AbstractWhen simulating the vertical drain method using a soil–water coupled finite-element analysis, we the macroelement method can be adopted to provide the water absorption function of drains to individual elements approximately. A previous study added the discharge function of vertical drains to the technique so that the occurrence of well resistance can be considered in accordance with its mechanism. This paper verified the approximation accuracy of this new macroelement method in quasi-static problems. The macroelement method was derived under the assumption that the axisymmetric pore water pressure distribution is equal Barron’s solution, but the first numerical example demonstrated that the technique has high approximation accuracy even for nonaxisymmetric problems by appropriately converting the permeability of a drain. The second calculation example included a number of drains and applied the vacuum consolidation method to clayey ground containing a middle sand layer. In this calculation, there was remarkable well resistance, and pore water flowed from the unimproved region to the improved region or between the effective areas of each drain. The calculation results indicated the superiority of the new macroelement method in terms of accurate approximations even for such a complex problem. Moreover, this study demonstrated the effectiveness of waterproof sealing, a countermeasure for sucking pore water from the middle sand layer, using the new macroelement method.IntroductionSimulating the vertical drains method in a soil–water coupled finite-element method requires determining how to model the drains. The simplest method is to model the vertical drains with finite elements by dividing the mesh finely in the improved region (e.g., Borges 2004). Although this is an ideal approach from the perspective of accurately incorporating the effects of the drains, it is problematic because it requires a three-dimensional (3D) analysis, and the total number of elements vastly increases. To resolve this problem, many studies have been conducted using homogenization methods to macroscopically take into account the consolidation promotion effects via drains without dealing with heterogeneity occurring around a single drain. The majority of such studies involved virtually expressing the effect of vertical drains by raising the permeability coefficient in the vertical direction of the ground to above its actual level (e.g., Ng and Tan 2015). Although this is simple and convenient, it does not successfully describe the various phenomena that occur in applying the vacuum consolidation method to a laminated ground containing highly permeable layers, as pointed out by Nguyen et al. (2015). This is because the method of macroscopically changing the permeability of the ground, which does not model the function of vertical drains directly, cannot describe the phenomenon in which pore water in highly permeable layers is pumped to the ground surface through drains, not via impermeable layers. On the other hand, using a more principled method to take into account the mechanism of the vertical drain method, Sekiguchi et al. (1986) attempted to express the consolidation promotion effects of the vertical drain method by providing the absorption function of drains for the individual elements in an improved region. They named this method the macroelement method. Even under two-dimensional (2D) plane strain conditions, this is an effective method of accurately incorporating the water absorption function of drains. Recently, this method has been used widely (e.g., Hirata et al. 2012, 2015).To extend the macroelement method function, Yamada et al. (2015a) proposed a method that considers the water pressure within the drain as an unknown quantity, which is provided as an analysis condition in the existing methods. By adding a continuity equation for drains to the governing equation, a discharge function of drains was added to the method while maintaining the absorption function of drains provided in the original macroelement method. This rational expansion allows generating well resistance depending on given conditions. However, the original macroelement method proposed by Sekiguchi et al. (1986) requires the drain spacing to be equal to or an integer multiple of the mesh partition width. Yamada et al. (2015a) made a formulation to avoid this restriction. Additionally, the original macroelement method restricts the horizontal exchange of pore water between adjacent macroelements, and treats the flow of interstitial water from the outside to inside the improved regions by designating the elements adjacent to the improved regions as transitional macroelements. On the other hand, Yamada et al. (2015a) removed these supplementary conditions, and left all the water flow through the pores of a ground to the inherent permeability function of the soil–water coupled analysis based on Darcy’s law. The innovation of this formulation and the abolition of the supplementary conditions allowed us to evaluate the effect of drain spacing on ground improvement by changing only the parameters without remeshing. Despite being simple, this improvement enhanced the user-friendliness of the macroelement method as a homogenization method drastically. Furthermore, these improvements are expected to promote the potential of the macroelement method to be applied to problems with significant exchanges of pore water between the inside and the outside of the improved region, for example, owing to the existence of middle sand layers. However, these improvements may decrease the approximation accuracy.The ultimate objective of this study was to verify the approximation accuracy of the extended macroelement method. Yamada et al. (2015a, b) verified the approximation accuracy under axisymmetric conditions by analyzing a consolidation problem involving a columnar drain and the effective area surrounding it. However, the verification did not cover the approximation accuracy in a large-scale problem with high heterogeneity in which multiple drains are installed. This study verified the approximation accuracy of the extended macroelement method in such challenging problems by modeling in two different ways, comparing the strictly expressed 3D analysis of drains using a mesh partition with 2D plane strain analysis applying the macroelement method to a comparatively rough mesh.We first investigated the situation in which band-shaped drains are arranged in a square configuration and selected one of these drains and its effective area as an analytical object. The approximation accuracy of the new macroelement method in a situation without axisymmetry was clarified through this analysis.Next, we addressed a multidimensional problem including a number of drains in which the vacuum consolidation method was applied to clayey ground containing a middle sand layer. In this case study, shear deformation owing to embankment and vacuum pressure loading caused nonuniform displacement in the horizontal direction. Furthermore, for this kind of problem, because the vertical drains typically absorb large quantities of water from the sand layer, the depressurization effect of the vacuum consolidation method is reduced. At the same time, as the water pressure outside the improved region decreases through the sand layer, settlement occurs in the surrounding ground. Therefore, in this problem, a calculation applying the new macroelement method cannot achieve high accuracy if the new method does not properly account for well resistance, and the soil–water coupled analysis does not bring about the expected benefits on horizontal water flow. In other words, the main points of the expansions were tested exhaustively in this case study.This study focused on the verification of the approximation accuracy of the new macroelement method. Nguyen et al. (2015) validated the method by comparing with field observation records and/or model experiment measurement results. Nguyen et al. (2015) also discussed the advantages of the extended macroelement method by comparing it with the mass permeability method (Asaoka et al. 1995; Tashiro et al. 2015), which expresses the effect of drains as a change in the macroscopic permeability of the ground.Outline of Soil–Water Finite-Deformation Analysis with Macroelement MethodThe authors implemented the macroelement method with water absorption and discharge functions of drains into a soil–water coupled finite-deformation code with an inertial term (Yamada et al. 2015a; Noda et al. 2015). Using this analysis code, the approximation accuracy of the macroelement method was verified for quasi-static problems.Prior to the verification, we present an overview of the aforementioned analysis method. The governing equations are the following three formulas: (1) M{v¨N}+K{vN}−LTu˙={f˙}(2) kgL{v˙N}−L{vN}=∑i=1mαi(h−hi)ρwg+κ(h−hD)ρwg(3) κ(h−hD)ρwg=∑j=12βj(hD−hDj)ρwgwhere an overdot denotes the material time derivative; {vN} = nodal velocity vector; u = representative value of pore-water pressure of each element; M = mass matrix; K = tangent stiffness matrix; L = matrix for converting {vN} to element volume change rate; {f˙} = nodal force rate vector; h and hi = total heads corresponding to representative values of water pressure for an element and adjacent elements, respectively; k = permeability coefficient of ground; g = magnitude of gravitational acceleration; αi = coefficient of pore-water flow to adjacent elements; ρw = density of water; and m = number of boundary surfaces for each element. Eq. (1) is a rate-type equation of motion; Eq. (2) is a soil-water-drain coupled equation; and Eq. (3) is a continuity equation of drains. Each matrix and its coefficients are given based on the theory of finite deformation. Their concrete forms were detailed by Noda et al. (2008, 2015) and Yamada et al. (2015a).The second term on the right-hand side of Eq. (2) is the term added to apply the absorption function of drains to each macroelement. Eq. (3) is the formula introduced to express the discharge function of drains. With the soil–water coupled analysis used in this study, {vN} and u are unknown in each element, but with the new macroelement method, the representative value hD for total head within the drain (or representative value for water pressure within the drain uD) also is treated as unknown. Eq. (3) compensates for the increased number of unknowns in the problem. In Eqs. (2) and (3), κ is the coefficient of pore-water flow from the soil to drain (i.e., the coefficient governing the absorption function), and βj is the coefficient of water flow through the virtual drain (i.e., the coefficient governing the discharge function); in Eq. (3), hDj is the representative value of total head within the drain included in the elements on the upper and lower sides of the macroelement discussed here (Fig. 1). Parameters κ and βj are expressed as follows: (4) (5) F(n)=n2n2−1lnn−3n2−14n2,n=dedw(6) where V = current volume of each element; de = equivalent diameter of effective area corresponding to a single drain; dw = diameter of a circular drain; kw = permeability coefficient of a circular drain; lj = vectors connecting element centers; lj = magnitudes thereof; nj and sj = outward-facing normal vector and area of boundary surface j of element; and a dot denotes the inner product of vectors.Eqs. (4)–(6) are derived from the assumption that the water pressure distribution around the drain is equivalent to the solution by Barron (1948) of the radial linear elastic consolidation theory around the drain under the equal strain condition. However, in the real problem, the axisymmetric condition does not strictly hold around a drain. This was taken into account by converting the parameters de, dw, and kw (section “Analysis of Non-Axisymmetric Problem Including a Single Drain”). Furthermore, because the plane strain condition and the mesh width is separated from the drain spacing, the flow coefficients κ and the βj are formulated by converting them into per unit volume or per unit area, respectively. Specifically, κ is formulated so that the pore-water flow per unit volume from the soil to the drain in each element coincides with that in Barron’s solution, and βj is formulated so that the area ratio of the cross section of a virtual drain to the boundary between vertically connected macroelements is equal to the area ratio of a cylindrical drain to the effective circle. Yamada et al. (2015a) detailed the derivation process of the extended macroelement method.The first term on the left-hand side of Eqs. (1) and (2) is the inertial term. The presence of these terms means that the analysis code also can be used to solve dynamic problems. Therefore, this code can be applied to the pore water pressure dissipation method, which is a liquefaction countermeasure method (Noda et al. 2015; Nonaka et al. 2017a, b). On the other hand, if hardly any acceleration occurs, the analysis code generates the same solution as the quasi-static analysis, which does not treat these inertial terms. In this study, the examples applied for the verification were quasi-static consolidation problems, but we used the aforementioned analysis code that employs the inertial terms.Analysis of Non-Axisymmetric Problem Including a Single DrainAs mentioned in the section “Outline of Soil–Water Finite-Deformation Analysis with Macroelement Method,” the macroelement method was developed using the solution by Barron (1948) obtained under the assumption of axisymmetry with the drain as the central axis. However, when drains are arranged in a square pattern or triangle pattern, the effective area of a single drain does not become columnar. Furthermore, in recent years, the majority of artificial drains have been band-shaped. For this reason, it is necessary to convert and determine the equivalent diameter of an effective area de, the diameter of a circular drain dw, and the permeability coefficient of a circular drain kw. In particular, there has been much discussion regarding the method of converting a band-shaped drain into a circular drain (e.g., Hansbo 1979; Rixner et al. 1986). This section verified the approximation accuracy of the new macroelement method in a problem targeting a single band-shaped drain and its effective area by comparing a 2D analysis using the macroelement method applying several conversion formulas with 3D analysis.Analysis ConditionsFor the constitutive model of a soil skeleton, the SYS Cam-clay elastoplastic constitutive model based on the soil skeleton structure concept (Asaoka et al. 2002) was applied. This constitutive model can reproduce the mechanical behavior of a wide range of soil materials, including natural deposited clay and sand. In this model, the material parameters common to the modified Cam-clay model basically are determined from normally consolidated and fully remolded samples. Moreover, the initial values of the internal variables and the material parameters of their evolution rules are determined by fitting from the results of laboratory tests of undisturbed samples.Figs. 2 and 3 show the respective finite-element mesh and boundary conditions of a 3D model (exact model) and a 2D plane strain model (approximate model) using the macroelement method. The exact model designated a quarter section as an analysis region and modeled the vertical drain with normal finite elements. On the other hand, the approximate model imposed plane strain conditions on a one-dimensional (1D) mesh and applied the macroelement method. The macroelement method considered the water absorption and discharge functions of drains but ignored their stiffness. In the exact model, whereas a permeability coefficient higher than that of the ground is applied to elements equivalent to the drain, the same mechanical properties as of the soil elements are applied to them.We envisaged a situation in which band-shaped drains (width a=100  mm, and thickness b=2  mm) spaced at d=1  m in a square configuration were installed vertically into a clay layer with a thickness of 20 m. The soil surface was exposed to the atmosphere, and the bottom of the clay layer was assumed to be impermeable. A uniformly distributed load of 150 kPa was applied vertically at the ground surface, and increased at a constant rate over a 10-day period. After that, analysis continued until consolidation was almost completed while keeping the surface load constant. Table 1 presents the material constants and initial values of the ground. These are the values for a typical alluvial clay determined by Yamada et al. (2015a). The initial stress state was determined by considering the self-weight of the ground, with a small ground surface load (9.81  kN/m2) applied, and the initial water pressure was set to the static water pressure distribution. The overconsolidation ratio was calculated based on the conditional equation that the state variables must satisfy (Noda et al. 2005).Table 1. Material constants and initial values of groundTable 1. Material constants and initial values of groundParameterClay Critical state index, M1.60 NCL intercept, N2.51 Compression index, λ˜0.300 Swelling index, κ˜0.020 Poisson’s ratio, ν0.3 Ratio of −Dvp to Dsp, cs0.3 Degradation index of structure, a0.8 Degradation index of overconsolidation, m5.0 Rotational hardening index, br0.001 Limitation of rotational hardening, mb1.00 Soil particle density, ρs (g/cm3)2.754 Permeability index, k (cm/s)1.0×10−7 Specific volume, v03.20 Stress ratio, η00.375 Degree of structure, 1/R0*12.0Degree of anisotropy, ζ00.107The equivalent diameter de was determined by equalizing the effective area (i.e., the improved cross-sectional area corresponding to a single drain) before and after conversion. Because this study assumed that the drains were arranged in a square configuration, de was calculated as follows: (7) de=2dπ: square patternIn order to convert a band-shaped drain to a columnar one, this study uses the following three methods (8) dw=2(a+b)π: based on equivalenet outer circumference of drain(9) dw=a+b2: one-half the sum of width and thickness of drain(10) dw=2abπ: based on equivalent cross-sectional area of drainEq. (8) was derived by equalizing the outer circumference of each drain (Hansbo 1979), Eq. (9) assigns one-half of the sum of the width and thickness of the band-shaped drain to the diameter of the circular drain (Rixner et al. 1986), and Eq. (10) was obtained by equalizing the cross-sectional area of each drain. Eqs. (8) and (9) are proposed mainly from the viewpoint of fairly evaluating the absorption function of drains, and changes in the cross-sectional area then can be associated with an altered impact on the discharge function of drains. With the new macroelement method equipped with the absorption and discharge functions of vertical drains, it is necessary to convert the permeability coefficient of a drain to satisfy the following equation to ensure that the drain discharge function does not change through conversion to a columnar drain: (11) where a˜w(=ab) and k˜w = cross-sectional area and permeability coefficient of band-shaped drain before conversion, respectively; and a˜w(=πdw2/4) and kw = cross-sectional area and permeability coefficient of columnar drain after conversion.Table 2 presents the analysis cases and material constants for the macroelement method. The permeability coefficient of a drain was set to a relatively small value, corresponding to a significant degree of well resistance. Case B involved application of the conversion in Eq. (11), whereas Case A did not. The flow coefficient κ, which expresses the absorption function of a drain, is in inverse proportion to F(n) [Eq. (4)]. Therefore, the smaller F(n), the greater is the absorption function of a drain. On the other hand, the flow coefficient βj, which represents the drainage function of a drain, is proportional to kw/n2 [Eq. (6)]. Therefore, the higher kw/n2, the greater is the drainage ability per unit improved region of the drain.Table 2. Case studies and material constants of macroelement methodTable 2. Case studies and material constants of macroelement methodConversion method of dwde (m)dw (m)F(n)kw (cm/s)kw/n2 (cm/s)kw (cm/s)kw/n2 (cm/s)Equal circumference [Eq. (8)]1.130.0662.105.0×10−2 (kw=k˜w)1.72×10−45.80×10−31.99×10−5Half of sum of width and thickness [Eq. (9)]1.130.0522.341.06×10−49.42×10−31.99×10−5Equal cross-section area [Eq. (10)]1.130.0233.171.99×10−55.0×10−21.99×10−5In the vertical drain method using a band-shaped drain, several previous studies (e.g., Aboshi et al. 2001; Shinsha et al. 2016) pointed out that ground settlement is affected by a virtual increase in the conversion diameter of a drain and a decrease in the water discharge ability of the drain due to bending deformation. Although such problems could occur in reality, they were not considered within the scope of this study.Analysis ResultsFigs. 4–7 present the settlement of the ground surface and the isochrones of excess pore-water pressure and water pressure within the drain after leaving the load fixed for a period of time (10, 100, and 400 days after application) in Case A and Case B. The ground surface settlement in the exact model indicates the value at the node of the center of the drain.In Case A, the approximate model using Eq. (10), based on the area conversion, achieved higher approximation accuracy than did the approximate model applying other equations. In Eqs. (8) and (9), the value of dw was higher than that in Eq. (10); therefore, when not using Eq. (11), the discharge ability of the drain was higher than that of the band-shaped drain (Table 2). This caused an underestimation of the well resistance; thus, the progress of consolidation of the approximate model using Eqs. (8) and (9) was faster than that of the exact model.In Case B, in which the permeability coefficient of the drain kw was converted, the approximate models reproduced the result of the exact model better than in Case A regardless of the conversion formulas. Consolidation proceeded increasingly slowly in the order of Eqs. (8)–(10). This was because the absorption function of the drain decreased slightly in the same order. As explained previously, the absorption function κ is in inverse proportion to F(n).The macroelement method proposed by Yamada et al. (2015a) can produce accurate approximations even in problems without axisymmetry (Figs. 4–7). However, when Eqs. (8) and (9) are applied as a conversion method of dw, the conversion of the permeability of a drain by Eq. (11) must be performed to avoid an underestimation of well resistance.Because there were relatively few differences due to the conversion method, it is difficult to assign clear superiority or inferiority among the three conversion equations, but under the conditions dealt with here, the results of analysis from the approximate model using the conversion method of Eq. (9) were the closest to those of the exact model. For this reason, conversion Eq. (9) was used for calculation in the next section.Analysis of Complicated Problem Including Multiple DrainsThis section verified the approximation accuracy of the macroelement method for large-scale problems that include multiple drains. When embankment loading and vacuum consolidation are performed on ground in which a number of drains are installed, the situation causes several complex phenomena, such as nonuniform ground deformation even in the horizontal direction, inflow and outflow of pore water from the improved region, and exchange of pore water between the effective area of each drain.The original macroelement method proposed by Sekiguchi et al. (1986) assumes the following conditions: 1.The mesh division of width and depth must be matched to the drain spacing when drains are deployed in a square pattern.2.Macroelements situated transversely adjacent to unimproved regions are designated transitional macroelements, whereas the remaining macroelements are designated basic macroelements. The transitional elements mediate the pore water flow from regular elements in the unimproved region to the basic macroelements in the improved region.3.There is no exchange of pore water via soil between transversely adjacent macroelements; hence, αi=0 in the horizontal direction.On the other hand, in the macroelement method, the formulation is implemented so as not to be restricted by Condition 1. Because we considered that the influences of pore-water inflow from the unimproved region to the improved region and water pressure drop outside the improved region are considered naturally, based on the permeability in accordance with Darcy’s law which is inherent in the soil–water coupled analysis, transitional macroelements from Condition 2 were not used. We expect that the effect of soil–water coupled analysis also produces the exchange of pore water between the effective areas of each drain automatically; therefore, the restrictions from Condition 3 are removed. As previously mentioned, these improvements make it possible to evaluate the effect of drain spacing on a single mesh. Although Yamada et al. (2015a) confirmed that mesh division has little influence on analysis results, further verification is required from the viewpoint of securing the adequacy of removing the constraints of Conditions 2 and 3. The following problem is suitable for the verification of removing these supplementary conditions.A vacuum consolidation method often is applied along with the use of vertical drains. The vacuum consolidation method is a method that can be expected to offset the outward lateral displacement due to embankment loading, with an inward displacement initiated via the application of vacuum pressure. Active research into embankment construction and stability management when employing vacuum consolidation has been conducted (e.g., Matsumoto et al. 2003; Ong and Chai 2011), and application in the field is proceeding as well. The problem is that if a continuous middle sand layer exists in the ground, the drains will suck up a large amount of pore water from this sand layer. Therefore, the effect of decompression from vacuum consolidation decreases due to the occurrence of well resistance, and the surrounding ground settles widely because the water pressure of the sand layer decreases even outside the improved region. As a countermeasure to this, a method of attaching an impermeable seal around a vertical drain passing through a middle sand layer has been developed, and research on the countermeasure effect also has been conducted (e.g., Chai et al. 2006). To represent adequately the phenomena that occur when a vacuum consolidation method is applied to a clayey ground containing a middle sand layer, together with the aforementioned countermeasure, an analysis tool must be able to model accurately the well resistance and movement of pore water inside and outside the improved region. We examined the approximation accuracy of the macroelement method using a case study of such a situation.Analysis ConditionsThe same analysis code as in the previous section was used. As the target of analysis, we used the aforementioned problem. Figs. 8 and 9 show the finite-element mesh and boundary conditions of a 3D model (exact model) and a 2D plane strain model (approximate model) using the macroelement method, respectively. A sand layer 3 m thick with high permeability was interposed between the clay layers. Symmetry was assumed around the center of the embankment. In the exact model, as in the previous section, the permeability of the element corresponding to the drain was increased. Assuming that the same pattern was repeated regularly in the out-of-plane direction of the embankment, the analysis region in the y-direction was set to one-half the length of the drain spacing (0.6 m). By fixing the motion of the nodes in the y-direction on the z, x-plane, the plane strain condition was satisfied macroscopically. In the approximate model, the macroelement method was applied to the elements designated as the improved region in Fig. 9.In both models, the ground surface was set as a permeable boundary (exposed to atmospheric pressure). A permeable boundary condition (constant water level) also was specified on the bottom surface. The undrained condition was given to the left-hand end because of the assumption of a half section. On the other hand, the drained condition (constant water level) was given on the right-hand end due to the continuity of the ground.Table 3 presents material constants and initial values of the ground and embankment. The values for typical alluvial clay and silica sand determined by Yamada et al. (2015a) were used for each layer. For the embankment, the values for crushed and compacted mudstone determined by Sakai and Nakano (2011) were used. The method of assigning the initial stress state and water pressure was the same as in the previous section.Table 3. Material constants and initial values of ground and embankmentTable 3. Material constants and initial values of ground and embankmentParameterClay 1SandClay 2Embankment Critical state index, M1.601.351.501.40 NCL intercept, N2.511.953.002.09 Compression index, λ˜0.3000.0860.2860.1 Swelling index, κ˜0.0200.0030.0240.03 Poisson’s ratio, ν0. Ratio of −Dvp to Dsp, cs0. Degradation index of structure, a0.85.00.350.3 Degradation index of overconsolidation, m5. Rotational hardening index, br0.0015.0000.0300.3 Limitation of rotational hardening, mb1.000.551.000.5 Soil particle density, ρs (g/cm3)2.7542.7872.7542.730 Permeability index, k (cm/s)1.0×10−75.0×10−21.5×10−81.0×10−4 Specific volume, v03.271.843.602.14 Stress ratio, η00.3750.3750.3750.663 Degree of structure, 1/R0*7.01.712.07.5 Degree of anisotropy, ζ00.1070.1070.1070.663As a ground improvement method, the vacuum consolidation method using a capped drain was assumed. The range of improvement in the horizontal direction was set to be equal to the embankment width. In the depth direction, the range of improvement was offset 2  m from the ground surface and the base of Clay layer 2 for sealing purposes. In the range of improvement, it was assumed that a band-shaped drain with a width a=0.1  m, a thickness b=0.005  m, and a permeability coefficient of the drain k˜w=1.0  cm/s was arranged in a square pattern with d=1.2-m spacing. The upper and lower ends of the drain were set as the permeable boundary (atmospheric pressure) and the impermeable boundary, respectively. From these values, de, dw, and kw were calculated using the conversion Eqs. (7), (9), and (11), respectively. Table 4 summarizes the material constants of the macroelement method.Table 4. Material constants of macroelement methodTable 4. Material constants of macroelement methodParameterValueEquivalent diameter, de (m)1.35Diameter of circular drain, dw (m)0.053Permeability coefficient of circular drain, kw (cm/s)2.31×10−1Table 5 presents the analysis cases, and Fig. 10 shows the load (vacuum pressure and embankment height) history of each case. The histories of the height of embankment show the calculation results of the approximate model. The embankment height of the approximate model and that of the exact model were the same immediately after the construction of the embankment at each stage, but the subsequent heights differed due to the influence of the settlement. In Case 1, unimproved ground was assumed. Case 2 presumed a situation in which vacuum consolidation was not performed, in order to estimate only the improvement effects of the drains. In Case 3, to apply the vacuum consolidation method, the water pressure at the upper end of the drains was reduced by 90  kN/m2 over the duration of 1 day before loading the embankment. After 140 days from the initiation of vacuum consolidation, the water pressure at the upper end of drains was restored. In Case 4, as a countermeasure for sucking pore water from the middle sand layer, it was assumed that a waterproof seal was applied to the drains in the vicinity of the sand layer (Figs. 8 and 9). In the exact model, the waterproof seal was represented by imposing the undrained condition at the boundaries between the drains and the ground over the height of the seal. In the approximate model, the waterproof seal was represented by setting κ=0 in Eq. (4) in the macroelements of the relevant part. In the new macroelement method, it is possible to lose only the water absorption function while maintaining the discharge function of the drains in such a simple manner. Conditions in areas other than the waterproof seal were the same as in Case 3. In each case, an embankment with a height of 6 m was constructed over 30 days, and it was left until consolidation was almost complete. The final height of the embankment differed among the cases (Fig. 10) due to the difference in generated settlement.Table 5. Case studiesCaseVertical drainVacuumSealingCase 1 (unimproved)NoNoNoCase 2 (only vertical drain)YesNoNoCase 3 (vacuum consolidation without sealing)YesYesNoCase 4 (vacuum consolidation with sealing)YesYesYesAnalysis ResultsFigs. 11–14 show contour maps comparing results of the exact and approximate models for the changes of pore water pressure and specific volume in Cases 1–4, respectively. The results were compared at four different times, namely 20 days after the initial state (before embankment construction), 50 days after the initial state (after embankment construction), 140 days after the initial state (at vacuum consolidation stoppage for Case 3 and Case 4), and after 84,000 days (after convergence of consolidation). In the exact model, the analysis results for the cross-section 0.4 m from drains in the y-direction is shown. In Case 1 (unimproved), the excess pore-water pressure induced by embankment loading in the clay layer hardly dissipated even after 140 days. On the other hand, in Case 2 (only vertical drain), there was a clear progression of consolidation after 140 days. Clay layer 2 dissipated water pressure faster than did Clay layer 1 due to the higher overburden in this layer. In Case 3 (vacuum consolidation without sealing), Clay layer 1 was depressurized markedly due to vacuum consolidation before embankment loading, but because of the influence of a large amount of water being sucked from the middle sand layer, Clay layer 2 did not depressurize sufficiently. In addition, in the middle sand layer, depressurization occurred outside the improved region. However, the dissipation of positive excess pore-water pressure in Clay layer 1 was faster than in Case 2. In Case 4 (vacuum consolidation with sealing), suction of pore water from the middle sand layer was suppressed, and as a result, the decompression effect reached Clay layer 2. The dissipation of water pressure was faster than in Case 3, and the speed of consolidation increased in both Clay layer 1 and Clay layer 2. Moreover, there was no more decompression in the sand layer. The approximate model using the macroelement method had values that were close to those of the exact model, in any case and at any time.Next, we considered the deformation of the ground. Fig. 15 shows the time–settlement relationship at the ground surface at the center of the embankment. The result of the exact model was the value at the node in contact with the drain. Almost the same results were obtained at other nodes in the y-direction. In Case 1, it took about 10,000 days after embankment loading for the majority of consolidation settlement to occur. In Case 2, consolidation was promoted by the effect of drains, and the time required for appreciable consolidation settlement was shortened to about 1/10 that in Case 1. In Cases 3 and 4, consolidation was promoted further, and the final settlement also was suppressed, although slightly. This was because of the suppression of shear deformation due to embankment loading, resulting from the consolidation promotion effect by vertical drains and the preloading effect by vacuum pressure. In Case 3 and Case 4, compared with Case 1 and Case 2, the approximate model tended to estimate faster settlement than the exact model. This was because Case 3 and Case 4 both had high heterogeneity in the radial direction of a drain in the exact model compared with that in the other cases. Because the change in material properties is treated as the average of the element in the macroelement method, the higher the heterogeneity around a single drain, the lower may be the approximation accuracy. Nevertheless, the difference in the settlement between the exact model and the approximate model was less than 5% at any given time.Next, we considered the deformation of the improved region. Fig. 16 shows the horizontal displacement at the right-hand end of the improved region. The horizontal displacement is relative to the center node at the bottom of the analysis region. Results at four time points—20, 50, and 140 days after the initial state, and after the completion of consolidation—are shown. In unimproved Case 1, a maximum lateral displacement of about 60 cm occurred when consolidation converged. In Case 2 of improvement with drain only, lateral displacement was suppressed because the soil had higher shear resistance following accelerated consolidation. In Case 3 and Case 4 with the vacuum consolidation method, the ground was dragged to the inward side of the improved region by the vacuum consolidation effect before the embankment loading, and the outward lateral displacement due to the embankment loading was suppressed compared with that in the other cases. In particular, Case 4 with a waterproof seal had the greatest reduction in lateral displacement. The approximate model captured the characteristics of these lateral displacements and quantitatively had results similar to those of the exact model.Fig. 17 shows the ground surface settlement. In Case 3 and Case 4, the approximate model tended to predict the progress of consolidation somewhat earlier than did the exact model, but the ground surface shape was almost the same in both models. In addition, comparing the ground surface shapes of Case 3 and Case 4 after 20 days showed that the settlement in Case 4 exceeded that in Case 3 within the improved region, whereas the settlement in Case 3 exceeded that in Case 4 outside the improved region. Due to the intervention of the sand layer, in Case 3 the decompression effect in Clay layer 2 was reduced, and the water pressure in the sand layer outside the improved region decreased (Figs. 13 and 14). In Case 4, this did not occur due to the function of the waterproof seal. The approximate model accurately captured these phenomena.The preceding results demonstrate that the proposed macroelement method can approximate water pressure change and ground deformation with high accuracy, even in such a large-scale problem under a complicated loading condition. Furthermore, because the problem dealt with in this section was a problem that caused a flow of pore water over the improvement region and the effective area of each drain, it was possible to confirm the validity of removing supplementary conditions imposed by the original macroelement method.Comparison of Computation TimeFinally, we compared the computation time in the exact model and the approximate model. In both cases, it took 8,400 time steps to calculate 84,000 days until the end of appreciable consolidation settlement. A convergence calculation was performed for each step. In both cases, the calculation environment in Table 6 was used. Table 7 presents the calculation time for both models. These calculation times are average values of Case 2 to Case 4 improved by drains. The calculation time of the approximate model was about 1/180 that required for the exact model. The macroelement method, including the associated reduction in mesh size, brings about significant improvement in calculation cost.Table 6. Computing environmentTable 6. Computing environmentCentral processing unitMemoryIntel Core i7-3970X 3.50 GHz16.0 GBTable 7. Average computation timeTable 7. Average computation timeModelNumber of elementsDegrees of freedomComputation time [s (days)]Exact77,000347,2161,323,700 (15.3)Approximate4,04013,5387,330 (0.08)SummaryTo verify the approximation accuracy of the new macroelement method developed by the authors for quasi-static problems, we compared the results of 2D plane strain analysis using the macroelement method with those of 3D analysis in which drains were represented exactly by dividing finite-element meshes finely. The main findings are as follows: 1.The macroelement method was derived assuming a water pressure distribution equal to Barron’s solution, which is obtained under the axisymmetric condition around a drain, but by converting the model parameters appropriately, high approximation accuracy can be achieved even in non-axisymmetric problems. It is necessary to convert the permeability coefficient of the drain so that the drainage capacity does not change by conversion, because the new macroelement method has a discharge function in addition to the water absorption function of drains.2.Even in large-scale problems involving multiple drains and complicated loading histories, the proposed macroelement method has high approximation accuracy. Its accuracy is high even in complicated ground environments in which, due to the existence of a middle sand layer, the improvement effect of vacuum consolidation decreases, and surrounding ground settlement occurs.3.The method performs with high accuracy in complicated problem because, in addition to the discharge function that was added to the macroelement method, the supplementary conditions imposed on the original macroelement method also were removed, and water flow was made to observe Darcy’s Law, which the soil–water coupled analysis possesses inherently.4.Application of the macroelement method results in a significant reduction of computation time. In the analysis case in the section “Analysis of Complicated Problem Including Multiple Drains,” the calculation time of the approximate model was about 1/180 that required for the exact model. Although not discussed in detail in this study, because the authors formulated the macroelement method so that the mesh division width can be separated from the drain spacing, this method can evaluate quantitatively the effects of drain spacing while using a single mesh. The macroelement method with the function enhanced cuts calculation cost significantly.Nguyen et al. (2015), Noda et al. (2015), Nonaka et al. (2017a, b), Yamada et al. (2019), and Tashiro et al. (2021) presented for applications of the extended macroelement method.Data Availability StatementAll data, models, and code generated or used during the study appear in the published article.AcknowledgmentsThe contents of this paper are recompilation of part of the dissertation by the third author (Nonaka 2017), which was written under the supervision of the first and second authors.References Aboshi, H., Y. Sutoh, T. Inoue, and Y. 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