M O D E L I N G E L A S T I C W A V E V E L O C I T I E S I N P O R O U S M E D I A : F R E Q U E N C Y - D E P E N D E N T E F F E C T S O F H E T E R O G E N E I T Y A T T H E P O R E - A N D P A T C H - S C A L E By Richard Taylor B. Sc. (Hons. Physics) University of British Columbia A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F E A R T H A N D O C E A N S C I E N C E S and I N S T I T U T E O F A P P L I E D M A T H E M A T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1999 © Richard Taylor, 1999 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Earth & Ocean Sciences The University of British Columbia Vancouver, Canada Date: Abstract The prediction of relationships between elastic wave velocities in a porous medium and the properties of the fluid and solid constituents therein is a longstanding problem in geophysical exploration. Previous authors have shown that such relationships depend on both the wave frequency and degree of heterogeneity present. This frequency depen-dence arises via the state of relaxation of the pore fluids. At sufficiently low frequencies, deformation-induced flow leads to equilibration of the fluid pressure, and the pore fluids are said to be in a "relaxed" state. At sufficiently high frequencies, there is insufficient time for equilibration to occur, and the fluids are said to be "unrelaxed". Current models of elastic wave velocities in porous media are, for the most part, confined to either the relaxed limit (e.g., the poroelastic Biot-Gassmann theory) or the unrelaxed limit (e.g., inclusion-based effective medium theory). In this thesis we incor-porate an explicit description of the relaxation mechanism into inclusion-based effective medium theory, so as to extend the theory toward the relaxed limit. Analysis of the mechanisms of relaxation leads to a description of the effective elastic behavior of the porous medium in terms of effective complex elastic moduli of the medium's constituents. Previous authors have identified two distinct scales of fluid distribution heterogene-ity: the pore scale and the patch scale. Accordingly, we treat these scales separately, describing relaxation in terms of Poiseuille flow at the pore scale, and Darcy's law at the patch scale. The results of our analyses are effective medium theories that provide a consistent approach to the prediction of elastic wave velocities, as well as attenuation due to the relaxation mechanism, over a broad range of frequencies and length scales of heterogeneity. In particular, our model is applicable to the regime where the pore fluids are in a state of relaxation intermediate between the completely relaxed and completely unrelaxed end members. ii Table of Contents Abstract ii List of Tables v List of Figures vi Acknowledgement ix 1 Introduction 1 2 Heterogeneity and Relaxation at the Pore Scale 8 2.1 Introduction 8 2.2 Introduction to Inclusion-Based Effective Medium Theory 13 2.3 Defining Effective Elastic Moduli for Inclusions 17 2.4 A Simple Pore-to-Pore Relaxation Model 20 2.4.1 Model Development 21 2.4.2 Modeling Results and Discussion 31 2.4.3 Incorporating Pore Compliance Effects 39 2.4.4 Limitations of the Pore-to-Pore Relaxation Model 42 2.5 Incorporating Long-Range Fluid Pressure Communication 43 2.5.1 Model Development 43 2.5.2 Modeling Results and Discussion 45 2.6 An Explicit Pore Network Model 47 2.6.1 Model Development 47 2.6.2 Modeling Results and Discussion 50 2.7 Discussion and Comparison of the Pore-Scale Relaxation Models 54 iii 3 Heterogeneity and Relaxation at the Patch Scale 57 3.1 Introduction 57 3.2 Equations of Pressure Diffusion in a Porous Medium 62 3.2.1 Pressure Diffusion Equation for Imposed Strain 64 .3.2.2 Pressure Diffusion Equation for Imposed Stress 65 3.3 Effective Bulk Modulus of a Patch 67 3.3.1 Solution of the Pressure Diffusion Equation for a Patch 67 3.3.2 Definition of the Effective Bulk Modulus of a Patch 74 3.3.3 Accounting for Partial Saturation 77 3.4 Discussion 80 3.4.1 Frequency Response of a Fluid-Saturated Patch 80 3.4.2 Effect of Permeability Contrast Across a Patch Boundary 81 3.4.3 Effect of Patch Size 84 3.4.4 Multiple Scales of Heterogeneity 87 3.5 An Application: Modeling Saturation Hysteresis 90 3.6 Summary 99 4 Conclusion 101 References 109 Appendices 112 A Probability Distribution of Distances Between Neighboring Pores 112 List of Notation 117 iv List of Tables 2.1 Physical properties of the constituents of a model sandstone 32 2.2 Parameters used to quantify the pore space and hydraulic properties of a model sandstone 32 3.1 Physical parameters for Spirit River sandstone saturated wi th water and air 85 v List of Figures 1.1 Computed P-Wave and S-wave velocities vs. water saturation for a model sandstone, illustrating the end members corresponding to the poroelas-tic model of Gassmann (1951) and the inclusion-based model of Berry-man (1980a; 1980b) 4 2.1 A sample of a porous solid as conceptualized in the inclusion-based effective medium theory approach 13 2.2 Incorporating fluid pressure communication into an inclusion-based model. 18 2.3 Schematic illustration of the replacement of hydraulically connected inclu-sions with equivalent inclusions having effective bulk moduli 20 2.4 Schematic illustration of an idealized pore-to-pore relaxation model: two fluid-filled inclusions connected by a narrow duct 21 2.5 Graphs of the real and imaginary parts of the effective bulk modulus K* vs. ui/u)0 for the pore-to-pore relaxation model 24 2.6 Phase angle 6 (by which pressure leads dilatation) vs. u>/u0 , for the pore-to-pore relaxation model 24 2.7 Probability distribution for the distance r from the center of a given water-filled inclusion to the center of its nearest gas-filled neighbor, for various levels of water saturation Sw 27 2.8 Shear-induced flow between two plates 28 2.9 Computed effective elastic moduli vs. water saturation at various frequen-cies for a model sandstone, using the pore-to-pore relaxation model. . . . 32 2.10 Computed P-wave velocity vs. water saturation at various frequencies for a model sandstone, using the pore-to-pore relaxation model 34 2.11 Computed P-wave Q'1 vs. water saturation at various frequencies for a model sandstone, using the pore-to-pore relaxation model 34 2.12 Computed S-Wave velocity vs. water saturation at various frequencies for a model sandstone, using the pore-to-pore relaxation model 37 vi 2.13 Computed S-wave Q 1 vs. water saturation at various frequencies for a model sandstone, using the pore-to-pore relaxation model 37 2.14 Computed P-wave velocity vs. frequency at various saturations for a model sandstone, using the pore-to-pore relaxation model 38 2.15 Computed Q'1 vs. frequency at various saturations for P-waves in a model sandstone, using the pore-to-pore relaxation model 38 2.16 Computed P-wave velocity vs. water saturation at various frequencies for a model sandstone, using the modified pore-to-pore relaxation model. . . 46 2.17 Computed P-wave Q"1 vs. water saturation at various frequencies for a model sandstone, using the modified pore-to-pore relaxation model. . . . 46 2.18 Schematic illustration of the explicit pore network relaxation model: every inclusion is connected to a number of other inclusions by cylindrical ducts through which fluid can flow 48 2.19 Computed P-wave velocity vs. water saturation at various frequencies for a model sandstone, using the explicit pore network relaxation model. . . 52 2.20 Computed P-wave Q~l vs. water saturation at various frequencies for a model sandstone, using the explicit pore network relaxation model. . . . 52 2.21 Computed P-wave velocity vs. frequency at 80% water saturation for a model sandstone, using the explicit pore network relaxation model. . . . 53 3.1 Schematic illustration of a fluid-saturated porous medium, of which we consider a particular small volume V 62 3.2 Schematic illustration of a model porous medium with a spherical patch of radius embedded in an infinite background medium 68 3.3 Graphs of the real part of the solution Pf(r)/po at various phases during a wave cycle, for UT = 103 72 3.4 Graphs of the real part of the solution Pf(r)/po at various phases during a wave cycle, for LOT = 10 _ 1 73 3.5 Graph of the real part of K*/Kd vs. (2-13) and follow immediately from the constitutive elastic relation for the fluids. We assume that the dilatations are equal, so 9\ — 92 = 9 (i.e., the inclusions are deformed under isos-train conditions—this assumption wi l l be justified later by the analysis in Section 2.4.3). Assuming a time dependence of the form eiut for the quantities pi, p2 and 9, equations (2.12)-(2.13) become iujPl = - K l (iu9 + (2.14) - K 2 ( i u 9 - ^ , (2.15) which, upon substitution of q from equation (2.10) can be solved to yield an expression of the form P i = —K*9, (2.16) where K* is given by Chapter 2. Heterogeneity and Relaxation at the Pore Scale 23 and (2.18) U0 = — ( « ! + K 2 ) . (2.19) Equation (2.17) gives the expression for the effective bulk modulus, K * , of inclusion 1. The limiting behavior of K* for high and low frequencies is as expected from the dis-cussion of Section 2.3. In the high-frequency limit, expression (2.17) reduces to «* = /«!. That is, the addition of hydraulic connectivity has no effect on the elastic behavior of inclusion 1, and it behaves as if it were hydraulically isolated from the other inclusion. This is consistent with the usual high-frequency assumption used in inclusion-based mod-eling. In the low-frequency limit, expression (2.17) reduces to K* = «, which is identical to equation (2.9). We investigate graphically the details of the frequency dependence of K* as defined in equation (2.17). Figure 2.5 presents graphs of the real and imaginary parts of ~ vs. ^ for the particular case where we take ^ = 0.1 (i.e., inclusion 2 is filled with a fluid that is much more compressible than that in inclusion 1). The behavior exhibited in Figure 2.5 is typical for a viscous relaxation mechanism (Bourbie et al, 1987, pp. 117-123), with a distinct transition from low-frequency to high-frequency behavior (evident in the graph of the real part of «*) occurring about the characteristic frequency u0. There is also a characteristic peak in viscous dissipation (represented by the imaginary part of K*) near u!Q, corresponding to a maximum near 45° of the phase angle by which the pressure in the pore leads the dilatation. Figure 2.6 shows a graph of this phase angle vs. tu/u>0 for the particular case considered above. It is clear from this figure that the pressure and dilatation are in phase only in the low- and high-frequency limits; otherwise the dilatation lags behind the pressure by a phase angle between 0° and 45°. A critical parameter in the effective pore fluid bulk modulus defined by equation Chapter 2. Heterogeneity and Relaxation at the Pore Scale 24 Figure 2.6: Phase angle 8 (by which pressure leads dilatation) vs. U/UQ , for the pore-to-pore relaxation model. Chapter 2. Heterogeneity and Relaxation at the Pore Scale 25 (2.17) is the inter-pore distance L which enters through the "conductivity coefficient" 7. As adjacent inclusions become further apart and L increases, the rate of flow between them decreases, thus limiting the degree to which fluid pressure is communicated between them. In the particular case described above, at a given frequency an increase in L would cause an increase in the effective bulk modulus of inclusion 1. Clearly, before equation (2.17) can be of practical use in defining effective inclusion moduli for an effective medium computation, we require an estimate of the parameter L for our inclusion-based medium. If inclusions are placed in a regular cubic array with number density n (inclusions per unit volume), the edge length of each unit cell, or equivalently the distance between the centers of inclusions, is given by n - 1 / 3 . For inclusions of equal volume V, the number density n is conveniently specified in terms of the porosity 0 by the relation n = 0/V. While L n - 1 / / 3 provides an estimate of the average pore-to-pore distance, in a more realistic porous medium with inclusions distributed randomly throughout the matrix, L would have a different value for each inclusion. It is therefore more realistic to treat L, and by extension the effective bulk moduli of inclusions, as random variables. If the probability distribution of L is known, the distribution of effective bulk moduli K* of inclusions follows immediately from the relationship between L and K* given by equations (2.17) and (2.11). The distribution of moduli K* is incorporated naturally into Berryman's IBEMT via the terms Q in equations (2.3) and (2.4). In Appendix A we derive the probability distribution for the distance r from the center of a given inclusion to the center of its nearest neighbor. This distribution is given by the probability density function P(r) defined by P(r) = ±Tmr2e-^nr\ (2.20) such that P(r)dr specifies the probability that the center-to-center distance is between r and r + dr. The distribution of duct lengths L is obtained from equation (2.20) by Chapter 2. Heterogeneity and Relaxation at the Pore Scale 26 subtracting from r the length corresponding to the sizes of the inclusions. With equations (2.17) and (2.20), along with estimates of the other physical parame-ters of the porous medium, we can now compute effective bulk moduli for the inclusions and then apply an inclusion-based effective medium theory to compute effective elastic properties for our model porous medium. We now proceed to modeling the saturation-and frequency-dependence of elastic wave velocities for a model sandstone saturated with water and air. We assume a saturation distribution such that each inclusion is filled with either water or air, and vary saturation by increasing the number of water-filled inclu-sions. We assume for now that for a given water-filled inclusion, the relaxation mechanism is dominated by fluid pressure communication with the nearest gas-filled inclusion. At low levels of saturation, gas-filled pores will be abundantly available. Most water-filled pores will be hydraulically connected to a gas-filled pore that is near by, so that the relaxation occurs relatively quickly. As the level of saturation is increased, gas-filled inclusions become less abundant and are typically further away from a given water-filled inclusion, and the relaxation process slows down. At a given frequency, this will result in less relaxed (i.e., elastically stiller) inclusions and hence an increase in their effective bulk moduli. This in turn causes an increase in the elastic stiffness of the composite. The change in abundance of gas-filled inclusions with changing saturation, and its effect on the state of relaxation of inclusions, is accounted for heuristically through the number density n9 of gas-filled inclusions. If the water saturation is Sw, then the number density of gas-filled pores is simply ng = (1 — Sw)n, where n is the number density of inclusions, as before. The number density ng is then used in equation (2.20) to compute distances to gas-filled inclusions. The effect of saturation on the distance from a given liquid-filled inclusion to its nearest gas-filled neighbor is illustrated in Figure 2.7 with a graph of the probability Chapter 2. Heterogeneity and Relaxation at the Pore Scale 27 2 0.5 h 1.5h 0 1h 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 rn 1/3 Figure 2.7: Probability distribution for the distance r from the center of a given wa-ter-filled inclusion to the center of its nearest gas-filled neighbor, for various levels of water saturation Sw. distribution (2.20) for various values of n9 corresponding to different levels of saturation. This figure clearly illustrates the expected tendency of water-filled inclusions to be further away from their nearest gas-filled neighbor as saturation is increased. The distribution also becomes more spread out with increasing saturation, while it is more narrowly peaked for lower levels of saturation. This suggests that accounting for the random distribution of L (rather than using one estimate for every pair of inclusions) becomes more important at higher levels of saturation. Effective Shear M o d u l i of Inclusions A fluid cannot, by definition, support a static shear stress, and fluids therefore have static shear moduli of zero. However, a viscous fluid can support a dynamic shear stress, and for sinusoidal forcing in time this dynamic shear stress gives rise to a non-zero effective complex shear modulus. To estimate the effective shear modulus of a pore fluid, we consider an idealized model consisting of two infinite parallel plates at y — 0 and y = L, between which is a fluid of viscosity n and density p (see Figure 2.8). The plate at y = L is held stationary while the plate at y = 0 is forced sinusoidally to that its displacement Chapter 2. Heterogeneity and Relaxation at the Pore Scale 28 y = L y = 0 Figure 2 .8 : Shear-induced flow between two plates. x(t) perpendicular to the y-axis is given by x(i) = x0eiu}t. The velocity of the fluid in the x-direction is given by u(y,t). The equation of motion for the fluid in the ^-direction is du d2u (2 .21 ) Assuming a velocity of the form u(y,t) — uQ(y)elujt and defining the kinematic viscosity v = n/p, equation ( 2 . 2 1 ) yields the following equation for the velocity profile u0(y), d2u0 u dy* = I - U Q . V (2 .22) The boundary conditions which u0(y) must satisfy are the no-slip conditions UQ(L) = 0 and ito(O) — U = iuxQ. The general solution of equation ( 2 . 2 2 ) is My) = c^-v^y + c2eV^y, (2 .23 ) for arbitrary constants c i , c2. Imposing the boundary conditions and solving for these constants yields the velocity profile sinh (v^R(l - D) where the dimensionless parameter R is given by (2 .24) R=J-L. v (2 .25 ) Chapter 2. Heterogeneity and Relaxation at the Pore Scale 29 The required force r(t) per unit area on the plate at y — 0 is given by du(y,t)\ r(t) = - 7 7 -dy y=o (2.26) r0e where . . . . ViR T 0 = IUIT] fXo\ y/XHtanh We now imagine that the fluid is replaced by a homogeneous solid having a shear modulus such that when a sinusoidal stress of amplitude r 0 is applied to the boundary at y = 0, the same 'displacement amplitude x0 is induced. The shear modulus of this effective material defines the effective complex shear modulus of the fluid. The shear strain e(£) in this effective material is given by x{t) e(t) = L ' (2.28) Defining the effective dynamic bulk modulus p* by the relation r = p*e, (2.29) it follows from equation (2.27) that [i* is given by T0L V = x0 V~iR = luin-(2.30) tanh \/iR By equation (2.29) we mean that under the specified sinusoidal shear stress, the displacement of the plate at y = 0 is the same as if the fluid were replaced by a solid material having shear modulus p* given by equation (2.30). That is, the fluid could be Chapter 2. Heterogeneity and Relaxation at the Pore Scale 3 0 replaced by this effective medium without altering the elastic behavior observed at the material boundaries. The dimensionless parameter R is a measure of the degree to which the momentum transferred from the plate at y — 0 to the fluid diffuses toward the other plate. In the limit —> 0, the momentum diffusion is complete, and the velocity profile between the plates becomes linear. The velocity shear becomes constant across the fluid layer, and p* —> iun. This is the expression used by Kuster and Toksoz (1974b) and Berryman (1980a; 1980b) for the dynamic shear modulus of pore fluids. For R 3> 1, the velocity shear is localized in a boundary layer at the plate at y = 0, and for increasing R the magnitude of p* grows without bound. We see that there are two physical processes that contribute to "stiffening" of the fluid layer with increasing frequency: (i) increasing u> increases the overall magnitude of the velocity shear, and (ii) increasing u> increases R, which concentrates the velocity shear near the plate at y = 0, giving rise to a greater shear stress at the plate. The expression (2.30) is obviously specific to the geometry of the idealized model being considered. A fluid in the pore space of a particular porous medium is subject to different (and much more complex) geometrical constraints, and the expression for the effective shear modulus will be different. Nevertheless, we regard expression (2.30) as a reasonably accurate estimate for the effective shear modulus of a pore fluid, where the parameter L is interpreted as representing a characteristic length scale of the pore space (e.g., the pore diameter). While the preceding development serves to demonstrate how frequency-dependent effective shear moduli can be defined for pore fluids, in practice we have found that the effects on the elastic behavior of model porous composites are negligible. This can be attributed to the fact that the magnitude of the effective shear modulus as defined in equation (2.30) is always very small relative to the shear modulus of the solid matrix, Chapter 2. Heterogeneity and Relaxation at the Pore Scale 31 as well as to the fact that the contribution of viscous shear to dissipation is dominated by losses due to the fluid pressure relaxation mechanism. Thus, in the modeling results that follow, in order to focus on the effects of fluid pressure relaxation we will entirely neglect the effects of non-zero effective shear moduli of fluids discussed here. 2.4.2 Model ing Results and Discussion Using the model developed in Section 2.4.1, we compute effective elastic moduli and wave velocities over a range of frequencies and levels of saturation for a "model sandstone". Our model sandstone is composed of water- and air-filled spherical inclusions in a solid quartz matrix, and is similar to that considered by Endres and Knight (1989), except that where they have used a distribution of inclusion shapes, for illustrative simplicity we consider only spherical inclusions. Pore-to-pore fluid pressure relaxation is permitted between inclusions filled with different fluids, and equation (2.17) is used to compute the resulting effective bulk moduli of the inclusions. Berryman's (1980a; 1980b) self-consistent effective medium theory is then used to compute estimates of the effective elastic moduli of the composite. Data describing the physical properties of the constituent materials are given in Table 2.1; other parameters used in the model are given in Table 2.2. The results of the self-consistent estimates of effective moduli for the model sandstone are presented in Figure 2.9 as graphs of the real parts of the effective bulk and shear moduli vs. water saturation, at various frequencies. A different curve is plotted for each of eight logarithmically sampled frequencies between 40 Hz and 150 kHz. As expected, at every level of saturation except Sw = 0 and Sw = 1, the high-frequency moduli of the composite are greater than the low-frequency moduli, owing to the greater restriction of pore pressure relaxation. The effective moduli are frequency-independent at 0 and 100% saturation because the pore fluid is homogeneous (either all-air or all-water) and Chapter 2. Heterogeneity and Relaxation at the Pore Scale 32 Matrix Brine Air bulk modulus, K [Pa] shear modulus, p [Pa] density, p [kg/m3] 2.14 x 101 0 3.05 x 109 2.42 x 103 2.10 x 109 0 1.08 x 103 1.38 x 106 0 1.17 x IO1 Table 2.1: Physical properties of the constituents of a model sandstone. (From Endres & Knight (1989)). Parameter Numerical Value porosity 0 (2.42) Chapter 2. Heterogeneity and Relaxation at the Pore Scale 45 with 7«i (2.43) UJ0 = v : Equation (2.42) is of the same form as equation (2.17), with the only difference being that here re = re^ff. Graphs of the real and imaginary parts of re* vs. u therefore have the same form as those in Figure 2.5. The effect of inclusion 2 having infinite volume is that the low-frequency limit of re* as defined in equation (2.42) is now rey-ff, irrespective of the properties of the fluid in pore 1. As this will be true for every inclusion in the porous medium, this change to the model has the global effect of making re^ff the effective bulk modulus for every inclusion in the low-frequency limit, which is the assumption used in the low-frequency Biot-Gassmann theory. We therefore expect this model to be in agreement with the predictions of Gassmann's equation. 2.5.2 Model ing Results and Discussion As before, we have computed effective bulk moduli for inclusions using equation (2.42), used Berryman's IBEMT to compute effective elastic moduli for the composite, and then used equation (2.31) to compute the resulting elastic P-wave velocities. The details of the dependence of the effective moduli on frequency and saturation will not be presented for this model. Rather, we move directly to the consideration of the saturation- and frequency-dependence of the P-wave velocity. In Figure 2.16 we present graphs of vp vs. Sw at various frequencies. Once again, frequencies are sampled at logarithmically spaced intervals to illustrate the spectrum of behavior from the low-frequency limit to the high-frequency limit. Figure 2.16 is qualitatively similar to Figure 2.10, except that the low-frequency curve now agrees much more closely with the expected curve obtained using Gassmann's equation. The pore-to-pore relaxation model now accurately reproduces the Chapter 2. Heterogeneity and Relaxation at the Pore Scale 46 2.2 r •| 51 1 1 i i i 0 0.2 0.4 0.6 0.8 1 S w Figure 2.16: Computed P-wave velocity vs. water saturation at various frequencies for a model sandstone, using the modified pore-to-pore relaxation model. For comparison, the results obtained using Gassmann's equation are plotted as the dashed ( ) curve, which is nearly indistinguishable from the low-frequency limit of this model. Figure 2.17: Computed P-wave Q 1 vs. water saturation at various frequencies for a model sandstone, using the modified pore-to-pore relaxation model. Chapter 2. Heterogeneity and Relaxation at the Pore Scale 47 expected results in both the high- and low-frequency limits. The associated graphs of Q~1 vs. Sw are presented in Figure 2.17. Our modification to the pore-to-pore relaxation model apparently has little effect on the attenuation characteristics, as the curves in this figure are indistinguishable from those of the previous model, given in Figure 2.11. Although this modified pore-to-pore relaxation model has been successful in reproduc-ing the expected results in the appropriate limits, the details of the predicted frequency dependence of elastic waves velocities remains questionable. In particular, the charac-teristic frequency LOQ about which the transition from low- to high-frequency behavior occurs, has probably been over-estimated. This results from the under-estimation of the permeability of the sample, as discussed in Section 2.4.4. In the following section we de-velop a more general relaxation model which accounts for multiple hydraulic connections between inclusions, in order to allow for more realistic sample permeabilities. 2.6 A n Explicit Pore Network Mode l 2.6.1 M o d e l Development In order to model more accurately the frequency- and saturation-dependence of elastic wave velocities, incorporating the increased permeability of a multiply-connected pore-space and accounting for the details of pore fluid distribution, we develop here a relax-ation model for an network of inclusions. Consider a pore space composed of inclusions connected as illustrated in Figure 2.4, but allowing a given inclusion to be hydraulically connected to any number of its neighboring inclusions, forming a hydraulically connected network. We illustrate this schematically in Figure 2.18, which shows a particular inclu-sion (labeled 1) connected to a number of other inclusions (labeled by the indices 2, 3 and 4) by thin ducts through which fluid can flow in response to pressure gradients. The diagram indicates cylindrical ducts of radii r i 2 , r i 3 and r i 4 , and lengths L 1 2 , L 1 3 and L 1 4 . Chapter 2. Heterogeneity and Relaxation at the Pore Scale 48 Figure 2.18: Schematic illustration of the explicit pore network relaxation model: every inclusion is connected to a number of other inclusions by cylindrical ducts through which fluid can flow. However, in principle, ducts of any geometry might be considered. As in the Section 2.4.1, we assume Poiseuille flow through the ducts in response to pressure gradients, such that the volumetric flow rate ^ from inclusion i to inclusion j is given by where the coefficient 7^ is a function of the geometry of the duct, and for the case of a cylindrical duct is given by equation (2.11). The differential equation analogous to equations (2.12) and (2.13), describing the evolution of the fluid pressure pi in inclusion i, is where m, n, . . . are the indices of the inclusions to which inclusion i is hydraulically connected. Again, we have assumed that the volumetric dilatations, 9, imposed on each Qn = Iii iVi ~ Pj) (2.44) (2.45) Chapter 2. Heterogeneity and Relaxation at the Pore Scale 4 9 i n c l u s i o n b y d e f o r m a t i o n o f t h e p o r o u s m e d i u m , a r e e q u a l . A s s u m i n g a t i m e d e p e n d e n c e o f t h e form e l w i f o r t h e q u a n t i t i e s Pi, pj, . . . a n d 8, a n d s u b s t i t u t i n g for q^ f r o m e q u a t i o n ( 2 . 4 4 ) , w e c a n r e w r i t e e q u a t i o n ( 2 . 4 5 ) as lUJPi -iuKi8 - (qim{pi - pm) + qin(Pi - p « ) H ) • (2 .46 ) A n e q u a t i o n o f t h i s f o r m i s c o n s t r u c t e d f o r e v e r y i n c l u s i o n i n t h e p o r o u s m e d i u m s a m p l e ( p o t e n t i a l l y y i e l d i n g a v e r y l a r g e se t o f e q u a t i o n s ) . I n d e x i n g a l l i n c l u s i o n s i n t h e s a m p l e b y t h e n u m b e r s 1 t h r o u g h AT, w e d e f i n e f o r t h e z t h i n c l u s i o n t h e q u a n t i t y 7* b y li lp = —iunO — « i / V i 0 0 K2/V2 Gp. (2.50) We can then solve equation (2.50) for the vector of pore pressures p to yield an expression of the form p = -K*9, (2.51) which defines the vector of effective pore fluid moduli, K*, for every inclusion in the sample. This vector is given by / L , A/ , n 1 \ " i / V i 0 0 K2/V2 G (2.52) where I is the identity matrix. In the case of only two inclusions connected by a single duct, equation (2.52) reduces to the previous result, equation (2.17). Equation (2.52) merely generalizes equation (2.17) to the case of a multiply-connected pore space. Fur-thermore, it is immediately apparent that in the high-frequency limit equation (2.52) reduces to the expected results K* = K (i.e., the inclusions behave as is they were hy-draulically disconnected). 2.6.2 Model ing Results and Discussion As before, we compute effective bulk moduli for inclusions using equation (2.52), and use Berryman's IBEMT to compute elastic wave velocities for the composite. We present in Figure 2.19 graphs of vp vs. Sw at logarithmic frequency intervals, for a model sandstone. The pore network was constructed by placing 250 spherical inclusions randomly within Chapter 2. Heterogeneity and Relaxation at the Pore Scale 51 a solid sphere with the appropriate volume to create a sample with the desired porosity, and placing a narrow duct between each inclusion and its three nearest neighboring inclusions. The parameters describing the relevant dimensions and physical properties of the model sandstone are again as given in Tables 2.1 and 2.2. Saturation was varied by starting with all inclusions gas-filled, and filling random gas-filled inclusions with water one at a time. Because the number of inclusions in our conceptual sample is finite, the effects of increases in saturation are discrete, whereas our previous models have depended con-tinuously on saturation. This causes the curves plotted in Figure 2.19 to be jagged at intermediate frequencies. These curves can be made more smooth by increasing the number of inclusions in the sample. However, because of the matrix inversion required in equation (2.52), this rapidly increases the time it takes to compute the effective pore fluid moduli K*. For the illustrative purposes of this particular realization, we found 250 inclusions to be a satisfactory compromise between accuracy and computation time. As expected, the high-frequency vp vs. Sw curve shown in Figure 2.19 reproduces the curve that would have been computed if a conventional inclusion-based effective medium theory without pore pressure communication had been used. We also note that we obtain very close agreement with the Gassmann prediction in the low-frequency limit. However, the main improvement in this model over previous models is that the interconnectedness of the pore space is treated explicitly. Therefore, we expect that the details of the frequency dependence of vp will be more realistic than that predicted with the previous models (due to the more realistic connectivity of the pore space). Furthermore, this model is also suitable for investigating the effects of macroscopic fluid distribution, as explicit control is available over which inclusions are filled with which fluids. The associated graphs of Q~1 vs. Sw are presented in Figure 2.20. Again, these curves are necessarily less smooth than those computed with the previous models. However, the Chapter 2. Heterogeneity and Relaxation at the Pore Scale 52 2.2 r 1.5' 1 1 ' 1 1 0 0.2 0.4 0.6 0.8 1 S w Figure 2.19: Computed P-wave velocity vs. water saturation at various frequencies for a model sandstone, using the explicit pore network relaxation model. For comparison, the results obtained using Gassmann's equation are plotted as the dotted (• • •) curve, which is nearly indistinguishable from the low-frequency limit of this model. Figure 2.20: Computed P-wave Q 1 vs. water saturation at various frequencies for a model sandstone, using the explicit pore network relaxation model. Chapter 2. Heterogeneity and Relaxation at the Pore Scale 53 frequency [Hz] Figure 2.21: Computed P-wave velocity vs. frequency at 80% water saturation for a model sandstone, using the explicit pore network relaxation model. qualitative characteristics are identical to those of the curves presented in Figures 2.11 and 2.17. We illustrate in Figure 2.21 the details of the dependence of P-wave velocity on frequency for this model sandstone. This figure shows a graph of vp vs. frequency for the particular level of saturation Sw — 0.8. Figure 2.21 exhibits qualitative similarities with Figure 2.14, with the expected asymptotic limits for both high and low frequencies, and with a transition for low- to high-frequency occurring about a characteristic relaxation frequency. However, this figure differs substantially from Figure 2.14 in that the transition occurs over a much wider range of frequencies (over more than two decades of frequency rather than just one), and is less symmetric about the central relaxation frequency. Chapter 2. Heterogeneity and Relaxation at the Pore Scale 54 2.7 Discussion and Comparison of the Pore-Scale Relaxation Models In this chapter we have presented a means of incorporating fluid pressure communication (i.e., relaxation) into inclusion-based effective medium theory. We have developed a number of explicit models of the relaxation process, and the results have been shown to be consistent with both the previous high-frequency inclusion-based velocity estimates and the Biot-Gassmann low-frequency estimates. Of the three relaxation models presented here, we consider the explicit pore network model developed in Section 2.6 to be the most accurate and versatile. The previous two models can in fact be viewed as ad hoc attempts to reproduce the combination of local and global flow effects that are accurately represented in the network model. For example, the effects of the random distribution of duct lengths, which were derived analytically for the first two models, are implicitly incorporated in the construction of the inclusion network and need not be incorporated explicitly. Furthermore, the effects of long-range pore pressure communication (i.e., beyond just the nearest neighboring inclusion) and fluid distribution heterogeneity are also automatically accounted for by the network model, while they are approached in an ad hoc way in the model presented in Section 2.4, and not at all in the model of Section 2.5. The pore network model also allows one to explicitly model effects that are beyond the reach of the other models. The explicit construction of the pore network allows one to specify the distribution of fluids in the pore space, the sizes and shapes of inclusions, and the number and radii of ducts (hence the permeability of the medium). While the pore network model provides the means to investigate the effects of many more parameters, as well as correlation structures among these parameters, it has the associated disadvantage that all of these parameters must be supplied explicitly in any particular realization of the model. Unfortunately, too little information about the pore Chapter 2. Heterogeneity and Relaxation at the Pore Scale 55 space of a given porous medium is known in order to construct such a detailed model. At best, one must rely on estimate of the statistical properties of the medium. Another disadvantage of the complexity introduced by the inclusion network is its increased com-putational requirements. The time required to solve the system of equations for the effective moduli of inclusions for a realistic network is perhaps out of proportion with the limited realism achievable with the model. A related issue is that the interpretation of the analytical results is less transparent. While the previous two models provide simple equations for the frequency dependence of effective inclusion moduli, the more general equations provided by the network model are much more complicated and therefore less amenable to intuitive understanding. For these reasons, the modified pore-to-pore relax-ation model may be preferred for a number of applications, as it reproduces the same low-and high-frequency limiting behavior, without the added complexity of the full network model. The most substantial improvement of the network model over the other relaxation models presented here is that it allows for the distribution of pore fluids to be specified explicitly. For reasons already discussed, a heterogeneous distribution of pore fluids is expected to limit the degree of pore pressure relaxation and thereby increase the elastic stiffness of a porous composite. Because the network model allows the inclusion network to be specified explicitly, fluids can be distributed in the network with any desired arrangement and correlation structure (i.e., scale of heterogeneity). The computation of effective inclusion moduli and subsequent application of an inclusion-based effective medium theory allows one to investigate the effect of any partic-ular arrangement on elastic wave velocities and attenuation. However, the computational complexity of the network model makes it unwieldy for many applications, especially if one wishes to investigate the effect of patch sizes that require a network of many thou-sands of inclusions. In the following chapter we take a different approach to modeling the Chapter 2. Heterogeneity and Relaxation at the Pore Scale 56 r e l a x a t i o n o f f l u i d p r e s s u r e s w h e n t h e l e n g t h s c a l e o f s a t u r a t i o n h e t e r o g e n e i t y is o n t h e o r d e r o f m a n y p o r e d i a m e t e r s . A t t h i s s c a l e w e c a n u s e t o a d v a n t a g e a c o n t i n u u m d e s c r i p -t i o n o f t h e p o r o u s m e d i u m a n d i t s s a t u r a t i n g f l u i d s , e x p r e s s e d i n t e r m s o f m a c r o s c o p i c p a r a m e t e r s s u c h as p o r o s i t y a n d p e r m e a b i l i t y . T h i s a v o i d s t h e c o m p l e x i t y o f s o l v i n g fo r t h e fluid p r e s s u r e e v o l u t i o n i n m a n y i n d i v i d u a l i n c l u s i o n s , a n d t h e a n a l y t i c r e s u l t s a r e m o r e t r a n s p a r e n t l y i n t e r p r e t a b l e . Chapter 3 Heterogeneity and Relaxation at the Patch Scale 3.1 Introduction There is substantial evidence from laboratory experiments (Knight and Nolen-Hoeksema, 1990; Cadoret, 1993) and numerical modeling (White, 1975; Endres and Knight, 1989; Gist, 1994; le Ravalec et al., 1996) that elastic wave velocities in fluid-saturated porous media can depend strongly on the spatial distribution of pore fluids. In particular, it has been observed that a porous medium in which the fluids are distributed heterogeneously is typically less compliant—and therefore exhibits higher elastic wave velocities—than the same medium with a more homogeneous distribution of pore fluids. As discussed in greater detail in the introduction to Chapter 2, this effect of heterogeneity depends on the wave frequency. The elastic reinforcement provided by a heterogeneous distribution of pore fluids is due to the fact that the fluid pressure is initially heterogeneous when the medium is deformed by a passing elastic wave. If the wave frequency is low enough, there is sufficient time during a wave cycle for flow to occur, whereby the fluid pressure is equilibrated, and the reinforcing effect of the fluid is minimized. The effect of fluid distribution heterogeneity is therefore most pronounced at high frequencies, where there is insufficient time for fluid pressure relaxation (i.e., equilibration) to occur, and be-comes negligible at sufficiently low frequencies where the induced fluid pressures become completely relaxed. In the current literature (see, for example, Gist (1994)), two distinct scales—the pore 57 Chapter 3. Heterogeneity and Relaxation at the Patch Scale 58 scale and the "patch" scale—of spatial heterogeneity of the fluid distribution are identi-fied. With each of these scales is associated a characteristic time scale of fluid pressure relaxation. Pore scale heterogeneity, considered in detail in Chapter 2, is associated with changes in fluid properties or pore geometry over a length scale less than a few pore diam-eters. The time scale of fluid pressure relaxation at this scale depends on the pore-scale properties of the porous medium, and the properties of the pore fluids. In the present chapter we consider heterogeneity at the "patch" scale, which is as-sociated with changes in fluid and/or matrix properties over a length scale many times greater than a typical pore diameter. (We restrict ourselves, however, to the "effective medium" approximation, in which the length scale of heterogeneity is assumed to be much less than the elastic wavelength, so that scattering from heterogeneities can be ne-glected.) There is significant experimental and theoretical evidence that heterogeneity at this scale can occur in natural geologic systems. Cadoret et al. (1995) provide laboratory evidence that centimeter-scale saturation heterogeneity can arise in limestones during drainage by drying, and that the scale of saturation heterogeneity can depend strongly on the nature of the imbibition/drainage process. Knight et al. (1998) also suggest how patch-scale saturation heterogeneity might arise at much larger (e.g., reservoir) scales when the underlying lithology is heterogeneous. Consider, for example, a liquid/gas-saturated medium in which fully liquid-saturated macroscopic regions (i.e., liquid-saturated "patches") exist adjacent to fully gas-saturated regions. Under deformation by a passing elastic wave, the fluid pressure in the liquid-saturated patches (which is initially higher than in the gas-saturated regions) reinforces the porous matrix and contributes to the elastic stiffness of the sample. However, given a sufficiently long wave period, flow from liquid-saturated patches to the gas-saturated regions will be induced so as to equilibrate the fluid pressure between macroscopic regions, thus limiting the extent to which the liquid reinforces the matrix. The time scale of this Chapter 3. Heterogeneity and Relaxation at the Patch Scale 59 fluid pressure relaxation can be estimated using the analysis of Dvorkin et al. (1994), who derive a diffusion equation governing the spatial distribution of fluid pressure. In the case where the pore fluid is much more compressible than the porous matrix, the fluid pressure diffusivity D is given by D^'if, (3.1) where Kf is the bulk modulus of the pore fluid, K is the intrinsic permeability of the medium, 4> is the porosity, and n is the viscosity of the fluid. For pressure diffusion out of a liquid-saturated patch with radius R, dimensional analysis yields r>2 r « ^ (3-2) as an estimate of the relaxation time scale r. If the time scale of deformation (e.g., the wave period) is much less than r, there is insufficient time for significant relaxation to occur, and the liquid is effectively trapped within the patch. If the time scale of deformation is much greater than r, there is sufficient time for complete relaxation to occur, and the fluid becomes effectively homogeneous. In this limit, Gassmann's (1951) formula (equation (1.1)) can be used to compute the effective elastic moduli—and hence the elastic wave velocities—of the composite. However, for many realistic porous media, and for wave frequencies relevant to geo-physical exploration, the time scale of patch-scale relaxation can be of the same order as the wave period. Consider, for example, a water/gas-saturated Fontainbleau sandstone with Kf = 2.25 GPa, K = 670 mD, = 0.136 and n = I O - 3 Pa-s (material data from Lucet (1989)). For a patch radius of 5 cm, equation (3.2) yields r w 2 . 3 x 10 - 4 s, which corresponds to a frequency 1/r of about 4 kHz (i.e., within the sonic range typically relevant in field seismic surveys). Assuming that a time much greater than r is required for complete relaxation to occur, we infer that at seismic frequencies Gassmann's formula Chapter 3. Heterogeneity and Relaxation at the Patch Scale 60 is not applicable for the sandstone considered here if the saturation heterogeneity is of a length scale of 5 cm or greater. Clearly, a more general model is required, in order to account for the effects of heterogeneity at this scale. Le Ravalec et al. (1996) have proposed such a model, using an inclusion-based ap-proach to estimate elastic wave velocities in fluid-saturated porous media with patch-scale heterogeneity, in the unrelaxed (i.e., high frequency) limit. They consider the porous medium to be composed of distinct macroscopic regions having different levels of partial fluid saturation. A differential self-consistent inclusion-based formulation is used to individually estimate effective elastic moduli for each region. The same inclusion-based approach is then used, at the sample scale, to estimate effective elastic moduli for the composite, which is composed of these individual regions in appropriate volu-metric proportions. While their model incorporates effects of both pore- and patch-scale heterogeneity, it does not allow for fluid pressure relaxation, and therefore requires the as-sumption that the wave frequency is sufficiently high (or the length scale of heterogeneity is sufficiently great) that relaxation can be neglected. The two approaches mentioned above (i.e., the poroelastic approach of Gassmann (1951) and the inclusion-based approach of Le Ravalec et al. (1996)) to modeling the relationship between partial fluid saturation and elastic wave velocities, treat opposite extremes of the frequency spectrum, and hence are mutually inconsistent. Each is limited to either the completely relaxed or completely unrelaxed (i.e., high- or low-frequency) regime, and neither provides estimates of elastic and anelastic (e.g., wave attenuation attributed to the patch-scale flow mechanism) behavior in the intermediate regime. We desire a model that is sufficiently general to treat both the high- and low-frequency limits in a consistent way, as well as treat the intermediate ranges of frequency and length scale of heterogeneity. One such model is the "gas pocket" model of White (1975), in which an idealized Chapter 3. Heterogeneity and Relaxation at the Patch Scale 61 fluid distribution is assumed, consisting of a cubic array of spherical gas-saturated pock-ets embedded in an otherwise liquid-saturated medium. Effective elastic moduli for this idealized medium are estimated, taking into account the mechanism of flow induced be-tween the liquid- and gas-saturated regions. Within the limits of effective medium theory (i.e., in which the size of the gas pockets is assumed to be much smaller than the elastic wavelength), this model treats the full frequency range, and reduces to the expected re-sults in the low- and high-frequency limits. However, the White model assumes that the properties of the porous matrix are spatially uniform, and therefore cannot treat the case where the saturation heterogeneity arises because of lithologic heterogeneity as suggested in Knight et al. (1998). Furthermore, the White model allows for only one length scale of heterogeneity, whereas one might realistically wish to consider the case where liquid-or gas-saturated patches of a range sizes exist. The new model we present here contains elements of both the White (1975) and le Ravalec et al. (1996) models. As in le Ravalec et al., we use an inclusion-based approach to compute effective elastic moduli of the composite, which is considered to be composed of distinct regions (i.e., "patches") having different levels of partial saturation. However, rather than use an inclusion-based approach to compute the effective elastic moduli of the individual patches (which immediately restricts the model to the unrelaxed limit), we follow a similar approach to that of White (1975). Using the equations of poroelasticity and Darcy's law, we compute the frequency response of a spherical patch, allowing for fluid flow between regions, and derive expressions for the effective complex bulk moduli of the patch. We then use Berryman's (1980a; 1980b) inclusion-based formulation (rather than the differential self-consistent approach used by le Ravalec et al.) to estimate effec-tive elastic moduli of the composite composed of liquid- and gas-saturated patches. The advantage of our formulation is its ability to model the effects of patch-scale heterogene-ity over a wide range of frequencies and length scales of heterogeneity while remaining Chapter 3. Heterogeneity and Relaxation at the Patch Scale 62 fluid-saturated porous medium sample small volume V solid fluid Figure 3.1: Schematic illustration of a fluid-saturated porous medium, of which we con-sider a particular small volume V. consistent with the expected low- and high-frequency limiting behavior. 3.2 Equations of Pressure Diffusion in a Porous Medium In this section we derive partial differential equations describing the diffusion of fluid pressure in a porous medium. Our results are used in the following section to analyze the relaxation of fluid pressure in a "patch", and to derive an expression for the complex effective bulk modulus of a patch. We consider a sample of fluid-saturated porous medium, and focus on a particular small region within the sample, having volume V and saturated with a single pore fluid (see Figure 3.1). We assume that the porosity within this region is homogeneous, in that when a uniform pressure is applied at the region's boundary the induced fluid pressure is spatially uniform. In particular, this implies either that the pore-scale fluid pressure heterogeneity induced by spatial variation of pore space geometry is negligible, or that the flow generated by this heterogeneity leads to pore-scale fluid pressure equilibrium on a time scale that is much shorter than the time scale of deformation. This is just the assumption used by Gassmann (1951) and many later authors, except that we require Chapter 3. Heterogeneity and Relaxation at the Patch Scale 63 that it hold only for the local volume V rather than for the whole sample. Given this assumption, Gueguen and Palciauskas (1994) (following Nur and Byerlee (1971)) derive the following constitutive relation of poroelasticity, p-apf = -Kd9, (3.3) which relates the induced fluid pressure pf to the pressure p applied at the boundary of the region and the volumetric dilatation 9 of the region. Here Kd is the "dry frame" bulk modulus (i.e., the bulk modulus of the porous material in the absence of pore fluids), and the "poroelastic parameter" a is defined as a = 1 - ^ , (3.4) Ks where KS is the bulk modulus of the mineral solid of which the porous matrix is composed. The simple volumetric average 6 = f + (1 - 4>)Ps, (3.6) where ps is the pressure in the solid component. Assuming that the solid component is subject to a homogeneous pressure due to the pore fluid, the constitutive elastic relation for the solid component is given by ps = -Ks9s. (3.7) Chapter 3. Heterogeneity and Relaxation at the Patch Scale 64 The constitutive elastic relation for the fluid component can be written as dpf _ d9f KfQ ~dT ~ -Kf-dT ~ W ( } where Kf is the bulk modulus of the fluid. Here Q represents the volume rate of fluid flow out through the region's boundary dV, and can be written as the surface integral Q = I v-nds, (3.9) J&V where v is the volume-averaged volumetric flow rate and n represents the outward unit normal to the sample surface. We now consider the limit of equation (3.8) as the volume of the particular region under consideration goes to zero, yielding the equation describing the evolution of the fluid pressure throughout the sample. Darcy's Law (Bourbie et al, 1987, p. 31), v = - - V P / ) (3.11) V can be used to express the flow velocity v in terms of the fluid pressure gradient V p / , the intrinsic permeability K of the porous medium, and the viscosity rj of the fluid. Substituting expression (3.11) into equation (3.10), we arrive at the final form of the constitutive relation for the fluid, 3.2.1 P r e s s u r e D i f f u s i o n E q u a t i o n for I m p o s e d S t r a i n If we assume that the time evolution of the dilatation field 9 within a sample of porous medium is known, we can solve the system of equations (3.3), (3.5), (3.6), (3.7) and Chapter 3. Heterogeneity and Relaxation at the Patch Scale 65 (3.12) for the time derivatives of the unknown quantities p, ps, pf, 9S and 9f in terms of | | . This yields the equation dpf 2 a 89 — = D e V P f - - F 0 - (3.13) for the evolution of the fluid pressure throughout the sample. The constants Do and Fe are given by De = ^ (3.14) 077 F9 Kf Ks (pK,s If the dilatation in a given region of porous medium is held fixed (i.e., 89/dt = 0), equation (3.13) reduces to the simple diffusion equation 8Pf dt = DeV2Pf. (3.16) We therefore interpret Dg as the fluid pressure diffusivity under conditions of constant dilatation. De is, in fact, identical to the "rock diffusivity" K derived by Dvorkin et al. (1994), and equation (3.13) is the generalization of their diffusion equation for time-varying strain. 3.2.2 Pressure Diffusion Equation for Imposed Stress Alternatively, one can consider the pressure field p to be known throughout the sample, and solve the same system of equations for the time derivatives of the unknown quantities ps, Pf, 9, 9S and 9f in terms of This yields the equation Chapter 3. Heterogeneity and Relaxation at the Patch Scale 66 for the evolution of the fluid pressure. The constants DP and Fp are given by D>=^r (318) F p Kf Ks (j)Kd If the overall pressure in a given region of porous medium is held fixed (i.e., dp/dt — 0), equation (3.17) again reduces to a simple diffusion equation, with diffusivity DP. We therefore interpret DP as the fluid pressure diffusivity under conditions of constant overall pressure. Although the terms DP and Fp differ from DQ and FQ, in practice it is generally true that Kf region 2 (infinite porous background medium) region 1 (spherical "patch") A A A F i g u r e 3 .2 : S c h e m a t i c i l l u s t r a t i o n o f a m o d e l p o r o u s m e d i u m w i t h a s p h e r i c a l p a t c h o f r a d i u s R ( d e n o t e d as r e g i o n 1) e m b e d d e d i n a n i n f i n i t e b a c k g r o u n d m e d i u m ( d e n o t e d as r e g i o n 2 ) . Chapter 3. Heterogeneity and Relaxation at the Patch Scale 69 Since we regard the pressure field p as prescribed, we use equation (3.17) to describe the evolution of the fluid pressure in the sample, i.e., . Dp(1)V2Pf + r < R (inside the patch), £ P / = J R (outside the patch) ^ 02«d(2) at where r is the radial coordinate measured from the center of the patch, and the indices 1 and 2 denote material properties in the regions inside and outside the patch, respectively. The terms D p(i), Dp@), Fp^ and Fp(2) are defined for their respective regions as in equations (3.18)-(3.19). Defining the dimensionless variables f = i , f = T - (3.23) where the time scale r is given by R2 and defining the dimensionless parameters (3.24) _ Qi^p(i) r 0=2^2 n % faFpi?) Kxn2 01 « d ( l ) 02«d(2) £>p(2) 01 ^p(l) • ^ 2 ?7l we can rewrite equation (3.22) in the simpler form (3.26) Exploiting the assumed spherical symmetry, we can write the dimensionless Laplacian operator V 2 as f 2 <9r V df Chapter 3. Heterogeneity and Relaxation at the Patch Scale 70 Assuming a solution of the form Pf(f,i) = pf(r)elbJt, equation (3.26) becomes ' ^ + ! * £ + i i r l P o f < i . \ n \^FT + -^r + ^ r 2 P „ f > i , ^ \ ar 2 r dr J where the dimensionless angular frequency a) is given by u = UT. (3.28) The general solution of equation (3.27) is given by {T i + + %e-Jw f < 1, r (3.29) r r for arbitrary constants c i , C2, C3 and C4. The boundary conditions that Pf(f) must satisfy are dp/ dr = 0 (3.30) lr=0 (ie., no flow through r = 0, as required by symmetry), which implies that c 2 = — C i , and lim |p/(r)| < 00 (3.31) f—+00 (i.e., the solution must be bounded as r —> 00), which implies that c 3 = 0. At f = 1 we must also have continuity of the fluid pressure, i.e., pf(r)=pf(l+), (3.32) and of the fluid flux defined by equation (3.11), i.e., Kx dpf I _ K2dPf % dr f = i - rj2 dr (3.33) f = i + Chapter 3. Heterogeneity and Relaxation at the Patch Scale 71 Applying the conditions (3.32)-(3.33) and solving for the remaining coefficients Ci and c 4 yields the solution ( ( r 2 - Ti)( l + \/iCjQ) sinhVw^r „ 11 H j== j=— , j== r < 1, £viu> cosh vza) + (1 — £ + y/iuSl) sinh \J%CJ r " ^ f ( r i - r 2 ) ( V ^ cosh y/i&- sinh v 7 ^) e v^f2 ( i-r) 1 2 H / = = T = , 7= ^ r > 1. £vza) cosh Vza) + (1 — £ -f- vza)f2) sinh v i £ r (3.34) Figure 3.3 presents graphs of the real part of the solution (3.34) for the fluid pressure inside and outside the patch, at different phases during a wave cycle. The particular values Ti = 0.2, T 2 = 0.02, £ = 0.1 and Q = 0.5 have been chosen for the various dimensionless parameters of the model, corresponding to a patch saturated with a fluid that is ten times less compressible than the fluid saturating the surrounding medium.' In this case we expect the greater fluid pressure induced within the patch to cause diffusion of pressure out of the patch. The wave frequency is chosen so that ur = 103 (i.e., the wave period is about 103 times shorter than the time scale of fluid pressure relaxation in the patch). These graphs illustrate the kind of behavior expected from the discussion in Sec-tion 3.1. Since the time scale of deformation is too short for significant relaxation to occur during a wave cycle, the greater fluid pressure induced inside the patch remains al-most entirely localized within the patch throughout the wave cycle. In other words, there is little fluid pressure communication between the patch and the surrounding medium. Other than a small deviation from a uniform distribution of fluid pressure near the bound-ary of the patch (i.e., near r = 1), the patch and the surrounding medium behave as if they are hydraulically isolated from each other. The patch is almost completely "unre-laxed" , and the induced fluid pressure contributes maximally to reinforcing the porous matrix. Figure 3.4 presents similar graphs, but for the case UJT — 10 - 1 , corresponding to Chapter 3. Heterogeneity and Relaxation at the Patch Scale 72 Figure 3.3: Graphs of the real part of the solution Pf(r)/Po at various phases during a wave cycle, for o»r = 103. Note that r has been scaled so that the patch corresponds to the domain [0,1]. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 73 0.02 -0.02 0.02 -0.02 0.02 € o CL -0.02 0.02 -0.02 cor= 2TI/4 -0.02 Figure 3.4: Graphs of the real part of the solution P/(r)/p 0 at various phases during a wave cycle, for CUT — 10 - 1 . Note that r has been scaled so that the patch corresponds to the domain [0,1]. Note also that the vertical scale is different than that of Figure 3.3. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 74 a wave period that is much longer than the time scale of fluid pressure relaxation in the patch. In this case there is sufficient time during a wave period for much more fluid pressure communication between the patch and the surrounding medium to occur, again as expected from the discussion of Section 3.1. The degree to which fluid pressure is communicated between the two regions is clear from the fact that the fluid pressure inside the patch differs only slightly from that outside the patch. The more compressible fluid saturating the meclium surrounding the patch acts as a low-pressure reservoir into which the fluid pressure induced inside the patch can diffuse. Consequently the fluid pressure inside the patch is much less throughout the wave cycle than in the case LOT = 1 0 3 . The decreased fluid pressure within the patch reduces the degree to which the fluid reinforces the porous matrix, and we therefore expect the sample as a whole to be more compliant in this case. 3.3.2 Definition of the Effective Bulk Modulus of a Patch Using the solution (3.34) for the induced fluid pressure field, along with the poroelastic relation given by equation (3.3), we can write the dilatation field inside the patch as 9(f,i) = e{f)eiC>\ where 0(f) Po - aiPf(f) Po 1 - a i T i + £\fiCb~ cosh \fiCj + (1 — £ + ViuQ) sinh y/iCJ r (1?2 — T i ) ( l + y/ubti) sinh y/i&f )] 3.35) Chapter 3. Heterogeneity and Relaxation at the Patch Scale 75 The total dilatation of the patch, 90, which is just the average of 8(f) over the patch, is then given by 60 = 3 f f26(f) df Jo 10 Po i - a i r\ + -3 (r 2 — Ti)(l + \/i£bQ)(\/iili cosh \fiCj — sinh \fiCS) VJJ £y/iCJ cosh y/iCj + (! — £ + y/iuCt) sinh y/itj (3.36) By relating the total dilatation of the patch to the applied pressure, we can define a complex effective bulk modulus, K*, for the patch, i.e., Po = H*9Q. (3.37) In other words, under sinusoidal deformation with frequency u>, the patch behaves iden-tically to a homogeneous elastic solid with bulk modulus K * . It follows immediately from equation (3.36) that K* is given by n - i l - « i Ti + 3 (T2 — Ti)(l + \JiCjQ)(\/id) cosh — sinh \fiCS) ^Vidj cosh y/iod + (! — £ + y/iuiD) sinh Vioj~ (3.38) After some simplification, and rewritten in terms of the original wave frequency LU, equa-tion (3.38) can be written K « d ( l ) i - a i i r i + zMtr-a-ri)] (3.39) where 3 \ / (1 + V'iu>T£l)(y/'iu>T — tanhy/iur) ILOTJ \ £>yJuJT' + (1 - f + yWrft) tanh V^wr (3.40) Chapter 3. Heterogeneity and Relaxation at the Patch Scale 76 Limiting Forms of the Effective Bulk Modulus We note that in the limit u —> oo (i.e., when there is insufficient time for communication of fluid pressure between the patch and the rest of the sample to occur), f(u) —> 0 and equation (3.39) reduces, after simplification, to K* = KD(L) + 0 I A ^ _ ^ (UJ - C O ) . (3.41) «s(l) This is just Gassmann's formula (compare with equation (1.1)) for the bulk modulus of the patch when the boundary of the patch is taken to be impermeable, and is the expected high frequency limit of the model. The high-frequency limit of our model is therefore identical to the patch-scale heterogeneity model proposed by Knight et al. (1998), where Gassmann's formula is used to estimate the effective bulk modulus of individual patches, under the assumption that the wave frequency is sufficiently high that no fluid pressure communication between adjacent patches occurs. Equation (3.39) can be regarded as the extension of their model to lower frequencies for which fluid pressure communication does occur. We note, however, that our model reduces to Gassmann's formula in the high frequency limit only because we have assumed that the porosity is homogeneous, so that even at high frequencies the induced fluid pressure is spatially uniform within the patch. The presence of pore-scale heterogeneity would act to further stiffen the patch at sufficiently high frequencies. This limitation also applies to the model of Knight et al. (1998). In the limit uo —> 0 (i.e., in the static limit), f(ui) —> 1 and equation (3.39) reduces, after simplification, to ^-^ + ^r^EA {u^0) (3,42) Chapter 3. Heterogeneity and Relaxation at the Patch Scale 77 in the particular case where K S(X) = KS(2), «d(i) = «d(2) and 0i = 02- This is just Gassmann's formula for the bulk modulus of the patch when fluid 1 is replaced by fluid 2. Again, this is the expected low frequency limit of the model. The result is independent of the bulk modulus of the fluid saturating the patch because we have assumed that the region saturated with fluid 2 is infinite in extent. In the case of total fluid pressure communication throughout the sample, the induced fluid pressure (and therefore the contribution of the fluid to the elastic stiffness of the composite) is determined entirely by the bulk modulus of the fluid in region 2. 3.3.3 Accounting for Partial Saturation As noted above, the low- and high-frequency limits of equation (3.39) are in agreement with the known Gassmann predictions for our patch model. However, at low frequencies the model actually underestimates the effective bulk modulus of a liquid-saturated patch in a real finite porous medium sample. This error arises because in our model construction we assume that the background medium surrounding the patch is infinite in extent. Consequently, the overall level of saturation of fluid 1 is effectively zero, and the effective fluid bulk modulus in the case of complete fluid pressure communication is just ft/(2)> independent of In order to use our results to estimate effective patch moduli for partial saturation in realistic finite porous materials, we must refine the model in order to take the effect of finite saturation into account. There are two immediately apparent means of accomplishing this. One is to take the approach of White (1975), and embed the patch at the center of a finite spherical sample, the boundary of which is closed, and the radius R2 of which is determined by the desired level of saturation of fluid 1 . That is, as the overall level of saturation of fluid 1 in the sample approaches 1, R2 approaches R, while as the overall level of saturation approaches 0, R2 approaches infinity. In this case the only change to the model derivation would be Chapter 3. Heterogeneity and Relaxation at the Patch Scale 78 the replacement of the boundary condition (3.31) with the condition 0 (3.43) dPf df \f=R2/R (i.e., the boundary at r = R2 is impermeable). This change serves to simulate the effects of finite saturation, and in the low frequency limit (i.e., in the limit of complete fluid pressure communication) will yield an effective fluid bulk modulus Kef given by 4f = — + — > (3-44) where Si is the overall level of saturation of fluid 1 in the sample. The effective bulk modulus of the patch in the low-frequency limit will then be given by the Gassmann prediction = ""W + 2. This is the expected low-frequency result for partial saturation, and differs from equation (3.42) only in the replacement of Kj( 2 ) with K*f. Unfortunately, when the new boundary condition (3.43) is implemented and the rest of the derivation is carried out, the modified expression for K* becomes impractically complicated. The heuristic model we have proposed hardly justifies the use of such a complex result, and we prefer the following alternative method of incorporating finite saturation effects, which does not increase the complexity of the resulting expression for K*. As described above, we expect the low-frequency limit of our model to yield the modified result (3.45). However, finite saturation effects will not alter the high-frequency limit of the model, where no fluid pressure communication occurs and the elastic behavior Chapter 3. Heterogeneity and Relaxation at the Patch Scale 79 of the patch is independent of the components of region 2. Furthermore, we do not expect partial saturation effects to significantly alter the form of the frequency response, but only to alter the low-frequency limit determined by the parameter r2. We simulate this behavior by considering the patch to be embedded in an infinite background saturated with a fluid having bulk modulus K*f (as given by equation (3.44)), rather than «/ ( 2 ) . That is, we consider the patch to be embedded in an infinite sample having an overall level of saturation S\ of fluid 1 (rather than zero as before), and treat the region outside the patch as if its fluids were distributed homogeneously. This guarantees that the low-frequency limit of the model agrees with equation (3.45). Furthermore, when S\ = 1 (i.e., when the whole sample is completely saturated with the fluid saturating the patch) and the properties of the matrix are constant throughout the sample, we have the expected result that Ti = T 2 , so that equation (3.39) is independent of o>. That is, no patch-scale flow of fluid is induced by the deformation of the sample. The model is easily generalized to the case of n distinct pore fluids having bulk moduli « / ( 2 ) ) - • • iKf{n)- I n this case the generalization of equation (3.44) is given by (Domenico, 1977) K f i=i K / M where Sj is the level of saturation of the ith fluid (defined such that Yli=i & = -0- The final form of our model for the effect complex bulk modulus K* of a spherical patch saturated with fluid 1, in a porous medium having a level of saturation S% of fluid 1, is then summarized as follows, K* = M l ) fa A7\ l - a i t T i + Z M f T a - r O ] 1 ' } IUT — tanh y/iur) f(u) = IUJTJ \ £v/iZ~Jr~+ (1 - £ + Viurri) tanh Viurr Chapter 3. Heterogeneity and Relaxation at the Patch Scale 80 where the various dimensionless parameters are defined by aiFp(i) 0 ^ ( 2 ) n Dp{1) Kxr]2 R2 ^l«d(l) 02«d(2) A>(2) if2??l Dpii) and we have further defined the quantities 1 1 1 a i 01^ 71 ^p(l) Ks(l) 0i«d(l) n _. ^ 2 F p ( 2 ) 1 _ 1 1 a 2 02??2 Fp(2) K f KS(2) 02«d(2) 3.4 Discussion 3.4.1 Frequency Response of a Fluid-Saturated Patch In Figure 3.5 we present graphs of the real and imaginary parts of K*/KJ vs. LUT/(2IT), for the same particular choice of values of the various dimensionless parameters as used in Figures 3.3-3.4, and with c\i = 0.5. The parameter CUT/(2TV) can be interpreted as a measure of the wave frequency relative to the relaxation frequency of the patch, so that when LOT I (2ir) « 1 the time scale of deformation and the time scale of fluid pressure relaxation are of the same order. The graphs exhibit the expected qualitative behavior, with the patch stiffening with increasing frequency. As CUT approaches infinity, Re{«;*} approaches the asymptotic value associated with the upper Gassmann limit given by equation (3.41). As LOT approaches zero, Re{K*} approaches an asymptotic value associ-ated with the lower Gassmann limit given by equation (3.42). The transition between the low- and high-frequency limits occurs near the characteristic relaxation frequency, where LOT/(2TT) ?n 1. The imaginary part of K* (which represents the component of the stress in the patch that is out of phase with the dilatation, and is associated with viscous losses caused by fluid flow) exhibits a peak near the relaxation frequency, and goes to zero in the low- and high-frequency limits. In these limits the pressure and dilatation within Chapter 3. Heterogeneity and Relaxation at the Patch Scale 81 1.25 0.08 y oT 1.1 cr vi°1.15 1.05 1.2 10 - 2 COT/(2TC) on/(27t) Figure 3.5: Graph of the real part of K*/'nd vs. UIT/(2TT) for the same values of the various parameters as used in Figures 3.3-3.4, with ct\ = 0.5. the patch are in phase. We note that, in contrast to the pore-scale relaxation models developed in Chapter 2, the transition from low- to high-frequency behavior is much more gradual, occurring over more than three frequency decades. Significant relaxation of fluid pressure occurs even for u>r/(27r) FU 100, and the patch becomes essentially un-relaxed only for UJT/(2IT) > 103. Note that this was a feature observed when long-range fluid pressure communication was accounted for in the pore-scale relaxation model of Section 2.6. 3.4.2 Effect of Permeability Contrast Across a Patch Boundary The customary estimate for the time scale of pore fluid relaxation in a patch, as given by equation (3.2), assumes that the properties of the porous matrix and fluids are con-stant across the patch boundary. Consequently, only the properties of the fluids and porous matrix within the patch itself appear in equations (3.1)-(3.2). However, many naturally occurring porous media are composed of heterogeneous lithology. In particu-lar, the permeability of porous materials can vary over many orders of magnitude (e.g., from the sub-/xD scale in tight gas sandstones to the scale of Darcys in unconsolidated Chapter 3. Heterogeneity and Relaxation at the Patch Scale 82 sands), and in a porous medium composed of multiple lithologic units (as considered in Knight et al. (1998)) significant spatial heterogeneity of permeability can be expected. Furthermore, it has been suggested (Knight et ai, 1998) that patch-scale saturation heterogeneity might even be caused by the underlying lithologic heterogeneity. Spatial heterogeneity of permeability clearly must have an effect on the time scale of patch-scale fluid pressure relaxation. At one extreme, one can consider a permeable patch surrounded by an impermeable background medium, in which case the patch will be unrelaxed at all wave frequencies and the time scale of relaxation given by expression (3.2) becomes meaningless. This effect of permeability heterogeneity on the time scale of patch-scale fluid pressure relaxation has received little attention in the current literature. In this section we briefly investigate this effect using the patch relaxation model developed in Section 3.3. We consider a water-saturated sandstone patch with permeability K~i, embedded in a gas-saturated background porous medium having a permeability K2 that is different than that of the patch. The remaining parameters required in our patch-scale relaxation model are assigned the particular values 771/772 = 20, Fp^/Fp^) = 400, (j>i/(p2 — 1, I \ = 0.9 and T 2 = 0.002, which are typical for a water/air-saturated sandstone. We have used equation (3.39) to compute the effective bulk modulus, «*, of the patch. The results are presented in Figure 3.6 as graphs of the real part of K* / Kd vs. U)T/(2TT), for various values of the permeability contrast K\jK2. The figure illustrates that the patch is almost completely relaxed when ur/(2n) « 1 in the case where Ki/K2 = 1 (i.e., if there is no permeability contrast). This indicates that, in this case, r is in fact a reasonable estimate of the time scale of relaxation. The same is true for the case where K\jK2 = 102. However, when the permeability contrast is increased to 104, the relaxation of the patch becomes significantly affected by the decreased permeability of the medium in which it is embedded. The relaxation frequency, Chapter 3. Heterogeneity and Relaxation at the Patch Scale 83 C0T/(27C) Figure 3.6: Real part of K* / KD vs. LUT/(2IT), for a water-saturated patch in a typical sandstone that is otherwise saturated with air, and for various values of the permeability contrast Ki/K2 (each curve is labeled with its associated value of K~i/K2). about which the transition from relaxed to unrelaxed behavior occurs, is almost one decade less than when there is no permeability contrast. In this case r underestimates the actual time scale of relaxation by a factor of about 10. When the permeability contrast is further increased to 106 this effect becomes accentuated, and r underestimates the actual time scale of relaxation by a factor of more than 100. As K~i/K2 approaches infinity, the wave frequency required for significant relaxation of the patch to occur approaches zero, at which point relaxation can no longer occur at all. On the other hand, we find that when Ki/K2 < 1 (i.e., when the permeability of the background medium is greater than that of the patch) K* is not significantly altered from the case where there is no permeability contrast, even as K\jK2 approaches zero. We conclude that, at least for the particular medium considered here, equation (3.2) provides an accurate estimate of the time scale of relaxation of a liquid-saturated patch when then the permeability contrast K\jK2 is less than 102. However, care must be Chapter 3. Heterogeneity and Relaxation at the Patch Scale 84 taken when using this expression to estimate the state of relaxation of a patch when the permeability contrast is greater than 102, where the reduced permeability of the background medium begins to limit the relaxation of the patch. 3.4.3 Effect of Patch Size In this section we investigate the effect of patch size on our model for the effective bulk modulus of a patch. We consider the particular example of a spherical water-saturated patch in a Spirit River sandstone that is otherwise saturated with air. The input data corresponding to these materials are summarized in Table 3.1. We use equation (3.39) to estimate the effective bulk modulus of the water-saturated patch for a range of patch sizes and wave frequencies. Figure 3.7 presents the resulting graphs of the real and imaginary parts of the computed values of K*/KD vs. patch size, for wave frequencies of 100 Hz and 50 kHz. This figure illustrates the expected dependence on patch size and frequency. At a given frequency Re{re*} for a water-saturated patch increases with increasing patch size, owing to the decreased degree of fluid pressure relaxation that can occur during a wave period. When the wave frequency is increased (from 100 Hz to 50 kHz in Figure 3.7), a smaller patch size is required for significant relaxation to occur within the time scale of deformation. Figure 3.7 provides an explicit example of the observation made in Chapter 2 that the very term "heterogeneity", in describing the distribution of pore fluids, is a frequency-dependent concept. For a wave frequency of 100 Hz, we find that water-saturated patches with radius less than about 0.2 mm are completely relaxed (i.e., the fluid distribution is effectively homogeneous, and Gassmann's (1951) approach gives a valid estimate of the effective elastic properties of the sample). This is in contrast with the 50 kHz case, for which a patch radius of less than about 10 pm is required for complete fluid pressure re-laxation to occur. In other words, a water/air-saturated sample of Spirit River sandstone Chapter 3. Heterogeneity and Relaxation at the Patch Scale 85 Parameter Numerical Value Porous Matrix (Spirit River Sandstone) KD [GPa] 5.6a pd [Gpa] 12.6a 0 0.0526 K [pD] . \ 1*_ Mineral Solid (Quartz) KS [GPa]. 38c ps [kg/m3] 2630c Water K w a t e r [GPa] 2.2 fato [10~3 Pa-s] 1_ K a i r [MPa] 0.8 77air [IO'3 Pa-s] 0.05 "Estimated from the velocity measurements of Knight and Nolen-Hoeksema (1990) 6From Knight and Nolen-Hoeksema (1990) cFrom le Ravalec et al. (1996). Table 3.1: Physical parameters for Spirit River sandstone saturated with water and air. R [m] R [m] Figure 3.7: Graphs of the real and imaginary parts of K*/Kd vs. patch radius at wave frequencies of 100 Hz and 50 kHz, for a spherical water-saturated patch in a Spirit River sandstone that is otherwise saturated with air. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 86 with fluid distribution heterogeneity of length scale on the order of 0.2 mm is effectively homogeneously saturated at wave frequencies less than 100 Hz, while the same sample behaves heterogeneously if the wave frequency is increased to 50 kHz or greater. At 50 kHz, the sample can be regarded as effectively homogeneous—so that the Gassmann estimate of the effective elastic moduli is valid—only if the length scale of heterogeneity is decreased to less than about 10 pm. These observations can be simply understood in terms of the scaling of the relaxation time scale r (as defined in equation (3.2)) with patch size. Both LO and R appear in equation (3.39) only through the dimensionless term LOT, in which LO and R2 play identical roles. That is, for a given set of matrix and fluid properties, increasing the wave frequency LO has the same effect on the effective bulk modulus of a patch as increasing R2 by the same relative amount. It comes as no surprise, then, that the effect of increasing the length scale of heterogeneity is the same as increasing the wave frequency, and that whether or not a sample behaves as if the fluids were distributed homogeneously depends on the wave frequency as well as the length scale of heterogeneity. Figure 3.7 also provides an explicit illustration of the observation, made by Knight et al. (1998), that if the pore fluids are arranged in sufficiently large patches, the fluids can have a stiffening effect even at low frequencies. For the particular combination of porous medium and fluids considered here, and for a wave frequency of 100 Hz, this stiffening effect occurs as the patch radius is increased above 0.1 mm, and is maximized for a patch radius greater than 1 cm. Thus, the conventional assumption that the pore fluids are completely relaxed in this frequency range—and that Gassmann's (1951) approach can be used to estimate the elastic wave velocities—is not valid even for this modest degree of saturation heterogeneity. Clearly, this could have serious consequences for the interpretation of seismic data. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 87 3.4.4 Mult iple Scales of Heterogeneity In real porous media, instead of a single length scale of heterogeneity, we expect hetero-geneity to exist at a wide range of scales, from the pore scale (on the order of microme-ters) through the patch scale (millimeters to centimeters) and extending to the scale of lithologic variation (on the order of meters or greater). While the models developed in this thesis do not provide a consistent framework for treating the simultaneous effects of multiple scales of heterogeneity (where the treatment is complicated by the fact that successively smaller scales of heterogeneity may themselves be embedded within larger scales of heterogeneity), they do provide insight into the expected qualitative behavior. Consider the following thought experiment in which a number of distinct length scales of heterogeneity are present in a fluid-saturated porous medium. From the our previous discussion it should be clear that there will be a frequency threshold above which the pore fluids can be regarded as completely unrelaxed, i.e., above which there is no sig-nificant flow of the pore fluids during a wave period. (We assume that all of the scales of heterogeneity are much smaller than the elastic wavelength at this frequency, so that scattering is negligible.) Starting at this frequency, if the wave frequency is gradually decreased, eventu-ally the wave period will be sufficiently long for fluid pressure relaxation to occur at the smallest length scale of heterogeneity (e.g., the pore scale, or the smallest patch scale). From our previous discussion it should be clear that it is the smallest scale of heterogeneity—by virtue of the correspondingly short length over which fluid pressures must be communicated—that is the first to relax. The frequency threshold at which this relaxation occurs is predicted by the pore-scale relaxation model of Chapter 2. As the wave frequency is decreased below this threshold, relaxation of fluid pressures at the pore scale weakens the elastic response of the composite, and there is an associated decrease Chapter 3. Heterogeneity and Relaxation at the Patch Scale 88 in the wave velocities. If the wave frequency is decreased further, the wave period eventually becomes suf-ficiently long that fluid pressure at the next greater length scale of heterogeneity (e.g., the patch scale) has sufficient time to relax. Associated with this relaxation is a further decrease in the elastic wave velocities. The frequency threshold below which this relax-ation occurs is predicted by the patch-scale relaxation model of Section 3.3. As the wave frequency is further decreased a cascade of relaxations occur, as the relaxation frequency associated with each successively greater length scale of heterogeneity is traversed. Our patch-scale relaxation model predicts that this relaxation frequency decreases as R~2, where R is a characteristic size of a fluid-saturated patch. When the wave frequency becomes sufficiently low that the fluid pressure is completely relaxed at the greatest length scale of heterogeneity present, Gassmann's (1951) approach to modeling the elastic properties of the composite becomes valid. To explicitly illustrate the thought experiment described above, we consider the par-ticular example of a water/air-saturated porous rock in which the water is present in distinct patches of 0.2 mm and 1 cm. We use the same material properties given in Table 3.1, and assume that the medium is 80% water-saturated, with the water being distributed in equal volumetric proportions between the larger and smaller patches. We use the patch-scale relaxation model to estimate effective elastic moduli for the patches, and use Berryman's (1980a) inclusion-based effective medium theory to compute effective elastic moduli for the composite at various wave frequencies. Figures 3.8-3.9 present the resulting graphs of P-wave velocity and attenuation, re-spectively, vs. wave frequency. Both graphs distinctly show the two successive relaxations that occur as the wave frequency is decreased from the completely unrelaxed limit to the completely relaxed limit. First the smaller patches become relaxed as the frequency is decreased through 104 Hz (where there is a distinct drop in P-wave velocity as well as a Chapter 3. Heterogeneity and Relaxation at the Patch Scale 89 frequency [Hz] Figure 3.8: P-wave velocity vs. wave frequency, computed using the patch-scale relaxation model for a particular low-permeability sandstone having 80% water saturation, with half of the water in patches with radius 0.2 mm and the other half with radius 1 cm. 0.08 0.07 h frequency [Hz] Figure 3.9: P-wave attenuation vs. wave frequency for the same medium considered in Figure 3.8. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 90 peak in attenuation), then the larger patches relax as the frequency is further decreased through about 10 Hz (where there is again a drop in P-wave velocity and a peak in attenuation). 3 . 5 An Application: Modeling Saturation Hysteresis As an application of the patch relaxation model developed in Section 3.3, we consider the effect of saturation hysteresis on elastic P-wave velocity measurements. Laboratory measurements of elastic wave velocities have demonstrated hysteresis in the relationship between P- and S-wave velocities and the level of water saturation in porous rocks (Knight and Nolen-Hoeksema, 1990; Cadoret, 1993). This velocity hysteresis has been associated with hysteresis of the scale of fluid distribution heterogeneity resulting from an imbibition or drainage process. The imbibition process tends to result in a more homogeneous distribution of the pore fluids, while the drainage process tends to result in a more heterogeneous, "patchy" distribution of fluids; this in turn causes fluid pressures to be less relaxed and the elastic wave velocities in the medium to be greater than during imbibition. In this section we use the patch-scale relaxation model developed in Section 3.3 to extend the inclusion-based approach of le Ravalec et al. (1996) to a broader range of frequencies. Because our relaxation model relies on the assumption that the porosity is homogeneous on the scale of an individual patch, we must restrict our consideration to porous media which satisfy this assumption. That is, we cannot account for the effects of additional heterogeneity at the sub-patch scale. In this sense our model is less general than that of le Ravalec et al, which accounts for pore-scale heterogeneity. We consider a liquid/gas-saturated porous medium in which the liquid and gas phases are segregated into 100% liquid-saturated and 100% gas-saturated regions (i.e., patches). Chapter 3. Heterogeneity and Relaxation at the Patch Scale 91 We associate with each region a length scale R representing the patch size or "radius", and use equation (3.47) to estimate an effective complex bulk modulus for the patch. The effective shear modulus of each patch is defined using the usual assumption that the effective shear modulus of a fluid-saturated medium is identical to the dry frame modulus of that medium, i.e., equation (1.2) (although, as discussed in Section 2.4.2, this is only an approximation). With effective elastic moduli defined for all liquid- and gas-saturated regions, we then compute effective bulk and shear moduli for the composite, using Berryman's (1980a; 1980b) inclusion-based effective medium theory, assuming that the patch sizes are all much less than the elastic wavelength. (In le Ravalec et al. this step is carried out using a differential self-consistent approach. The advantage of using Berryman's effective medium theory, beyond its consistency with all known rigorous bounds on effective moduli, is that is allows for the elastic moduli of the constituents of the composite to be complex, which is critical in the implementation of our model.) Although Berryman's effective medium theory allows for the geometry of inclusions to be quite general, at present we restrict ourselves to the case of approximately spherical patches. P-wave velocities and attenuation for the composite are then computed from the effective elastic moduli of the composite with equation (2.31). There appear to be at least the three following mechanisms by which the level of liquid saturation in a sample can be varied: 1. Varying of the level of liquid saturation in existing patches. 2. Varying the number of liquid-saturated patches of a given size. 3. Varying the size of existing liquid-saturated patches. In the model of le Ravalec et al. (1996), mechanisms 2 and 3 are combined into a single mechanism. A single parameter—the volume fraction of the sample occupied by patches Chapter 3. Heterogeneity and Relaxation at the Patch Scale 92 of high liquid saturation—is used to simultaneously vary the size and number of patches, without distinguishing between these mechanisms. This is possible because their model treats only the high-frequency limit, in which only the volume fraction occupied by patches, not the size of individual patches, appears as a parameter in the inclusion-based effective medium theory. However, in our frequency-dependent patch-scale heterogeneity model, we must distinguish between mechanisms 2 and 3, because the effective bulk modulus of a patch depends on its size. Given the above three independent means of varying the level of liquid saturation in a sample of a porous medium, we can expect a wide variety of possible saturation distributions—and therefore a wide variety of functional forms for the dependence of elastic wave velocities on saturation—arising from a given imbibition or drainage process. To simplify matters, we eliminate mechanism 1 by assuming that the sample can be divided into distinct regions of 100% liquid saturation and 100% gas saturation (i.e., liquid-saturated patches and gas-saturated patches). We make this assumption only to more clearly elucidate the effects of the length scale of saturation heterogeneity on P-wave velocities measured during imbibition/drainage experiments, and in practice it is easily relaxed. For the same reasons, we also eliminate mechanism 2 by assuming that during an imbibition-drainage process the number of liquid-saturated patches in a given sample is fixed, while only the sizes of individual patches vary. To model the variation of patch size with the level of liquid saturation in a sample, we define a characteristic length, Ro, representative of the length scale of saturation heterogeneity. We then assume that the typical size, Rw (e.g., the radius), of a given liquid-saturated patch can be expressed in terms of the level of liquid saturation SW as Rw = SWRQ. (3.48) That is, we assume that the size of a given liquid-saturated patch increases linearly with Chapter 3. Heterogeneity and Relaxation at the Patch Scale 93 saturation. The size of a patch is assumed to be zero at Sw — 0, and is assumed to approach R0 as Sw approaches 1. This relationship is arbitrarily chosen as the simplest relationship between Sw and R^. By defining the following linear estimate of liquid saturation, S w = T ^ T T T ' ( 3 - 4 9 ) where Rg is the typical size of a gas-saturated patch, we can substitute for Ryj from equation (3.48), and rearrange equation (3.49) to write Rg = (l- Sw)Ro- (3.50) That is, as Sw approaches zero the size of gas-saturated regions approaches RQ, while as Sw approaches 1 the size of gas-saturated regions approaches 0. Equations (3.48) and (3.50) together provide a physically reasonable parameterization of the variation of the distribution of pore fluids as a function of Sw. The parameterization involves the single parameter R0 which determines the length scale of saturation heterogeneity. Following the discussion of Cadoret et al. (1995), we can characterize a given imbi-bition or drainage process by the length scale of heterogeneity that is associated with it. In our model this length scale is associated with the parameter RQ. In particular, we expect imbibition to result in a more homogeneous distribution of pore fluids, and we therefore model the imbibition process as one for which the associated value of R0 is relatively small. That is, during imbibition the level of liquid saturation is increased via the increasing size of a large number of relatively small liquid-saturated patches. By contrast, we expect the drainage process to result in a more heterogeneous distribution of fluids, and we therefore model the drainage process as one for which the value of Ro is relatively large. That is, during drainage the level of liquid saturation is decreased via the reduction in size of fewer and larger liquid-saturated patches. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 94 We consider the particular example of a sample of Spirit River sandstone saturated with water and air. Again, the relevant physical parameters are given in Table 3.1. Using the inclusion-based model described above, we compute the elastic P-wave velocity in the sample as a function of Sw, for various values of R0. In Figure 3.10 we present the resulting graphs of vp vs. Sw for a wave frequency of 100 Hz, and for logarithmically spaced values of R0 ranging from 79 pm to 1.8 cm (these particular values are chosen only in order to most clearly illustrate the full range of dependence on Ro). As expected, we find that a sample having a greater length scale of saturation het-erogeneity exhibits a higher elastic P-wave velocity at all levels of liquid saturation. This is due to the increasing time required for a patch to relax as the size of the patch is increased. For the Spirit River sandstone, we find that for a length scale of heterogeneity of about 79 pm or less, there is sufficient time for complete fluid pressure relaxation to occur at a wave frequency of 100 Hz. At this scale of heterogeneity the fluid pressure is equilibrated throughout the sample, and in fact the curve corresponding to Ro = 79 pm coincides with the Gassmann prediction for this sample, as would be expected. When the scale of heterogeneity is increased to about 1.8 cm or greater, the opposite is true: there is insufficient time for significant relaxation of fluid pressures to occur. Hence the curve corresponding to RQ = 1.8 cm coincides with the result that would be predicted by the high-frequency model of le Ravalec et al. (1996) (given the additional assumption of homogeneous porosity that we require). Figure 3.11 presents similar graphs, except that a wave frequency of 10 kHz, and logarithmically spaced values of Ro between 7.9 pm and 0.18 cm, were chosen. Because the time scale of relaxation r defined in equation (3.24) scales as R2, the curves in this figure are exactly the same as those in Figure 3.10, since we have increased the wave frequency by two orders of magnitude while decreasing the patch size by one order of magnitude. Complete relaxation of fluid pressures at this wave frequency requires a scale Chapter 3. Heterogeneity and Relaxation at the Patch Scale 95 4 r 3.8 ^ 3.6 JL 3 4 3.2 3 2.8 -1 . 8 / ^ ^ j -^ ^ ^ ^ ^ ^ - ^ 0 1 2 ^ ^ ^ ^ ^ ^^jsmX-^J 0.0079 1 1 i ; i i 0 0.2 0.4 0.6 S 0.8 w Figure 3.10: Elastic P-wave velocity vs. water saturation in Spirit River sandstone (data given in Table 3.1) at 100 Hz, computed using the patch-scale relaxation model, illus-trating the dependence on the length scale of saturation heterogeneity. Each curve is labeled with its corresponding value of Ro, in centimeters. 4 3.8 3.6 J 3.4 3.2 3 2.8 • 0 . 1 8 / 0 ^ J - ^ ^ ^ 0 0 4 6 ^ ^ - ^ 1 X 0 1 2 ^^^^ ^ o s m x ^ J 0.00079 1 1 1 1 1 0 0.2 0.4 0.6 s 0.8 w Figure 3.11: Elastic P-wave velocity vs. water saturation in Spirit River sandstone (data given in Table 3.1) at 10 kHz, computed using the patch-scale relaxation model, illus-trating the dependence on the length scale of saturation heterogeneity. Each curve is labeled with its corresponding value of R0, in centimeters. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 96 of heterogeneity of 7.9 pm or less. Gist (1994) estimates for this rock sample a pore diameter of 0.3 pm, so that the concept of a "patch" is valid even at this small length scale. However, it is clear that if the wave frequency is increased to 1 MHz (i.e., to ultrasonic levels), complete fluid pressure relaxation cannot occur unless the "patches" have a radius of about 0 . 7 9 ( i . e . , of the same order as pore diameter, for which the very concept of a "patch" obviously fails). At this frequency he assumption used by le Ravalec et al. (1996) that patches are completely unrelaxed becomes valid for all patches with radius greater than 0.18 mm. Turning now to the particular application of modeling saturation hysteresis associated with patch-scale heterogeneity, we interpret the curves in Figures 3.10-3.11 corresponding to smaller values of Ro as being those that would result from processes that yield a more homogeneous distribution of fluids (e.g., imbibition). The curves corresponding to larger values of Ro are interpreted as those which would result from processes that yield a more heterogeneous distribution of fluids (e.g., drainage by evaporative drying). As an explicit example, we again consider an imbibition/drainage experiment with a water/air-saturated Spirit River sandstone. We assume that the imbibition proceeds leads to a distribution of pore fluids with a length scale of heterogeneity, Ro, of 0.1 mm. This is consistent with the scale of heterogeneity observed by Cadoret et al. (1995) for imbibition of various limestones by depressurization. We assume that the drainage process leads to a more heterogeneous distribution of fluids, with an associated length scale of heterogeneity of 1 cm. This is also consistent with the scale of heterogeneity observed by Cadoret et al. (1995) for drainage of various limestones by evaporative drying. In Figure 3.12 we present the resulting computed graph of P-wave velocity vs. Sw, for both imbibition and drainage at wave frequencies of 100 Hz and 10 kHz. The greater heterogeneity of the fluid distribution during drainage causes the model rock sample to exhibit hysteresis, with P-wave velocities being higher during drainage than during Chapter 3. Heterogeneity and Relaxation at the Patch Scale 97 4r 2 . 8 1 ' 1 1 1 ' 0 0.2 0.4 0.6 0.8 1 s w Figure 3.12: Computed saturation hysteresis of elastic P-wave velocity in Spirit River sandstone, at 100 Hz (—) and 10 kHz (—). Assumed length scales of saturation hetero-geneity are RQ = 0.1 mm for imbibition and R0 = 1 cm for drainage. 0 . 1 4 r 0 .12h ^ \ w Figure 3.13: Computed saturation hysteresis of P-wave attenuation Q~x for'Spirit River sandstone, at 100 Hz (—) and 10 kHz (—). Assumed length scales of saturation hetero-geneity are R0 = 0.1 mm for imbibition and R0 = 1 cm for drainage. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 98 imbibition. The computed hysteresis is most pronounced at 100 Hz, at which frequency the 0.1 mm scale liquid- and gas-saturated patches are completely relaxed, while the 1 cm scale patches are almost completely unrelaxed. At 10 kHz the 0.1 mm scale patches are less relaxed and the exhibited saturation hysteresis is less pronounced, though significant. The 1 cm scale patches are completely unrelaxed at this frequency, and the computed velocities during drainage are therefore slightly higher than at 100 Hz. The associated graphs of Q~1 vs. Sw, presented in Figure 3.13, also exhibit significant saturation hysteresis at both of the sampled wave frequencies. Naturally, the greatest attenuation is exhibited by the imbibition curve at 10 kHz, where the state of relaxation of the patches is intermediate between completely relaxed and completely unrelaxed, as can be seen in Figure 3.12. The curves with minimum attenuation are the one for imbibition at 100 Hz (where the patches are completely relaxed) and the one for drainage at 10 kHz (where the patches are completely unrelaxed). Interestingly, each of these curves exhibits a maximum attenuation at some intermediate saturation, corresponding to the optimal level of saturation such that the viscous losses due to flow between water-saturated and gas-saturated regions are maximized. The hysteresis curves presented in Figure 3.12 do not accurately model the particular form of the experimentally measured curves given for various limestones in Cadoret et al. (1995) and for Spirit River sandstone in Knight and Nolen-Hoeksema (1990). These data are more accurately reproduced when pore-scale effects of the imbibition/drainage process are taken into account, as in the model of le Ravalec et al. (1996). However, our computations do illustrate that saturation hysteresis due to patch-scale heterogeneity alone can be significant in realistic and geophysically relevant situations. Furthermore, the predicted behavior exhibits a frequency dependence than cannot be predicted by the high-frequency model of le Ravalec et al. The model presented here also provides esti-mates of the wave attenuation caused by viscous losses due to the patch-scale relaxation Chapter 3. Heterogeneity and Relaxation at the Patch Scale 99 mechanism, which have previously been provided only by the White (1975) model. 3.6 Summary In this chapter we have developed a frequency-dependent model for estimating elastic wave velocities in fluid-saturated porous media with "patch"-scale saturation hetero-geneity (i.e., media in which the length scale of saturation heterogeneity is on the order of many pore diameters). Our approach is similar to that taken in Chapter 2, in that we have sought to describe the frequency-dependent elastic response of a patch in terms of frequency-dependent, complex effective elastic moduli. With effective elastic moduli defined for each patch, an inclusion-based effective medium theory is used to estimate the effective moduli of the composite. The model is similar to that of le Ravalec et al. (1996), who also use an inclusion-based formulation to estimate elastic moduli of the composite from the effective elastic moduli of the individual patches. However, their model is lim-ited to the unrelaxed (i.e., high frequency or large patch size) limit,'whereas the model presented here considers the full frequency range within the limits of effective medium theory. Our approach has been to derive partial differential equations describing the diffusion of fluid pressure between patches. We construct a solution of these equations for the particular case of a spherical patch in an infinite porous medium subject to deformation that is sinusoidal in time (in this our model is similar to that of White (1975)). This solution is then used to relate the pressure p applied to a patch to its dilatation 6, so as to define an effective bulk modulus K* = —p/6 for the patch. Our results are consistent with the expected low-frequency limit as given by Gassmann's (1951) formula, and with the expected high-frequency, "unrelaxed" limit where there is no communication of fluid pressure between patches. Chapter 3. Heterogeneity and Relaxation at the Patch Scale 100 The chief limitation of the model presented here is that we consider saturation het-erogeneity at the patch scale only. In fact we require that the fluid distribution be homogeneous at the pore scale (or at least effectively homogeneous, in the sense that the time scale of relaxation of an initial pore-scale fluid pressure inhomogeneity is much shorter than the wave period). This is a serious limitation, since we expect that in many realistic porous materials heterogeneity will exist at many length scales ranging from the pore scale to the patch scale. However, under appropriate circumstances the model should provide a useful correction to the usual Gassmann estimate of effective elastic moduli (and hence wave velocities), for the case of saturation heterogeneity on a length scale intermediate between the pore-scale and the elastic wavelength. Because the effective patch moduli defined by our model are complex, they can be used to predict not only the velocities of elastic waves but also their attenuation due to viscous losses associated with flow of the pore fluids. This requires that the effective medium theory used to form the composite of all patches must allow for the constituents of the composite to have complex moduli. One such inclusion-based effective medium theory is that of Berryman (1980a; 1980b) used here. As a particular application of our model of patch-scale heterogeneity, we have consid-ered the effects of saturation hysteresis on elastic P-wave velocities. This application is similar to the saturation hysteresis model of le Ravalec et al. (1996) except that we have attempted to incorporate fluid pressure relaxation, whereas le Ravalec et al. consider only the unrelaxed limit. Our patch-scale relaxation model allows us to estimate the saturation hysteresis of both the velocity and attenuation of elastic waves for a broad range of wave frequencies. Chapter 4 Conclusion The objective of this thesis has been to develop predictive models of elastic wave veloci-ties in fluid-saturated porous media, with the particular aim of modeling the frequency-dependent effects of fluid distribution heterogeneity at various length scales. The models presented here complement previous models—which provide end-member velocity esti-mates in the limits of high and low frequency—by explicitly incorporating the dynamics of the fluid phase that determine the elastic behaviour of the porous composite at interme-diate frequencies. Our approach has been to incorporate a description of the mechanism of fluid pressure communication and relaxation into existing inclusion-based effective medium theories, thus extending their range of validity to lower wave frequencies. Two distinct length scales of saturation heterogeneity have been identified: the pore scale and the "patch" scale. We have considered each of these scales separately in Chap-ters 2 and 3, respectively. Although we choose different models for the dynamics of the pore fluid at these two scales, our approach to incorporating the effects of fluid pressure relaxation has been similar. In each case, we identify those constituents of the porous composite that exhibit frequency-dependent behavior due to deformation-induced flow of the pore fluids. In the particular cases of pore-scale and patch-scale heterogeneity, these constituents are the pores and the patches, respectively. We proceed by defining effec-tive complex elastic moduli for these constituents, and consider an equivalent medium in which the pores or patches are conceptually replaced by homogeneous materials having these elastic moduli. Berryman's (1980a; 1980b) inclusion-based effective medium theory 101 Chapter 4. Conclusion 102 is then used to estimate the effective elastic moduli (and hence the elastic wave velocities and attenuation) of the composite. The majority of our modeling efforts have been in deriving expressions for the ef-fective elastic moduli of the frequency-dependent constituents. In each case we seek a relationship between the overall stress and strain of a pore or patch. Noting that the overall behavior of a pore or patch under sinusoidal deformation is identical to that of some equivalent homogeneous solid, we use this relationship to define the elastic moduli of an equivalent medium with which a given pore or patch is then replaced. The overall stress-strain relationships used to define these effective moduli are derived by assuming a particular mechanism of fluid pressure relaxation (Poiseuille flow at the pore scale, and Darcy's law at the patch scale), and accounting for the contribution of this flow to the elastic response of a pore or patch. At both scales of heterogeneity, the frequency-dependent behavior of the composite arises because the degree to which fluid pressure communication between adjacent re-gions (i.e., between pores or between patches) occurs depends on the wave period. This frequency dependence can be described in terms of a characteristic relaxation time scale. At the pore scale, we have analyzed the communication of fluid pressure between two pores that are saturated with different fluids and are hydraulically connected by a narrow duct. For the particular case where the pores have identical volume V and the duct is cylindrical, we find that the time scale of pore-scale fluid pressure relaxation, r p o r e , is given by 8nLV ''"pore 7 r r 4 ( « i + « 2 ) ' where K\ and K2 are the bulk moduli of the two fluids, r and L are the radius and length of the duct, and n is the viscosity of the fluid in the duct. Previous models have for the most part corresponded to the "end members" in which Chapter 4. Conclusion 103 the wave period, r, satisfies either r >^ r p o r e (e.g., the Gassmann (1951) theory), or T ^ r pore (e.g., the inclusion-based formulations of Kuster and Toksoz (1974a) and Berryman (1980a; 1980b)). These end members correspond to the limits in which fluid pressures induced by a passing elastic wave are either completely relaxed or completely unrelaxed. By explicitly accounting for the mechanism of pore-scale fluid pressure relax-ation, the models developed in Chapter 2 are able to treat the full range of frequencies between these end members. In particular, one can use these models to estimate the elas-tic wave velocities and attenuation in the previously untreated regime where r ~ r p o r e . At the patch scale, we have analyzed the relaxation of fluid pressure between in a spherical "patch", e.g., a macroscopic region where the compressibility of the saturating fluid differs from that of the fluid in the surrounding medium. Our analysis yields the time scale of patch-scale fluid pressure relaxation, r p a t c h, given by r p a t c h , so that induced fluid pressures in patches are completely relaxed (e.g., as in the Gassmann (1951) theory), or r n - 1 / 3 , and only a small fraction, equal to of inclusions have their nearest neighbor at a distance greater than n 1^3. The mean distance (r) from a given inclusion to its nearest neighbor, given by (A.ll) ( 3 ) 1/3 (r) = / rP(r) dr Ann (A.12) Jo significantly less than the naive estimate of n 1 / / 3 proposed in section 2.4.1. List of Notation Ci Volume fraction of the z t h constituent of a porous composite 15 D Diffusivity of fluid pressure 59 Dp Diffusivity of fluid pressure under constant sample pressure 66 De Diffusivity of fluid pressure under constant sample dilatation 65 / Wave frequency 104 F Parameter used in Berryman's (1980b) IBEMT 15 K Intrinsic permeability of a porous medium 59 L Length of the duct connecting two pores 21 n Number density of inclusions 25 V Pressure 18 Pf Pressure in the pore fluid 39 ps Pressure in the mineral solid component 63 P*1 A geometrical term used in Berryman's (1980b) IBEMT 15 q Volumetric flow rate between inclusions 21 QH A geometrical term used in Berryman's (1980b) IBEMT 15 r Radius of a cylindrical duct connecting two pores 21 R Radius of a spherical patch 59 Si Volume fraction of the pore space occupied by the ith pore fluid 19 Sw Volume fraction of the pore space occupied by liquid (usually water) 4 t Time 18 V Volume of a pore 21 vp Elastic P-wave velocity 14 117 List of Notation 118 vs Elastic S-wave velocity 14 a Poroelastic parameter equal to 1 — K d / K s 3 7 Poiseuille flow coefficient, equal to 7rr 4/(8nL) for a spherical duct 21 TJ Fluid viscosity 22 0 Volumetric dilatation 18 9f Volumetric dilatation of the pore fluid 63 6S Volumetric dilatation of the mineral solid component 63 K Bulk modulus of an isotropic medium 14 K* Effective complex bulk modulus of a pore or patch 19 k Parameter used in the pore-to-pore relaxation model 23 Kd Dry frame bulk modulus 3 fteff Effective bulk modulus of a porous composite 3 Kf Bulk modulus of the pore fluid 3 ttyff Bulk modulus of the "effective pore fluid" 4 Ki Bulk modulus of the z t h constituent of a porous composite 14 KS Bulk modulus of the mineral solid component of a porous matrix 3 A Elastic wavelength 106 p Shear modulus of an isotropic medium 14 p* Effective complex shear modulus of a pore 29 Pd Dry frame shear modulus 3 ^eff Effective shear modulus of a porous composite 3 Pi Shear modulus of the ith constituent of a porous medium 15 T Time scale of fluid pressure relaxation 59 v Kinematic fluid viscosity equal to n/p 28 p Density 14 List of Notation 119 0 Porosity (i.e., the volume fraction occupied by the pore space) 3 to Angular wave frequency equal to 2nf 18 UJQ Characteristic frequency of a relaxation mechanism 23