International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Single vs. multi-drain probabilistic analyses of soil consolidation via prefabricated vertical drains Bari, Mohammad W.; Shahin, Mohamed A.; Soubra, Abdul-Hamid Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  1 Single vs Multi-drain Probabilistic Analyses of Soil Consolidation via Prefabricated Vertical Drains Mohammad W. Bari Research Associate, Dept. of Civil Engineering, Curtin University, Perth, Australia Mohamed A. Shahin Associate Professor, Dept. of Civil Engineering, Curtin University, Perth, Australia Abdul-Hamid Soubra Professor, Dept. of Civil Engineering, University of Nantes, Saint-Nazaire, France ABSTRACT: Natural soils are one of the most inherently variable in the ground. Although the significance of inherent soil variability in relation to reliable prediction of consolidation rates of soil deposits has long been realized, there have been few studies that addressed the issue of soil variability for the problem of ground improvement by prefabricated vertical drains (PVDs). Despite showing valuable insights into the impact of soil spatial variability on soil consolidation by PVDs, availab le stochastic works on this subject are based on a single drain (or unit cell) analysis. In a spatially variable soil, however, the condition of unit cell may be violated. Therefore, in a probabilistic context, it is necessary to assess the feasibility of performing an analysis based on the unit cell concept as compared to the multi-drain analysis. In this study, a rigorous stochastic finite element modeling approach that allows the nature of soil spatial variability to be considered in a quantifiable manner, both for the single and multi-drain cases, is presented. It is shown that with proper input statistics representative of a particular domain of interest, both single and multi-drain analyses yield almost identical results. This study also highlights the importance of proper modeling of soil spatial variability in design of ground improvement by PVDs. 1. INTRODUCTION The use of prefabricated vertical drains (PVDs) in combination with pre-loading is becoming one of the most commonly used methods for promoting radial drainage and accelerating the time rate of soil consolidation. Natural soils, however, are highly variable in the ground due to the uneven soil micro fabric, geological deposition and stress history. Soil consolidation by PVDs is strongly dependent on several spatially variable soil properties, most significantly is the coefficient of consolidation. The review of relevant literature has indicated that although the significance of inherent soil variability in relation to reliable prediction of consolidation rates in soil deposits has long been realized (Rowe 1972), there have been a few studies (Hong and Shang 1998; Zhou et al. 1999; Bari et al. 2012, 2013; Bari and Shahin 2014) that used stochastic approaches to investigate the problem of ground improvement by PVDs for spatially variable soils. Despite showing valuable insights into the impact of soil spatial variability on soil consolidation, available stochastic works of PVD-improved ground are based on an idealized single-drain (or unit cell) analysis instead of considering the actual full multi-drain situation.  In practice, soil improvement via PVDs typically consists of hundreds of drains installed in square or triangular patterns, with spacing varying between 1–3m. This means that the area being treated including each drain in a numerica l analysis can be significantly large and computationally too intensive. In order to reduce the computational effort, full three dimensiona l (3D) multi-drain system, is usually modeled by 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  2 considering only a “unit cell” of soil positioned with a central vertical drain, so that the consolidation problem can be analyzed at the unit cell level. In deterministic context, the single-drain ‘‘unit cell’’ analysis is often sufficient to investigate the overall consolidation behavior of soil (Indraratna and Redana 2000). However, in a spatially variable soil, the condition of the unit cell may be violated. Therefore, the aim of this paper is to investigate the feasibility of performing an equivalent unit cell analysis to substitute the multi-drain analysis in a probabilistic context.  In order to treat the soil spatial variability in most geotechnical engineering problems, a stochastic computational scheme that combine the finite element (FE) method and Monte Carlo technique is often used (e.g., Fenton and Griffiths 2005; Huang et al. 2010; Bari et al. 2013; Bari and Shahin 2014). The same approach is adopted in the present study which allows the nature of soil spatial variability to be considered in a quantifiable manner, both for the single and mult i-drain analyses. The approach involves the development of advanced numerical models that merge the local average subdivision (LAS) technique (Fenton and Vanmarcke 1990) of the random field theory (Vanmarcke 1984) and the finite element method into a Monte Carlo frame work. For the case of PVDs, the overall consolidation is governed by the radial (horizontal) flow of water rather than the vertical flow for the fact that the drainage length in the horizontal direction is much less than that of the vertical direction and the horizontal permeability is often much higher than the vertical permeability (Hansbo 1981). Under such reasoning, the soil consolidation in the current study is considered as a perfectly radial drainage problem where the single-drain influence area is approximated by a square (equivalent to a circular area). The results obtained from both the multi-drain analysis and idealized unit cell model are used to establish probability density functions relating to the degree of consolidation. In the sections that follow, the stochastic finite element Monte Carlo (FEMC) approach is described in some detail followed by detailed demonstration and discussion of the obtained results. 2. STOCHASTIC FINITE ELEMENT MONTE CARLO APPROACH As indicated earlier, the equivalence between the single and multi-drain cases is examined by employing a stochastic FEMC approach. The procedure of the stochastic FEMC approach is as follows: 1. Create a virtual soil profile that contains realizations of the designated soil properties, allowing the inherent soil spatial variability to be realistically simulated; 2. Incorporate the generated realizations of soil profile into a FE modeling of soil consolidation by PVDs; and 3. Repeat these steps numerous times using the Monte Carlo technique by creating new realizations of virtual soil profile and performing the subsequent FE analysis so that a series of consolidation responses can be obtained from which the statistica l distribution parameters of the output quantities can be estimated.  The above steps, as well as the numerica l procedures, are described below. 2.1. Simulation of virtual soil profiles In order to warrant the true influence of soil spatial variability for the problem in hand, virtual soil profiles that allow the rational distributions of the designated spatially variable soil properties across the soil mass need to be generated (based on a predefined probability density function (PDF) and a prescribed spatial correlation function) which can then be implemented in the FEM modeling. Prior to proceeding with this step, it is necessary to identify the soil properties that have the most significant impact on soil consolidation by PVDs so that they can be treated as random fields when creating the virtual soil profiles. As indicated earlier, spatial variability of several soil properties can affect soil consolidation by PVDs. However, as far as the 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  3 radial drainage is concerned, the coefficient of horizontal consolidation, ch, is the most significant random soil property affecting the behavior of soil consolidation by PVDs, as indicated by many researchers (e.g., Hong and Shang 1998; Zhou et al. 1999). Accordingly, in the current study, ch, is considered to be the only spatially variable soil property, while the other soil properties are held constant and treated deterministically so as to reduce the superfluous complexity of the problem. The spatial variability of ch is assumed to be characterized by lognormal distribution because the observation obtained from field tests data reported by Chang (1985) suggested that the variation of ch can be adequately modeled by a lognormal distribution. Based on the random field theory, random fields of a spatially variable soil property with lognormal distribution can be characterized by the soil property mean value, µ, variance, σ2 (can also be represented by the standard deviation, σ) and correlation length or scale of fluctuation (SOF), θ. The value of θ describes the limits of spatial continuity and can simply be defined as the distance over which a soil property shows considerable correlation between two spatial points. Therefore, a large value of θ indicates strong correlation (i.e., uniform soil property field), whereas a small value of θ implies weak correlation (i.e., erratic soil property field).  In this study, the LAS method (Fenton and Vanmarcke 1990) extracted from the random field theory (Vanmarcke 1984) is used to generate 2D random fields of ch. The LAS algorithm generates realizations of ch in the form of a grid of cells that are assigned locally averaged values of ch different from one another across the soil mass, by taking full account of the finite element size in the local averaging process, albeit remained constant within each element of the soil domain. In the process of simulating the realization of ch, correlated local averages of standard normal random field G(x) are first generated with zero mean, unit variance and spatial correlation function using the 2D LAS technique. The required lognormally distributed random field of ch defined by hc and hc is then obtained using the following transformation function (Fenton and Vanmarcke 1990):   icch xGc hhi lnlnexp                    (1)                                    where, x i and ihc are, respectively, the vector containing the coordinates of the center of the ith element and the soil property value assigned to that element; hcln and hcln are, respectively, the mean and standard deviation of the underlying normally distributed ch, i.e., ln(ch). The correlation coefficient between ch measured at a point x1 and a second point x2 is specified by an exponentially decaying spatial correlation function, ρ(τ), as follows (Fenton and Vanmarcke 1990): hc 2exp)(                       (2)  where, τ = |x1 - x2|. It should be noted that the spatial correlation function in Eq. (2) is assumed to be statistically isotropic, i.e., SOF in the x and y directions on a horizontal plane are assumed to be the same (i.e.,hhh cycxc   )()( ). This means that on a horizontal (x-y) plane, ch is spatially variable but it is spatially constant with infinite SOF in the vertical (z) direction. Although the correlation structures in a naturally occurring soil stratum are usually different in any spatial direction (i.e., anisotropic), the reason for assuming ch as an isotropic random field is that the correlation structure is more related to the formation process (i.e., layer deposition). Therefore, on a horizontal plane the spatial correlation structure of ch would have similar SOF in any direction. It is worthy to note that the spatial correlation length is estimated with respect to the underlying normally distributed field, i.e., ln(ch). 2.2. Finite element modeling incorporating soil spatial variability The soil profile simulated in the previous step with the specified spatial variation of ch can now 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  4 be mapped onto a refined FE mesh. An uncoupled consolidation analysis is then followed. A modified version of the FE computational scheme ‘‘Program 8.6’’ as presented in the book by Smith and Griffiths (2004) is used in this study to carry out all the analyses in which soil consolidation is treated as a 2D uncoupled problem under an axisymmetric condition. Originally “Program 8.6” was for general two (plane) or three dimensional analyses of uncoupled consolidat ion using an implicit time integration with the ‘‘theta’’ method. The authors of the current paper modified the source code of “Program 8.6” to allow axisymmetric and repetitive Monte-Carlo analyses.  In this study, the potentially complicated multi-drain influence boundary is approximated by assuming a square grid pattern of sixteen drains enclosing an area equivalent to the sum of all single drain influence areas (see Figure 1a). The spacing, S, between the drains and the equivalent radius of the drain, rw, are assumed to be equal to 0.95m and 0.032m, respectively. On the other hand, the spacing, S, in the multi-drain analysis represents the side length (S) of the square influence area in the single drain “unit cell” analysis (see Figure 1b). It should be noted that, during mandrel installation of PVDs, a disturbed zone (i.e., smear zone) of reduced permeability is produced. However, in the present study, no smear zone is considered bearing in mind that the main characteristics of the stochastic equivalence between the single and multi-drain consolidat ion needs to be known for the simplest case first, and more complex disparities on the subject is left for future refinement. In addition, for simplicity, the well resistance which may affect the rate of consolidation is also not considered. This is due to the fact that the discharge capacities of most PVDs available in the market are relatively high; hence, the impact of well resistance can be ignored in most practical cases, as suggested by many researchers (e.g., Chu et al. 2004).   (a)  (b) Figure 1: Realizations of PVD-improved ground: (a) 16 drains in a square grid pattern; (b) single drain in a square geometry  Generally speaking, the more elements used to discretize the domain of the problem, the greater the accuracy of the FE solution. However, a trade-off between accuracy and run-time efficiency is necessary. In the current study, a sensitivity analysis on two different FE meshes with element sizes of 0.05m and 0.025m is conducted. For a certain SOF, two random fields of two selected meshes are generated using the same seed value, and FE analysis is conducted. The results obtained from the two meshes are then compared to see if they are identical, otherwise finer meshes are generated and the previous process is repeated. Several different random seeds and SOFs are tested, for the highes t 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  5 coefficient of variation of ch considered in this study. It is found that a ratio of SOF to FE size ≥ 2 gives a reasonable precision. Based on this observation and in order to comply with the minimum correlation length used, a more refined mesh with an element size of 0.05m × 0.05m is adopted in the current study. The initial condition for the uncoupled analysis (i.e., no displacement degrees of freedom and only pore pressure degrees of freedom) is such that the excess pore pressure at all nodes (except at the nodes of the drain boundary) is set to be equal to 100kPa, while the excess pore pressure at each node of the drain boundary is set to be equal to zero. After the generation of a given realization and the subsequent implementation of the finite-element analysis of that realization, the corresponding degree of consolidation, U(t), at any consolidation time, t, is calculated based on the excess pore pressure concept with the help of the following expression: 01)( uutU                     (3) where, u0 and ū are the initial uniform and average excess pore water pressures, respectively. It has to be emphasized that the average excess pore pressure (ū) at any time of the consolidat ion process is calculated by numerically integrat ing the pore water pressure across the entire area of the mesh and dividing by the total mesh area. 2.3. Repetition of process based on the Monte Carlo technique By applying the Monte Carlo simulat ion technique, the process of generating a realizat ion of ch and the subsequent implementation of the FE analysis is repeated numerous times until an acceptable accuracy of the estimated statistics of U(t) is achieved. It was found that 2000 Monte Carlo simulations are sufficient to yield reasonably reproducible estimate of the first two moments (i.e., mean, μU, and standard deviation, U) of U(t). Each simulation of the Monte Carlo process involves the same hc and hc (i.e., standard deviation and SOF of ch); however, the spatial distribution of ch varies from one simulation to the next. The obtained U(t) from the suite of 2000 realizations of the Monte Carlo process are collated and μU and U of the degree of consolidation over the 2000 simulations are estimated using the method of moments. 3. PARAMETIC STUDIES Following the stochastic FEMC procedure set out above, parametric studies are performed to investigate the equivalence between the single and multi-drain analyses in terms of μU and U of the degree of consolidation. For this purpose, two groups of FEMC analysis are performed. In the first group, the point mean, standard deviation and SOF are assumed to be the same for both the single and multi-drain cases, while in the second group the point statistics are derived based on assumed local average statistics associated with the soil domain of interest. It should be noted that, the random fields are characterized by their point statistics, meaning thathc , hc and hc of ch are defined at the point level. However, the soil properties are rarely measured at the point level and most engineering measurements concerned with soil properties are performed on samples of some finite volume. Therefore, the measured soil properties are actually locally averaged over the sample volume. The point statistics associated with the local average measurements depends on several factors, namely: (i) the size of the sample over which the measurement represents an average; (ii) the correlation coefficient between all points in the soil domain; and (iii) the type of averaging that the observations represent. The size of the averaging domain, D, is taken into account to compute point statistics of ch in the second group of analysis. The details of each group analysis and the results obtained are described below. 3.1. Results of parametric studies considering same point statistics for both single and multi-drain cases The results obtained from the single and mult i-drain FEMC analyses employing the same input 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  6 random field parameters are compared in this section for different combinations of hcand hcwhile hcis kept at a fixed value equal to 15 m2/ year.  It should be noted that hcis presented herein by the normalized coefficient of variation, hc, where hhh ccc  /. The following values of hcandhcare considered:  hc = 25, 50 and 100 (%)  hc = 0.5, 1.0, 4.0 and 100 (m)  The abovementioned selected range of hc is typical to those reported in the literature (e.g., Beacher and Christian 2003). The SOF is less well-documented, particularly in the horizonta l direction. Phoon and Kulhway (1999) reported that the horizontal SOF typically ranges between 3 and 80 m. Accordingly, a wide range of SOF is selected in this study where its minimum and maximum values are specified to be 0.5m and 100m, respectively. A series of FEMC analyses for various combination of hc and hc are performed. The sensitivity of μU and σU to the statistically defined input data (i.e., hc and hc ) is examined in Figures 2−3 by expressing them as functions of the consolidation time t. The comparison between μU derived via the single and multi-drain FEMC simulations is examined in Figure 2. The effect of increasing hc on μU at a fixed value of hc = 0.5m is illustrated in Figure 2a, which indicates that μU obtained from the single drain case agrees very well with that obtained from the multi-dra in counterpart, for all cases of hc . For both cases, µU decreases with the increase of hc , and the decreasing rate of µU consistently increases with the increase of hc . On the other hand, Figure 2b shows the variation of μU as estimated via the single and multi-drain FEMC analyses, for various values of hc and at a fixed value of hc  = 50%. In general, it can be observed that even though the results for various θ are drawn in Figure 2b, they are embodied into a single curve, implying that the obtained results at different θ are very close and cannot be distinguished. The virtually identical curves for all θ demonstrate that µU is largely independent of θ. This is expected because in principle θ does not affect the local average mean of the process.    Figure 2. Effect of: (a) hc for hc = 0.5m; (b) hc  for hc =50% on U. The equivalence between the single and multi-drain analyses is further examined via matching the estimated σU at different values of hc and hc , as shown in Figure 3. It can be seen that σU obtained from the single drain cases is significantly higher than that obtained from the multi-drain cases and the difference in σU between the two solutions increases as hcincreases (see Figure 3a). This behavior can be explained by noting that the total flow from the vicinity of PVD is effectively an averaging 00,20,40,60,810,0005 0,005 0,05 0,5μUt (years)θch = 0.5mυch = 25%υch = 50%υch = 100%1 drain                 16 drains(a)00,20,40,60,810,0005 0,005 0,05 0,5μUt (years)υch = 50%θch = 0.5mθch = 1.0mθch = 4.0mθch = 100.0m1 drain             16 drains(b)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  7 process where high flow rates in some regions are offset by lower flow rates in the other regions. Notice also that, U(t) is determined by averaging the excess pore pressure over the entire mesh. The main impact of the local averaging is to reduce the variance and damp the contribution from the high frequency components. As the averaging domain is significantly smaller for the single drain case compared to the multi-drain domain, there is less variance reduction, resulting in higher σU in the single drain case than the multi-drain solution.   Figure 3.Effect of: (a) hc for hc = 0.5m; (b) hc  for hc =50% on σU.  The influence of hc on the compliance between the single and multi-drain solutions in terms of σU at a fixed value of hc  = 50% is shown in Figure 3b. It can be seen that considerable differences in σU obtained from the two solutions are found particularly when hc is as low as 4.0. On the other hand, little or no differences in σU are found for very high hc (e.g., 100.0m). This is due to the fact that when hc>> D (where D is the size of the problem), the variance reduction factor γ(D) →1.0 implying no variance reduction. It can be seen that the maximum σU occur at an intermediate t, while σU is zero at t = 0 and at large t. This behaviour can be explained by noting that U(t) approaches 0 and 1 as t approaches 0 and ∞ regardless of the variability of ch. 3.2. Results of parametric studies considering same local average statistics for both single and multi-drain cases As indicated earlier, soil property measurements are generally averages over a volume (or area). In this group of parametric study, it is assumed that the local average statistics of ch corresponding to an average over the area of the single drain and over the area of the multi-drain are the same. The mean, µD, and coefficient of variation, υD, of the local average measurement of ch are assumed to be equal to 15 m2/ year and 0.2, respectively. The given local average statistics are now needed to be transformed to point statistics for generating the random field of ch. Assuming that each local average measurement is deemed to be a geometric average, the relationship between the local average statistics and the ideal point mean, hc , and standard deviation, hc , are as follows (Fenton and Griffiths 2008):        DDDDch  211lnexp 2          (4)      11lnexp22DDcc hh           (5) where, γ(D) is the variance reduction factor corresponding to the underlying normal random field ln (ch). Considering the geometry of the single and multi-drain problem, γ(D) for various hc is computed numerically from the corresponding variance reduction function of the correlation structure shown in Eq. (2), as presented by Fenton and Griffiths (2008) and summarized in Table 1.    00,050,10,150,20,250,30,0005 0,005 0,05 0,5σUt (years)θch = 0.5m1 drain                     16 drainsυch = 25%υch = 50%υch = 100%(a)00,050,10,150,20,250,30,0005 0,005 0,05 0,5σ Ut (years)υch = 50%θch = 0.5mθch = 1.0mθch = 4.0mθch = 100.0m1 drain                   16 drains(b)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  8 Table 1 Variance reduction factor for various SOF. hc Single drain γ(D) 16 drains γ(D) 0.5 0.206 0.023 1.0 0.413 0.076 4.0 0.786 0.413 100.0 0.99 0.9613 By substituting the given µD, υD and computed values of γ(D) in Eqs. (4) and (5), hcand hc are computed for both the single and multi-drain problems and the results are shown in Table 2. Employing the computed hcand hccorresponding to each hc , a series of FEMC analyses is performed for both the single and multi-drain cases and the equivalence between μU and σU obtained from the two solutions are examined (Figure 4). Table 2 Estimated point mean and standard deviation computed from the given local average statistics. hc  Single drain  16 drains hc  hc  hc  hc  0.5 16.18 7.41 34.5 73.2 1.0 15.4 4.87 19.04 15.65 4.0 15.08 3.41 15.42 4.87 100.0 15.003 3.01 15.01 3.06 It can be seen from Figure 4 that both μU (Figure 4a) and U (Figure 4b) obtained from the single drain analysis agree well with those obtained from the multi-drain analysis, for all cases of hc , despite the slight discrepancy in μU and U when hc is as low as 0.5. Overall, the good agreement between the single and mult i-drain analyses in terms of μU and U indicates that stochastic equivalence between the unit cell and multi-drain solutions can be established by assigning representative parameters for their corresponding domain.   Figure 4.Effect of (a) hc  on U; (b) hc  on U for different point hc and hc computed from the same given local average statistics. 4. CONCLUSIONS This paper used the random field theory and finite element modeling to investigate the stochastic equivalence between the single drain “unit cell” and multi-drain solution for PVD-improved ground. The horizontal coefficient of consolidation, ch, was treated as the only random field and an uncoupled 2D finite element analysis was applied. Two groups of stochastic finite element Monte Carlo (FEMC) analyses were performed. In the first group, the point input statistical parameters were assumed to be the same for both the single and multi-drain cases. It was found that the mean degree of consolidat ion, μU, obtained from the single drain analysis agrees reasonably well with that obtained from the mult i-drain counterpart irrespective of the input parameters. However, a considerable difference in σU obtained from the two solutions was found except for very high scale of fluctuation. In the 00,20,40,60,810,0001 0,001 0,01 0,1 1μUt (years)1 drain                 16 drainsθch = 0.5mθch = 1.0mθch = 4.0mθch = 100.0m(a)00,050,10,150,20,250,30,0001 0,001 0,01 0,1 1σ Ut (years)1 drain                        16 drainsθch = 0.5mθch = 1.0mθch = 4.0mθch = 100.0m(b)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12  Vancouver, Canada, July 12-15, 2015  9 second group, it was assumed that the local average statistics of ch corresponding to an average over the single drain area and over the area of the multi-drain are the same. By computing corresponding point statistics and performing FEMC analysis it was found that both μU and U obtained from the single drain analysis agrees very well with those obtained from the multi-drain analysis, for all selected scales of fluctuation. Overall, it was shown that to establish stochastic equivalence between the unit cell and multi-drain analyses, proper input statistics representative to the soil domain of interest need to be used. This study also demonstrated the importance of proper modeling of soil spatial variability in design of ground improvement by PVDs.     5. REFERENCES Bari, M. W., and Shahin, M. A., 2014. Probabilistic design of ground improvement by vertical drains for soil of spatially variable coefficient of consolidation. Geotextiles and Geomembranes, 42(1), 1-14. Bari, M. W., Shahin, M. A., and Nikraz, H. R., 2012. Effects of soil spatial variability on axisymmetric versus plane strain analyses of ground improvement by prefabricated vertical drains. 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