v i a t i o n s . I n F i g u r e 5-10<f, the head standard d e v i a t i o n s a s s o c i a t e d w i t h a s t a t i s t i c a l l y a n i s o t r o p i c covariance f u n c t i o n are p l o t t e d . The space law parameters are given i n the diagram. This p l o t should be compared w i t h F i g u r e 5-10b. The head standard d e v i a t i o n s have decreased f o r the s m a l l e r \ s i n c e the z - d i r e c t i o n g r a d i e n t s dominate the flow f i e l d . R ecognition o f the reduced i n t e g r a l s c a l e i n the z - d i r e c t i o n i s more important i n t h i s problem than i n the u n i d i r e c t i o n a l mean flow problem d i s c u s s e d e a r l i e r . e) d i s c r e t i z a t i o n e r r o r i n numerical s o l u t i o n The d i s c r e t i z a t i o n e r r o r i n the f i n i t e element s o l u t i o n was evaluated f o r the nonuniform g r a d i e n t f i e l d . T h i s problem has the l a r g e s t expected g r a d i e n t s considered and thus the g r e a t e s t probable e r r o r . I t i s not s u f f i c i e n t t o compare s o l u t i o n s i n uniform media 143 f o r d i f f e r e n t d i s c r e t i z a t i o n i n t e r v a l s because the e r r o r w i l l v a r y w i t h each r e a l i z a t i o n , depending upon the p a r t i c u l a r arrangement of c o n d u c t i v i t y b l o c k s and the subsequent head g r a d i e n t s . Twenty one c o n d u c t i v i t y b l o c k s i n the r e g i o n o f the l a r g e s t expected g r a d i e n t s were d i v i d e d i n t o 4 t r i a n g u l a r elements by i n s e r t i n g a nodal p o i n t i n t h e ce n t e r o f each b l o c k . T h i s d i v i s i o n i s s c h e m a t i c a l l y i l l u s t r a t e d i n F i g u r e 5-11. A comparison was made f o r the problem shown i n F i g u r e 5-10b. On averaging over the same 300 r e a l i z a t i o n s , the estimated mean head d i d change s l i g h t l y w i t h the a d d i t i o n a l elements i n c l u d e d . The d i f f e r e n c e was g r e a t e s t where the expected g r a d i e n t was g r e a t e s t (rraximum d i f f e r e n c e o f 0.015, most l e s s than 0.001). D i f f e r e n c e s i n the estimated standard d e v i a t i o n s were s m a l l e r . The contoured standard d e v i a t i o n p l o t remains e s s e n t i a l l y unchanged. I t i s f e l t the i n t e r -p r e t a t i o n s o f the s t o c h a s t i c problem are soundly based on the d i s c r e t i -z a t i o n i n t e r v a l used. f) f l u x v a r i a b i l i t y The f l u x through the flo w domain w i l l be concentrated near the r i g h t v e r t i c a l boundary. Estimated moments o f the f l u x d i s t r i b u t i o n a t s e l e c t e d nodal p o i n t s on the constant head boundaries are summarized i n Table 5-1. The numbered l o c a t i o n s are given on the head standard d e v i a t i o n p l o t s . As the i n t e g r a l s c a l e i n c r e a s e s , so too does the f l u x v a r i a b i l i t y across the constant head boundaries. The skew o f the 144 nodes (1 ' ' ' 6) elements (A ' " ' D) nodes (1 ' ' • 8) elements (A ' " " H) Figure 5-11 F i n i t e element gri d for evaluation of the d i s c r e t i z a t i o n error 145 Integral Figure Scales Location Mean St. Dev. Coeff. Var. 1 0.041 0.018 0.44 X = 10.4 2 0.092 0.039 0.42 5-10d X X = 10.4 z 3 0.20 0.082 0.41 4 0.15 0.054 0.36 1 0.036 0.025 0.69 2 0.061 0.044 0.72 3 0.085 0.060 0.70 4 0.12 0.096 0.80 \ = 17-6 5 0.46 0.29 0.63 5-10b x ~X = 16.9 z 6 0.19 0.12 0.63 7 0.15 0.09 0.60 8 0.14 0.08 0.57 1 0.038 0.035 0.92 2 0.067 0.060 0.89 3 0.11 0.12 1.09 X = 25.6 4 0.17 0.26 1.53 5-10c x X = 26.3 z 5 0.59 0.59 1.00 6 0.18 0.15 0.83 7 0.18 0.15 0.83 8 0.16 0.13 0.81 1 0.042 0.029 0.69 X = 18.1 2 0.087 0.052 0.60 5-10e X ~X = 13.3 z 3 0.19 0.095 0.50 4 0.15 0.083 0.55 Note: flux cm/s Table 5-1 Flux Variability in Two-Dimensional, Nonuniform Gradient Field d i s t r i b u t i o n i n c r e a s e s f o r l a r g e r mean f l u x e s . This leads t o g r e a t e r c o e f f i c i e n t s o f v a r i a t i o n f o r those nodes where the mean f l u x i s gr e a t e r . g) g r a d i e n t v a r i a b i l i t y Estimated moments o f the g r a d i e n t d i s t r i b u t i o n are summarized i n Table 5-2. The g r a d i e n t v a r i a b i l i t y was formed along the boundary AC (Figure 5-10a) where J w i l l be n e g l i g i b l e . F or a uniform g r a d i e n t f i e l d , the c o e f f i c i e n t o f v a r i a t i o n i n the g r a d i e n t i n c r e a s e d f o r l a r g e r i n t e g r a l s c a l e s i n the d i r e c t i o n o f flow. I n s p e c t i o n o f the t a b l e i n d i c a t e s t h a t t h i s h o lds i n moving from a v e r t i c a l i n t e g r a l s c a l e o f 10.4 t o 16.9 u n i t s b u t not from 16.9 t o 26.3 u n i t s . Rather than i n d i c a t i n g a fundamental change i n the behavior o f the g r a d i e n t response, the d i f f e r e n c e i s probably due t o s t a t i s t i c a l ' f l u c t u a t i o n i n the s i m u l a t i o n . h) e f f e c t o f two s t a t i o n a r y u n i t s w i t h i n the f l o w domain I f two s t a t i o n a r y u n i t s w i t h d i f f e r e n t mean co n d u c t i v i t i e s occur w i t h i n the f l o w domain, the s p a t i a l v a r i a t i o n i n the h y d r a u l i c head standard d e v i a t i o n s w i l l depend upon the arrangement o f the u n i t s w i t h r e s p e c t t o the imposed boundary c o n d i t i o n s . Consider again a c l a y o v e r l y i n g a h o r i z o n t a l sand, as shown i n F i g u r e 5-12a. A l s o p l o t t e d i s the d e t e r m i n i s t i c head s o l u t i o n u s i n g s i n g l e c o n d u c t i v i t y 147 F i g u r e 5-10d 5-10b 5-10e 5-10c X , X x « 10.4, 10.4 17.8, 16.9 18.1, 13.3 25.6, 26.3 C 20.0 40.0 20.0 40.0 20.0 40.0 20.0 40.0 J X 1.15 1.43 1.16 1.41 1.15 1.42 1.13 1.35 S J x . 0.45 0.34 0.59 0.36 0.54 0.36 0.53 0.30 v J x 0.39 0.24 0.50 0.26 0.47 0.25 0.47 0.22 Table 5-2 Gradient V a r i a b i l i t y i n Two-Dimensional, Nonuniform Gradient F i e l d 148 a) x.815 x.446 - 078 <j>= 1.0 x.812 .807 x x .435 .422 x -.060 .037-.804 >, .607 .409 >' .210 V 007 7 7 7 7 T 7 7 7 7 ~/ / / / / / / . 0= 0.0 Y:N[ M y ay] m= 10 n = 20 p = 200 s=10.0 MC = 300 deterministic solution layer 1 a y = 0.44 V 1 7 . 0 layer 2 a y = 0.44 ^ = 1 6 . 7 Figure 5-12 Influence of two stationary units within the flow domain on the standard deviations i n hydraulic head, nonuniform gradient f i e l d 149 values i n each l a y e r . A p p l y i n g equation 3-29 f o r values o f -3.00 and -1.00 y i e l d s c o n d u c t i v i t y v a l u e s o f 0.001 and 0.1 cm/s, r e s p e c t i v e l y . Because o f the c o n d u c t i v i t y c o n t r a s t , almost a l l o f the head drop occurs i n the c l a y ; c r e a t i n g a n e a r - v e r t i c a l flow f i e l d toward the sand l a y e r . The moments i n the {Y } sequences f o r the two l a y e r s are gi v e n i n F i g u r e 5-12. The moments o f the lognormal c o n d u c t i v i t y d i s t r i b u t i o n , determined from equations 3-30 and 3-31 are: V k 2.77 x 1 0 _ 1 cm/s y' k 6.59 x 1 0 _ 1 cm/s l a y e r 1 _ 2 l a y e r 2 _ 1 a k 5.30 x 10 cm/s cr ^  1.27 x 10 cm/s The standard d e v i a t i o n s i n h y d r a u l i c head are gi v e n i n F i g u r e 5-12b f o r 10 nodal p o i n t s . The hi g h e r mean g r a d i e n t s i n the c l a y l e a d t o the l a r g e s t standard d e v i a t i o n s along the c e n t r a l h o r i z o n t a l a x i s o f t h a t l a y e r . Because o f t r u n c a t i o n i n the p o s s i b l e range o f v a l u e s , the v a r i a b i l i t y decreases towards the boundaries o f the l a y e r . The mean gr a d i e n t s i n the sand l a y e r are much lower, l e a d i n g t o s m a l l e r head standard d e v i a t i o n s . T h i s occurs even though the v a r i a b i l i t y o f the c o n d u c t i v i t y d i s t r i b u t i o n i s g r e a t e r i n the sand. The r e s u l t s o f F i g u r e 5-12b imply i t would be more important t o o b t a i n good estimates o f the space law parameters i n the c l a y than i n the sand. Any e r r o r s i n e s t i m a t i n g the space law parameters i n the 150 sand w i l l not g r e a t l y i n f l u e n c e the p r e d i c t i o n u n c e r t a i n t i e s t h a t a r i s e due t o the heterogeneity o f the medium. Because the expected g r a d i e n t s are much g r e a t e r i n the c l a y , the output v a r i a b i l i t i e s are much more s e n s i t i v e t o the parameters o f i t s space law. 5) Nonstationary C o n d u c t i v i t y F i e l d , Nonuniform Gradient F i e l d a) s p a t i a l t r e n d i n mean c o n d u c t i v i t y A simple form o f n o n s t a t i o n a r i t y can be considered by a l l o w i n g the mean c o n d u c t i v i t y t o vary w i t h i n the flow domain. I n t h i s study, the v a r i a t i o n i s r e s t r i c t e d t o r e p r e s e n t a t i o n by a d e t e r m i n i s t i c t r e n d . A t r e n d can be d e f i n e d whenever: 1) The mean o f the c o n d u c t i v i t y d i s t r i b u t i o n v a r i e s across the flow domain. 2) The space law i s s t a t i o n a r y but i n c l u d e s l o n g p e r i o d f l u c t u a t i o n s t h a t are i n the order o f o r g r e a t e r than the flow domain bei n g considered. By s e p a r a t i n g a t r e n d from the p o i n t - t o - p o i n t a u t o r e g r e s s i v e component, the assumption i s made t h a t the t r e n d i s c h a r a c t e r i s t i c o f the ensemble and not j u s t the s i n g l e r e a l i z a t i o n a v a i l a b l e t o us. Freeze (1977) analyzed the e f f e c t o f a l i n e a r l y t r e n d i n g mean i n a one-dimensional c o n s o l i d a t i o n problem and found i t s r e c o g n i t i o n t o s i g n i f i c a n t l y a f f e c t p r e d i c t i o n u n c e r t a i n t i e s . 151 A l i n e a r t r e n d i n the z coordinate d i r e c t i o n can be represented as: p = p + 3 ( z - p ) 5-6 y • z y y v *z where P z i s the c o n d i t i o n a l mean o f the random v a r i a b l e Y, given the l o c a t i o n z; £^ i s the slop e o f the r e g r e s s i o n curve o f Y on z,;; and p- i s the l o c a t i o n o f the center o f the flow domain i n the z coordinate. A two-dimensional l i n e a r t r e n d o r a hi g h e r order v a r i a t i o n are simple extensions o f equation 5-6. The s p a t i a l s t r u c t u r e i n the c o n d u c t i v i t y f i e l d can then be w r i t t e n : Yij = H y + ey H + [WJ{Y} + { e } 5-7 where the l a s t two terms d e s c r i b e the aut o r e g r e s s i v e component, the f i r s t term i s the non-zero mean o f the random f u n c t i o n {Y } , and the second term d e f i n e s a l i n e a r t r e n d i n the z coordinate. The m a t r i x pz] contains the d e v i a t i o n s o f the z coordinates o f the centers o f the c o n d u c t i v i t y b l o c k s from p ., There are now two vari a n c e s t o consider. The f i r s t , 0 2 , y • z i s a p o i n t v a r i a n c e a r i s i n g from the aut o r e g r e s s i v e component and the random term. The p o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n s are given as ~\ The second v a r i a n c e i s denoted a y N 2 The second v a r i a n c e i s denoted a and d e f i n e s the P' > o y • z ' y • z l _ _ v a r i a b i l i t y over the f l o w domain. They are r e l a t e d by 2 2 ^ . 2 2 CT = CT + $ CT 5_o y y • z y z J O 2 where CTZ i s given by a2 = ( ? ( L ) - z ( O ) ) 2 z _ _ _ _ _ _ 5-9 12 f o r a domain w i t h r e c t a n g u l a r boundaries. Here, z(-L) i s the maximum z coordinate i n the domain, z (0) the iirinimum. I f no t r e n d i s present, 2 . 2 a equals cr y ^ y • z 2 In generating a r e a l i z a t i o n w i t h a given v a r i a n c e CT^ , 2 equation 5-8 i s f i r s t s o l v e d f o r cr by assuming a value f o r 3 . -1 y • z y Then the va r i a n c e parameter n i s determined from equation 3-21 o r 2 • 2 3-22. E i t h e r CT o r 3 can be v a r i e d t o achieve a gi v e n cr y • z y y The i n t e g r a l s c a l e must be c a l c u l a t e d before a p p l y i n g the t r e n d t o a generated sequence. b) u n i d i r e c t i o n a l f low, nonuniform g r a d i e n t f i e l d With a t r e n d i n g mean c o n d u c t i v i t y , the g r a d i e n t f i e l d w i l l be nonuniform, although the mean flow may s t i l l be u n i d i r e c t i o n a l . The head standard d e v i a t i o n s w i l l r e f l e c t the s p a t i a l v a r i a t i o n i n the mean head g r a d i e n t . F i g u r e 5-13 shows the i n f l u e n c e o f a l i n e a r l y t r e n d i n g c o n d u c t i v i t y i n the d i r e c t i o n o f mean f l o w f o r the boundary valu e problem i l l u s t r a t e d i n F i g u r e 3-2. Curves A and B represent trends w i t h d i f f e r e n t 3 values. A l s o p l o t t e d are the head standard d e v i a t i o n s 153 Y: N[-2.00,a y] m = 10 n = 20 p = 200 s = 10.0 MC = 300 0(0) = 100 0(Lx)=O A CTy.x = 0.15 a y = 0.43 ^=17 .7 \ - 17.8 B oy.x = 0.32 a y = 0.43 = 18.0 X 2=18.1 •c Vx = ay = a_43 Xx-17.4 \ = 17.5 0.02 0.06 0.10 0.14 F i g u r e 5-13 I n f l u e n c e o f a l i n e a r l y t r e n d i n g mean h y d r a u l i c c o n d u c t i v i t y on t h e s t a n d a r d d e v i a t i o n s i n h y d r a u l i c h e a d f o r a case w i t h the same a y but a mean c o n d u c t i v i t y equal t o the average o f the t r e n d i n g c o n d u c t i v i t i e s above (curve C). The skewed shape o f the curves r e f l e c t s the s p a t i a l v a r i a t i o n o f the mean con-d u c t i v i t y . For the problem here, V y . z i s lower f o r s m a l l e r values o f dimensionless d i s t a n c e . Comparing curves B and C, i t can be seen t h a t the standard d e v i a t i o n s i n head are reduced, i n a l a r g e p a r t o f the flow domain. This p a t t e r n i s e x p l a i n e d by the i n t e r a c t i o n o f the s p a t i a l v a r i a t i o n 2 i n the mean head g r a d i e n t s , the reduced p o i n t v a r i a b i l i t y a y # z , and the i n f l u e n c e o f the constant head boundaries. As the s t r e n g t h o f 2 2 the t r e n d i n c r e a s e s (and a decreases t o keep 0 c o n s t a n t ) , the • y • z y head v a r i a b i l i t y decreases. Of course, t h i s does not imply t h a t media w i t h t r e n d i n g c o n d u c t i v i t i e s a r e s u b j e c t t o s m a l l e r p r e d i c t i o n v a r i a -2 . . . b i l l t i e s . I f as had been h e l d f i x e d , l a r g e r standard d e v i a t i o n s i n y «°z head would a s s o c i a t e w i t h s t r o n g e r trends. The f l u x v a r i a b i l i t i e s across two nodal p o i n t s on each o f the i n f l o w and out f l o w boundaries are summarized i n Table 5-3. Because the mean flow i s u n i d i r e c t i o n a l , the mean f l u x i s the same across both boundaries. Note however t h a t as the s t r e n g t h o f the t r e n d decreases, the estimated mean f l u x i n c r e a s e s . T h i s occurs even though both media have the same mean c o n d u c t i v i t y when averaged over the flo w domain. The d i f f e r e n t mean f l u x e s r e f l e c t s a response t o the d i f f e r e n t s p a t i a l v a r i a t i o n s i n the mean head g r a d i e n t s . This behavior w i l l have p a r t i c u l 155 implications in defining an equivalent uniform medium, as will be discussed in Chapter 7. Vz = 0- 1 5 V z = 0 - 3 2 a y =0.43 cy =0.43 mean s t d . dev. c o e f f . v a r. mean s t d . dev. c o e f f . v a r. i n f l o w 0.034 0.0070 0.21 0.044 0.020 0.46 D =0.0 X o u t f l o w 0.034 0.0066 0.20 0.041 0.017 0.43 D =1.0 X Note: f l u x cm/s Table. 5-3 F l u x V a r i a b i l i t y , Trend i n Mean C o n d u c t i v i t y I n s p e c t i o n o f Table 5-3 i n d i c a t e s the f o l l o w i n g : 1) F o r a given t r e n d , the c o e f f i c i e n t o f v a r i a t i o n i n the f l u x does not depend upon the d i f f e r e n t magnitudes o f the expected head gr a d i e n t s i n the v i c i n i t y o f the two boundaries. 2) The standard d e v i a t i o n s i n f l u x decrease w i t h a 156 reduction in the point variance of conductivity, 0 1 • y • z 6) Additional Examples Up to this point, idealized representations of flow systems have- been constructed in order to isolate the controlling factors in the stochastic analysis. To conclude this chapter, two flow problems wil l be presented that represent typical field situations amenable to a steady state, saturated analysis. The f i r s t will consider a tunnel beneath a horizontal constant head boundary, and the second, flow beneath an impermeable dam. a) inflow to a tunnel The boundary conditions and the deterministic solution using a single conductivity value everywhere within the flow domain is plotted in Figure 5-14a. To maintain a head drop of 1.0 m between the inflow and outflow surfaces, the boundary condition is set to a head of zero rather than using its actual elevation head. The conductivity field is a realization from a single stationary, statistically isotropic space law. The spatial variation of the standard deviations in head are contoured in Figure 5-14b. The parameters of the conductivity space law are given in the Figure. A large region of the flow domain is subject to head standard deviations of between 10 and 14% of the total 157 F i g u r e 5-14 I n f l o w t o a t u n n e l 158 head drop. Moments o f the f l u x d i s t r i b u t i o n a t s i x nodal p o i n t s on the t u n n e l are summarized i n Table 5-4. The node l o c a t i o n s are d e f i n e d i n Fig u r e 5-14a. Note the l a r g e c o e f f i c i e n t s o f v a r i a t i o n a s s o c i a t e d w i t h the f l u x d i s t r i b u t i o n . The estimated t o t a l f l u x i n t o the t u n n e l was 2 2 2.16 cm / s / u n i t t h i c k n e s s . I t s standard d e v i a t i o n was 0.66 cm / s / u n i t t h i c k n e s s . The v a r i a t i o n i n the t o t a l f l o w i s due e n t i r e l y t o the ho n e r g o d i c i t y o f each o f the r e a l i z a t i o n s . The logarithms o f the absolute v a l u e o f the f l u x f i t a normal d i s t r i b u t i o n a t the 95% s i g n i -f i a n c e l e v e l . L o c a t i o n 1 2 3 4 5 6 Mean 0. 54 0. .27 0. .60 0. .33 0. .10 0. .33 Std. Dev. 0. 28 0. .16 0. ,35 0. .16 0. .05 0. .16 Coeff. Var. 0. 52 o, .59 0. .58 0. .48 0. .50 0. .48 Note: f l u x cm /s Table 5-4 F l u x V a r i a b i l i t y F or In f l o w t o the Tunnel 159 b) flew beneath an inpermaeble dam The boundary value problem is illustrated in Figure 5-15a. Also plotted is the deterministic solution for hydraulic head using a single conductivity value everywhere within the flow domain. Constant head boundary conditions are imposed on the inflow and outflow surfaces. . The head standard deviations are contoured in Figure 5-15b. The maximum variability occurs in the central region beneath the dam immediately below its impermeable base. It is in this region that the nraximum uncertainties in our model predictions of hydraulic head (and " therefore pore pressures) occur. In Figure 5-15b the maximum standard deviations in head are over 15% of the total head drop. The estimated mean and standard deviation of the flux d i s t r i -bution calculated at 4 nodal points on the outflow surface are given in Table 5-5. Again, large variabilities exist in the flux predictions. The logarithms of the absolute value of the flux f i t a normal di s t r i -bution at the 95% significance level. The estimated total flux beneath-2 the dam was 0.61 cm /s/unit thickness. Its standard deviation was 2 0.23 cm /s/unit thickness. In the sense that the boundaries are imposed by us in our model of the actual field situation, this variability is real. If the conductivity field were stationary beyond our imposed model boundaries, extending those boundaries to model a larger region of the inflow and outflow surfaces will lead to a smaller uncertainty in the total flow beneath the dam. 160 a) 1 2 3 4 • t i t t m = 10 n = 20 p = 200 deterministic solution for hydraulic head Figure 5-15 Flow beneath an impermeable dam 161 L o c a t i o n 1 2 3 4 Mean 0. .27 0. .091 0. .070 0. .052 Std. Dev. 0. .19 0. .047 0. .036 0. .030 Coeff. Var. 0. .70 0. .52 0. .51 0. .58 Note: f l u x cm'/s .Table; 5-5 F l u x V a r i a b i l i t y on Outflow Surface Below Dam 7) Conclusions 1) I n the case o f u n i d i r e c t i o n a l f l o w i n a two-dimensional uniform g r a d i e n t f i e l d , the s o l u t i o n behaves i n an analogous manner t o the one-dimensional problem. The output u n c e r t a i n t i e s must be i n t e r p r e t e d i n l i g h t o f the i n t e g r a l s c a l e / l e n g t h r a t i o s along the two coordinate axes ( A J J / L » ^^^z)' I t w a s ° k s e r v e < 3 t h a t : a) The standard d e v i a t i o n s i n head (S ) i n c r e a s e w i t h an i n c r e a s e i n A r X o r a , where a i s X 2 y y the standard d e v i a t i o n i n the logarithms o f c o n d u c t i v i t y b) The g r a d i e n t v a r i a b i l i t y i n c r e a s e s w i t h l a r g e r values o f x . "A and/or a The behavior is explained by considering the probability of like conductivity values following each other and their departure from the mean conductivity of the stationary medium. In turn,this determines the departure of the head solution on any realization from the expected head solution in that region of the flow domain. 2) On comparing one and twci-dimensional analyses for media with equivalent space law parameters, the standard deviations in head are approximately halved in the t^-dimensional model. The maximum observed values of S (expressed as a percent of the total head drop) in the two-dimensional simulations were 4.8, 10.4 and 19.4%, where these values refer to"\^/L ratios of approximately 0.09, 0.087 and 0.07 and for o y values of 0.21, 0.43 and 0.86, respectively. The X g/L g ratios are one half of those in the x direction. 3) In a two-dimensional model, the flux variability is due to the combined effects of: a) the arrangement of conductivity values within the flow domain b) the nonergodicity of each realization. : The flux variability increases with an increase in ) , x _ X • 2 and/or a . For 'X /L , X /L ratios of 0.10 and 0.05, respectively, y XL X 2 X the coefficient of variation in flux is estimated as 0.31 and 0.60 for 163 0 cr values of 0.21 and 0.43, respectively. These results were obtained y for a uniform gradient field. 4) The main conclusion to be drawn from considering a 'nonuniform gradient field is that the spatial distribution of depends upon both the heterogeneity of the porous medium and the nature of the flow system operating within that medium. is greatest for that region of the flow domain where the expected gradients are relatively large, yet the region is far enough removed from the constant head boundaries so that truncation of the permissible range of values is not significant. For egual to 0.43, the neximum observed value of in the nonuniform gradient field was approximately 17% of the total head drop. In a uniform flow field of the same size and with the same space law for the conductivity distribution, the iraximum S ^  was 10.5% of the total head drop. 5) Statistically anisotropic covariance functions influence the output variabilities in terms of the respective ratios of X/L along the two coordinate axes. The integral scale aligned with the coordinate axis in which the expected gradients are the greatest wi l l have the most marked effect on S^  . Care must be taken in interpreting the effects of anisotropy introduced through block expansion. 6) If a conductivity field is layered such that the dimensions of the individual layers are not significantly smaller than the dimensions 1 6 4 of the flo w domain, then t h i s determinism must be accounted f o r and i n s e r t e d i n t o the e n s e m b l e s t r u c t u r e . Recognition o f two s t a t i o n a r y u n i t s i n a uniform g r a d i e n t f i e l d leads t o the ob s e r v a t i o n s : a) S i s g r e a t e r i n the l a y e r w i t h t h e l a r g e r 9 X A _ and/or a X z y b) The v a r i a b i l i t y i n head a t any p o i n t i s dependent upon the v a r i a b i l i t y a t a l l p o i n t s w i t h i n the flow domain. In a nonuniform g r a d i e n t f i e l d , the s p a t i a l v a r i a t i o n s i n S depend 9 upon: a) The arrangement o f the two u n i t s w i t h r e s p e c t t o the imposed boundary c o n d i t i o n s . b) The d i f f e r e n c e s i n the space law parameters o f the two l a y e r s . Layers w i t h d i f f e r e n t mean c o n d u c t i v i t i e s w i l l have p a r t i c u l a r i m p l i c a t i o n s f o r sampling theory. 7). r e c o g n i t i o n o f a s p a t i a l t r e n d i n the mean c o n d u c t i v i t y i s c r i t i c a l i n determining the magnitude and s p a t i a l v a r i a t i o n o f . The t r e n d i n g mean w i l l : a) C o n t r o l S through i t s e f f e c t on the s p a t i a l 9 distribution of the mean head gradient. Determine the split between the point variance 2 ( ° y. z) of the conductivity distribution and the 2 total variance over the flow domain, 0 y . For 2 a fixed a y , S^ is generally reduced the 2 smaller the value of cr . . 166 i CHAPTER 6 The Question o f Sca l e : A D i s c u s s i o n l j I n t r o d u c t i o n The groundwater flo w equation i s a p p l i e d t o f i e l d problems a t many d i f f e r e n t s c a l e s . For example, we may be i n t e r e s t e d i n modeling the flow i n t o an i n d i v i d u a l i r r i g a t i o n d i t c h on the s c a l e o f a few meters, o r the flow through an e a r t h f i l l dam over tens o f meters, o r the flow i n a l a r g e - s c a l e r e g i o n a l system o p e r a t i n g over many k i l o m e t e r s . I t i s u s u a l p r a c t i c e i n d e t e r m i n i s t i c modeling t o de s c r i b e the c o n d u c t i v i t y v a r i a t i o n s by means o f s i n g l e - v a l u e d g e o l o g i c u n i t s t h a t r e f l e c t the s c a l e o f the model. A t h i n c l a y l a y e r f o u r f e e t below the wa t e r t a b l e may be c r i t i c a l t o the drainage problem y e t i t i s n ot considered as a separate e n t i t y i n a r e g i o n a l f l o w a n a l y s i s . I n the p r a c t i c a l a p p l i c a t i o n o f s t o c h a s t i c techniques t o modeling any o f the above f i e l d problems, s c a l e e f f e c t s must a l s o be considered. We must co n s i d e r how our d e s c r i p t i o n o f the c o n d u c t i v i t y space law i s dependent upon the i n t e r a c t i o n between the s c a l e o f the flow problem and the nature o f the s p a t i a l heterogeneity. I n e a r l i e r chapters, the importance o f the v a r i o u s space law parameters i n determining the output v a r i a b i l i t i e s i n a t h e o r e t i c a l f l o w model was demonstrated. For the steady s t a t e problem, the flow net i s independent o f the s c a l e o f the a n a l y s i s . By s u i t a b l e expansion, the model r e s u l t s c o u l d e q u a l l y w e l l apply t o a s e t o f b l o c k s a meter 167 on a s i d e as they c o u l d t o bl o c k s 100 meters on a s i d e . I n a f i e l d s i t u a t i o n , such an expansion i s v a l i d o n l y i f the d i s c r e t e b l o c k c o n d u c t i v i t y model preserves the covariance s t r u c t u r e i n the r e s p e c t i v e flow domains. Thus, we must con s i d e r how the d i s c r e t i z a t i o n i n t e r v a l depends upon the s c a l e o f the flow domain i n a p p l y i n g s t o c h a s t i c t e c h -niques t o f i e l d problems. I n the succeeding s e c t i o n s , a q u a l i t a t i v e d i s c u s s i o n o f v a r i o u s c o n s i d e r a t i o n s regarding the s c a l e o f the flow a n a l y s i s w i l l be presented. L i t t l e work has been done on t h i s problem as y e t , as i t s s o l u t i o n depends upon the c o l l e c t i o n o f f i e l d data on numerous s c a l e s and i n d i f f e r e n t hydrogeologic environments. B o r e l i e t a l (1977) comment on s c a l e e f f e c t s i n a s t o c h a s t i c a n a l y s i s o f flo w i n f i s s u r e d rock masses. T h e o r e t i c a l s o l u t i o n s can o n l y h i g h l i g h t the dependence o f the p r e d i c t i o n v a r i a b i l i t i e s on a p o s t u l a t e d space law s t r u c t u r e . I n t e g r a t i o n o f these techniques w i t h w e l l - d e f i n e d f i e l d problems i s a p r e s s i n g need i n f u t u r e research. 2) S c a l e C l a s s i f i c a t i o n Many d e s c r i p t i o n s o f the heterogeneity w i t h i n a sedimentary sequence are a v a i l a b l e i n the l i t e r a t u r e (eg. Johnson and Greenkorn (1963), Polaesk and Hutchinson (1967), Alpay (1972)). Perhaps the most u s e f u l c l a s s i f i c a t i o n o f the s c a l e s o f heterogeneity f o r our purposes i s t h a t o f Dagan ( i n p r e s s ) , who adopts the parameters used by Lumley and Panofsky (1964). T h i s c l a s s i f i c a t i o n i s based on the p r o b a b i l i s t i c s t r u c t u r e o f a heterogeneous medium t h a t was o u t l i n e d i n Chapter 2. Three d i f f e r e n t s c a l e s are recognized: 1) A nonhomogeneity s c a l e ( L * ) , c h a r a c t e r i z i n g the s c a l e over which the average p r o p e r t i e s o f the porous medium change s i g n i f i c a n t l y . 2) The i n t e g r a l s c a l e ( \ ) , the average d i s t a n c e over which p o i n t values o f c o n d u c t i v i t y are c o r r e l a t e d . 3) A m i c r o s c a l e ( & ) , d e f i n e d as a f u n c t i o n o f the r a t e o f change o f the s l o p e o f the c o r r e l a t i o n f u n c t i o n a t the o r i g i n . I t i s i n t e r p r e t e d i n terms o f the d i s t a n c e over which c o n d u c t i v i t y values are almost f u l l y c o r r e l a t e d . The m i c r o s c a l e and i n t e g r a l s c a l e a r e d e f i n e d n u m e r i c a l l y i n terms o f the a u t o c o r r e l a t i o n f u n c t i o n . The m i c r o s c a l e can be r e l a t e d t o the b l o c k s i z e i n a d i s c r e t i z e d model o f c o n d u c t i v i t y v a r i a t i o n s . The nonhomogeneity s c a l e i s a conceptual term, r a t h e r than i t having a p r e c i s e numerical valu e . I t can be viewed as an upper bound f o r the i d e n t i f i c a t i o n o f s t a t i o n a r y u n i t s w i t h i n the flo w domain. L o c a l s t a t i o n a r i t y i s assumed i f A^'L*. The nonhomogeneity s c a l e depends upon e i t h e r the s t r e n g t h o f s p a t i a l trends i n the moments o f the flo w 1 6 9 parameters o r the e x i s t e n c e o f zones w i t h h i g h l y c o n t r a s t i n g mean c o n d u c t i v i t i e s . 3) Block S t r u c t u r e i ) assumptions Th i s study i s based on modeling macroscopic v a r i a t i o n s i n c o n d u c t i v i t y . The p o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n s apply, i n p r a c t i c e , t o values averaged over s m a l l volumes such as a sample core o r the s m a l l volume which i n f l u e n c e s a f a l l i n g head t e s t i n a piezometer. A b l o c k model o f the s p a t i a l v a r i a t i o n s i n c o n d u c t i v i t y r e q u i r e s the assumption t h a t s i n g l e values can be d e f i n e d t o represent the h y d r a u l i c behavior o f the heterogeneity w i t h i n each b l o c k . I n g e n e r a l , the area average o f a b l o c k w i l l be much l a r g e r than the area average o f the p o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n s . By u s i n g e q u i v a l e n t values i n the d i s c r e t i z e d domain, s p a t i a l v a r i a t i o n s on a s c a l e s m a l l e r than the b l o c k do not c o n t r i b u t e t o the estimates o f the s o l u t i o n v a r i a b i l i t y . I t f o l l o w s t h a t s p a t i a l h e t e r o g e n e i t i e s a t s c a l e s equal t o o r l a r g e r than the b l o c k s i z e determine the estimated p r e d i c t i o n v a r i a b i l i t i e s i n the d i s c r e t i z e d model. I n modeling a f i e l d s i t u a t i o n , the b l o c k s i z e must be s p e c i f i e d i n accord w i t h t h i s d i s c r e t i z e d behavior. I n t e g r a t i o n o f t h i s concept w i t h the s c a l e o f the flo w a n a l y s i s f o l l o w s i n a succeeding s e c t i o n . 170 For the c o n d u c t i v i t y r e a l i z a t i o n s generated i n e a r l i e r chapters, the c o n d u c t i v i t y values i n s e r t e d i n t o t h e b l o c k s were s e l e c t e d from the same p r o b a b i l i t y d e n s i t y f u n c t i o n as d e s c r i b e d the p o i n t values v o f c o n d u c t i v i t y . The correspondence between the p r o b a b i l i t y d i s t r i -b u t i o n s o f the s i n g l e e q u i v a l e n t values and the p o i n t values o f c o n d u c t i v i t y bears f u r t h e r i n v e s t i g a t i o n . The d i s c r e t e b l o c k model assumes t h a t s i n g l e e q u i v a l e n t values e x i s t and t h a t they can be s e l e c t e d by choosing values from the p o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n s . However i t does not r e q u i r e t h a t the f u n c t i o n a l dependence o f these s i n g l e values on the porous medium heterogeneity be s p e c i f i e d . T h i s problem w i l l be b r i e f l y considered i n Chapter 7. There i s a need f o r f u t u r e research i n t o the p r e c i s e dependence o f the s i n g l e e q u i v a l e n t values on the b l o c k s i z e and shape, the f l o w f i e l d , the heterog e n e i t y o f the medium and the s c a l e o f the f l o w a n a l y s i s . i i ) m i c r o s c a l e The m i c r o s c a l e s o f a heterogeneous medium can be a s s o c i a t e d w i t h the b l o c k dimensions i n the nearest neighbour model. For a domain d i s c r e t i z e d i n t o square b l o c k s , there i s no p r e f e r r e d o r i e n t a t i o n i n re p r e s e n t i n g the s p a t i a l heterogeneity by e q u i v a l e n t s i n g l e v a l u e s . Rectangular b l o c k s were considered e a r l i e r i n Chapter 5 as a means o f i n t r o d u c i n g a s t a t i s t i c a l l y a n i s o t r o p i c covariance f u n c t i o n i n t o the 171 c o n d u c t i v i t y f i e l d . I n a s t r a t i f i e d d e p o s i t , t h e r e i s an i n t u i t i v e appeal t o u s i n g r e c t a n g u l a r b l o c k s i n o r d e r t o match the b l o c k s i z e and shape t o the sedimentary p a t t e r n . I f the s t r a t i f i c a t i o n cannot be r e f l e c t e d i n a s e t o f a n i s o t r o p i c a u t o r e g r e s s i v e parameters, r e c t a n g u l a r b l o c k s provide an a l t e r n a t i v e model w h i l e s t i l l r e l y i n g on the nearest neighbour r e l a t i o n . I n a two-dimensional r e c t a n g u l a r b l o c k system, t h e r e would be an a n i s o t r o p y i n the m i c r o s c a l e as w e l l as i n the i n t e g r a l s c a l e o f the medium. I n t e r p r e t i n g the c o r r e l a t i o n s t r u c t u r e o f the model i s now more complicated because the c o r r e l a t i o n between c o n d u c t i v i t y values i n the b l o c k s t r u c t u r e i s on a d i f f e r e n t s c a l e i n the two coordinate d i r e c t i o n s . The d i f f e r e n t m i c r o s c a l e s i n the two coordinate d i r e c t i o n s w i l l i n f l u e n c e the er g o d i c behavior o f each o f the generated c o n d u c t i v i t y f i e l d s . For a s e t o f square b l o c k s the nearest neighbour weights (eqn. 3-3) were symmetric i f the process had a s t a t i s t i c a l l y i s o t r o p i c covariance f u n c t i o n . T h i s symmetry would be preserved on a r e c t a n g u l a r l a t t i c e , even though the lengths o f the contiguous boundaries along the coordinate d i r e c t i o n s would no longer be equal. The c o r r e l a t i o n -l a g r e l a t i o n (eqn. 3-i2) would be the same f o r both square and r e c t a n g u l a r b l o c k s . However, on a p p l y i n g the m i c r o s c a l e s , the c o r r e l a t i o n - d i s t a n c e r e l a t i o n would d i f f e r . 172 3) S c a l e o f the Flow A n a l y s i s a) general c o n s i d e r a t i o n s I t i s reasonable t o assume t h a t the s c a l e a t which the hetero-geneity o f the c o n d u c t i v i t y f i e l d i s d e s c r i b e d i s somehow r e l a t e d t o the s c a l e o f the flow a n a l y s i s . For example, i n a r e g i o n a l f l o w , the model would preserve the r e g i o n a l s t r u c t u r e s and p a t t e r n s o f c o n d u c t i v i t y v a r i a b i l i t y . On the o t h e r hand, i n c o n s i d e r i n g f l o w t o a t i l e d r a i n ; the space law would r e f l e c t the l o c a l dependence i n the r e g i o n about the d r a i n . By chance, i t may happen t h a t the f l e w domains are d i s c r e t i z e d i n t o a s i m i l a r number o f b l o c k s . However, i n the r e g i o n a l model, s i n g l e values would be a p p l i e d over much l a r g e r areas. Rodriquez-Iturbe and M e j i a (1974) d i s c u s s a s i m i l a r problem i n p r e s e r v i n g the c o r r e l a t i o n s t r u c t u r e o f r a i n f a l l over b a s i n s o f v a r i a b l e s i z e . The s p a t i a l v a r i a t i o n s i n c o n d u c t i v i t y a t a f i e l d s i t e l e a d t o a t h e o r e t i c a l s e t o f u n c e r t a i n t i e s i n our attempt t o model t h a t medium. The t h e o r e t i c a l v a r i a b i l i t y i s b u i l t upon the c o n t r i b u t i o n s o f a l l o f the s c a l e s o f v a r i a t i o n i n the c o n d u c t i v i t y f i e l d , from mi c r o s c o p i c through t o the l a r g e s c a l e macroscopic h e t e r o g e n e i t i e s . A d i s c r e t e b l o c k model approximates the heterogeneity o f the r e a l medium. I t f o l l o w s the Monte C a r l o s o l u t i o n o n l y approximates the t r u e v a r i a b i l i t y i n the p r e d i c t i o n o f h y d r a u l i c head. I n o r d e r t o o b t a i n an accurate s o l u t i o n i n modeling a f i e l d s i t u a t i o n , the b l o c k model must reproduce 173 a l l o f the s i g n i f i c a n t s c a l e s o f heterogeneity t h a t c o n t r i b u t e t o the t h e o r e t i c a l s e t o f p r e d i c t i o n u n c e r t a i n t i e s . What c o n s t i t u t e s a s i g n i f i c a n t heterogeneity must be i n t e r p r e t e d i n l i g h t o f the s c a l e o f the flow a n a l y s i s . I n t u r n , t h i s can be c a s t i n terms o f the i n f l u e n c e o f the h e t e r o g e n e i t i e s on the expected g r a d i e n t f i e l d w i t h i n the flow domain. The g r a d i e n t f i e l d w i l l respond t o s p a t i a l v a r i a t i o n s i n c o n d u c t i v i t y i n terms o f both the magnitude o f t h a t v a r i a t i o n from the mean c o n d u c t i v i t y o f the medium and the s i z e and c o n t i n u i t y o f the r e g i o n over which the v a r i a t i o n s occur. Because the g r a d i e n t f i e l d r e f l e c t s most s t r o n g l y the l a r g e s t hetero-g e n e i t i e s w i t h i n the flow domain, t h e i r s c a l e w i l l dominate the p r e d i c t i o n v a r i a b i l i t i e s . C o n d u c t i v i t y v a r i a t i o n s over a few meters w i l l dominate the u n c e r t a i n t y i n the g r a d i e n t f i e l d i n a s m a l l s c a l e f l o w a n a l y s i s . These s t r u c t u r e s must then be preserved i n the system o f b l o c k s . I n a r e g i o n a l model, v a r i a t i o n s on t h i s s c a l e are not s i g n i f i c a n t i n p e r t u r b i n g the g r a d i e n t f i e l d from i t s expected behavior. Rather, i t i s the s t r u c t u r e o f the l a r g e s c a l e macroscopic v a r i a t i o n s which must be preserved i n the d i s c r e t e b l o c k model. The modeling approach used i n t h i s study i s based on i d e n t i f y i n g s t a t i o n a r y u n i t s w i t h i n the f l o w domain. S p a t i a l averages must be formed over a l a r g e enough area t o assume s t a t i o n a r i t y , y e t not so l a r g e as to i n c l u d e gross heterogeneity. A d i s t i n c t i o n i s made between the 174 s t o c h a s t i c components o f the heterogeneity and t h a t which w i l l be d e s c r i b e d d e t e r m i n i s t i c a l l y . Whenever a d e t e r m i n i s t i c s t r u c t u r e i s s p e c i f i e d , i t becomes p a r t o f the ensemble s t r u c t u r e and not j u s t the s i n g l e r e a l i z a t i o n a t the f i e l d s i t e . Layered systems may be considered under a completely s t o c h a s t i c framework i n a r e g i o n a l f l o w a n a l y s i s ; y e t i n a s m a l l e r s c a l e a n a l y s i s , d e t e r m i n i s t i c d e s c r i p t i o n o f the l a y e r s may be r e q u i r e d . I n t h i s case, o n l y the c o n d u c t i v i t y v a r i a t i o n s w i t h i n the l a y e r would be random. A p r a c t i c a l example o f t h i s c o n s i d e r a t i o n can be drawn from the well-known models o f m u l t i p l e a q u i f e r systems. Each o f the a q u i f e r s and aq u i t a r d s would be .recognized as a separate stationary'unit'.. ,..The. c o n d u c t i v i t y v a r i a t i o n s w i t h i n each o f the l a y e r s would be modeled s t o c h a s t i c a l l y , l e a d i n g t o p r o b a b i l i t y d i s t r i b u t i o n s on the flow w i t h i n and leakage across each o f the u n i t s . I t i s assumed the number o f l a y e r s and t h e i r boundary l o c a t i o n s are known from w e l l - l o g data. b) g e o l o g i c c o n s i d e r a t i o n s The i n t e r a c t i o n between s c a l e and heterogeneity can a l s o be discussed i n terms o f the primary d e p o s i t i o n a l f e a t u r e l e a d i n g t o the s p a t i a l v a r i a b i l i t y a t the s c a l e o f the flo w a n a l y s i s . Consider the 4 c r o s s - s e c t i o n s shown i n F i g u r e 6-1. For each o f these media, the f o l l o w i n g comments can be made: 175 1) 6 - l a This diagram d e p i c t s a 'homogeneously heterogeneous' crossbedded sand. The sedimentary s t r u c t u r e s l e a d i n g t o the s p a t i a l v a r i a t i o n s i n c o n d u c t i v i t y occur on a s c a l e much s m a l l e r than t h a t o f the flo w domain. Consequently, the e n t i r e medium would be modeled i n a s t o c h a s t i c framework. I f the medium s a t i s f i e s the s t a t i o n a r i t y assumption, any sample i n t e r v a l l e s s than the i n t e g r a l s c a l e w i l l show a c o r r e l a t i o n between neighbouring p o i n t s . I f the sample i n t e r v a l i s g r e a t e r than the i n t e g r a l s c a l e ; there w i l l be no c o r r e l a t i o n between sample p o i n t s , whatever the l e n g t h o f the sample s e r i e s . 2) 6-lb The medium here i s complexly bedded a t the s c a l e o f o b s e r v a t i o n . The primary d e p o s i t i o n a l f e a t u r e i s the bedding i t s e l f . The arrangements and dimensions o f the beds c o n t r o l the s i g n i f i c a n t c o n d u c t i v i t y v a r i a t i o n s ; not the v a r i a t i o n w i t h i n any bed. I t i s the c o r r e l a t i o n s t r u c t u r e i n these bedding p a t t e r n s which must be preserved i n the s t o c h a s t i c model. 3) 6 - l c Here a l a y e r e d medium i s modeled a t a s c a l e such t h a t o n l y a few l a y e r s occur w i t h i n t h e flow domain. The l a y e r i n g would be de s c r i b e d d e t e r m i n i s t i c a l l y ; each l a y e r b e i n g modeled as a s t a t i o n a r y u n i t . 176 a) / / / / / / / / / , / / / / / / / / / . / / / / / / / / / . / / / / / / / / / . / / / / / / / /. / / / / / / / / / . / / / / / / / / / . / / / V / / / / b) c) 7 / 7 7 7 7 7 K~ / d) V / / / / / / / y J / / / / ' • A / / / / V / / / < / N /V Figure 6-1 Schematic diagram of geologic considerations for the interaction between scale and heterogeneity 177 4) 6-ld A bimodal model will occur when lenses of low conductivity are dispersed in a higher conductivity matrix (or vice versa). It may be appropriate to describe the ccntinuity of the lenses in terms of their own stochastic process. The heterogeneity of the medium would depend upon the arrangement, size and density of the inclusions, the conductivity contrast between the two modes, and the variability about each mode. The above hypothetical patterns are part of an infinite sequence of possible arrangements. No scale was set to these diagrams as i t is the patterns of variation within the flow domain, whatever its size, that determines the relationship between heterogeneity and scale. An additional example of a scale consideration is provided by the transition between two stationary units. The transition can range from gradational through to a sharp contact. The boundaries must be considered in light of the scale at which the units themselves are considered stationary. For a sharp or abruptly gradational contact, i t is reasonable to assume there is no continuity in the spatial covariance function across the contact. Gradational boundaries pose more of a problem, as i t is not clear what stochastic model parameters 178 apply in the transition zone between the two distinct units. As the extent of the transition zone increases with respect to the scale of the stationary units, i t will be more difficult to separate them or to have confidence in the applicability of the stationarity assumption. Separate identification of the transition zone may be possible but one is s t i l l left with the problem of modeling the resulting nonstationary unit. c) model considerations The nearest neighbour model describes one of the simplest forms of spatial heterogeneity. One scale is responsible for the entire pattern of the conductivity variation within the stationary field. That scale is defined by the block size and the covariance function. The entire structure results from a local stochastic interaction between neighbouring conductivity values. The probability distribution for conductivity is monomodal. Therefore, the nearest neighbour relation is most suited to modeling media that can be characteristically described as 'homogeneously heterogeneous'. Provided these conditions are satisfied at the field site, the model is equally applicable to a small scale drainage problem as for the regional analysis of flow. It is important to recognize that the nearest neighbour model is not a general model encompassing a l l forms of spatial heterogeneity. For example, i t is not suited to describing randomly dispersed low 179 conductivity lenses within a higher conductivity matrix. It would be more appropriate to describe the conductivity variations within the matrix itself. Because of the monomodal conductivity distribution and the local nature of the spatial dependence, a low conductivity lens.with a ccntinuity on the same scale as the entire conductivity field will not be generated. Other probabilistic models may be more suited to describing these field conditions. 180 CHAPTER 7 Comparison Among Models 1) I n t r o d u c t i o n I n t h i s study, Monte C a r l o techniques are used t o estimate the p r o b a b i l i t y d i s t r i b u t i o n s on the p r e d i c t i o n v a r i a b l e s . Other mathematical techniques are a l s o a v a i l a b l e t o s o l v e the s t o c h a s t i c flow equations. The r e l a t i o n s h i p s between these v a r i o u s approaches w i l l be considered i n t h i s chapter. As w e l l , s e v e r a l p r e d i c t i o n s t h a t a r i s e from the r e s p e c t i v e s o l u t i o n s w i l l be compared. I t i s shewn t h a t t h e Monte C a r l o and s p e c t r a l a n a l y s i s techniques p r e d i c t a s i m i l a r response t o f l o w modeling i n heterogeneous media. There are d i f f e r e n c e s however i n the degree o f dependence o f the h y d r a u l i c head v a r i a b i l i t y on the dimension o f the a n a l y s i s . The i n t e n t o f t h i s chapter i s not t o fa v o r one approach over the o t h e r s , as each has a r o l e t o p l a y i n u n r a v e l i n g the c o m p l e x i t i e s o f the s t o c h a s t i c approach t o flow modeling. Attempts have been made t o d e s c r i b e a heterogeneous medium by an e q u i v a l e n t , s i n g l e valued uniform medium. The geometric mean has been suggested as a p o s s i b l e v a l u e f o r t h i s " e f f e c t i v e " parameter. I t i s shown t h a t there are s t r i c t c o n d i t i o n s i n a two-dimensional, steady s t a t e f l o w problem f o r the e f f e c t i v e c o n d u c t i v i t y t o be given by the geometric mean c o n d u c t i v i t y . 2) Comparison w i t h S p e c t r a l Techniques a) d i f f e r e n c e s i n the approach D i r e c t comparisons o f the p r e d i c t i o n v a r i a b i l i t y determined from the numerical Monte C a r l o technique and the a n a l y t i c a l s p e c t r a l approach as used by Bakr (1976) and Gelhar (197;?) are not p o s s i b l e . T h i s i s due t o d i f f e r e n c e s i n the assumptions made about both the form o f the random v a r i a b l e s and the f l o w domain i n the r e s p e c t i v e s o l u t i o n s . The major d i f f e r e n c e s can be summarized as f o l l o w s : 1) The c o n d u c t i v i t y f i e l d i s represented here by a s e r i e s o f d i s c r e t e b l o c k s w h i l e the s p e c t r a l approach i s a continuum a n a l y s i s . 2) The p e r t u r b a t i o n s o f the random v a r i a b l e s (K, 9 ) r e q u i r e d i n s e t t i n g up the s p e c t r a l r e p r e s e n t a t i o n o f the flow equations must be s m a l l t o o b t a i n an accurate s o l u t i o n . No such assumption i s made i n the Monte C a r l o technique. 3) S p e c t r a l a n a l y s i s provides i n f o r m a t i o n on the mean and covariance o f the output v a r i a b l e s . Because the Monte C a r l o technique assumes a c o n d u c t i v i t y d i s t r i b u t i o n , p r o b a b i l i t y d i s t r i b u t i o n s o f the output v a r i a b l e s can be estimated. 182 4) The s p e c t r a l approach r e q u i r e s t h a t the mean, var i a n c e and covariance o f the head p e r t u r b a t i o n s be constant everywhere w i t h i n the flow domain. I f constant head boundaries are d e f i n e d , the va r i a n c e o f the estimated head d i s t r i b u t i o n s i s a f u n c t i o n o f p o s i t i o n ( i e . n o n s t a t i o n a r y ) . 5) As a consequence o f both the s t a t i o n a r i t y assumption i n the head p e r t u r b a t i o n s and the r e p r e s e n t a t i o n o f c o n d u c t i v i t y and head by continuous s p e c t r a , the f l o w domain cannot be bounded i n the d i r e c t i o n o f mean flow. S p e c t r a l s o l u t i o n s f o r head v a r i a b i l i t y may c o n t a i n l e n g t h parameters (eg. Gelhar (1977)) b u t these r e l a t e t o boundaries o r i e n t a t e d p a r a l l e l t o the mean flow d i r e c t i o n . 6) The s p e c t r a l approach e f f e c t i v e l y assumes the parameter X/L (introduced i n Chapter 4) i s zero s i n c e the system i s not bounded i n the d i r e c t i o n o f mean flow. The Monte C a r l o output can o n l y be i n t e r p r e t e d i n l i g h t o f t h i s parameter. Length s c a l e s i n the d i r e c t i o n o f mean flow are intro d u c e d i n s p e c t r a l techniques o n l y when a p p l y i n g the s p e c t r a l s o l u t i o n s t o 183 network design problems o r some such e r r o r a n a l y s i s . 7) To date, the s p e c t r a l approach i s r e s t r i c t e d t o the a n a l y s i s o f a s i n g l e s t a t i o n a r y u n i t i n a uniform g r a d i e n t f i e l d . More general problems are e a s i l y considered u s i n g Monte C a r l o a n a l y s i s . The e x t e n t o f the d i f f e r e n c e s between the two techniques should come as no s u r p r i s e . I n d e t e r m i n i s t i c s o l u t i o n s o f d i f f e r e n t i a l equations, numerical techniques are more f l e x i b l e than an a n a l y t i c approach. Although more r e s t r i c t e d i n t h e i r a p p l i c a t i o n t o f l o w problems, the s p e c t r a l techniques do have the advantage o f p r o v i d i n g f u n c t i o n a l s o l u t i o n s i n terms o f the model parameters. Inferences i n the Monte C a r l o technique must be made by comparison o f r e s u l t s from a s e r i e s o f s i m u l a t i o n experiments. b) s o l u t i o n f o r the head v a r i a n c e Bakr (1976) o b t a i n s an approximate s o l u t i o n o f the one-domensional, steady s t a t e flow equation (4-1) w i t h o u t the c o n s t r a i n t o f the f i x e d head boundary c o n d i t i o n (4-2). He u t i l i z e d s p e c t r a l techniques w i t h p e r t u r b a t i o n s i n the n a t u r a l l o g a r i t h m o f c o n d u c t i v i t y . The v a r i a n c e o f the h y d r a u l i c head f l u c t u a t i o n i s g iven by Bakr (1976) as: 2 2 2 % = q °y "7-1 2 2 a py 184 where q i s a f i x e d f l u x along the f l o w l i n e (which can be regarded as t a k i n g the r o l e o f a boundary c o n d i t i o n ) , a i s the decay parameter 2 . i n an e x p o n e n t i a l a u t o c o r r e l a t i o n f u n c t i o n , a i s the va r i a n c e 2 . of the l o g c o n d u c t i v i t y p e r t u r b a t i o n s , and y^ i s the square o f the mean o f the l o g c o n d u c t i v i t y d i s t r i b u t i o n . L e t the parameters o f the porous medium be constant. Then equation 7-1 can be w r i t t e n : 2 2 d , =• c q 9 Taking logarithms y i e l d s 2 l o g a = l o g c + 2 l o g q 7-2 Thus, a l o g - l o g p l o t o f the v a r i a n c e i n head versus the f l u x w i l l show a s t r a i g h t l i n e w i t h slope two and i n t e r c e p t c. We can observe whether t h i s same r e l a t i o n s h i p i s p r e d i c t e d by the Monte C a r l o s i m u l a t i o n s . The f l u x i s f i x e d i n the s p e c t r a l s o l u t i o n , i n our s o l u t i o n i t i s random on any one r e a l i z a t i o n because o f non-erg o d i c e f f e c t s . The expected f l u x can be v a r i e d by changing the t o t a l g r a d i e n t across the system. Fi g u r e 7-1 i s a l o g - l o g p l o t o f the head va r i a n c e versus mean f l u x . The s i m u l a t i o n parameters are l i s t e d on the diagram. The Figure 7-1 Dependence of the hydraulic head variance on the mean flux, one-dimensional model 186 Monte C a r l o s o l u t i o n s p r e d i c t a slope o f 2.04. Th i s i s i n agreement w i t h the s p e c t r a l s o l u t i o n when the v a r i a b i l i t y i n the estimated standard d e v i a t i o n i n head i s recognized. A s i m i l a r agreement i s obtained f o r the r e l a t i o n determined a t o t h e r p o s i t i o n s along the f l o w l i n e , a t o t h e r r a t i o s o f X/L, and o t h e r i n p u t v a r i a b i l i t i e s i n c o n d u c t i v i t y . c) network design problem S e v e r a l network design problems were considered by Bakr (1976) t o determine the i n f l u e n c e o f the s p a t i a l a u t o c o r r e l a t i o n f u n c t i o n on e r r o r s i n p r e d i c t i o n s made from the s t o c h a s t i c s o l u t i o n o f the f l o w equation. I n one case, Bakr computed t he e r r o r i n the estimate o f flow across an a q u i t a r d given two i n t e r n a l head measurements and a c o n d u c t i v i t y measurement a t an intermediate d i s t a n c e between the head measurements. A normalized e r r o r term i s d e f i n e d : e = 7-3 q where q i s the mean f l u x and q i t s estimate. Bakr (1976) shows the expected e r r o r i s a f u n c t i o n o f both the v a r i a n c e i n the h y d r a u l i c r e s i s t i v i t y (1/K), and the c r o s s -c o r r e l a t i o n between the head p e r t u r b a t i o n s a t the two measurement p o i n t s and the r e s i s t i v i t y p e r t u r b a t i o n . The e r r o r decreases as the spacing 187 between the head measurements decreases. For a f i x e d s e p a r a t i o n , the minimum e r r o r i n a f l u x estimate occurs when the h y d r a u l i c r e s i s t i v i t y i s measured a t the midpoint between the head measurements. An e q u i v a l e n t s t a t i s t i c can be computed from the Monte C a r l o s o l u t i o n s by averaging the e r r o r over a s e r i e s o f r e a l i z a t i o n s . Using the c o n d u c t i v i t y v a l u e i n a b l o c k between two p r e s e l e c t e d head measurements a d i s t a n c e C a p a r t , an estimated f l u x (qjj can be computed by Darcy's Law. The normalized expected e r r o r i s estimated by: n 1 t q i %" 7-4 e = — . , n i = l qp where q^, i s the estimated mean f l u x over the s i m u l a t i o n . The e r r o r d i s t r i b u t i o n was skewed t o the r i g h t , e s p e c i a l l y f o r q^ estimated u s i n g c o n d u c t i v i t y b l o c k s adjacent t o the p o i n t s o f head measurement. For t h i s reason, the e r r o r i s not zero-meaned on averaging over the s i m u l a t i o n . Numerous s l i g h t l y negative and p o s i t i v e e r r o r s are o f f s e t by a s m a l l b u t s i g n i f i c a n t number o f l a r g e r p o s i t i v e e r r o r s . T his i s a numerical e f f e c t i n the e r r o r r a t i o . A f l u x estimate an order o f magnitude above the mean y i e l d s an absolute e r r o r much l a r g e r than a f l u x estimate an order o f magnitude below the mean. The maximum p o s i t i v e errors- always occurred f o r c o n d u c t i v i t y b l o c k s adjacent t o the head.observation p o i n t s . As the r a t i o o f the i n t e g r a l s c a l e t o 188 the s e p a r a t i o n between the ob s e r v a t i o n p o i n t s decreased o r the c o n d u c t i v i t y v a r i a b i l i t y i n c r e a s e d , the e r r o r s i n c r e a s e d markedly and the histogram was more h i g h l y skewed. A p l o t o f t h i s e r r o r s t a t i s t i c versus the p r o p o r t i o n a t e p o s i t i o n o f the c o n d u c t i v i t y measurement between the head o b s e r v a t i o n p o i n t s (a/Cj, i s shown i n F i g u r e 7-2. Th i s p l o t agrees i n form w i t h Bakr (1976).. The i n s t a b i l i t y a t hi g h e r r a t i o s o f C/X. i s due t o the numerical behavior o f the estimated e r r o r term and the s m a l l e r c o r r e l a t i o n between the head g r a d i e n t and the inte r m e d i a t e c o n d u c t i v i t y v a l u e s . For l a r g e r standard d e v i a t i o n s i n the c o n d u c t i v i t y d i s t r i b u t i o n , a l l the curves are s h i f t e d upwards. 3) E f f e c t i v e C o n d u c t i v i t y a) i n t r o d u c t i o n Frequent reference has been made t o the concept o f an e f f e c t i v e c o n d u c t i v i t y o f a heterogeneous medium Under t h i s concept, i t should be p o s s i b l e t o r e p l a c e a heterogeneous medium by a medium w i t h a s i n g l e c o n d u c t i v i t y , w h i l e p r e s e r v i n g the h y d r a u l i c behavior o f the o r i g i n a l medium. Equivalence i n f e r s the f o l l o w i n g (Freeze, 1975): 1) The mean value o f the h y d r a u l i c head a t any p o i n t , as determined from a s t o c h a s t i c , s o l u t i o n r e c o g n i z i n g the s p a t i a l h e t e r o g e n e i t i e s , must Figure 7-2 Normalized error i n the flux estimate, network design problem 190 equal the head val u e a t t h a t p o i n t , as determined from a s i n g l e d e t e r m i n i s t i c s o l u t i o n based on the mean o f the c o n d u c t i v i t y d i s t r i -b u t i o n . 2) The mean value o f any i n t e g r a t e d f l o w measure determined from the s t o c h a s t i c s o l u t i o n must equal the s i n g l e v a l u e provided by the d e t e r m i n i s t i c run, as d e f i n e d above. I n t h i s s e c t i o n , the concept o f an e f f e c t i v e c o n d u c t i v i t y i s considered i n l i g h t o f the two-dimensional f l o w model presented i n Chapter 5. Wood (1976) considers the equivalence concept in. a s u r f a c e water model. Warren and P r i c e (1961), Bouwer (1969) and Freeze (1975) have p r e v i o u s l y used Monte C a r l o a n a l y s i s t o evaluate proposed values f o r the e f f e c t i v e c o n d u c t i v i t y . Dagan ( i n press) presents an a n a l y t i c d e termination o f the e f f e c t i v e p e r m e a b i l i t y . H i s model i s based on t h a t o f Beran (1968). Both bounding values and b e s t estimates f o r the e f f e c t i v e p e r m e a b i l i t y are computed f o r a uniform, steady f l o w i n an i n f i n i t e domain. Dagan shows t h a t i n two dimensions, the b e s t estimate o f the e f f e c t i v e p e r m e a b i l i t y i s given by the geometric mean, assuming p e r m e a b i l i t y i s lognormally d i s t r i b u t e d . I n a Monte C a r l o a n a l y s i s , the geometric mean can be computed e i t h e r d i r e c t l y from the lognormal ( d i s t r i b u t i o n o r as averages formed over the s e t o f c o n d u c t i v i t y r e a l i z a t i o n s . F or p measurements o f K, the geometric mean K i s d e f i n e d as: K_ = ( Z K.) P 7-5 G i = l 1 The l a t t e r method was adopted i n t h i s study. b) e v a l u a t i o n o f the geometric mean as the e f f e c t i v e c o n d u c t i v i t y Equivalence i n the steady s t a t e f l o w problem must be based on an i n t e g r a t e d f l o w measure as any s i n g l e c o n d u c t i v i t y value w i l l reproduce the mean head s o l u t i o n o f the s t o c h a s t i c problem. Checks were made a g a i n s t both the t o t a l f l u x through the system and the f l u x across f i x e d nodal p o i n t s . The r e s u l t s can be summarized as f o l l o w s : 1) I n a uniform g r a d i e n t f i e l d , w i t h u n i d i r e c t i o n a l mean flow, the geometric mean a c c u r a t e l y p r e d i c t s an e q u i v a l e n t behavior. N e i t h e r i n c r e a s e d c o n d u c t i v i t y standard d e v i a t i o n s o r l a r g e r i n t e g r a l s c a l e s a f f e c t e d the match between the uniform and s t o c h a s t i c s o l u t i o n s . 2) I n the two l a y e r c o n f i g u r a t i o n , w i t h d i f f e r e n t means i n each l a y e r , a geometric mean formed over the e n t i r e domain d i d not y i e l d an e q u i v a l e n t medium. Only i f the two l a y e r s are recognized and geometric means formed i n each would an e q u i v a l e n t medium be achieved. 3) In nonuniform g r a d i e n t f i e l d s , the geometric mean c o n s i s t e n t l y underestimated the t o t a l f l u x , as shown i n the f i r s t 4 rows o f Table 7-1. As g r a d i e n t f i e l d s i n r e a l w o r l d f l o w systems are almost always nonuniform, the v a l i d i t y o f d e s c r i b i n g an e f f e c t i v e c o n d u c t i v i t y from a geometric mean formed on the s e t o f a v a i l a b l e f i e l d data must be questioned. 4) I n a medium w i t h a t r e n d i n g c o n d u c t i v i t y , the geometric mean formed over the e n t i r e domain d i d not l e a d t o t h e mean s t o c h a s t i c s o l u t i o n (Table 7-1, row 5). T h i s i s because the s i n g l e c o n d u c t i v i t y value i s not capable o f reproducing the s p a t i a l v a r i a t i o n o f the head g r a d i e n t s and t h e r e f o r e the f l u x through the system. I t can be concluded t h a t t h e r e a r e s t r i c t c o n d i t i o n s t o be s a t i s f i e d i n order f o r the e f f e c t i v e c o n d u c t i v i t y t o be g i v e n by the geometric mean. I t s s u i t a b i l i t y depends upon both the space law o f the c o n d u c t i v i t y f i e l d and the flow system o p e r a t i n g w i t h i n t h a t domain. I n c o n s i d e r i n g e f f e c t i v e parameters o f heterogeneous media, a numerical Figure "X~ X X z a y f Total KG f Total Comments 5-lOc 25.6 26.3 0.42 0.98 1.23 nonuniform flow 5-10d 10.4 10.4 0.43 0.88 1.04 nonuniform flow 5-13b 17.2 17.6 0.43 1.91 2.16 tunnel problem 5-14b 17.2 17.0 0.43 0.56 0.61 dam problem 5-12 18.0 18.1 0.43 0.48 0.41 trend in, mean Note: flux cm /s Table 7-1 Comparison of Flux Estimate Based on the Geometric Mean Conductivity with the Mean Flux in a Stochastic Solution. i — 194 solution can only be used to evaluate a proposed value. No alternative is suggested i f the equivalence criteria are not satisfied. An analytical approach to defining an effective parameter, such as that used by Dagan (in press), appears to be more appropriate. 4) Dimension Dependence In this section, the relation between the prediction varia-bilities and the dimension of the flow equation is investigated in terms of-the'different solution teciiniques. The dimension dependence arises from the way in which the flow can be routed through the system. Bakr (1976) found an order of magnitude reduction in the head variability in comparing one and three-dimensional flows using spectral techniques. Again using spectral theory, Gelhar (1977) considered unidirectional mean flow through a phreatic aquifer and found over an order of magnitude difference betweeen the one and two-dimensional formulations. Dagan (in press) computed both the gradient and specific discharge variability for uniform flow in an infinite domain. The divergence between the variabilities predicted by models of different dimension increased as a nonlinear function of the conductivity standard deviation. For 0^=1, the one-dimensional variance is slightly more than an order of magnitude greater than in an equivalent two-dimensional model. The difference is going from two to three dimensions was much smaller. 195 In comparing the head standard deviations for one and two-dimensional flow through a bounded domain (section 5-3), i t was found that the variability was about halved for a set to 0.86 (Figure 5-4). y Similar reductions were observed for smaller values of a . The y smaller divergence between dimensions is probably due to the bounded nature of the conductivity realizations and the nonstationary head field. If the flow domain was expanded along the coordinate axis perpendicular to the mean flow direction, i t is presumed that the differences between the one and two-dimensional analysis would increase. The resulting reduction in the two-dimensional variability may lead to a limiting value equal to that of Bakr (1976) and Gelhar (1977), although this possibility was not investigated. 196 CHAPTER 8 Parameter A n a l y s i s o f Quadra Sands Data 1) I n t r o d u c t i o n I n Chapter 2, a p r o b a b i l i s t i c model d e s c r i b i n g the s p a t i a l heterogeneity o f a porous medium was o u t l i n e d . The i n f l u e n c e o f the parameters c h a r a c t e r i z i n g these s p a t i a l v a r i a t i o n s i n determining the output v a r i a b i l i t y i n a steady s t a t e f l o w a n a l y s i s was considered i n Chapters 4 and 5. I n t h i s chapter, the t h e o r e t i c a l models o f s p a t i a l v a r i a b i l i t y w i l l be a p p l i e d t o a r e a l d a ta s e t . Four porous media parameters are considered; these being c o n d u c t i v i t y , p o r o s i t y , c o m p r e s s i b i l i t y and g r a i n s i z e d i s t r i b u t i o n curves. P o r o s i t y and c o m p r e s s i b i l i t y are s t u d i e d as they w i l l be r e q u i r e d f o r a t r a n s i e n t flow a n a l y s i s . G r a i n s i z e d i s t r i b u t i o n curves were measured t o com-pare t h e i r p a t t e r n s o f s p a t i a l v a r i a t i o n w i t h those i n the c o n d u c t i v i t y and p o r o s i t y data. A f t e r d e s c r i b i n g the u n i t from which the samples were taken and b r i e f l y r e v i e w i n g the sampling and measurement procedures, a v a r i e t y of s t a t i s t i c a l techniques w i l l be a p p l i e d t o c h a r a c t e r i z e the observed v a r i a b i l i t y . The o b j e c t i v e s o f t h i s chapter can be s t a t e d as f o l l o w s : 1) To apply a f i r s t o r d e r , nearest neighbour model t o the data s e t . 2) To present a more general a n a l y s i s o f the s p a t i a l 197 v a r i a b i l i t y to provide estimates required for alternative approaches to the stochastic flow problem. 2) Sample Site, Sample Plan, Measurement Technique a) description of the unit The porous medium from which several sets of saturated con-ductivity, porosity, compressibility and grain size measurements were taken belongs to the Quadra Sands unit. It i s exposed i n a series of c l i f f faces near the University of B r i t i s h Columbia, Vancouver. A map of the sample si t e i s given i n Figure 8-1. Clague (1976, 1977) presents a detailed geologic description of the Quadra Sand. The unit i s described characteristically as a well sorted, medium grained, horizontally s t r a t i f i e d sand with relatively few s i l t or gravel inter-beds. It has neither a significant s i l t - c l a y matrix or intergranular cement, although i t i s cohesive enough to stand i n near ver t i c a l c l i f f s . At the sample s i t e , the Quadra Sand i s approximately 60 m thick. The unit can be subdivided into two zones. In the lower 18 m the sand i s interbedded with horizontal layers of organic-rich s i l t . These layers are absent i n the upper zone. The contact i s easily recognized by a seepage line developed just above the top of the lower zone. Samples were taken only from the upper zone. The sands are overlain by a stony t i l l , the base of which defines the upper boundary of the sample 198 100 0 100 200 I •—I •—I I I I I M e i e r * Figure 8-1 Sample s i t e , Quadra Sands data 199 domain. A photograph o f the l a r g e s t o f the c l i f f exposures i s g i v e n i n F i g u r e 8-2. Talus covers most o f the lower zone here. Cross-bedding and c u t and f i l l s t r u c t u r e s dominate the s m a l l s c a l e sedimentary p a t t e r n . S i l t and c l a y occur as t h i n beds and laminae i n t e r s t r a t i f i e d w i t h the sand. The sand i t s e l f has o n l y a s m a l l percentage o f f i n e r p a r t i c l e s . Clague (1977) concludes t h a t the Quadra Sands were most probably deposited as reworked d i s t a l outwash from advancing g l a c i e r s t o the n o r t h about 20,000 years ago. D e p o s i t i o n a t any one s i t e was f a i r l y r a p i d . The a x i a l o r i e n t a t i o n o f the c r o s s s t r a t a i n d i c a t e a l o c a l flow v a r i a b i l i t y t h a t r e f l e c t s s h i f t s i n channel p a t t e r n s , although there i s a d e f i n i t e r e g i o n a l t rend. b) sample p l a n A l a r g e s c a l e random sample o f 100 p o i n t s , d i s t r i b u t e d over 12 c l i f f exposures ( i n an approximate p r o p o r t i o n t o the area o f each exposure) was c o l l e c t e d f o r parameter c o r r e l a t i o n and frequency d i s t r i -b u t i o n a n a l y s i s . The exposures sampled are numbered on F i g u r e 8-1. In a d d i t i o n , three s e t s o f non-random samples were c o l l e c t e d i n order t o estimate the correlogram, the power spectrum, the variogram, and the nearest neighbour a u t o r e g r e s s i v e parameters. A 10 p o i n t by 10 p o i n t square g r i d on 1.52 m centers and two 100 p o i n t l i n e samples w i t h a spacing o f 0.30 m were taken i n the face o f the c l i f f . One l i n e sample 200 F i g u r e 8 - 3 More d e t a i l e d view o f Quadra Sand, sample l o c a t i o n 5 201 was h o r i z o n t a l t o the bedding, the o t h e r v e r t i c a l . F o r t y o f the random samples, both l i n e samples, and the two-dimensional g r i d were l o c a t e d w i t h i n the exposure p i c t u r e d i n F i g u r e 8-2. c) measurement techniques A t each sample p o i n t , a d i s t u r b e d sample and a s m a l l undisturbed core were c o l l e c t e d . The p o r o s i t y was determined from the undisturbed core by computing the weight o f s o l i d s i n the known volume and assuming a s p e c i f i c g r a v i t y o f 2.65 f o r the s o l i d s . F a l l i n g - h e a d permeameters were used t o measure the s a t u r a t e d c o n d u c t i v i t y o f the d i s t u r b e d samples (Lambe, 1951). The lab-determined c o n d u c t i v i t i e s were measured a t s e v e r a l p o r o s i t i e s f o r each sample and then c o r r e c t e d t o the f i e l d p o r o s i t y t o estimate the f i e l d c o n d u c t i v i t y . C o m p r e s s i b i l i t i e s were measured f o r the two-dirrensional gridded sample u s i n g a standard con-s o l i d a t i o n t e s t (Lambe, 1951). G r a i n s i z e d i s t r i b u t i o n curves were determined f o r the random sample and the two l i n e samples from a s i e v e a n a l y s i s (Lambe, 1951). Lopaschuk (1976)and M i l n e (1977) pr o v i d e d e t a i l e d accounts of the sampling and t e s t i n g procedure. A l i s t i n g o f the data i s given i n Appendix TV. I t i s recognized t h a t c o n d u c t i v i t y determinations on d i s t u r b e d cores, c o r r e c t e d t o the f i e l d p o r o s i t y , y i e l d o n l y approximate estimates o f the f i e l d c o n d u c t i v i t y a t each sample p o i n t . The i n s i t u pore c h a r a c t e r i s t i c s and the very s m a l l s c a l e sedimentary s t r u c t u r e s such as 202 t h i n c l a y laminae and i n t e r g r a n u l a r contacts cannot be r e p l i c a t e d i n the l a b permeameter. A sampling e r r o r i s then intr o d u c e d i n t o the d a t a and a l l the subsequent s t a t i s t i c a l analyses o f t h a t data. E r r o r s i n the measurement technique on the f a l l i n g head apparatus are a l s o i n c l u d e d . By u s i n g d i s t u r b e d c o r e s , the e f f e c t s o f the s m a l l s c a l e h e t e r o g e n e i t i e s i n c o n t r i b u t i n g to-the estimates o f the s p a t i a l v a r i a -b i l i t y i n c o n d u c t i v i t y are destroyed. As such, the recorded conduct-i v i t y v a r i a t i o n s r e f l e c t o n l y the l a r g e r s c a l e t e x t u r a l changes w i t h i n the porous medium. d) v i s u a l i n f o r m a t i o n I n a n a l y z i n g the Quadra Sand dat a , we have the advantage o f seeing the p o p u l a t i o n from which the sample i s taken. Of course, such i s not the case f o r borehole data. T h i s a d d i t i o n a l v i s u a l i n f o r -mation can be summarized as f o l l o w s : 1) The sample i s r e s t r i c t e d t o a s i n g l e g e o l o g i c u r i i t . 2) I d e n t i f i a b l e h o r i z o n t a l l a y e r s are up t o tens o f meters long. The l a y e r s are recognized by t h e i r v a r y i n g r e s i s t a n c e t o e r o s i o n and/or s l i g h t t e x t u r a l v a r i a t i o n s i n g r a i n s i z e . 3) A s i d e from the l a y e r i n g , there are no apparent 203 l a r g e s c a l e s t r u c t u r e s w i t h i n the u n i t . F or example, a t the s c a l e o f a s i n g l e exposure, there i s not a s e r i e s o f h i g h e r c o n d u c t i v i t y lenses d i s p e r s e d w i t h i n a lower c o n d u c t i v i t y m a t r i x . 4) W i t h i n any bed, the dominant fea t u r e s are crossbeds and c u t and f i l l s t r u c t u r e s , up t o about one meter long. A more d e t a i l e d view o f the Quadra Sand i s provided i n F i g u r e 8-3. I f the u n i t were s a t u r a t e d , i t would be reasonable t o i d e n t i f y i t as a s i n g l e a q u i f e r and con s i d e r i t 'homogeneously heterogeneous'. I t i s probably g e o l o g i c a l l y r e p r e s e n t a t i v e o f many unconsolidated and a q u i f e r s t h a t are e x p l o i t e d f o r groundwater s u p p l i e s . 3) Assumptions Required R e c a l l from Chapters 2 and 3 t h a t , i n o r d e r t o s o l v e the s t o c h a s t i c flow equation, f a i r l y s t r i c t c o n d i t i o n s were r e q u i r e d on the nature of the heterogeneity o f the porous medium. I n a n a l y z i n g r e a l media, a s i m i l a r s e t o f assumptions must be made. Both s t a t i o n a r i t y and e r g o d i c i t y must apply i f the s p a t i a l averages are t o estimate the space law parameters. For example, the s p a t i a l mean would estimate the mean o f each o f the p o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n s . The d e n s i t y f u n c t i o n i t s e l f would be estimated from a frequency a n a l y s i s on the s p a t i a l l y d i s t r i b u t e d data. I n order t o o b t a i n estimates o f the correlogram, the s p a t i a l dependence must be a f u n c t i o n o n l y o f the v e c t o r s e p a r a t i o n , i r r e s p e c t i v e o f i t s l o c a t i o n i n the sample domain. S t a t i s t i c a l t e s t s can be designed t o con s i d e r these assumptions although they can never be proved. A s t a t i o n a r y r e g i o n must have a common mean and v a r i a n c e . I f the data r e c o r d i s s u f f i c i e n t l y l o n g , subsets o f the sequence can be formed and moments compared. Even though a s e r i e s may be s t a t i o n a r y , d i f f e r e n t p a r t s o f the r e c o r d need not look e x a c t l y the same nor are the s p a t i a l averages formed over the subsets n e c e s s a r i l y equal ( J u l i a n , 1967). I f the sample s t a t i s t i c s f o l l o w a well-known d i s t r i b u t i o n , comparisons can be made w i t h the a i d o f confidence i n t e r v a l s . Correlograms and s p e c t r a formed on data sub-s e t s can a l s o be compared i n t h i s manner. The s t a t i s t i c s must always be i n t e r p r e t e d i n terms o f the s c a l e a t which the sample i s taken. Smaller o r l a r g e r s c a l e v a r i a t i o n s are i n h e r e n t l y i n c l u d e d i n the sample but t h e i r i n f l u e n c e cannot be i d e n t i f i e d u niquely. U s u a l l y they are lumped as e i t h e r a t r e n d ( l a r g e r s c a l e ) o r random v a r i a t i o n ( s m a l l e r s c a l e ) . 4) Moment and Histogram A n a l y s i s The p r o b a b i l i t y d e n s i t y f u n c t i o n o f a random v a r i a b l e i s desc r i b e d by the form o f the d i s t r i b u t i o n and the values o f i t s moments Estimates o f the mean and standard d e v i a t i o n f o r each o f measured parameters w i l l be presented f i r s t . Then t h e i r p r o b a b i l i t y d i s t r i -b u t i o n s w i l l be considered. The moment estimates are given i n Tables 8-1 and 8-2. Note t h a t : 1) The standard d e v i a t i o n s are g r e a t e r f o r the v e r t i c a l l i n e sample than f o r the h o r i z o n t a l sample. The standard d e v i a t i o n s are a l s o g r e a t e r i n the v e r t i c a l sample than those estimated f o r the l a r g e s c a l e random sample. 2) The standard d e v i a t i o n o f the c o n d u c t i v i t y d i s t r i b u t i o n i s a t the low end o f the range coromonly report e d i n the l i t e r a t u r e . 3) There i s a g r e a t e r c o e f f i c i e n t o f v a r i a t i o n ( C J / U ) i n the s i z e o f the co a r s e r - g r a i n e d f r a c t i o n s than i n the f i n e r f r a c t i o n . 4) The moment estimates on the g r a i n s i z e f r a c t i o n s based on fewer than 100 samples are not completely r e p r e s e n t a t i v e o f the Quadra Sands. I n these cases, the m i s s i n g data are due e i t h e r t o a Dc f r a c t i o n s m a l l e r o than a 200 mesh o r i n s u f f i c i e n t f r a c t i o n a t i o n o f the coarser g r a i n s i z e component. Re-SAMPLE PARAMETER MEAN STD. DEV. K 0.044 0.018 RANDOM l o g K -1.403 0.219 n 0.424 0.026 K 0.055 0.010 HORIZONTAL L I N E l o g K n -1.272 0.418 0.087 0.015 K 0.044 0.022 VERTICAL LINE l o g K n -1.443 0.420 0.364 0.028 K 0.052 0.016 SQUARE GRID l o g K n * a -1.306 0.435 0.0027 0.151 0.018 0.0012 * l o g a -2.60 0.16 Note: K c o n d u c t i v i t y cm/s n * a p o r o s i t y 2 c o m p r e s s i b i l i t y cm /Kg Table 8-1 Estimated Moments o f Flow Parameters D 10 D 16 D 50 '60 D 84 '95 Horizontal Line Number 100 100 100 Mean 0.161 0.190 0.216 Std:. Dev. 0.011 0.017 0.018 100 100 100 94 0.322 0.357 0.483 0.648 0.032 0.039 0.063 0.098 Vertical Line Random Number Mean St. Dev. 83 98 98 100 99 80 50 0.151 0.176 0.206 0.379 0.392 0.517 0.642 0.032 0.045 0.052 0.402 0.126 0.125 0.141 Number 86 99 100 100 100 100 76 Mean 0.143 0.167 0.192 0.302 0.340 0.507 0.691 Std. Dev. 0.026 0.036 0.041 0.074 0.089 0.252 0.583 Note: grain size in mm. Table 8-2 Estimated Moments of Percent Finer Grain Size Fractions 208 sieving or hydrometer analysis would be required to include a l l 100 samples. Histograms of the sample distributions are plotted in Figure 8-4. They a l l appear monomodal, implying a variation about a single mean rather than there being two or more significantly different populations. In contrast, Madsen (1974) collected a set of 47 conductivity measurements from both the lower and upper zones of the Quadra Sand on one c l i f f exposure and found the histogram to be bimodal. A sample restricted to the lower zone would probably have revealed a bimodal distribution due to the presence of both sand and s i l t layers. The frequency histograms were compared with both normal and lognormal distributions using a chi-square goodness of f i t test (Krumbein and Graybill, 1965). Two comments should be made. First, this test assumes a series of independent observations from the population distribution. In a spatial domain, a l l of the sample points must be separated from each other by a distance greater than the corre-lation length of the medium. Secondly, as Agterberg (1974) notes, i t is not possible to distinguish between normal and lognormal distributions for small coefficients of variation because the forms of the two distributions become similar (ie. the expected class frequencies are nearly the same). The random sample conductivity distribution could be accepted as fitting either a normal or lognormal distribution at the 95% 209 a) porosity random sample frequency 30 n 25 H 20 15 1 10 5H (O T in to co o 8 5 !?.. 9 1- • ci porosity b) conductivity random sample 30 i 25 -frequency 20 15 A 10 o r> r- o CO CO in o> «- -t conductivity x 10 (cm/s) Figure 8-4 Frequency d i s t r i b u t i o n s for conductivity, porosity, and compressibility 210 p o r o s i t y h o r i z o n t a l l i n e 30 25 H 20 d) c o n d u c t i v i t y h o r i z o n t a l l i n e > o c CO o- 15 10 o co 25 n 20 A >• u co 1 0 CT 10 O) 00 CO to co g CN T co 'f. •* d d o d in tn tn r- CM n ri -f ui in t CO porosity 1 conductivity x 10 (cm/s) ) p o r o s i t y v e r t i c a l l i n e 25 f ) o c cu 3 CT 20 15 10 H 5H CO CO d o CO CO o CD CN CD co -^ *r d d d porosity c o n d u c t i v i t y v e r t i c a l l i n e 25 20 > o c CO tx CO 15 10 5 H oo * CN 1-' O CO CO conductivity x 10 (cm/s) igure 8-4 Cont. 211 g) porosity 2D g r i d 30 25 o c a> 3 cr co 20 15 H 10 5 H h) conductivity 2D g r i d 30 -25 -20 > u c cc 3 cr CP 10 5 H S cn to * CN 2 — co in r» 9 d o d o co tD TT 05 co co (O l(i CN O o r-porosity conductivity x 10 (cm/s) i) 40 -i 30 H > u c a> or 20 H 10 H compressibility 2D grid I D -S' o ir> in 3 0 compressibility x 10 (cm /kg) Figure 8-4 Cont. 212 s i g n i f i c a n c e l e v e l . The chi-sguare s t a t i s t i c f o r the normal d i s t r i -b u t i o n was s m a l l e r . The r e l a t i v e ' u n i f o r m i t y ' o f the Quadra Sands i n the upper zone may mask the t y p i c a l skewed form o f the lognormal d i s t r i b u t i o n . The p o r o s i t y d i s t r i b u t i o n was accepted as normal a t the 95% s i g n i f i c a n c e l e v e l . I f i t i s assumed t h a t the parameter values i n the two-dimensional g r i d are a t most o n l y weakly a u t o c o r r e l a t e d , chi-squared t e s t s can be a p p l i e d without m o d i f i c a t i o n . The r e s u l t w i l l o n l y be approximate. C o n d u c t i v i t y values c o u l d be f i t t e d t o b o t h normal and lognormal d i s t r i b u t i o n s . P o r o s i t y was not accepted as normal because o f i t s peakedness and i r r e g u l a r i t y a t lower p o r o s i t y v a l u e s . The frequency d i s t r i b u t i o n o f ranpressibility has a ' t y p i c a l ' lognormal form, b e i n g peaked and skewed t o the r i g h t . A n a l y s i s o f v a r i a n c e techniques can be used t o i d e n t i f y d i s t i n c t p o p u l a t i o n s w i t h i n the frequency histogram due t o s p a t i a l groupings i n the sample data. These r e s u l t s are not presented here. 5) C o n d u c t i v i t y V a r i a b i l i t y W i t h i n a Sample In o r d e r t o c o n s i d e r the r e p r o d u c i b i l i t y o f the c o n d u c t i v i t y values determined f o r the d i s t u r b e d samples, measurements were made on two separate samples taken from each sample p o i n t . D e v i a t i o n s are bound t o occur b u t t o keep them i n p e r s p e c t i v e , the v a r i a b i l i t y can" be compared t o t h a t w i t h i n the e n t i r e s e t o f samples u s i n g an a n a l y s i s 213 o f v a r i a n c e model. T h i s t e s t i s o u t l i n e d i n Appendix V. The check was performed on the s e t o f randomly l o c a t e d samples. The r e p l i c a t i o n mean square was 1.6% o f the mean square f o r the v a r i a t i o n among samples. I t i s concluded t h a t the c o n d u c t i v i t y v a r i a t i o n s r e f l e c t l a r g e r s c a l e d i f f e r e n c e s between p o i n t s and not a l o c a l v a r i a t i o n w i t h i n each d i s t u r b e d sample. 6) Trend Surface A n a l y s i s I d e n t i f i c a t i o n o f s p a t i a l trends i n the f l o w parameters i s important f o r two reasons. F i r s t , the s t a t i s t i c a l estimates b e i n g used t o c h a r a c t e r i z e the s p a t i a l heterogeneity assume t h a t the v a r i a t i o n occurs about a constant valu e . I f t h i s i s not the case, the est i m a t o r s must be mod i f i e d t o account f o r a nonconstant mean. Secondly, the magnitude and s p a t i a l d i s t r i b u t i o n o f the h y d r a u l i c head v a r i a b i l i t i e s were shown i n Chapter 5 t o be a f u n c t i o n o f the t r e n d i n the mean c o n d u c t i v i t y . S p a t i a l v a r i a t i o n s i n the mean can be accounted f o r by assuming the v a r i a t i o n s a r i s e from a d e t e r m i n i s t i c t r e n d . Then, the p o s t u l a t e d f u n c t i o n can be f i t t e d t o the data by l e a s t squares theory. A p l o t o f the s p a t i a l v a r i a t i o n s i n the p o r o s i t y and c o n d u c t i v i t y values along the l i n e t r a n s e c t s i s given i n F i g u r e 8-5. Note the d i f f e r e n t h o r i z o n t a l s c a l e s . The d i f f e r e n t c h a r a c t e r i n the p a t t e r n of the v a r i a t i o n s w i t h i n the h o r i z o n t a l and v e r t i c a l s e r i e s i s immediately apparent, e s p e c i a l l y f o r the c o n d u c t i v i t y data. 2 1 4 conductivity (cm/s) conductivity (cm/s) horizontal line vertical line Figure 8-5 Spatial v a r i a t i o n i n conductivity and porosity along the horizontal and v e r t i c a l l i n e transects 215 porosity o o o o o O O Q O O co r» co gj o — N fl » in n o n ft * * * <<r d d o d o d d d d d J I I I i i i i i i horizontal line porosity 8 L £ > Q i r > O m o t n o i r > 2 < o r - e n o CN ro i n c o CO CO CO CO « *t • * » d d o © d d d d d d i i i i i i i i i i vertical line Figure 8-5 Cont. A two-dimensional t r e n d can be de s c r i b e d by a polynomial: K ( x i f z . ) = e Q + B lx. + B 2 z . + 8 _ ! up t o order n . S e v e r a l comments can be made w i t h regard t o t r e n d s u r f a c e a n a l y s i s : 1) I t i s p o s s i b l e t h a t the s p a t i a l v a r i a t i o n i n the mean c o n d u c t i v i t y i s i t s e l f a random process. However, we f o r c e i t s r e p r e s e n t a t i o n as a d e t e r m i n i s t i c f u n c t i o n i n f i t t i n g the above polynomial. 2) I n bounded systems, s p a t i a l v a r i a t i o n s i n the mean cannot be a s c r i b e d t o the l a r g e s c a l e f l u c t u a t i o n s i n a s t a t i o n a r y space law. Where imaginary boundaries are imposed on the c o n d u c t i v i t y f i e l d by d e f i n i t i o n o f the flow boundaries, the lon g - p e r i o d v a r i a t i o n s must be t r e a t e d as a non s t a t i o n a r y component. 3) L e a s t squares e s t i m a t i o n assumes t h a t the r e s i d u a l s from the t r e n d are u n c o r r e l a t e d . I n f a c t , r e s i d u a l s represent c o r r e l a t e d f l u c t u a t i o n s on a s c a l e s m a l l e r than the trend. S t a t i s t i c a l i n f e r e n c e i s then no longer s t r a i g h t f o r w a r d . The s t r e n g t h o f a t r e n d can be evaluated by i t s percent ex-p l a i n e d sum o f squares. T h i s i s d e f i n e d : 217 0 c c _ sum o f squares due t o r e g r e s s i o n „ „ -6 Hioo — j . --, ~p X XUU o—z t o t a l sum o f squares I t i n d i c a t e s the percent o f the v a r i a t i o n i n the sample t h a t i s due t o a t r e n d i n g mean as opposed t o t h a t p r o p o r t i o n t h a t i s r e s i d u a l about t h a t trend. The % ESS f o r the gridded data f o r surfaces up t o a cub i c are l i s t e d i n Tables 8-3 and 8-4. I n s p e c t i o n i n d i c a t e s t h a t over the sample s c a l e o n l y a weak t r e n d e x i s t s i n the parameter means. This might be expected on obser v i n g the p a t t e r n s o f v a r i a t i o n shown i n F i g u r e 8-5. Of the trends i n d i c a t e d , the st r o n g e s t occur i n the v e r t i c a l c o m p r e s s i b i l i t y sequence i n the twci-dimensional g r i d and the h o r i z o n t a l l i n e sample i n the c o n d u c t i v i t y sequence. The a n a l y s i s does not preclude the e x i s t e n c e o f s i g n i f i c a n t trends on s u b i n t e r v a l s w i t h i n the sequence o r as p a r t o f a l a r g e r s c a l e v a r i a t i o n . 7) Correlograms The s p a t i a l dependence between neighbouring c o n d u c t i v i t y values can be c h a r a c t e r i z e d by the a u t o c o r r e l a t i o n f u n c t i o n . For a s t a t i o n a r y medium, the a u t o c o r r e l a t i o n f u n c t i o n i s d e f i n e d by equation 2-3. I n one dimension, i t s e s t i m a t o r i s g i v e n by equation 3-34. The two-dimensional a u t o c o r r e l a t i o n f u n c t i o n i s estimated u s i n g equation 3-36. In t h i s s e c t i o n , the estimated correlograms f o r the one and two-dimensional gridded samples w i l l be presented. I n a succeeding s e c t i o n , they w i l l be compared w i t h the correlograms f o r a f i r s t o r d e r , nearest Horizontal Line Vertical Line Degree Conductivity Porosity Conductivity Porosity of Surface % ESS pure % ESS total pure total pure total pure total Linear 9.88 9.88 1.24 1.24 2.93 2.93 5.79 5.79 Quadratic 11.45 21.33 11.51 12.75 2.29 5.22 3.25 9.04 Cubic 0.28 21.61 0.27 13.02 5.28 10.50 1.04 10.08 Residual 78.39 -86.98 89.50 89.92 Table 8-3 Percent Explained Sum of Squares for Trend i\nalysis, Line Samples M CO CONDUCTIYITY POROSITY COMPRESSIBILITY Degree o f % ESS % ESS % ESS Surface each term pure t o t a l each term pure t o t a l each term pure t o t a l L i n e a r X 2 1.38 0.46 1.84 1.84 0.26 4.76 5.02 5.02 0.58 13.30 13.88 13.88 Quadratic X2 0.45 3.97 1.09 5.51 7.35 2.10 0.66 5.33 8.08 13.10 0.47 1.60 5.82 7.89 21.77 Cubic X 3 x^z xa^ a3 0.20 0.63 0.28 8.89 9.99 17.34 1.74 0.52 0.30 0.10 2.67 15.77 0.33 2.25 0.00 3.06 5.64 27.41 R e s i d u a l 82.66 84.23 72.59 Table 8-4 Percent E x p l a i n e d Sum o f Squares f o r Trend A n a l y s i s , Square G r i d 220 neighbour model. a) one-dimensional correlograms Correlograms f o r the l i n e samples are given i n F i g u r e s 8-6a through 8-6d. No t r e n d was removed o r f i l t e r a p p l i e d t o the s e r i e s . N e i t h e r were the c o n d u c t i v i t y values l o g transformed. Because neighbour-i n g l a g estimates i n the correlogram are c o r r e l a t e d , i t i s d i f f i c u l t t o c o n s t r u c t confidence i n t e r v a l s on the correlogram. Instead, the 95% confidence i n t e r v a l f o r an u n c o r r e l a t e d sequence o f the same l e n g t h can be used as a guide i n i n t e r p r e t i n g a t which l a g s the c o r r e l a t i o n may be s i g n i f i c a n t l y d i f f e r e n t from zero (Jenkins and Watts, 1968, pg. 187). Although the t a i l s o f the correlograms show i n t e r e s t i n g t r e n d s , the s e r i e s are too s h o r t t o a l l o w i n t e r p r e t a t i o n t h e i r s i g n i f i c a n c e . The drop from l a g zero t o l a g one i s due t o both the decay i n s p a t i a l dependence -w i t h d i s t a n c e and t o v a r i a t i o n on a s c a l e s m a l l e r than the sample i n t e r v a l . From these p l o t s , i t can be concluded: 1) The correlogram f o r the v e r t i c a l c o n d u c t i v i t y s e r i e s does not have a simple form. I t decays toward zero and then i n c r e a s e s again t o what may be a s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n . T h i s may be a response t o a c y c l i c p a t t e r n i n the l a y e r i n g w i t h i n the v e r t i c a l sequence. a) porosity horizontal l i n e r (I) I 10 I 12 I 14 I 16 I 18 20 lag number b) conductivity horizontal l i n e r (D I - lag number Figure 8-6 One-dimensional correlograms for conductivity and porosity q) porosity v e r t i c a l l i n e d) conductivity v e r t i c a l l i n e 1.0 i r(l) I - lag number Figure 8-6 Cont. 223 2) The p o r o s i t y and c o n d u c t i v i t y s e r i e s along the h o r i z o n t a l l i n e have very s i m i l a r correlograms. 3) Of the f o u r s e r i e s , p o r o s i t y values along the v e r t i c a l l i n e are p o s i t i v e l y c o r r e l a t e d over the g r e a t e s t d i s t a n c e . 4) Both fl o w parameters have s t a t i s t i c a l l y a n i s o t r o p i c c o r r e l a t i o n f u n c t i o n s . I f the form o f the v e r t i c a l c o n d u c t i v i t y correlogram i s r e a l , i t suggests t h a t a n i s o t r o p y i n the f u n c t i o n a l form o f the s p a t i a l c o r r e l a t i o n should be considered. The correlograms f o r the DJ-Q g r a i n s i z e f r a c t i o n i n the h o r i z o n t a l (curve H) and v e r t i c a l (curve V) l i n e samples are p l o t t e d i n F i g u r e 8-7. Other s i z e f r a c t i o n s i n the h o r i z o n t a l l i n e sample a l l have s i m i l a r c o r r e l a t i o n - l a g r e l a t i o n s . Note the marked c o n t r a s t between the h o r i z o n t a l and v e r t i c a l D,-Q correlograms. The v e r t i c a l c o r r e l a t i o n decays t o near-zero values f o r a l l l a g s because o f the i n f l u e n c e o f a s i n g l e g r a v e l l e n s i n t e r s e c t e d by the sequence. The s i z e w i t h i n the g r a v e l l a y e r i s 10.5 standard d e v i a t i o n s from the s e r i e s mean. I f t h i s v a l u e i s removed and the average o f i t s two neighbouring values i n the s e r i e s i n s e r t e d , the c o r r e l a t i o n f u n c t i o n l a b e l e d B i s obtained. Horton e t a l . (1964) removed s c a t t e r e d , s h a r p l y 224 r ( l ) I - lag number Figure 8-7 One-dimensional correlograms for the D,-n grain size f r a c t i o n 225 defined maxima in their data by smoothing across the base of the peak. The finer grain size fractions in the gravel lens are in a similar range to that occurring in the rest of the vertical sequence. The modified correlogram (B)1 is similar to that obtained for the finer grain size fractions in the vertical series. The above behavior results from the nonstationarity of the sample sequence due to the presence of the single gravel layer. If i t were not present, the series could be approximated by the stationarity assumption. Further research is needed to characterize the space law df a porous medium which can be represented as a stationary unit with the inclusion of a small number of zones with parameter values far removed from the mean. These inclusions could not be ignored in a flow analysis i f they were to exceed a certain size and/or continuity with respect to the flow model. b) two-dimensional correlograms Selected coefficients of the two-dimensional sample autocorrelation function for the conductivity, porosity, and compressibility data are listed in Table 8-5. The origin of the function is set in the top left hand corner of the sample grid. Conductivity values along the layers are significantly correlated at a 1.52 m spacing, whereas at an equal separation across the layering they are not correlated. The correlation does not decay for larger separations along the layers, an indication Horizontal Lag Vertical Parameter 0 Lag 0 1.00 0.48 0.51 0.33 1 -0.06 0.03 Conductivity 2 -0.15 3 -0.03 0 1.00 0.04 0.00 0.19 1 0.06 -0.07 Porosity 2 -0.02 3 -0.03 0 1.00 0.35 0.30 0.26 Log 10 1 0.24 0.14 Compressibility 2 0.14 3 0.11 Table 8-5 Two-Dimensional Sample Autocorrelation Matrices 227 o f n o n s t a t i o n a r i t y a t the sample s c a l e . P o r o s i t y values are e s s e n t i a l l y u n c o r r e l a t e d along and across the l a y e r s . Log transformed compressi-b i l i t y values a l s o show p o s i t i v e c o r r e l a t i o n s . The behavior o f the estimated c o r r e l a t i o n s i s s t r o n g l y c o n t r o l l e d by the l i m i t e d e x t e n t o f the sample g r i d . The c o r r e l a t i o n a l o n g the l a y e r s can be viewed as the average o f t e n , 10 p o i n t one-dimensional a u t o c o r r e l a t i o n f u n c t i o n s . However, w i t h i n each l a y e r , d i f f e r e n c e s are formed a g a i n s t the o v e r a l l g r i d mean, r a t h e r than the l a y e r mean. For a s m a l l sample s i z e , the divergence o f the l a y e r mean from the o v e r a l l mean may be r e l a t i v e l y l a r g e . This c o u l d impart an apparent, c o r r e l a t i o n t h a t r e f l e c t s the l a y e r i n g w i t h i n the sampled r e g i o n , r a t h e r than the p o i n t - t o - p o i n t c o r r e l a t i o n s . The v a r i a b i l i t y i n the l a y e r means i s lower f o r the p o r o s i t y sequence than t h a t f o r the c o n d u c t i v i t y d a t a . One i s l e d t o conclude t h a t the c o n d u c t i v i t y estimates cannot be i n t e r p r e t e d i n terms o f a s t a t i o n a r y model f o r t h i s sample g r i d . T h i s e f f e c t i s s i m i l a r t o t h a t observed by Webster and Cuanalo (1975) i n t h e i r one-dimensional correlograms formed on v a r i o u s s o i l p r o p e r t i e s . T h e i r t r a n s e c t s crossed a s e r i e s o f d i s c r e t e g e o l o g i c u n i t s . The correlograms r e f l e c t e d the average l e n g t h o f these u n i t s r a t h e r than the p o i n t - t o - p o i n t c o r r e l a t i o n . On removing the u n i t means from the data, the correlograms decayed t o zero a t much s m a l l e r l a g s . For the Quadra Sands c o n d u c t i v i t y d a t a , i f the l a y e r means are su b t r a c t e d from the values i n the r e s p e c t i v e l a y e r s ; then the c o r r e l a t i o n goes t o near 228 zero values f o r l a g one. This would then be i n agreement w i t h the c o r r e l a t i o n lengths p r e d i c t e d from the one-dimensional correlogram. The above d i s c u s s i o n has i m p l i c a t i o n s f o r the i n t e g r a l s c a l e estimates i n the twc>-dimensional models presented i n Chapter 5. Here the data m a t r i x was (10 x 20) observations. D i f f e r e n c e s i n the row and column means may l e a d t o an overestimate o f the i n t e g r a l s c a l e . However, s i n c e i t i s kncwn t h a t the g r i d was generated w i t h a s i n g l e mean and t h a t the estimated mean sample c o r r e l a t i o n f u n c t i o n c l o s e l y f o l l o w e d i t s t h e o r e t i c a l curve, t h i s i s not seen as ca u s i n g problems i n the e a r l i e r a n a l y s i s . 8) Power Spectra f o r the L i n e Samples Mthough s p e c t r a l theory does not. enter i n t o t h e s t o c h a s t i c flow a n a l y s i s based on the nearest neighbour models, p r e s e n t a t i o n o f the power s p e c t r a i s warranted i n terms o f a general a n a l y s i s o f the s p a t i a l s t r u c t u r e w i t h i n c o n d u c t i v i t y o r p o r o s i t y f i e l d s . The power s p e c t r a f o r c o n d u c t i v i t y form the b a s i s o f the s p e c t r a l s o l u t i o n s o f the s t o c h a s t i c f l o w problem. The power spectrum was d e f i n e d i n equation 2-5 and.shown t o represent the d i s t r i b u t i o n o f v a r i a n c e over frequency. Various aspects Of s p e c t r a l a n a l y s i s are f i r s t reviewed. The one-dimensional s p e c t r a obtained f o r the Quadra Sands are then presented. a) d e t a i l s o f s p e c t r a l a n a l y s i s S p e c t r a l a n a l y s i s o f f i e l d data i n v o l v e s c o n s i d e r a t i o n o f the f o l l o w i n g : 1) The power s p e c t r a l e s t i m a t o r i s given by: L - l •S- ( f ) = 2A{C (0)+2 E C (h)w(h)cos2TrAfh} 0<f< 8-3 A A A A i •'~ t A X /A k = l where A sample i n t e r v a l cx x ( h ) autocovariance a t l a g h w(h) w e i g h t i n g f u n c t i o n ( l a g window) f frequency L l a g l e n g t h o f window (number o f l a g s used i n the autocovariance function) 2) A we i g h t i n g f u n c t i o n i s used t o reduce the v a r i a n c e o f the s p e c t r a l e s t i m a t o r . I n choosing i t s form and l e n g t h , a t r a d e o f f i s made between the b i a s o f the es t i m a t o r and the v a r i a n c e o f S ( f ) . The b i a s w i l l xx i n c r e a s e and the v a r i a n c e decrease f o r longer l a g windows. 3) I n order t o get a good estimate o f a peak i n the spectrum, the bandwidth- o f the s p e c t r a l window must 230 be o f the same order as the w i d t h o f the peak. The bandwidth i s the frequency domain r e p r e s e n t a t i o n o f the l a g window. 4) The l e n g t h o f the sample s e r i e s determines both the extent t o which peaks i n the spectrum can be d i s -t i n g u i s h e d and the v a r i a n c e o f the s p e c t r a l e s t i m a t o r . 5) The h i g h e s t frequency a t which power can be determined i s 1/2 A . Frequencies g r e a t e r than t h i s are fo l d e d back (alaised) i n t o the spectrum. T h i s leads t o an overestimate o f the power a t higher frequencies. 6) The confidence i n t e r v a l on the l o g o f the t r u e spectrum i s g i v e n by: log S x x ( f ) + l o g - p - log;P(f) - log S (f)+log -j- • 8-4 *v(2> X X X > l - 6 , -2-where v = 2.667 N/L f o r the we i g h t i n g f u n c t i o n used here. The s e r i e s i s o f l e n g t h N. When the spectrum i s p l o t t e d on a l o g a r i t h m i c s c a l e , the confidence i n t e r v a l f o r a l l frequencies i s the same and can be represented by a s i n g l e l i n e . 7) S p e c t r a l theory assumes the r e g i o n i s not bounded, 231 so t h a t there i s a continuous range o f s p a t i a l frequencies ( J u l i a n , 1967). More d e t a i l e d d i s c u s s i o n s o f s p e c t r a l e s t i m a t i o n are given by Jenkins and Watts (1968). The s p e c t r a were computed u s i n g program BMD:02T (Biomedical Computer Program, Autocovariance and Power S p e c t r a l A n a l y s i s , W.J. Dixon (ed.), 1974). A Tukey w e i g h t i n g f u n c t i o n was used t o smooth the spectrum (Jenkins and Watts, pg. 244, 1968). b) r e s u l t s The power s p e c t r a f o r the c o n d u c t i v i t y , p o r o s i t y and h o r i z o n t a l DJ-Q g r a i n s i z e s e r i e s are p l o t t e d i n Figu r e s 8-8a through 8-8f. On each p l o t , two s p e c t r a l estimates are presented f o r two d i f f e r e n t l a g windows. The s p e c t r a w i t h a l a g w i d t h o f 20 shew more d e t a i l but the v a r i a b i l i t y i n the estimate i s g r e a t e r . The v e r t i c a l c r ossbar i d e n t i -f i e s an 80% confidence i n t e r v a l on the s p e c t r a l estimate. A l l the s p e c t r a show higher power a t s m a l l e r frequen c i e s , t y p i c a l o f a p o s i t i v e l y c o r r e l a t e d a u t o r e g r e s s i v e model. However, i n three o f the s e r i e s , t h ere i s a tendency t o an i n t e r m e d i a t e peak a t frequencies between 0.6 and 0.9 cycles/m. I t i s most s t r o n g l y developed i n the v e r t i c a l c o n d u c t i v i t y s e r i e s . The int e r m e d i a t e peak perhaps r e f l e c t s a response t o a c y c l i c p a t t e r n i n the l a y e r i n g w i t h i n the Quadra Sand. A d d i t i o n a l l y , both v e r t i c a l s e r i e s show i n c r e a s e d power a t h i g h frequencies. This c o u l d e i t h e r represent another c y c l i c 232 a) porosity horizontal l i n e b) conductivity horizontal l i n e m = 20 m = 10 m = 20 m = 10 0.1x10 °-J u >• o O.lxlO^H E -5 , o 0.1x10 DH 0.1x10 -i 1 1 r 0.328 0.656 0.984 1.311 1.639 f (cycles/m) 0.1x10' 1 1 1 1 1 0.328 0.656 0.984 1.311 1.639 f (cycles/m) 0.1x10""'-, m = 20 m = 10 o >• u _ o.ixio" JH 0.1x10 0.1x10'"'-m = 20 m = 10 ™ , o.ixio' J-J 1 1 1 1 1 0.328 0.656 0584 1.311 1.639 0.1x10 -i 1 1 1 l 0.328 0.656 0.984 1.311 1.639 f (cycles/m) c) porosity v e r t i c a l l i n e f (cycles/m) d) conductivity v e r t i c a l l i n e Figure 8-8 Power spectra for conductivity, porosity and the D ^Q grain size f r a c t i o n e) conductivity, v e r t i c a l l i n e , l i n e a r p l o t Figure 8-8 Cont. 234 v a r i a t i o n o r be due t o a l a i s i n g . The v e r t i c a l p o r o s i t y s e r i e s does not have an int e r m e d i a t e peak. There i s an abrupt i n c r e a s e i n power f o r frequencies l e s s than 0.3 cycles/m. The D^^ g r a i n s i z e spectrum d i f f e r s from the h o r i z o n t a l c o n d u c t i v i t y and p o r o s i t y spectrums i n t h a t an intermediate peak occurs a t h i g h e r frequencies o f about 1.3 cycles/m. The p r o p o r t i o n a l i t y o f v a r i a n c e t o t h e area under the spectrum does not h o l d on a p l o t o f the l o g o f s p e c t r a l power a g a i n s t l i n e a r frequency. A l s o presented i n F i g u r e 8-8f i s a l i n e a r s p e c t r a l p l o t f o r the v e r t i c a l c o n d u c t i v i t y s e r i e s . Here, we see more c l e a r l y the d i s t r i b u t i o n o f v a r i a n c e w i t h frequency and the r e l a t i v e importance o f the intermediate peak. 9) F i t t e d Nearest Neighbour Models a) e s t i m a t i o n technique I n e a r l i e r chapters, a f i r s t order,' nearest neighbour model was used t o generate c o n d u c t i v i t y r e a l i z a t i o n s w i t h a known s p a t i a l s t r u c t u r e . T h i s model w i l l now be a p p l i e d t o the Quadra Sands data t o cons i d e r the adequacy o f the model i n d e s c r i b i n g the s p a t i a l dependence w i t h i n t h e c o n d u c t i v i t y and p o r o s i t y s e r i e s . Estimates o f the auto-r e g r e s s i v e parameters w i l l a l s o be obtained. Departures from the model may suggest a broader c l a s s o f s p a t i a l models w i l l be necessary. 235 A general c o n s i d e r a t i o n o f the s u i t a b i l i t y o f nearest neighbour models i n r e p r e s e n t i n g the p r o b a b i l i s t i c s t r u c t u r e o f porous media w i l l r e q u i r e the a n a l y s i s o f data s e t s c o l l e c t e d i n many d i f f e r e n t g e o l o g i c a l environments and over d i f f e r e n t sample s c a l e s . An e s t i m a t i o n technique based on the j o i n t p r o b a b i l i t y f o r m u l a t i o n o f the s p a t i a l dependence i s g i v e n by Ord (1975). Besag (1974) and W h i t t l e (1954) present a l t e r n a t i v e approaches. Because o f the m u l t i -l a t e r a l nature o f the s p a t i a l dependence, both the l e a s t squares and maximum l i k e l i h o o d e s t imators a r i s i n g from a u n i l a t e r a l model must be modifie d f o r a s p a t i a l domain. The f o l l o w i n g e s t i m a t i o n procedure i s summarized from Ord (1975). The e s t i m a t o r & o f the aut o r e g r e s s i v e parameter a: i s t h a t v a l u e which irouiimizes the l i k e l i h o o d f u n c t i o n : R e c a l l the nearest neighbour system was d e f i n e d i n equation 3-2 as: {Y} = ct [WJ { Y } + { e } n 2 T T {n i = i (1 - <*A. )•} ri ( {Y} .{Y} - 2a {Y} . { Y L } + 8-5 a 2 { Y L } T • { Y L } ) where the n o t a t i o n i s as above w i t h the a d d i t i o n a l terms: {Y } is the vector product [Wj {Y} LJ X ^  eigenvalues of [w] The likelihood function assumes the uncorrelated residuals 1 E } are normally distributed with mean zero and constant variance. For a one-dimensional, first order system, the eigenvalues of [w] are given by: X^ = cos{ iri/n+1} •-! = 1,..., n g. for an n block system. Equation 7-5 i s solved iteratively for a using a Newton-Raphson procedure. Ord (1975) suggests the starting value: a Q = {Y} T{Y L}/{Y} T {Y} In fitting a stochastic process model to a time series, fairly exact procedures have been developed to guide in model identi-fication and diagnostic checking (Box and Jenkins, 1976). A similar approach would apply in a spatial domain. As a fi r s t analysis, less rigorous checks are applied here. Residuals from the fitted auto-regression can be computed from equation 3-7. The residuals should have a mean of zero. By computing a correlogram on the residuals, their internal correlation can be considered. 237 b) fitted autoregressive parameters The estimated autoregressive parameters for a fi r s t order model fitted to the line samples taken in the Quadra Sands are listed in Table 8-6. The nearest neighbour model was also fitted to the permeability and porosity sequences considered by Bakr (1976). Bakr's measurements were made on cores taken from a deeply buried ortho-quartzite aquifer in Illinois. The variability in the permeability distribution was much larger than in the Quadra Sands. A description of the unit and presentation of the correlograms and power spectra for these series of vertical line samples can be found in the original paper. c) comparison of sample and theoretical correlograms In Figure 8-9, the sample correlograms (curves labeled s) for the Quadra Sands data are compared with the theoretical correlograms (labeled T) of a f i r s t order nearest neighbour model based on the estimated autoregressive parameter. Care must be used in interpreting the differences in the two curves. Because neighbouring lag estimates can be correlated, the sample correlogram may not decay as rapidly as its true correlogram. This leads to a divergence in the tails of the correlograms that does not reflect a real structure in the sample series. Therefore, comparisons based solely on the correlogram can be misleading (Julian, 1967). Also plotted on Figure 8-9 are the 95% confidence intervals on a correlogram for a white noise series of the same length as the Quadra Sands Bakr ( 1 9 7 6 ) Data Sample a Sample a Horz. n 0.45 IL036 H-K 0.50 Horz. K 0.41 V-K 0.33 Vert, n 0.49 -n 0.29 Vert. K 0.31 IL056 H-K 0.53 Horz. Dj. 5 0.57 V-K 0.57 D10 0.53 m 0.28 D16 0.54 IL066 H-K 0.62 D50 0.58 V-K 0.59 D60 0.60 -n 0.40 D84 0.55 Note: i n Bakr's data, H denotes horizontal permeability, V denotes v e r t i c a l permeability i n Quadra Sands data, Horz.is horizontal line sample, Vert, i s vertical l i n e sample Table 8-6 Fitted Autoregressive Parameters-First Order Nearest Neighbour Model a) r ( D -0.2 H C) r (D 1.0 0.8 0.6 0.4 0.2 H 0.0 •0.2 e) 1.0 0.8 A 0.6 A 0.4 H 0.2 0.0 -0.2 H porosity horizontal l i n e I - lag number - \ porosity \ v e r t i c a l l i n e W - s \ \ r ' s t s N \ \ \ T 1 I \ T 4 6 8 10 I T lag number - \ D,-Q grain size \ horizontal l i n e \\ \ ^ \ \ \ \ - T V i / / • - — r - i i i b) •d) -0.2 d) (I) conductivity horizontal l i n e 239 I - lag number conductivity v e r t i c a l l i n e Figure 8-9 Comparison of sample correlograms for the Quadra Sands data with the th e o r e t i c a l correlograms for a f i t t e d f i r s t order, nearest neighbour model 8 10 I - lag number 240 sample s e r i e s . With these c a u t i o n s i n mind, the d i f f e r e n c e s i n the correlograms are o u t l i n e d below. The l a g one a u t o c o r r e l a t i o n c o e f f i c i e n t i s approximately preserved i n f i t t i n g a f i r s t o r d e r , nearest neighbour model. I t i s apparent i n the c o n d u c t i v i t y and p o r o s i t y s e r i e s f o r the h o r i z o n t a l l i n e sample t h a t a f i r s t order model does not account f o r the e l e v a t e d s p a t i a l dependence frcm l a g s 2 t o 5. The match t o the v e r t i c a l conduct-i v i t y s e r i e s i s l e s s s a t i s f a c t o r y . A f i r s t o r d er equation w i t h a p o s i t i v e a u t o r e g r e s s i v e parameter i s not capable o f e x p l a i n i n g the i n c r e a s e d c o r r e l a t i o n s a t l a g s 4, 5 and 6. As mentioned e a r l i e r , t h i s c o u l d r e f l e c t a c y c l i c p a t t e r n i n the l a y e r e d sequence. The correlograms f o r the h o r i z o n t a l D^Q g r a i n s i z e f r a c t i o n and the p o r o s i t y sequence along the v e r t i c a l decay a t a slower r a t e than do t h e i r t h e o r e t i c a l curves. Comparisons i n the frequency domain are not considered here. I t should be noted t h a t the f i r s t o r d e r , nearest neighbour model w i t h a p o s i t i v e a u t o r e g r e s s i v e parameter cannot e x p l a i n the intermediate peaks observed i n sane o f the sample s p e c t r a . In summary, a f i r s t o rder, nearest neighbour model d i d not represent the s p a t i a l s t r u c t u r e i n the v e r t i c a l l i n e samples. Higher order models may account f o r the s t r a t i f i e d nature o f the d e p o s i t . I t f o l l o w s t h a t i n a p p l y i n g s t o c h a s t i c techniques t o a g i v e n f i e l d problem, a broader c l a s s o f models may be necessary. Again, d a t a s e t s c o l l e c t e d 241 i n d i f f e r e n t g e o l o g i c a l environments and over d i f f e r e n t sample s c a l e s / need t o be considered. 10) Variograms The c h a r a c t e r i z a t i o n o f the s p a t i a l dependence w i t h i n a random f i e l d by the variogram was int r o d u c e d i n Chapter 2. Variograms f o r the Quadra Sands data are presented here f o r two reasons. F i r s t , s p a t i a l e s t i m a t i o n o r averaging w i t h i n a c o n d u c t i v i t y f i e l d based on the variogram i s more f l e x i b l e than t h a t which r e q u i r e s s p e c i f i c a t i o n o f the c o r r e l a t i o n f u n c t i o n . Thus, i t i s o f i n t e r e s t t o see the form o f the variograms f o r our data s e t . Secondly, the variogram may be used as a b a s i s f o r the generation o f c o n d u c t i v i t y r e a l i z a t i o n s (Delhomme, 1976). The variogram (equation 2-8) measures the s p a t i a l dependence as the square o f the expected d i f f e r e n c e between the random v a r i a b l e measured a t two p o i n t s . For separations taken along the rows i n a two-dimensional variogram, the variogram w i l l not depend on the d i f f e r -ences i n the row means as i t d i d i n the correlogram. (/Assuming the s p a t i a l dependence i s the same i n each row, a s i d e from a v a r y i n g mean). Vector separations c r o s s i n g rows w i l l be dependent on d i f f e r e n c e s i n the row means. On a r e g u l a r g r i d , t h e variogram i s estimated by (Delhomme, 1976) : 242 1 2N(h) N Ch.) T O O 1=1 8-7 where value o f the random v a r i a b l e a t l o c a t i o n x. l h v e c t o r s e p a r a t i o n N(h) number o f p a i r s o f p o i n t s separated by h. The variogram can be i n t e r p r e t e d i n terms o f the d i s t r i b u t i o n o f v a r i a n c e over d i s t a n c e . From t h i s , one can estimate the l e n g t h over which the s p a t i a l dependence between c o n d u c t i v i t y values tends t o reduce the estimated v a r i a n c e formed from a s p a t i a l average. This d i s t a n c e i s c a l l e d the range. Estimated variograms f o r the l i n e samples are p l o t t e d i n 1) Both p o r o s i t y and c o n d u c t i v i t y have a s t a t i s t i c a l l y a n i s o t r o p i c s p a t i a l dependence. 2) For both these parameters, the expected d i f f e r e n c e s are much g r e a t e r v e r t i c a l l y than h o r i z o n t a l l y . f o r a given s e p a r a t i o n (Note the d i f f e r e n t s c a l e s on YL(h) f o r the p l o t s ) . 3) The variograms have a bounded behavior over the sample le n g t h . The range estimates f o r the c o n d u c t i v i t y and p o r o s i t y s e r i e s f o r the h o r i z o n t a l F i g u r e 8-10. The p l o t s suggest: 243 a) porosity horizontal l i n e b) conductivity horizontal l i n e 2— -( 1 1 1 1 -i 1 1 1 1 1 H \ 1 1 1 1 1 1 1 1 1 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 h - lag number h - lag number Figure 8-10 One-dimensional variograms for conductivity and porosity 244 l i n e sample are both 1.8 m. I n the v e r t i c a l l i n e sample, the range f o r c o n d u c t i v i t y i s 0.6 m, w h i l e f o r p o r o s i t y i t i s 4.0 m. The variogram f o r the two-dimensional g r i d , g iven i n F i g u r e 8-11 shows very l i t t l e s p a t i a l s t r u c t u r e . There i s a s l i g h t i n c r e a s e i n the expected d i f f e r e n c e o f the h o r i z o n t a l c o n d u c t i v i t y values w i t h d i s t a n c e . The l a c k o f s t r u c t u r e suggests a t most a weak s p a t i a l dependence between the sample p o i n t s a t a spacing o f 1.52 m. Th i s c o n c l u s i o n c o u l d o n l y be reached i n the correlogram by accounting f o r the d i f f e r e n c e s i n the l a y e r means by modifying the c o r r e l a t i o n s t a t i s t i c s p e c i f i c a l l y t o s u i t the sample domain. 11) C o r r e l a t i o n Between Flow Parameters T r a n s i e n t flow a n a l y s i s r e q u i r e s the s p e c i f i c a t i o n o f the m u l t i -v a r i a t e p r o b a b i l i t y d e n s i t y f u n c t i o n between c o m p r e s s i b i l i t y , p o r o s i t y , and c o n d u c t i v i t y . The m u l t i v a r i a t e d i s t r i b u t i o n i s d e f i n e d , i n p a r t , by the c o r r e l a t i o n c o e f f i c i e n t between each o f the v a r i a b l e s . The c o r r e l a t i o n c o e f f i c i e n t between two random v a r i a b l e s , measured a t the same l o c a t i o n , i s estimated by: r(x,y) = n-1 1 n E i = l ( x ± - x) ( y ± - y) 8-8 horizontal lag h - lag number igure 8-11 Variograms for conductivity, two-dimensional grid 246 Care must be taken i n working w i t h s p a t i a l l y d i s t r i b u t e d data s i n c e e i t h e r s i m i l a r p a t t e r n s i n the s p a t i a l a u t o c o r r e l a t i o n o f the two v a r i a t e s o r i n trends i n t h e i r means can l e a d t o o v e r e s t i m a t i o n o f the c o r r e l a t i o n (Krumbein, 1970). In Table 8-7, the estimated c o r r e l a t i o n s between the f l o w parameters f o r the random sample and the two-dimensional gridded sample are l i s t e d . Note t h a t none o f the p a i r s o f the v a r i a b l e s are h i g h l y c o r r e l a t e d . The s i g n on the c o r r e l a t i o n i n d i c a t e s an i n c r e a s e i n c o n d u c t i v i t y and c o m p r e s s i b i l i t y w i t h p o r o s i t y (sand range) and an i n c r e a s e i n c o m p r e s s i b i l i t y w i t h a decrease i n c o n d u c t i v i t y . A s e t o f percent f i n e r g r a i n s i z e f r a c t i o n s were c o r r e l a t e d w i t h the c o n d u c t i v i t y and p o r o s i t y values f o r the l a r g e s c a l e random sample. The r e s u l t s are l i s t e d i n Table 8-8. These c o e f f i c i e n t s i n -d i c a t e : 1) That h i g h e r c o n d u c t i v i t i e s tend t o occur w i t h a c o a r s e r sand ( i e . a l a r g e r percent f i n e r g r a i n s i z e ) . 2) That h i g h e r p o r o s i t i e s a s s o c i a t e w i t h a f i n e r sand ( i e . a s m a l l e r percent f i n e r g r a i n s i z e ) . 3) That the s t r e n g t h o f the c o r r e l a t i o n i n c r e a s e s f o r the s m a l l e r g r a i n s i z e f r a c t i o n s . To compute t h i s c o r r e l a t i o n s t a t i s t i c f o r the l i n e samples, the s e r i e s would have t o be f i l t e r e d t o remove the p o i n t - t o - p o i n t c o r r e l a t i o n s Sample Parameters C o r r e l a t i o n Randcm k,n 0.26 k,n 0.25 Square k, a -0.20 G r i d A n, a 0.16 l o g k, l o g a -0.23 l o g a , n 0.25 Note: k c o n d u c t i v i t y n p o r o s i t y * c o m p r e s s i b i l i t y a Table 8-7 Estimated C o r r e l a t i o n C o e f f i c i e n t s Between Flow Parameters % f i n e r g r a i n s i z e P o r o s i t y C o n d u c t i v i t y f r a c t i o n D 1 6 -0.31 0.61 D 5 0 -0.31 0.58 D 6 0 -0.30 0.55 D84 -0.20 0.41 Table 8-8 Estimated C o r r e l a t i o n C o e f f i c i e n t s Between Flow Parameters and G r a i n S i z e F r a c t i o n , Random Sample w i t h i n each s e r i e s . For the raw s e r i e s , the c o r r e l a t i o n c o e f f i c i e n t between c o n d u c t i v i t y and p o r o s i t y values i n the h o r i z o n t a l sample i s 0.67, and i n the v e r t i c a l sample i t i s -0.02. C r o s s - c o r r e l a t i o n a n a l y s i s would p r o v i d e estimates o f the l a g c o e f f i c i e n t s i n the j o i n t m u l t i v a r i a t e p r o b a b i l i t y d e n s i t y f u n c t i o n . 12) Conclusions 1) The s p a t i a l heterogeneity of. the f l o w parameters a t the sample s i t e i s a t the very low end o f the report e d range o f values f o r porous media, as found i n the l i t e r a t u r e . 2) Because of the low v a r i a b i l i t y i n the parameter h e t e r o g e n e i t y , i t was not p o s s i b l e t o d i s t i n g u i s h between a normal and lognormal d i s t r i b u t i o n f o r the c o n d u c t i v i t y histogram. P o r o s i t y values from the l a r g e s c a l e random sample were f i t t e d t o a normal d i s t r i b u t i o n . C o m p r e s s i b i l i t i e s measured from samples on a two-dimensional g r i d f i t a lognormal d i s -t r i b u t i o n . 3) A p l o t o f the v a r i a t i o n s i n c o n d u c t i v i t y and p o r o s i t y along the h o r i z o n t a l and v e r t i c a l l i n e samples v i s u a l l y i l l u s t r a t e s the d i f f e r e n c e s i n the p a t t e r n s o f s p a t i a l v a r i a t i o n along and across a s t r a t i f i e d sequence. 249 4) The cxanductivity and p o r o s i t y sequences have a s t a t i s t i c a l l y a n i s o t r o p i c s p a t i a l s t r u c t u r e . This i s demonstrated i n : a) the crie-dimensional correlograms b) the variograms formed on the l i n e samples. For the c o n d u c t i v i t y sequence, the f u n c t i o n a l form o f the c o r r e l a t i o n d ag r e l a t i o n was d i r e c t i o n dependent. 5) I t was found a f i r s t o r d e r , nearest neighbour model may not adequately represent the s p a t i a l s t r u c t u r e i n a s t r a t i f i e d sequence. D i f f e r e n c e s i n the Quadra Sands d a t a from the t h e o r e t i c a l s t r u c t u r e o f a f i r s t o r d er model were noted i n t h a t : a) A f i r s t o r d er model cannot generate i n c r e a s e d c o r r e l a t i o n s a t g r e a t e r l a g s as was observed i n the v e r t i c a l c o n d u c t i v i t y sequence. b) S e v e r a l o f the power s p e c t r a had peaks a t intermediate f r e q u e n c i e s , r a t h e r than being monotonically decaying. Departures from a f i r s t o r d er model are probably due t o a c y c l i c component i n the s t r a t i f i e d sequence. T h i s suggests a broader range of nearest neighbour models may be necessary i n a p p l y i n g s t o c h a s t i c process models t o f i e l d problems. 250 6) Departures from stationarity were observed in two of the gridded samples. Because of layering within the Quadra Sands, the small two-dimensional grid did not average over a large enough area to achieve stationarity in the mean conductivity. This occurred even though the Sands are relatively uniform. The correlogram of the DJ-Q grain size fraction illustrates the problem of characterizing the space law for media which, at the sample scale, can be assumed to be stationary except from a small number of inclusions with parameter values far removed from the mean (eg. clay or gravel lenses in a sand matrix). CHAPTER 9 Conclusions, Recoinriiendations f o r Future Research Conclusions The Model a) The s t o c h a s t i c , steady s t a t e , groundwater f l o w equation can be solved n u m e r i c a l l y u s i n g Monte C a r l o s i m u l a t i o n techniques i n connection w i t h a d i s c r e t e b l o c k model o f the c o n d u c t i v i t y f i e l d . A f i r s t o r d e r , nearest neighbour model can be used t o b u i l d m u l t i l a t e r a l s p a t i a l dependence between c o n d u c t i v i t y values i n adjacent b l o c k s . The model i s designed s p e c i f i c a l l y t o simulate a l o c a l s t o c h a s t i c i n t e r a c t i o n between c o n d u c t i v i t y v alues i n a s t a t i o n a r y medium. The covariance s t r u c t u r e o f the medium i s known. The model a p p l i e s e q u a l l y w e l l " t o one o r two-dimensional bounded domains. b) Input t o the c o n d u c t i v i t y model c o n s i s t s o f the parameters d e s c r i b i n g the b l o c k system, the au t o r e g r e s s i v e parameters t h a t s p e c i f y the s t r e n g t h o f the dependence between c o n d u c t i v i t y values i n neighbouring b l o c k s , and the moments of the c o n d u c t i v i t y d i s t r i b u t i o n . The primary output from the s o l u t i o n o f the flow equation i s the s t a t i s t i c a l d i s t r i b u t i o n o f h y d r a u l i c head 9 , p a r t i c u l a r l y the mean head "^"(x) and the standard d e v i a t i o n S ^  ( x ) . Both are a f u n c t i o n o f p o s i t i o n w i l d i i n the f l o w domain. Qne-Dimensional A n a l y s i s a) The r a t i o o f the i n t e g r a l s c a l e t o the d i s t a n c e between the boundary p o i n t s i s a fundamental parameter i n modeling the s t o c h a s t i c behavior o f a bounded, s t a t i o n a r y medium. The output d i s t r i b u t i o n s on the p r e d i c t i o n v a r i a b l e s ( h y d r a u l i c head, head g r a i d e n t and f l u x ) must be i n t e r p r e t e d i n l i g h t o f t h i s parameter. b) H y d r a u l i c head standard d e v i a t i o n s are a f u n c t i o n o f i ) the standard d e v i a t i o n i n the c o n d u c t i v i t y d i s t r i b u t i o n i i ) the mean h y d r a u l i c head g r a d i e n t i i i ) the r a t i o o f the i n t e g r a l s c a l e t o the l e n g t h o f the system. T h i s dependence can be e x p l a i n e d i n terms o f the p r o b a b i l i t y o f l i k e c o n d u c t i v i t y values f o l l o w i n g 253 each other along the flowline and the departure of that alignment from the mean conductivity of the medium. c) For a standard deviation in the log-conductivity of 0.84 and integral scale/length ratios ranging from 0.01 to 0.036,. the standard deviation in head (expressed as a percent of the total head drop) at the midpoint of the flowline ranged from 15 to 29%, respectively. d) In the one-dimensional model, the variability in the estimated flux is due entirely to the nonergodic behavior of each of the conductivity realizations. Large coefficients of variation in the flux estimate can occur. e) Aside from a minor variation, output variabilities will not depend upon the block size used, provided a sufficiently accurate representation of the in-tegral scale is made. f) Standard deviations in hydraulic head are more sensitive to the integral scale of the autocorrelation function in conductivity than to its precise form of decay with increasing lag (for a monotonically decaying f u n c t i o n ) . Two-Dimensional A n a l y s i s a) Standard d e v i a t i o n s i n the h y d r a u l i c head d i s t r i b u t i o n are dependent upon: i ) the s p a t i a l d i s t r i b u t i o n o f expected head gra d i e n t s i i ) the space law o f the c o n d u c t i v i t y d i s t r i b u t i o n i i i ) the r a t i o s o f the i n t e g r a l s c a l e s t o the lengths o f the f l o w domain i n the two coordinate d i r e c t i o n s i v ) the arrangement o f s t a t i o n a r y u n i t s w i t h i n the f l o w domain. The standard d e v i a t i o n s i n head are g r e a t e s t i n t h a t r e g i o n o f the f l o w domain where the expected gr a d i e n t s are r e l a t i v e l y l a r g e , y e t the r e g i o n i s f a r enough removed from the constant head boundaries so t h a t t r u n c a t i o n o f the p e r m i s s i b l e range o f head values i s not s i g n i f i c a n t . b) For a standard d e v i a t i o n i n the l o g - c o n d u c t i v i t y o f 0.43, the maximum standard d e v i a t i o n i n head was approximately 17% of the total head drop. This result applies to the nonuniform gradient field. In a uniform gradient field of the same size and with the same space law describing the heterogeneity in conductivity, the maximum standard deviation in head was 10.5% of the total head drop. In both cases, the total head drop was the same. c) The standard deviations in head are approximately halved in a two-dimensional model from those in a one-dimensional model with equivalent space law parameters and the same uniform gradient field. For a standard deviation in the log-conductivity of 0.86 and an integral scale/length ratio of 0.07, the maximum standard deviation in head was 31.0% of the total head drop for a one-dimensional analysis. In two dimensions, the observed maximum was 19.4%. d) In two dimensions, the variability in the estimated flux depends upon both nonergodic effects and the arrangement of the conductivity values within the flow domain. e) The effects of a statistically anisotropic covariance f u n c t i o n can be i n t e r p r e t e d i n terms o f the r e s p e c t i v e r a t i o s o f the i n t e g r a l s c a l e t o the l e n g t h o f the flow system along the two c o o rdinate axes. f) R ecognition o f a s p a t i a l t r e n d i n the mean c o n d u c t i v i t y i s c r i t i c a l i n determining the magnitude and s p a t i a l v a r i a t i o n o f the standard d e v i a t i o n s i n h y d r a u l i c head. Scal e Considerations a) Estimated p r e d i c t i o n u n c e r t a i n t i e s are o n l y as good as the s t o c h a s t i c model used t o represent e the a c t u a l s t r u c t u r e o f the c o n d u c t i v i t y v a r i a t i o n s w i t h i n the f l o w domain. A d i s c r e t e b l o c k model must preserve the s t r u c t u r e o f those h e t e r o g e n e i t i e s which c o n t r i b u t e s i g n i f i c a n t l y t o the p e r t u r b a t i o n o f the expected g r a d i e n t f i e l d a t the s c a l e o f the f l o w a n a l y s i s . b) The modeling approach used i n t h i s study i s based on i d e n t i f y i n g s t a t i o n a r y u n i t s w i t h i n the f l o w domain. I n doing so, both s t o c h a s t i c and d e t e r m i n i s t i c components o f the heterogeneity may be recognized. Comparison w i t h Other Models a) Although d i r e c t comparisons cannot be made, both the Monte C a r l o approach and the s p e c t r a l a n a l y s i s s o l u t i o n p r e d i c t a s i m i l a r behavior. E f f e c t i v e C o n d u c t i v i t y a) Only i n the case o f uniform f l o w through a s i n g l e s t a t i o n a r y u n i t i s the e f f e c t i v e c o n d u c t i v i t y o f a two-dimensional model given by the geometric mean. I f the mean g r a d i e n t f i e l d i s nonuniform, o r i f the mean c o n d u c t i v i t y has a s p a t i a l t r e n d , p r e d i c t i o n s based on the geometric mean conduct-i v i t y do not s a t i s f y the equivalence c r i t e r i a . A n a l y s i s o f Quadra Sands Data a) The s p a t i a l heterogeneity o f c o n d u c t i v i t y i s a t the very lew end o f the re p o r t e d range o f values f o r porous media. Standard d e v i a t i o n s i n the logarithms o f c o n d u c t i v i t y f o r the d i f f e r e n t sample plans ranged from 0.09 t o 0.36. b) The frequency histograms f o r c o n d u c t i v i t y , p o r o s i t y and c o m p r e s s i b i l i t y are monomodal. The c o n d u c t i v i t y histogram i s not skewed; but 258 because o f i t s low c o e f f i c i e n t o f v a r i a t i o n , i t can be f i t t e d t o both normal and lognormal d i s -t r i b u t i o n s . P o r o s i t y values can be f i t t e d t o a normal d i s t r i b u t i o n , whereas c o m p r e s s i b i l i t y values are lognormal. c) The d a t a show a s t a t i s t i c a l l y a n i s o t r o p i c c o r r e l a t i o n s t r u c t u r e , both i n the i n t e g r a l s c a l e s and i n the f u n c t i o n a l form o f the a u t o c o r r e l a t i o n . d) A f i r s t o r d e r , nearest neighbour model prov i d e s an approximation t o the s p a t i a l dependence i n t h e h o r i z o n t a l l i n e sample. I t may not adequately represent the s p a t i a l s t r u c t u r e along l i n e t r a n s e c t s c r o s s i n g a s t r a t i f i e d sequence. e) Departures from s t a t i o n a r i t y were observed i n s e v e r a l o f the gridded samples, e i t h e r because o f i n s u f f i c i e n t averaging o r i n c l u s i o n s w i t h parameter values f a r removed from the mean. Th i s occurs even though the Quadra Sands are r e l a t i v e l y uniform. B) Recoirmendations f o r Future Research The o b j e c t i v e o f t h i s s e c t i o n i s t o suggest v a r i o u s avenues o f fu t u r e research o r extensions o f e x i s t i n g methods. limitations on the number of conductivity blocks included within a stationary unit caused by computer storage limitations may be reduced by one of the following means: a) Storage requirements for many of the matrices utilized in the generating algorithm could be reduced by exploiting their property of sparseness. b) Intermediate storage on tape could be used to release computer storage to hold expanded matrices. c) Several stationary units with identical space law parameters could be joined into a single stationary unit. In order to preserve the covariance relation across the connecting boundaries, a second set of nearest neighbour equations could be applied to blocks in the vicinity of the boundary. . Con-ductivity values in these blocks would then be recalculated, approximately preserving the correlation across the boundary. Convergence of the finite element solution of the two-dimensional flow equation was sensitive to the standard deviation in the log-conductivity (a ) • Investigation of the solution y a l g o r i t h m may shed l i g h t on extending the a n a l y s i s t o a values g r e a t e r than one. A l t e r n a t e models t o the s t a t i o n a r y , f i r s t o r d er nearest neighbour system can be suggested. These i n c l u d e the f o l l o w i n g a) Higher o r d e r nearest neighbour models may be used t o account f o r c y c l i c p a t t e r n s i n s t r a t i f i e d sedimentary sequences. b) A two-scale nearest neighbour model may be appr o p r i a t e i n a l a y e r e d medium i n a s i t u a t i o n where there are more l a y e r s i n the flo w domain than can be d e s c r i b e d d e t e r m i n i s t i c a l l y . C o n d u c t i v i t y v a r i a t i o n s w i t h i n each l a y e r would be generated from t h e i r own nearest neighbour process. However, the mean c o n d u c t i v i t y o f each o f the l a y e r s would be t i e d together by a second nearest neighbour r e l a t i o n . I n such a sequence, the v a r i a n c e i n c o n d u c t i v i t y computed along a v e c t o r c r o s s i n g the l a y e r s w i l l be g r e a t e r than t h a t f o r a v e c t o r remaining w i t h i n one o f the l a y e r s . T h i s again has i n t u i t i v e appeal i n modeling a s t r a t i f i e d sedimentary secruence. 261 c) Simple nonstationary process models may be used to describe conductivity variations. These techniques are well known in time series analysis. It should be possible to extend these models into the spatial domain, certainly i f a unidirectional structure is allowed. d) Two stochastic processes could be coupled to model a bimodal medium in which one stationary unit occurred as inclusions in a second stationary unit. One model would describe the spatial structure in the matrix; the other the dispersion of the inclusions. e) By accounting for the tensor form of conduct-ivity, hydraulic anisotropy could be allowed in the point probability density function. f) The nearest neighbour model may be extended to the multivariate problem, in which several random variables are simultaneously generated while respecting the cross-correlation structure. Such a model is needed for a transient flow analysis. 2 6 2 4) The steady s t a t e f l o w eguation has been considered i n t h i s study. Other flow problems may be s u i t a b l y i n v e s t i g a t e d u s i n g comparable techniques. For example: a) P h y s i c a l l y based macroscopic d i s p e r s i o n i n h e t e r o -geneous media can be t a c k l e d by generating r e a l i z a t i o n s o f the s p a t i a l h e t e r o g e n e i t i e s , f o l l o w e d by a d e s c r i p t i o n o f the d i s p e r s i v e c h a r a c t e r o f the medium. b) E f f o r t s t o i n c l u d e confidence l i m i t s on s a f e t y f a c t o r s i n many g e o t e c h n i c a l problems can be analyzed u s i n g t h i s p r o b a b i l i s t i c approach c) S t o c h a s t i c f l o w equations t h a t d e s c r i b e the t r a n s i e n t response o f porous media may be handled u s i n g the nearest neighbour model. However, s o l u t i o n o f a two o r three-dimensional t r a n s i e n t equation u s i n g Monte C a r l o techniques w i l l r e q u i r e very l a r g e amounts o f computing time. 5) A t h e o r e t i c a l a n a l y s i s o f p r e d i c t i o n u n c e r t a i n t i e s i n modeling groundwater f l o w systems based on a known space law i s o n l y p a r t o f the s t o c h a s t i c problem. The u l t i m a t e o b j e c t i v e o f t h i s approach i s t o apply these models t o p r a c t i c a l f i e l d problems. This introduces an a d d i t i o n a l u n c e r t a i n t y i n t o the s o l u t i o n when the parameters o f the space law must be estimated from sample data. I n a Monte C a r l o a n a l y s i s , the e r r o r s are twof o l d . The f i r s t a r i s e s from the inadequacy o f the s t o c h a s t i c process model t o d e s c r i b e the c o n d u c t i v i t y v a r i a t i o n s a t the f i e l d s i t e . A second e r r o r i s introduced i n e s t i m a t i n g the model parameters from the sample data. L i s t e d below are s e v e r a l o f the sampling problems which must be considered. a) A sample p l a n must be designed t o estimate the parameters o f the space law. The p h y s i c a l c o n s t r a i n t s on o b t a i n i n g subsurface data would be i n c l u d e d i n the a n a l y s i s . b) S t a t i o n a r y u n i t s w i t h i n the flow domain must be i d e n t i f i e d and t h e i r boundaries l o c a t e d . c) A sample i n t e r v a l must be chosen f o r gridded data. d) I t was shown i n Chapter 5 t h a t t he p r e d i c t i o n uncer-t a i n t i e s w i l l have a v a r i a b l e s e n s i t i v i t y t o e r r o r s i n the i n p u t data. An e f f i c i e n t sample p l a n must take t h i s dependence i n t o account. S e v e r a l complex t h e o r e t i c a l problems remain t o be i n v e s t i g a t e d . These can be summarized as f o l l o w s : a) The v a l i d i t y o f the concept o f an e q u i v a l e n t medium i n a nonuniform f l o w domain has y e t t o be demonstrated. 264 b) The h y d r a u l i c behavior o f a d i s c r e t i z e d model o f the c o n d u c t i v i t y v a r i a t i o n s i n i t s r e p r e s e n t a t i o n o f a continuous f i e l d bears f u r t h e r study. c) The c o n s i d e r a b l e problem o f i d e n t i f y i n g the s t o c h a s t i c - d e t e r m i n i s t i c boundary i n c o n c e p t u a l i z i n g the s p a t i a l heterogeneity Of porous media was not considered here. 7) A f i e l d study i s needed t o i n t e g r a t e the s t o c h a s t i c approach w i t h r e a l w o r l d problems. Such a study would i n v o l v e the i n v e s t i g a t i o n o f both the f l o w domain and the c o n d u c t i v i t y f i e l d . Data s e t s c o l l e c t e d t o o b t a i n estimates o f the heterogeneity o f a porous medium (such as i n Chapter 8) are not s u f f i c i e n t t o handle t h i s problem. 8) The need remains f o r d a t a s e r i e s c o l l e c t e d i n d i f f e r e n t hydro-g e o l o g i c environments and on d i f f e r e n t s c a l e s , as our understanding of the p r o b a b i l i s t i c s t r u c t u r e o f c o n d u c t i v i t y v a r i a t i o n s i s f a r from complete. I f the Quadra Sands were resampled, appro-p r i a t e sampling plans would i n c l u d e : a) A s e t o f undisturbed core measurements; both i n the upper zone already sampled and i n the lower zone. b) Attempts a t i n s i t u d e t e r n i n a t i o n s o f c o n d u c t i v i t y . c) F u r t h e r i n v e s t i g a t i o n o f c y c l i c p a t t e r n s i n the v e r t i c a l sequence. REFERENCES Agterberg, F., 1970. 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APPENDICES 1) Appendix I: Derivation of Variance Expansion Factor, TWo-Dimensional Model 2) Appendix I I : S o l u t i o n o f the Flow Equations 3) Appendix I I I : Flowchart f o r the So l u t i o n o f the Two-Dimensional Stochastic Equation 4) Appendix TV: Quadra Sands Data 5) Appendix V: Analysis of Variance, R e p l i c a t i o n Variance Versus Sample Variance 6) Appendix VI: Glossary 274 A P P E N D I X I The two-dimensional nearest neighbour a u t o r e g r e s s i v e model i s d e f i n e d by eguation 3-1 as: 1-1 The v a r i a n c e f a c t o r n i s c a l c u l a t e d frcm equation 3-19: 2 v. y [R] [M] -1 1-2 The m a t r i x JM i s d e f i n e d : { ( [i] - [w] ) ( [i] - [w]. ) T} T, -1 1-3 where [w] i s the m a t r i x o f s c a l e d s p a t i a l l a g operators. Therefore [M] ^ i s simply the product m a t r i x : [M-] 1 = ( [i] - [w] ) ( [ i W ,T 1-4 For a system o f p b l o c k s , the matrices are dimensioned (pxp). By c a r e f u l bookkeeping, n can be deterrnined from 1-2 w i t h o u t c a r r y i n g out the operations on the e n t i r e m a t r i c e s . We must work on a l a r g e enough b l o c k system so t h a t the general m u l t i l a t e r a l term i s c a r r i e d through w i t h o u t b e i n g confounded by t r u n c a t i o n o f dependence a t the boundaries. F or the twc»-dimensional model t h i s corresponds t o a (7x7) b l o c k system, y i e l d i n g matrices dimensioned (49x49). Consider the block system shown i n F i g u r e 1-1. Each row i o f the weight m a t r i x jw] expresses the dependence o f b l o c k i on a l l o t h e r b l o c k s j . For a c e n t r a l b l o c k , there are 4 nonzero e n t r i e s corresponding t o i t s 4 contiguous neighbours (equation 3-3). Row 25 o f [w] can be w r i t t e n as r e l a t i o n 1-5 i n Table 1-1. Then f o r ( [ij -[w] ) , row 25 can be w r i t t e n as r e l a t i o n 1-6 i n Table 1-1. A l l b l o c k s w i t h 4 contiguous neighbours w i l l have a s i m i l a r s t r u c t u r e , w i t h non-zero e n t i r e s i n t h e i r a p p r o p r i a t e columns. On forming the transpose m a t r i x ( [i] - [w] ) T , row 25 becomes column 25. We now m u l t i p l y ( [ij - [w] ) by i t s transpose, ( [i] - [w]) T The 25th row o f the product m a t r i x i s formed by m u l t i p l y i n g row 25 o f ( Ji] - [w] ) w i t h a l l columns o f ( [i] - [w] ) T. Since the nearest neighbour equation 1-1 f o r b l o c k 25 shares no common b l o c k s w i t h t h a t f o r b l o c k 1, there w i l l be no conrnon non-zero e n t r i e s i n row 25 o f ( [i] - jw] ) and column 1 o f ( [i] - [w] ) T. Therefore, the f i r s t e n t r y i n row 25 o f the product m a t r i x w i l l be zero. S i m i l a r i l y , i t can be shown o n l y columns 11, 17, 18, 19, 23, 24, 25, 26, 27, 31, 32, 33, 39 o f row 25 w i l l have non-zero e n t r i e s . These terms are g i v e n i n Table 1-2. A l l b l o c k s w i t h 4 contiguous neighbours, which i n t u r n have 4 contiguous neighbours, w i l l have the same s t r u c t u r e , w i t h e n t r i e s i n the a p p r o p r i a t e columns. Because o f symmetry i n the nearest neighbour equation f o r each b l o c k , we know column 25 o f the product m a t r i x w i l l have the same non-zero e n t r i e s i n rows 11, 17, 18, 32, 33, 39 as 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Figure 1-1 Block system used in deriving relation 1-9 c o l . 1-5 1-6 1-7 1-8 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 4 0 0 0 0 a 4 0 0 a 2 4 0 -a z 4~ 0 0 0 0 0 -a -a 4 0 0 0 0 -a z 4 0 25,17 25,18 (•25,19 ^25,20 p25,21 P25,22 p25,23 P25,24 p p25,25 P25,26 p25,27 P25,28 p25,29 p25,30 P25,31 p25,32 P25,33 p(i;i> p(l,0) p ( l , l ) Pd,2) p(l,3) p(0,3) P(0,2) P(0,1) p(0,0) P(0,1) P(0,2) P(0,3) P(l,3) Pd,2) P(1,D P(1,0) PCI.D Table 1-1 Relations Required in Deriving Equation 1-10 278 Row Golumn Entxy Row Column Entxy 2 25 11 « Z / 1 6 25 26 -a • /2 X 25 17 25 27 a x A 6 25 18 " « z / 2 25 31 . a x a z / 8 25 19 a ' x a 25 32 - 0 z / 2 25 23 a x / 1 6 25 33 a z a x / 8 25 24 - a x /2 25 39 a 2 /16 25 25 a 2 / 8 + a 2 / 8 + l z X Table 1-2 Non-Zero Terms i n Row 25 o f Product M a t r i x •{( [I] " [W] ) ( [ l ] - jw]- ) T} d i d row 25 i n the p r e v i o u s l y l i s t e d columns. That i s , f o r example, the e n t r y i n (25, 11) i s i d e n t i c a l t o t h a t i n (11, 25). To compute |~l] n i n terms o f equation 1-2, the above product m a t r i x must be m u l t i p l i e d by the c o r r e l a t i o n m a t r i x [ R ] . Row 25 o f the c o r r e l a t i o n m a t r i x i s w r i t t e n as r e l a t i o n 1-7 i n Table 1-1, where p 2 5 17 ' ^ o r e x a m P x e ' l S ^ e c o r r e l a t i o n c o e f f i c i e n t between b l o c k s 25 and 17. For a s t a t i o n a r y model and the g r i d shown i n F i g u r e T-1, row 25 can be reduced t o r e l a t i o n 1-8 i n Table 1-1. Here, p ( . , . ) 279 is the correlation coefficient for a row lag of i , and a column lag of j (if j = 0, 1, 2...). On multiplying row 25 of the correlation matrix by column 25 of the product matrix ( [i] - jw] ) ( [ij - [w] ) T , we obtain the following expression for the entry in position (25, 25.) of the product matrix: 2 x z 16 % Pz(2) + az ax p ( l , l ) - az pz(l) + V x p ( l , l ) + _\ Px(2) - V P x ( l ) + a\ + a\ + 1 i Px(l) + 16 2 8 8 2 1-9 2 t x 16 ax Px(2) + axa:z p ( l , l ) - az Pz(l) + V x p ( l , l ) + z Kz(2) 16 Here we have set p(0,2)= P (2); p (0,1) = p (1); p (1,0). = p o ( l ) X X s and p (2,0) = P g(2) to simplify the notation. Gombining terms yields: •_Z P Z(2) +-V^ P ( l , l ) - az Pz(l) - ax Px(l) + 8 2 2 2 2 \ + \ + 1 + \ p x (2) 8 8 8 1-10 Identical expressions are obtained for a l l diagonal elements of the product matrix JRJ |MJ ^ for which the nearest neighbour equations are not confounded by boundary effects. Using equation 1-2, r\ can then be defined: where Q is equal to relation 1-10. This is then the same as equation 3-21. 281 APPENDIX I I 1) A n a l y t i c a l S o l u t i o n o f the One-Dimensional, Steady S t a t e Flow Equation i n a Layered Medium The a n a l y t i c a l s o l u t i o n o f the boundary val u e problem 4-1, 4-2 i s given by Freeze (1975). By s o l v i n g the flow equation i n each o f say, 4 l a y e r s , and deterrruuiing the unknown c o e f f i c i e n t s o f each s o l u t i o n through the i n t e r l a y e r boundary c o n d i t i o n s ; i t i s p o s s i b l e t o extend the s o l u t i o n t o the n l a y e r problem by i n d u c t i o n . The i n t e r l a y e r boundary c o n d i t i o n r e q u i r e s c o n t i n u i t y o f flo w across a l a y e r boundary. That i s , 1 " i + 1 ~ where <f> j_ ( x. ) and <j> ( x . ) are the head s o l u t i o n s a t the p o i n t developed f o r the l a y e r s i and i+1, r e s p e c t i v e l y . For the l a y e r s i = l , 2, ...m: •, - ^ ax + • i = l II-2 • t = £ - + TT 0-« <^ > & " « +*0 1 j=2 3 3-1 l<i<m 282 • =• ax •- aL + 9 m i=m where 9 - 9 L o a = L m 'm-l . v. s ( j - l ) f e (JL j=2 k J k j " l 1) + (m-l) (__ - 1) + m km-1 f o r m > 3 and 9 - 9 L o a - L m /km (m-l) x - 1) + m f o r m = 1, 2 and where i = _ + 1 L the r i g h t hand s i d e being truncated t o the lower i n t e g e r . 2) F i n i t e Element S o l u t i o n o f the Two-Dimensional, Steady S t a t e Flow Eguation i n a Heterogeneous, I s o t r o p i c Medium The f i n i t e element method i s used t o o b t a i n a numerical s o l u t i o n o f the boundary value problem d e s c r i b e d by equations 5-3, 5-4. Remson e t a l (1971) present a b a s i c f o r m u l a t i o n o f the f i n i t e element method from the p r i n c i p l e s o f v a r i a t i o n a l c a l c u l u s . P i n d e r and Gray (1977) d i s c u s s an a l t e r n a t e G a l e r k i n approach t o formulate the i n t e g r a l eqxiations 283 and s e t up the f i n i t e element approximations. The development here f o l l o w s the R a l e i g h - R i t z method based on the c a l c u l u s o f v a r i a t i o n s . Instead o f s o l v i n g the boundary value problem d i r e c t l y , we seek the f u n c t i o n which w i l l minimize a corresponding i n t e g r a l equation. This then i s a l s o the s o l u t i o n o f the boundary value problem. The i n t e g r a l equation we must minimize f o r two-dimensional, steady s t a t e f l o w i n a heterogeneous, i s o t r o p i c medium i n a domain R i s (Remson e t a l (1971), p. 323): * [ * ] - m i We seek the f u n c t i o n <f> (x,y) which ininimizes l|<j>J . We proceed by s o l v i n g f o r an approximate mini m i z i n g f u n c t i o n d e f i n e d on a d i s c r e t e s e t o f subdomains i n the r e g i o n R. The subdomains, o r f i n i t e elements, are chosen t o be t r i a n g u l a r . See F i g u r e I I - l . W i t h i n each element, a l i n e a r approximation t o the unknown f u n c t i o n i s used, g i v e n as: 4>e = Ax + By + C II-4 Because the p l a n a r s u r f a c e 9 must be c o i n c i d e n t w i t h the nodal values of the 9' s on any element, we can w r i t e K ( | i ) 2 + K ( | i , 2 }dxdy I I - 3 F i g u r e I I - l F l o w d o m a i n and f i n i t e e l e m e n t d i s c r e t i z a t i o n 285 r •> e • i *k 1 x. y. 1 J i 1 x. y . 3 3 r ^ r \ A 1 I I - 5 - i v y B CN solving for the unknown coefficients, equation I I - 4 can be rewritten: cj>e = (N. 4 . + N. 9 . + N, <j>. ) e i Y i k T k I I - 6 where N i = [ ( x j y k " V j } + ( y j - y k ) x + ( xk - X j ) y ] / 2 A N. = 3 [ ( x k y i " x i y k } + ( yk - y i ) x + ( x i - V y ] 7 2 A \ = [ ( x i y j - x j y k > + ( y i - y j ) x + ( x j - x i ) y ] / 2 A and where 2A = 1 X. l Y i = 2 x (area triangle i j. k ) 1 X . 3 Y J 1 yk We must niijiimize the i n t e g r a l eguation w i t h r e s p e c t t o each o f the element c o n t r i b u t i o n s over R. To do t h i s , the d e r i v a t i v e o f the i n t e g r a l equation I I - 3 i s s e t t o zero w i t h r e s p e c t t o each o f the nodal values o f f . That i s , — — = 0 (m=l, n) I I - 7 Y m where n i s the number o f nodal p o i n t s within.R. Since s e v e r a l elements may be common t o each node, the m i n i m i z a t i o n i s w r i t t e n : 1 H~ m=l, ...,n X 1 _ 8 e Tm t h where l e i s the <x>ntribution o f the e element t o the i n t e g r a l equation f o r a g i v e n node. T h i s can be determined by s u b s t i t u t i n g the l i n e a r e approximating f u n c t i o n 9 over each element i n t o the i n t e g r a l equation I I - 3 . Thus, ,r 9N. ,3N. m 2 l e = / / { - ^ ( ^ i . + ^ j + _ m + I X _ 9 e. - 9 N - 9 N - 9 N , x, ^ 1 U J 0 For the gen e r a l element ( i , j,k) and the nodal p o i n t i ; , 287 differentiation yields: where ale 8 9 i jEM±, M . , iv^} J 9 i 9 j <f> k 11-10 (^-^k)2 + ^ - v 2 M. = { A fa ~ y±) ^ - yk> + <x± - V <\ - *j>]>' \ = { A T <y± - ^  - y k } + ( x j - x i ) ( xk - x j ) The system of equations over the entire domain R can be written: v UJ. e 1 He. L 8<j> e n 11-11 = 0 238 The left hand side of 11-11 can be expanded to form an nxn coefficient matrix j ^ . .j multiplied by an (n x 1) vector } . To set upjp^jj , we move through a l l the nodes, one at a time, and sum up a l l the element contributions (defined by , M_., M^ ) of which that node is a part. The matrix JP^^Jwill be sparse since nodes not connected through a common element with the pivotal node 9 ^  will have a zero coefficient. We then have the system of equations: P i j * * i * = 0 ifj=l» n 11-12 The boundary conditions are easily handled by the finite element method. If left unspecified, impermeable boundaries are automatically included in the integral equation (Remson et al. (1971), p. 305). Head values along a constant head boundary are directly inserted into { 9 ^ } . By numbering these nodes last, they can be partitioned from the matrix solution. Thus, eguation 11-12 becomes: P-', {9 J = " ( V i = l f t . J. II-13 j=l, ..., n where t is the number of unknown head values and {B ± } l l 1 1 / m m • f t j=t+l, n 289, For nodes i = t + 1 t o n a t which head values are s p e c i f i e d , Neuman e t a l (1970) s t a t e s the i n f l o w o r o u t f l o w across the element boundary can be c a l c u l a t e d from: - {Q±} P. . ID { c f , - } i = t + l , n J 11-14 j = l , n The approximate itdnimizing f u n c t i o n , and t h e r e f o r e the s o l u t i o n o f the boundary value problem, i s obtained by s o l v i n g equation 11-13 t o o b t a i n the nodal values o f 9 ^ . Pinder and Gray (1977) p r o v i d e a good review o f the techniques a v a i l a b l e t o s o l v e these systems o f equations. 290 Appendix I I I A fl o w c h a r t o f the computer program w r i t t e n t o c a r r y out a twci-ciimensional s t o c h a s t i c a n a l y s i s o f steady s t a t e groundwater fl o w i n a bounded domain i s given i n F i g u r e I I I - l . The parameter MC denotes the number of c o n d u c t i v i t y r e a l i z a t i o n s s p e c i f i e d f o r a g i v e n s i m u l a t i o n . READ IN AND ECHO DATA CALL WGT2 TO GENERATE WEIGHT MATRIX INVERT NEAREST NEIGHBOUR OPERATOR MATRIX MCCN = 1 CALL GENER TO GENERATE CONDUCTIVITY REALIZATION CALL ACF2 TO CALCULATE SAMPLE AOTCCORRELATION FCN. INSERT REALIZATION INTO FINITE ELEMENT GRID CALL FT,KM TO SOLVE FLOW EQUATION SAVE HEAD SOLUTION AT PREDEFINED NODES MCCN = MCCN + 1 YES MCCN < MC ^ t NO COMPUTE SUMMARY STATISTICS ON SAMPLE ACF COMPUTE SUMMARY STATISTICS ON HEAD DISTRIBUTION COMPUTE GRADIENT VARIABILITY AT PREDEFINED NODES I COMPUTE SUMMARY STATISTICS ON FLUX DISTRIBUTION WRITE OUT RESULTS F i g u r e I I I - l F l o w c hart f o r fe'o-dimensional s o l u t i o n Appendix IV Quadra Sands Data 293 HORIZONTAL LINE SAMPLE CONDUCTIVITY MEASURED IN CM/S SAMPLE CONDUCTIVITY POROSITY 1 0.0288 0.406 2 0.0396 0.423 3 0.0368 0.423 4 0.0361 0.408 5 0.0546 0.42 5 6 0. 03 58 0.391 7 0.0483 0.425 8 0. 0540 0.421 9 0.0507 0.409 10 0.0396 0.401 11 0.05 5 0 0.42 3 12 0.0612 0.431 13 0.0477 0.416 14 0.0506 0.425 15 0.0500 0.421 16 0.0522 0.415 17 0.0452 C.420 18 0.0509 0.412 19 0.049 4 C.4C7 20 0.0416 0.402 21 0.0421 0.396 22 0.C432 0.409 23 0.0534 0.420 24 0.0601 0.428 25 0.0476 0.414 SAMPLE CONDUCTIVITY POROSITY 26 0.04 56 0.407 27 0.0524 0.422 28 0.0588 0.427 29 0. 0494 0.426 30 0.0588 0.437 31 0.0499 0.421 32 0.0614 0.425 33 0.0677 0.438 34 0. 05 75 0.429 3 5 0.0701 0.436 36 0.0688 0.438 3 7 0.0652 0.431 38 0.05 80 0.426 39 0.0635 0.432 40 0. 0606 0.423 41 0 .0554 0.433 42 0.0477 0.432 43 0.04G7 0.412 44 0.0417 0.408 45 0.04 62 0.404 46 0.0642 C.440 47 0.05 82 0.432 48 0.0484 0.408 49 0.0570 0.424 50 0.0642 0.421 294 HORIZONTAL LINE SAMPLE CONT. SAMPLE CONDUCTIVITY POROS ITY 51 0.0684 0.434 52 0.0470 0.408 53 0.0554 0.414 54 0.04 66 0.403 55 0.055 3 0.413 56 0.0602 0.424 57 0.0532 0.414 58 0.0616 0.431 59 0.0600 0.435 60 0.0660 0.428 61 0.0527 0.402 62 0.0568 0.409 63 0.0636 0.410 64 0.0440 0.419 65 0.0550 0.423 66 0.0574 0.434 67 0.0586 0.435 68 0.0630 0.431 69 0.06 02 0.443 70 0.0659 0.430 71 0.0694 0.441 72 0.0662 0.438 73 0.0819 0.450 74 0.0706 0.437 75 0.0637 0.426 SAMPLE CONDUCTIVITY POROSITY 76 0.0744 0.428 77 0.0625 0.412 78 0.0784 0.425 79 0.0470 0.379 80 0.0320 0.366 81 0.03 82 0.386 82 0.05 33 0.417 83 0.0503 0.420 84 0.0392 0.410 85 0.0490 0.403 86 0.0612 0.436 87 0.0490 0.402 88 0.038G 0.390 89 0.0623 0.424 90 0.07 04 0.434 91 0.07 83 G. 409 92 0.0521 0.419 93 0.0476 0.405 94 0.0538 0.426 95 0.0566 0.418 96 0.G43 8 0.391 97 0.0514 0.401 98 0 . 05 72 0.413 99 0.0542 0.404 100 0.0612 0.397 295 VERTICAL LINE SAMPLE CONDUCTIVITY MEASURED IN CM/S SAMPLE CONDUCT IVITY POROSITY 1 0.0334 0.425 2 0.0534 0.42 0 3 0.0358 0.422 4 0.0344 0.459 5 0.0376 0.453 6 0.0476 0.466 7 0.0406 0.444 8 0.0538 0.441 9 0.0573 0.459 10 0.0118 0.439 11 0.0290 0.427 12 0.0493 0.414 13 0.0508 0.443 14 0.0774 0.446 15 0.0190 0.463 16 0.012 4 0.386 17 0.0622 0.432 18 0.0845 G.388 19 0.0626 0.424 20 0.0640 0.415 21 0. 0590 0.410 22 0.0158 0.391 23 0.0550 0.422 24 0.0514 0.448 25 0.0524 0.454 SAMPLE CONDUCTIVITY POROSITY 26 0.0773 0.442 27 0.0281 0.469 28 0.0341 0.452 29 0.03 84 0 .460 30 0.0432 0.412 31 0.0344 0.449 32 0.0380 0.420 33 0. 02 93 0.448 34 0.0382 0.413 35 0.0801 0.466 36 0.03 76 0.445 37 0.0582 0.413 3 8 0.0560 0.425 39 0.0658 0.435 40 0.0638 0.431 41 0.0703 0.430 42 0.0758 0.433 43 0.0386 0.397 44 0.0320 0.3 85 45 0.0574 0.410 46 0.0606 0.3 82 47 0.0560 0.325 48 0.0642 0 .396 49 0.0852 0.411 50 0.0823 0.358 296 VERTICAL LINE SAMPLE CONT. SAMPLE CONDUCTIVITY POROSITY 51 0.0859 0.348 52 0,0153 0.400 53 0.0542 0.391 54 0.0518 0.389 55 0.0824 0.409 56 0.0398 0.334 57 0.0432 0.409 58 0.0413 0.397 59 0.0438 0.431 60 0.0423 0. 431 61 0.0583 0.445 62 0-04 71 0.422 63 0, 05 00 0.422 64 0.0352 0.412 65 0.0642 0.383 66 0.0194 0.43 0 67 0.0294 0.442 68 0.0362 0.445 69 0.0003 0.447 70 0.0017 0.423 71 0.0450 0.43 7 72 0.0412 0.448 73 0.046 8 0.435 74 0.0200 0.404 75 0.0405 0.422 SAMPLE CONDUCT I V I TV POROSITY 76 0.0095 0.400 77 0 .03 58 0.437 78 0.0408 0.468 79 0.0262 0-448 80 0.0052 0.414 81 0.0171 0.416 82 0.0108 0.379 83 0.0420 0.404 84 0 . 03 9 8 0.440 85 0.0602 0,411 86 0.0148 0.412 87 0.0140 C.440 88 0-0370 0.418 89 0.0526 0.427 90 0.0317 0 .4 16 91 0.03 74 0.404 92 0.0887 0.420 93 0.07 54 0.423 94 0.0708 0.439 95 0.0095 0.419 96 0 .0051 0.383 97 0.0862 0.441 98 0.0650 0,411 99 0.0247 0,3-94 100 0.0106 0.364 RANDOM SAMPLE CONDUCTIVITY MEASURED IN CM/S SAMPLE 1 2 3 4 5 6 COND. RUN 1 0.0176 0.0472 0.0256 0.0632 0.0295 0.0327 COND. RUN 2 0.0222 0.0368 0.0275 0.0630 0.0248 0.0328 POROSITY 0.481 0.472 0.489 0.498 0.416 0.450 2- 1 2- 2 2- 3 2- 4 0.0794 0.0729 0.0236 0.0342 0.072 3 0.0738 0.0270 0.0334 0.463 0.449 0.409 0.458 3- 1 3- 2 3- 3 3- 4 3- 5 3- 6 0.0454 0.0749 0.0402 0.0464 0.0284 0.0340 0.0402 0.0863 0.0453 0.0470 0.0313 0.0340 0.430 0.408 0.443 0.426 0.477 0.438 4- I 4- 2 0.0348 0.0243 0.0498 0.0252 0.481 0.413 5- 1 5- 2 5- 3 5- 4 5- 5 5- 6 5- 7 0.0424 0.0980 0.0714 0.0643 0.0806 0.0265 0.0494 0.0474 0.0983 0.0769 0.0707 0.0772 0.0 274 0.0504 0.404 0.456 0.4 19 0.466 0.440 0.4 36 0.447 RANDOM SAMPLE» CONT SAMPLE 5- 8 5- 9 5-10 5-11 5-12 5-13 5-14 5-15 6- 1 6- 2 6- 3 6- 4 6- 5 6- 6 6- 7 6- 8 6- 9 6-10 6-11 6-12 6-13 6-14 6-15 6-16 6-17 COND. RUN 1 0.0808 0.0359 0.0344 0.0564 0.0442 0.0372 0.0642 0.0408 COND. RUN 2 0.0817 0.0364 0.0364 0.0620 0.0444 0.0400 0.0712 0.0414 POROSITY 0.4 29 0,418 0.407 0.414 0.415 0.438 0.448 0.409 0.0407 0.0394 0.0558 0.0235 0.0440 0.0512 0.0540 0.0582 0.0618 0.0517 0.032 5 0.0195 0.0604 0.0299 0.0227 0.0415 0.0396 0.0493 0.0383 0.0585 0.0249 0.0496 0.0498 0.0526 0.0 554 0.0567 0.0550 0.0344 0.0214 0.0609 0.0324 0.0215 0.0389 0.0376 0.444 0.474 0.425 0.443 0.415 0.438 0.4 23 0.4 06 0.429 0.442 0.394 0.451 0.434 0.404 0.418 0.4 20 0.410 RANDOM SAMPLE, CONT. SAMPLE CONO. RUN 1 COND. RUN 2 POROSITY 6-18 0.0425 0.0422 0.410 6-19 0.0509 0.0 498 0.381 6-20 0.0362 0.0428 0.425 6-21 0. 0600 0.0616 0.4 29 6-22 0.0417 0.0404 0.427 6-23 0.0460 0.0501 0.423 6-24 0.0313 0.0235 • 0.372 6-25 0.0626 0.0703 0.407 7- 1 0.0524 0.0508 0.419 7- 2 0.0273 0.0 304 0.3 83 7- 3 0.0387 0.0444 0.448 7- 4 0.0447 0.0472 0.422 7- 5 0.0494 0.0 522 0.436 7- 6 0.0351 0.0376 0.420 8- 1 0.0462 0.0480 0.441 8- 2 0.0124 0.0125 0.424 8- 3 0.03 84 0.0363 0.420 8- 4 0.0596 0.0530 0.396 8- 5 0. 003 9 0.0029 0.3 55 8- 6 0.0450 0.0395 0.381 8- 7 0.0240 0.0212 0.392 9- 1 0.0356 0.0388 0.422 9- 2 0.0591 0.0604 0.437 9- 3 0.0911 0.0875 0.404 9- 4 0.0456 0.0501 0.429 RANDOM SAMPLE, CONT. SAMPLE COND. RUN I COND. RUN 2 POROSITY 9- 5 0.0467 0.0497 0.416 9- 6 0.0396 0.0375 0.402 10- 1 0.0140 10- 2 0.0470 10- 3 0.0503 10- 4 0.0134 10- 5 0.0424 0.0169 0.421 0.0482 0.438 0.0560 0.413 0.0154 0.391 0.0483 0.422 11- 1 0.0108 11- 2 0.0450 11-3 0.0318 11- 4 0.0250 11- 5 0.0415 11- 6 0.0320 0.0116 0.381 0.0475 0.429 0.0350 0.428 0.0255 0.394 0.0448 0.425 0.0355 0.400 12- 1 12- 2 12- 3 12- 4 12- 5 12- 6 12- 7 12- 8 12- 9 12-10 12-11 12-12 0.0540 0.0662 0.0602 0.0240 0.0208 0.0583 0.0408 0. 013 0 0.0180 0.0592 0.0446 0.0236 0.0560 0.0768 0.0594 0.0404 0.0212 0.0 553 0.0378 0.0107 0.0290 0.0 594 0.0458 0.0210 0.414 0.434 0.412 0.399 0.396 0.405 0.4 35 0.408 0.389 0.406 0.425 0.390 TWO-DIMENSIONAL GRIDDED SAMPLE CONDUCTIVITY MEASURED IN CM/S COMPRESSIBILITY MEASURED IN CM*CM/S w CONDUCTIVITY POROSITY COMPRESS I8ILITY 0.0174 0.397 0.0025 0.0320 0.445 0.0026 0.0472 0.475 0.0025 0. 0310 0.445 0.0021 0.0362 0.468 0.0065 0.0620 0.434 0.0026 0.0697 0.438 0.0025 0.0746 0.45 3 0.0025 0.0494 0.408 0.0015 0. 03 66 0.426 0.0024 2 0.0542 0.426 0.0040 2 0.0500 0.399 0.0012 2 0.04 91 0.448 0.0030 2 0.0715 0.419 0.00 20 2 0.0517 0.423 0.0027 2 0.0515 0.404 0.0024 2 0.0635 0.423 0.0027 2 0.0912 0.475 0.0021 2 0.0770 0.434 0.0012 2 0.0786 0.423 0.0013 3 0. 0494 0.419 0.0020 3 0.06 24 0.43 8 0.0030 3 0.0766 0.445 0.0019 3 0.0648 0.430 0.0022 3 0.0526 0.430 0.0016 -DIMENSIONAL GRIDDED SAMPLE» CONT. CONDUCTIVITY POROSITY 3 0.0486 0.442 3 0.0482 0.434 3 0.0541 0.423 3 0.0625 0.438 3 0.0488 0.423 4 0.0446 0.426 4 0.0667 0.445 4 0.0664 0.4 34 4 0.0479 0.419 4 0.05 72 0.453 4 0.0589 0.419 4 0.0479 0.438 4 0.0531 0.430 4 0.0422 0.397 4 0.0571 0.423 5 0.0492 0.423 5 0.0442 0.442 5 0.0670 0.438 5 0.0640 0.445 5 0.0696 0.434 5 0.0576 0.423 5 0.0755 0.430 5 0.0646 0.438 5 0.0479 0.426 5 0.0684 0.457 COMPRESSIBI LITY 0.0020 0.0015 0.0021 0.0017 0.0022 0.0016 0.0028 0.0021 0.0014 0.0019 0.0017 0.0013 0.0016 0.0016 0.0018 0.0030 0.0025 0.0033 0.0020 0.0020 0.0018 0.0019 0.0018 0.0017 0.0023 TWO-DIMENSIONAL GRIDDED SAMPLE, CONT• ROW CONDUCTIVITY POROSITY COMPRESS I 81LITY 6 0. 04 60 0.438 0.0036 6 0.0341 0.411 0.0019 6 0.0367 0.434 0.0022 6 0.0308 0.415 0.0019 6 0.0348 0.426 0.0019 6 0.0262 0.411 0.0052 6 0.0421 0.430 0.0026 6 0.0394 0.426 0.0033 6 0.04 78 0.430 0.0023 6 0.0430 0.419 0.0019 7 0.02 76 0.438 0.0031 7 0.02 70 0.448 0.0028 7 0.0346 0.453 0.0031 7 0.0186 0.404 0.0034 7 0.0346 0.438 0.0054 7 0.0156 0.438 0.00 26 7 0.0440 0.43 8 0.0019 7 0.0376 0.434 0.0027 7 0.0533 0.445 0.0032 7 0.0306 0.442 0.0038 8 0.0695 0.4 38 0.0039 8 0.0591 0.438 0.00 25 8 0.0825 0.464 0.0026 8 0.0575 0.399 0.0021 8 0.0809 0.438 0.0024 TWO-DIMENSIONAL GRIDDED SAMPLE, CONT. ROW CONDUCTIVITY POROSITY COMPRESSIBILITY 8 0. 0640 0.445 0.00 21 8 0.0733 0.430 0.0023 8 0.04 86 0.423 0.0021 8 0.0630 0.4 77 0.0026 8 0.0604 0.442 0.0024 9 0.0437 0.404 0.0039 9 0.0597 0.438 0.0024 9 0.0210 0.434 0.0029 9 0.0486 0.453 0.00 25 9 0.02 86 0.4 80 0.0041 9 0.0757 0.448 0.0036 9 0.0491 0.423 0.0026 9 0.0430 0.453 0.0037 9 0.0546 0.419 0.0097 9 0.0360 0.442 0 .0054 10 0.0642 0.430 0.0054 10 0.0658 0.438 0.0033 10 0.0676 0.445 0.0022 10 0.0616 0.485 0.0038 10 0.0392 0.430 0.0056 10 0,0490 0.434 0.00 22 10 0.07 80 0.472 0.0035 10 0.0500 0.453 0.0027 10 0.0446 0.453 0.0034 10 0.0540 0.445 0.0022 305 PERCENT FINER GRAIN SIZE FRACTION HORIZONTAL LINE SAMPLE GRAIN SIZE IN MM PLE 05 1 0. 155 2 0-150 3 0-155 4 0. 155 5 0.150 6 0.155 7 0.155 8 0.155 9 0. 160 10 0.155 11 0.165 12 0.175 13 0.165 14 0.155 15 0.155 16 0.160 17 0.155 18 0.155 19 0.155 20 0.155 21 0.165 22 0.165 23 0.165 24 0.165 25 0. 16 5 DiO D16 0.165 0.185 0.165 0.185 0.165 0.185 0.170 0.19C 0.165 0.18 5 0.180 0.205 0.175 0.205 0.180 0.210 0.180 0.210 0.175 0.200 0.195 0.220 0.215 0.230 0.190 0.220 0.180 0.215 0.180 0.210 0.185 0.215 0.170 0.190 0.180 0.205 0.175 0,200 0.180 0.205 0.200 0.225 0.190 0.220 0.195 0.220 0.195 0.220 0.190 C.215 050 D60 0.265 0.290 0.265 0.290 0.265 0.290 0.270 0.29 5 0.26 5 0.28 5 0.290 0.32 5 0.28 5 0.320 0.295 0.325 0.305 0.335 0.28 5 0.315 0.320 0.350 0.350 0.385 0.320 0.355 0.300 0.333 0.280 0.310 0.305 0.340 0.270 0.295 0.300 0.330 0.295 0.330 0.340 0.370 0.330 0.36 0 0.320 0.350 0.320 0.350 0.320 0.350 0.310 0.340 084 DS5 0.390 0.520 0.385 0.515 0.380 0.505 0.390 0.530 0.380 0.47 5 0.41C 0.555 0.405 0.550 0.420 0.565 0.445 0.580 0.415 0.565 0.455 0.590 0.525 0.475 C. 590 0.420 0.56 5 0.405 0.555 0.460 0.590 0.385 0.490 0.415 0.550 0.415 0.570 0.520 0.805 0.495 0.690 0.470 0.590 0.455 0.590 0.455 0.585 0.430 0.560 HORIZONTAL LINE SAMPLE CONT. SAMPLE D5 26 0.160 27 0.160 28 0.165 29 0.165 30 0.165 31 0.165 32 0.170 33 0.175 34 0.175 35 0.185 36 0.175 37 0.175 38 0.170 39 0.16 0 40 0.165 41 0.155 42 0.150 43 0.155 44 0.160 45 0.160 46 0.160 4 7 0.170 48 0.165 49 0.16 5 50 0.185 DIO D16 0.190 0.220 0.180 0.205 0.190 0.220 0.190 0.220 0.195 0.22 0 0.195 0.220 0.200 0.225 0.215 0.235 0.215 0.235 0.220 0.240 0.215 0.230 0.210 0.230 0.200 0.225 0.190 0.215 0. 195 0.220 0.175 0.200 0.165 0.180 0.170 0.195 0.180 0.215 0.180 0.210 0.185 0.210 0.205 0.230 0.200 0.225 0.200 0.225 0.220 0.240 D 50 D60 0.315 0.340 0. 305 0.33 5 0.315 0.345 0.310 0.340 0.325 0.360 0.330 0.360 0.335 0.37C 0.355 0.395 0.360 0.40 0 0. 355 0.390 0.350 0.380 0.345 0.380 0.330 0.360 0.330 0.36 5 0.330 0.360 0.300 0.330 0.270 0.300 0.290 0.325 0.305 0.335 0.300 0.330 0.305 0.340 0.350 0.385 0.335 0.370 0.345 0.38 0 0.36 5 0.40 0 084 095 0.435 0.570 0.435 0.590 0.450 0.575 0.445 0.570 0.480 0.840 0.505 0.670 0.510 0.680 0.535 0.740 0.540 0.750 0.530 0.720 0.525 0.760 0.530 0.840 0.490 0.690 0.500 0.710 0.480 0.590 0.435 0.575 0.405 0.550 0.415 0.550 0.430 0.565 0.425 0.560 0.445 0.570 0.530 0.750 0.505 0.700 0.525 0.740 0.545 0.790 HORIZONTAL LINE SAMPLE CONT SAMPLE 05 51 0.165 52 0.160 53 0.165 54 0.165 55 0.160 56 0.170 5 7 0. 160 58 0.160 59 0.160 60 0.180 61 0.180 62 0.195 63 0.170 64 0.155 65 0.165 66 0.165 67 0.160 68 0.165 69 0.165 70 0.16 5 71 0.165 72 0.165 73 0.175 74 0.170 75 0.170 010 D16 0.200 0.230 0.190 0.220 0.205 0.225 0. 190 0.215 0.185 0.215 0.205 0.225 0.190 0.220 0.190 0.220 0.190 0.220 0.220 0.245 0.220 0.250 0.225 0.255 0.215 0.245 0.175 0.205 0.205 C.225 0.200 0.225 0.190 0.215 0.200 0.225 0.195 0.220 0.210 0.23C 0.200 0.220 0.200 0.225 0.215 0.240 0.210 0.235 0.205 0.2 30 0 50 060 0.370 0.415 0.325 0.360 0.340 0.3 70 0.32 0 0.35 0 0.325 0.360 0.335 0.36 5 0.32 5 0.36 0 0.350 0.390 0.340 0.375 0. 38 5 0.43 5 0.385 0.43C 0.400 0.45 0 0.410 0.470 0.305 0.340 0.335 0.365 0.330 0.36 0 0.320 0.355 0. 340 0.375 0.330 0.36 5 0.345 0.380 0.330 0.360 0.335 0.370 0.365 0.405 0.350 0.390 0.340 0.375 D84 095 0.580 0.840 0.480 0.620 0.500 0.660 0.475 0.610 0.5 00 0.700 0.50C 0.700 0.495 0.700 0.53 5 0.730 0.505 0.610 0.570 0.840 0.570 0.840 0.630 0.720 0.455 C.590 0.495 0.640 C.480 0.615 0.460 0.575 0.510 0.650 0.500 0.720 0.510 0.710 0.485 0.620 0.500 0.640 0.555 0.820 0.540 0.760 0.525 0.820 308 HORIZONTAL L INE SAMPLE CONT. SAMPLE 05 76 0.175 77 0.180 78 0.185 79 0.155 80 0.130 81 0.120 82 0.150 83 0.145 84 0.130 85 0.155 86 0.150 87 0.160 88 0.120 89 0.150 90 0.165 91 0.155 92 0.155 93 0.160 94 0.160 95 0.165 96 0.160 97 0.155 98 0.150 99 0.155 100 0.160 010 016 0.215 0.235 0.225 0.245 0.220 0.250 0. 175 0.20 5 0.160 0.175 0.155 0.165 0. 170 0.190 0.165 0.185 0.160 0.175 0.215 0.23 5 0.165 0.185 0.190 0.220 0.155 0.175 0.175 0.210 0.210 0.235 0.180 0.210 0.180 0.210 0.180 0.2 10 0.185 0.215 0.195 0.220 0.180 0.210 0.180 0.215 0.180 0.210 0.180 0.215 0.195 0.225 0 50 060 0.360 0.400 0.380 0.42 0 0.390 0.440 0.355 0.410 0.275 0.310 0.250 0.280 0.295 0.335 0.29 5 0.33 0 0.275 0.310 0.355 0.395 0.300 0.340 0.355 0.395 0. 285 0.320 0.325 0.360 0.355 0.400 0.300 0.33 5 0.305 0.33 5 0. 295 0.330 0.310 0.340 0.320 0.350 0.300 0.33 5 0.320 0.355 0.320 0.350 0.330 0.36 5 0.345 0.380 084 D95 0.545 0.770 G.560 0.840 0.610 0.700 0.445 0.700 0.395 0.570 0.495 0.840 0.460 0.650 0.420 0.570 0.550 0.740 0.475 0.650 0.550 0,810 0.450 0.590 0.500 0.695 0.565 0.43 5 0.590 0.435 0.570 0.43 0 0.580 0.42 5 0.560 0.460 0.580 0.435 0.590 0.480 0.600 0.485 0.620 0.5 GO 0.700 0.530 0.770 PERCENT FINER GRAIN SIZE FRACTION VERTICAL LINE SAMPLE GRAIN SIZE IN MM SAMPLE 05 010 D16 050 D60 084 095 1 0,129 0.150 G.160 0.227 0.250 0.348 0.500 2 0.165 0.195 0.220 0.320 0.350 0.455 0.590 3 0.120 0.154 0.170 0.260 0.289 0.410 0.540 4 0.116 0.138 0.207 0.230 0.310 0.440 5 0.120 0.150 0.249 0.283 C.459 0.740 6 0.129 0.158 0.307 0.380 0.720 7 0.113 0.151 0.175 0.310 0.370 0.619 8 0.145 0.170 0.202 0.370 0.450 9 0.125 0. 159 0.182 0.320 0.375 0.619 10 0.105 0.123 0.225 0.260 0.435 11 0.122 0.150 0.250 0.291 0.528 12 0. 155 0.186 0.220 0.414 0.488 0.820 13 0.135 0.163 0.188 0.320 0.370 0.560 14 0.167 0.212 0.240 0.411 0.470 0.750 15 0.079 0. 1C6 0. 182 0.209 0.385 0.660 16 0.141 0.168 0.334 0.399 0.630 17 0.175 0.220 0.248 0.415 0.470 0.680 18 0.193 0.2 3 8 0 . 28 3 0 . 562 0 . 732 19 0.168 0.210 0.231 0.368 0.415 0.575 G.840 20 0.172 0.212 0.280 0.370 0.420 0.588 21 0.154 0.187 0.220 0.367 0.421 0.648 22 0.156 0.215 0.264 0.500 0.578 23 0.127 0.168 0.213 0.390 0.450 0.642 24 0.127 0.160 0.182 0.336 0.390 0.580 25 0.105 0.125 0.149 0,203 0.224 0.280 0.360 310 VERTICAL LINE SAMPLE CONT. SAMPLE 05 26 0.175 27 28 29 30 0 .14 5 31 0.105 32 0.125 33 34 0.127 35 0. 160 36 0.126 37 0.162 3 8 0.161 39 0.162 40 0. 16 2 41 0.165 42 0.168 43 0.151 44 0.155 45 0.156 46 0.166 47 0.216 48 0.187 49 0.230 50 0.208 DIO D16 0.222 0.250 0.114 0.133 0.120 0.150 0.130 0.156 0.170 0.200 0.133 0.158 0. 158 0.182 0.135 0.164 0.160 0.190 0.197 0.22 9 0.140 0.162 0.190 0.222 0.195 0.225 0.200 0.230 0.200 0.230 0.214 0.240 0.214 0.240 0. 167 0.190 0.172 0.195 0.184 0.219 0.218 0.260 0.280 0.342 0.242 0.280 0.282 0.330 0.257 0.320 050 D60 0.390 0.441 0.230 0.271 0.252 0.291 0.280 0.332 0.335 0.380 0.254 0.290 0.330 0.390 0.425 0.52 5 0.400 0.480 0.382 0.439 0.280 0.355 0.360 0.40 8 0.360 0.402 0.36 8 0.410 0.371 0.42 0 0.394 0.440 0.385 0.430 0.320 0.378 0.320 0.370 0.381 0.440 0.470 0.53 5 4.250 0.510 0.590 0. 550 0.630 0.645 0.835 084 095 0.678 0.530 0.836 G.540 0.590 0.555 0.475 0.840 0.770 0.550 0.560 0.800 0.570 0.740 0.545 0.740 0.560 0.800 0.565 0.760 0.558 C.760 0.590 0.560 0.720 VERTICAL LINE SAMPLE CONT. SAMPLE D5 DIO D16 D50 D60 D84 C95 51 0.196 0.250 0.306 0.570 0.675 52 0.107 0.130 0.152 0.206 0.230 0.315 0.570 53 0.156 0. 180 0.215 0.437 0.520 54 0.180 0.230 0.272 0.490 0.560 55 0.230 0.285 0.33 5 0.565 0.645 56 0.177 0.220 0.250 0.565 0.740 57 0.152 0.180 0.213 0.290 0.330 0.461 0.580 58 0.152 0. 183 0.217 0.345 0.400 0.570 0.800 59 0.150 0.162 0.180 0.261 0.285 0.385 0.500 60 0.150 0.170 0.200 0.307 0.347 0.490 0.660 61 0.155 0.180 0.214 0.320 0.353 0-477 0.580 62 0.153 0.176 0.209 0.318 0.356 0.505 0.670 63 0.155 0.180 0.211 0.311 0.345 0.460 0.570 64 0.126 0.156 0.114 0.285 0.330 0.490 0.690 65 0.170 0.215 0.250 0.505 0.620 66 0.107 0.122 0.142 0.190 0.206 0.273 0.380 67 0.112 0.130 0.151 0.201 0.226 0.332 0.560 68 0.122 0.150 0.160 0.231 0.255 0.366 0.620 69 0.025 0. 131 0.157 0.209 0.290 70 0.072 0.220 0.262 0.436 0.720 71 0.107 0.152 0.177 0.317 0.371 0.645 72 0.137 0.160 0.178 0.275 0.316 0.565 73 0.152 0.170 0.196 0.274 0.301 0.410 0.565 74 0.118 0.150 0.162 0.236 0.255 0.320 0.402 75 0.135 0.171 0.210 0.341 0.385 0.535 0.710 VERTICAL LINE SAMPLE CONT SAMPLE 05 76 0.107 77 0.122 78 79 0.112 80 81 0.105 82 0.116 83 0.149 84 0.145 85 0.200 86 87 88 0.152 89 0.157 90 0.155 91 0. 138 92 0.229 93 0 .180 94 0.166 95 0,110 96 0.108 97 0.220 98 0.220 99 0.115 100 0.114 D I O 016 0.151 0.170 0.159 0.180 0.143 0.168 0.151 0.169 0.145 0,165 0.141 0.162 0.152 0.175 0.175 0.211 0.160 0.180 0.250 0.285 0.129 0.157 0.116 0.141 0.171 0.198 0.190 0.220 0.190 0,221 0.170 0.2CC 0.300 0,335 0.229 0.267 0.213 0.240 0,149 0.168 0.132 0.154 0.270 0.313 0.262 0.3C8 0.150 0.170 0.140 0.163 D50 060 0.265 0.294 0.280 0.315 0-323 0.391 0.280 0.319 0,265 0.294 0.268 0-301 0.320 0.371 0.3 70 0.42 0 0.278 0.312 0.440 0.486 0.261 0.292 0.216 0,240 0.294 0.331 0.340 0.37 5 0.33 5 0.36 2 0.330 0.370 0. 530 0.585 0.480 0.550 0.385 0.434 0.282 0.328 0.251 0.297 0. 480 0.52 5 0.446 0.485 0.310 0.350 0.277 0.314 084 095 0-440 0.700 0.43 8 0.700 0.750 0.485 C.770 0.401 0.585 0.457 0.725 0.555 0.570 0.830 0.414 0.550 0.663 0.395 0.525 0.329 0.440 0.470 0.610 0.510 0.670 0.470 0.57 5 0.535 0.835 0.585 0.5CC 0.835 0.73 0 G.590 0.800 0.494 0.660 0.430 0.570 PERCENT FINER GRAIN SIZE FRACTION RANDOM SAMPLE GRAIN SIZE IN MM SAMPLE D5 1- 1 1- 2 l - 3 l - 4 0, 115 1- 5 0.125 1- 6 2- 1 0.150 2- 2 0.165 2- 3 0.105 2- 4 0.105 3- 1 0.130 3- 2 0.155 3- 3 0.110 3- 4 0.150 3- 5 3- 6 0. 100 4- 1 4- 2 5- 1 0.150 5- 2 0.175 5- 3 0.175 5- 4 0.115 5- 5 0.140 5- 6 0.155 5- 7 0.115 010 016 0.090 0.105 0.120 0.150 0.105 0.130 0.150 0.16 0 0.165 0.185 0.115 0.145 0.180 0.215 0.215 0.240 0.130 0.155 0.125 0.150 C.160 0.175 0.200 0.240 0.150 0.165 0.165 0.190 0.110 0.130 C.160 0.110 0.135 0.115 0.140 0.175 0.205 0.225 0.265 0.230 0.285 0.160 0.185 0.165 0.190 0.200 0.230 0.150 0.165 D50 D60 0. 185 0.210 0.240 0.270 0.215 0.240 0.22 0 0.24 5 0.265 0.290 0.260 0.310 0.340 0.38G 0.380 0.425 0.250 0.285 0.205 0.225 0.255 0.275 0.475 0.530 0.240 0.260 0.270 0.295 0.190 0.225 0.265 0,300 0.210 0.200 0.330 0.365 0.435 0.485 0.465 0.520 0.325 0.36 5 0.290 0.325 0.375 0.415 0.250 0.280 D84 095 0.325 0.525 0.430 0.700 0.325 0.420 0.310 0.400 0.390 0.540 0.560 G. 560 0.840 0.570 0.805 0.440 0.710 0.2 90 0.390 0.365 0.470 1.370 3.600 0.355 0.470 0,375 0.415 0.415 0.470 0.730 0.530 0.840 0.740 0.770 0.510 0.835 0.405 0.530 0.555 0.815 0,410 0,620 0.245 0.385 C.590 0.220 0.275 0.340 314 RANDOM SAMPLE CONT. SAMPLE D5 5- 8 0.120 5- 9 0.165 5-10 0. 145 5-11 0.125 5-12 0.190 5-13 0.135 5-14 0.115 5-15 0.150 6- 1 0.145 6- 2 0.115 6- 3 0.165 6- 4 6- 5 0.155 6- 6 0.155 6- 7 0.130 6- 8 0.150 6- 9 0.165 6-10 0.125 6- 11 0.140 6-12 6-13 0.160 6-14 0.120 6-15 0.115 6-16 0. 125 6-17 0.135 DIO D16 0.155 0.175 0.205 0.245 0.165 0.180 0.155 0.175 0.255 0.275 0.160 0.180 0.150 0.160 0.195 0.235 0.165 0.185 0.145 0.160 0.195 0.220 0.115 0.135 0.170 0.195 0.165 0.185 0.165 0.185 0. 175 0.205 0.195 0.220 0.160 0.185 0.165 0.190 0.078 0.105 0. 190 0.220 0.155 0.170 0.135 0.155 0.160 0.180 0.165 0.195 D50 060 0.295 0.33 0 0.490 0.585 0.260 0.285 0.275 0.310 0.400 0.445 0.265 0.290 0.230 0.250 0.375 0.415 0.330 0.405 0.250 0.275 0. 330 0.35 5 0.185 0.200 0.355 0.43 5 0.260 0.280 0.325 0.370 0.320 0.35 0 0.330 0.365 0.325 0.375 0.275 0.305 0.200 0.245 0.325 0.355 0.275 0.315 0.200 0.220 0.300 0.335 0. 325 0.3 70 084 C95 0.435 0.560 1.250 2.650 0.375 0.475 0.445 0.730 0.570 0.800 0.375 0.470 0.315 0.395 0.535 0.670 0.830 0.375 0.495 0.42 0 0.680 0.260 0.300 0.820 0.36 5 0.420 0.580 0.465 0.590 0.515 0.740 0.575 0.385 0.465 0.400 0.570 0.485 0,650 0.510 0.310 0.495 0.455 0.740 C. 590 RANDOM SAMPLE CONT SAMPLE D5 6-18 0.15 0 6-19 0.170 6-20 0.125 6-21 0. 170 6-22 0.125 6-23 0.125 6-24 0. 120 6-25 0.175 7- 1 0. 170 7- 2 0.115 7- 3 0.125 7- 4 0.140 7- 5 0.150 7- 6 0.105 8- 1 0.105 8- 2 8- 3 0.115 8- 4 0.170 8- 5 8- 6 0.215 8- 7 0. 105 9- 1 0.150 9- 2 0.165 9- 3 0.210 9- 4 0.135 DiO 016 0.170 0.190 0.215 0.240 0.155 0.170 0.215 0.240 0.155 0.170 0.165 0.190 0.155 0.180 0.220 0.245 0.215 0.225 0.150 0.165 0.160 0.185 0. 165 0. 190 0.165 0.190 0.140 0.165 0.150 0.170 0.110 0.130 0.155 0.170 0.215 0.250 0.110 0.155 0.255 0.3G5 0.170 0.215 0.165 0.18 5 0.200 0.225 0.255 0.310 0.165 0.190 D50 D60 0.275 0.300 0.375 0.415 0.260 0.290 0.360 0.390 0.250 0.270 0.305 0.340 0.325 0.380 0.385 0.435 0.315 0.34 0 0.250 0.275 0.290 0.320 0. 295 0.335 0.280 0.315 0.275 0.315 0.285 0.330 0.225 0.260 0.255 0.280 0.410 0.455 0.305 0.37 0 0.430 0.470 0.375 0.425 0.270 0.295 0.330 0.36 0 0.600 0.770 0. 285 0.315 084 095 0.385 0.490 0.555 0.800 0.530 0.510 0.585 0.365 0.465 0.485 0.800 0.630 0.625 0.405 0.515 0.420 0.465 0.395 0.495 0.455 0.580 0.405 0.570 0.465 0.650 0.520 0.395 0.630 0.380 0.525 0.575 0.800 0.700 0.575 0.780 0.650 0.390 0.500 0.480 0.660 2.350 4.000 0.405 0.545 RANDOM SAMPLE CONT. SAMPLE D5 9- 5 0. 155 9- 6 0.155 10- 1 10- 2 0.160 10- 3 0.195 10- 4 0.105 10- 5 0.155 11- 1 11- 2 0.150 11- 3 0.105 11- 4 0.125 11- 5 0.150 11- 6 0. 145 12- 1 0.180 12- 2 0. 180 12- 3 0.170 12- 4 0.155 12- 5 12- 6 0.155 12- 7 0. 150 12- 8 0.105 12- 9 0.145 12-10 0. 165 12-11 0.155 12-12 0.150 DIO 016 0. 180 0.210 0.180 0.210 0.120 0.155 0.200 0.230 0.230 0.270 0.150 0.170 0.180 0.215 0.105 0.120 0.165 0.185 0.130 0.155 0.155 0.175 0.165 0.185 0.160 0.180 0.220 0.245 0.220 0.250 0.210 0.235 0.200 0.225 0.125 0.155 0.190 0.230 0. 165 0.190 0.130 0.155 0.165 0.185 0.215 0.250 0.175 0.205 0.165 0.190 D50 060 0.285 0.310 0.305 0.34 5 0.255 0.300 0.360 0.400 0.390 0.430 0.305 0.360 0.325 0.355 0. 180 0.20 0 0.270 0.300 0.220 0.240 0.255 0.280 0.285 0.320 0.270 0.300 0.355 0.390 0.400 0.450 0.36 5 0.40 0 0.350 0.385 0.250 0.280 0.445 0.505 0.270 0.295 0.225 0.250 0.280 0.310 0.405 0.465 0.280 0.305 0.275 0.305 084 D95 0.430 0.510 0.505 0. 540 0. 740 0.575 0.580 0.460 0.575 0.27C 0.370 0.390 0.500 0.2 95 0.370 0.3 75 0.48 5 0.440 0.600 0.390 0.500 0.520 0.680 0.705 0.540 0.710 0.510 0.610 0.395 0.540 0.780 0.395 0.535 0.320 0.405 0.405 0.5 35 0.730 0.39C 0.490 0.405 0.550 317 APPENDIX V A one-way a n a l y s i s o f v a r i a n c e model i s used t o compare the r e p l i c a t i o n v a r i a n c e i n c o n d u c t i v i t y values measured a t each sample p o i n t w i t h the v a r i a n c e measured over the s e t o f sample p o i n t s . L e t k^ _. be a c o n d u c t i v i t y measurement a t the i t h sample p o i n t , j t h r e p l i c a t e . L e t the number o f r e p l i c a t e measurements on each sample be a, the number o f sample p o i n t s be n. The o v e r a l l mean c o n d u c t i v i t y i s c a l c u l a t e d : a n k = — ^ k, . V - l 3 1 1 j = l i = l X J The t o t a l sum o f squares i n the data i s then given as: a 2 SS = E 1 ( k ± i - K) V-2 m XJ j = l i = l The sum o f squares f o r the v a r i a t i o n among the sample p o i n t s i s given as: a _ 2 SS, = E : E (k.. - k V-3 A j = 1 i = l ^ * J where l c i ^ i s the average c o n d u c t i v i t y o f the j t h s e t o f r e p l i c a t e s , formed over a l l the sample p o i n t s . The sum o f sguare due t o the r e p l i c a t i o n of measurements i s given as: _ 2 SS„ = n E (k°. - k) V-4 j = l ' 3 318 Davis (1973) provides equations f o r these sums of squares which are more s u i t a b l e f o r computer c a l c u l a t i o n . The mean squares are formed from the sum o f squares by d i v i d i n g the degrees o f freedom f o r each estimate. Thus, the mean square f o r v a r i a t i o n among the sample p o i n t s i s : MS, = SS / n - l V-5 A A The mean square f o r the r e p l i c a t i o n v a r i a n c e i s : MS R = SS^/ n ( a - l ) V-6 The r a t i o o f MS^/MS^ w i l l i n d i c a t e the r e l a t i v e importance o f the va r i a n c e i n the r e p l i c a t e s t o the v a r i a n c e over the sample g r i d . The t o t a l v a r i a n c e i n both the r e p l i c a t e s and over the sample g r i d i s estimated by: MS T = S S ^ an-1 V-7 APPENDIX VT Glossary 320 APPENDIX VI Listed below are definitions of various statistical terms used through the text. A reader not familiar with probabilistic or statistical concepts may find i t useful to refer to this glossary as the development in the body of the text proceeds. In keeping with its intended purpose, descriptive definitions are given without the added rigor of the mathematics of probability. Those readers desiring more precise definitions are referred to any standard text on probability and statistics, such as Mood, Graybill and Boes (1974). 1) random variable: Any numerical quantity whose value is determined by the outcome of a random experiment. 2) probability: Let X be a random variable. Probability is a measure of the relative frequency with which the outcome of a random variable assumes a given value, say c. A simple definition can then be given: i \ - number of experiments in which x=c P ~ total number of experiments 3) probability density function: Consider some random variable X and let X = •'{xlf ... x^.} be 321 the s e t o f d i f f e r e n t values t h a t x can take on. The f u n c t i o n which a s s o c i a t e s w i t h each element x.eX , 1 the p r o b a b i l i t y p(x^) t h a t x=x^ i s c a l l e d the p r o b a b i l i t y f u n c t i o n f o r the random v a r i a b l e X. I f x can take on every p o s s i b l e v a l u e i n some i n t e r v a l , then X i s c a l l e d a continuous random v a r i a b l e and p(x^) i s a p r o b a b i l i t y d e n s i t y f u n c t i o n . 4) j o i n t p r o b a b i l i t y : Consider 2 random v a r i a b l e s X and Y which both can take on values ranging from 0 t o 1. L e t c and d be f i x e d values i n t h i s range. Then the j o i n t p r o b a b i l i t y o f X and Y can be d e f i n e d : . . _ number o f experiments i n which x=c and y?d pix=c, y~ - t o t a l number o f experiments A j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n p(x^, y^) a s s o c i a t e s a p r o b a b i l i t y w i t h the outcome t h a t x takes on the valu e x^ and y takes on the valu e y^, f o r the e n t i r e range o f p o s s i b l e values x^, y^. 5) c o n d i t i o n a l p r o b a b i l i t y : Consider the j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n p(x^, y^) o f 2 random v a r i a b l e s X and Y. The e o n i i i t i o n a l v p r o b a b i l i t y o f x, g i v e n y, d e f i n e s the p r o b a b i l i t y t h a t x takes on 322 the value x\ , given t h a t y has already assumed the valu e y^. This p r o b a b i l i t y i s denoted: •<* I y) - ^ P , A I " " :><•,•> where p(y) i s the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r y alone. 6) independence: Two random v a r i a b l e s X and Y are independent i f p(x | y) = p(x) That i s , knowing the valu e which y assumes does not i n any way determine the p r o b a b i l i t y w i t h which the random v a r i a b l e X assumes v a l u e s . 7) e x p e c t a t i o n : Consider a random v a r i a b l e X which can assume any v a l u e i n the range from a t o b. L e t i t s p r o b a b i l i t y d e n s i t y f u n c t i o n be p ( x ) . The mean o r average val u e o f X i s the expected valu e o f X, d e f i n e d as: X = j x p(x)dx = y x The expected val u e o f a random v a r i a b l e i s a measure o f the c e n t r a l l o c a t i o n o f i t s p r o b a b i l i t y d e n s i t y f u n c t i o n . 323 8) moments: Consider a random v a r i a b l e X. Then the expected value o f X3" i s the r t h moment o f the random v a r i a b l e . The f i r s t moment o f X i s the mean. The second moment about the mean (E j j X - y x ) 2 J ) i s equal t o the v a r i a n c e . The f i r s t j o i n t moment between 2 random v a r i a b l e s X and Y, taken about t h e i r means (E £ (X - y ^ ) (Y - Vy)J" ) d e f i n e s the covariance. 9) skewness: The skewness o f a p r o b a b i l i t y d e n s i t y f u n c t i o n i s a measure o f i t s asymnetry about i t s c e n t r a l maximum. I f i t i s . symmetrical, the skew i s zero. A p r o b a b i l i t y d e n s i t y f u n c t i o n w i t h a r i g h t skew has a longer t a i l t o the r i g h t o f i t s c e n t r a l maximum than t o i t s l e f t . I f the reverse i s t r u e , the p r o b a b i l i t y d e n s i t y f u n c t i o n i s l e f t skewed. 10) k u r t o s i s : The k u r t o s i s o f a p r o b a b i l i t y d e n s i t y f u n c t i o n i s a measure o f i t s peakedness (or f l a t n e s s ) about i t s c e n t r a l maximum. A p r o b a b i l i t y d e n s i t y f u n c t i o n w i t h a r e l a t i v e l y h i g h peak i s c a l l e d l e p t o k u r t i c . 324 11) normal distribution: Consider a continuous random variable X with a mean y and standard deviation a • The probability density function for X is normally distributed i f i t is of the form: - (x-y) 2 P(x) = e 2a 2 O-V|2TT . 12) lognormal distribution: Consider the transformation Y = ln X on the random variable X. If the new random variable Y is normally distributed, X is said to be lognormally distributed. < 13) realization: The outcome of a single random experiment. A realization may be defined on a single random variable (x) or on a set of joint random variables (x, y...). 14) ensemble: The ensemble is defined by the set of a l l possible realizations of a random variable or set of joint random variables. 15 random function: In definitions 13 and 14, reference was made to a set of 325 joint random variables. If this set has an ordered position in some coordinate system (such as time or space), then the set of random variables is referred to as a random function. 16) space law: The space law of a random function defines the joint probability density function between a set of random variables distributed in a spatial domain. 17) stochastic process: A phenomena that evolves in time and/or space according to probabilistic laws. A stochastic process model describes, in mathematical terms, our abstraction of that probability law. 18) stationarity: A stochastic process is stationary i f the same probability law holds at every point in space. The less restrictive assumption of second order stationarity requires that the stochastic process has the same expected value at each point and that the covariance between the random variables at any two points must depend only upon the vector separating them and not their absolute position. 

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