# One-Dimensional Large-Strain Nonlinear Consolidation of Overconsolidated Clays with a Threshold Hydraulic Gradient

Authors: Chuanxun Li; Jinyang Xiao; Yang Yang; Wenbing Wu
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The existence of the threshold hydraulic gradient in clays under a low hydraulic gradient has been recognized by many studies. Meanwhile, most nature clays to some extent exist in an overconsolidated state more or less. However, the consolidation theory of overconsolidated clays with the threshold hydraulic gradient has been rarely reported in the literature. In this paper, a one-dimensional large-strain consolidation model of overconsolidated clays with consideration of the threshold hydraulic gradient is developed, and the finite differential method is adopted to obtain solutions for this model. The influence of the threshold hydraulic gradient and the preconsolidation pressure of overconsolidated clay on consolidation behavior is investigated. The consolidation rate under large-strain supposition is faster than that under small-strain supposition, and the difference in the consolidation rate between different geometric suppositions increases with an increase in the threshold hydraulic gradient and a decrease in the preconsolidation pressure. If Darcy’s law is valid, the final settlement of overconsolidated clays under large-strain supposition is the same as that under small-strain supposition. For the existence of the threshold hydraulic gradient, the final settlement of the clay layer with large-strain supposition is greater than that with small-strain supposition.


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## Body

## 1. Introduction

The clay whose present effective overburden stress is equal to that in the nature state is called normally consolidated clay. Normally consolidated clays are usually found in young geological deposits or prepared resedimented samples in the laboratory. The clay whose present effective overburden stress is less than that in the nature state is called overconsolidated clay. In practice, most nature clays to some extent exist in an overconsolidated state more or less. There are some differences in consolidation parameters between normally consolidated and overconsolidated clays. The coefficient of volume compressibility (mv) of overconsolidated clays is smaller than that of normally consolidated clays corresponding to the same pressure, for instance, and the coefficient of consolidation (cv) of overconsolidated clays may be larger than that of normally consolidated clays corresponding to the same pressure. Therefore, there has been some progress in the theory of consolidation of clays with consideration of the overconsolidated state [1–3]. These studies show that the overconsolidated state certainly has great influences on consolidation behavior.The estimation of settlement is fundamental to all the designs of civil structures [4]. The consolidation theory of clays, which reveals the constitutive relationship of the deformation and the dissipation of excess pore water pressure, plays a vital role in the calculation of the settlement of clays [5–7]. The consolidation also has great influences on the temporal and spatial variation of clay properties [8]. Therefore, since Terzaghi’s theory of consolidation was developed [9], the theory of consolidation is one of the most important theories in geomechanics. Darcy’s law is usually adopted to describe the water flow in these theories of consolidation. For fine-grained overconsolidated clays under low hydraulic gradients, however, the deviation of water flow from Darcy’s law has been confirmed in some studies [10–14]. The water flow obeying Darcy’s law is usually called as the Darcian flow, while the water flow in clays deviating from Darcy’s law may be named as the non-Darcian flow. The model of the water flow in clays with a threshold gradient [11] is the simplest one in these non-Darcian flow laws. The existence of the threshold gradient in an absolute sense still is a controversy. However, just like the reasonable presentation [15], the effective permeability of the clay with other non-Darcian flows may decrease rather abruptly under low gradient. Under this condition, an approximation to the effective flow velocity described by non-Darcian flow [10, 13] can be provided by the model of water flow in clay with a threshold gradient. In this way, there has been some progress in consolidation theories of clays with a threshold gradient [16–20]. The small-strain supposition was incorporated in all above consolidation theories with consideration of a threshold gradient, while the characteristics of large strain and overconsolidated state in nature soft clays were also ignored in these theories.Since the governing equation of one-dimensional large-strain consolidation was developed [21], in which the void ratio of clay was adopted as a variable, several studies have been developed on the theory of large-strain consolidation in numerical and analytical methods [22–26]. However, all these studies on one-dimensional large-strain consolidation essentially based on Darcy’s law may not be applicable to fine-grained overconsolidated clays under low gradients. As stated above, for fine-grained clays, there is a theoretical significance in large-strain consolidation theory with consideration of a threshold gradient.The main objective of this paper is to develop a new model for one-dimensional large-strain consolidation of overconsolidated clays with a threshold hydraulic gradient. Moreover, the influences of the threshold hydraulic gradient and overconsolidated state on large-strain consolidation behavior will be analyzed.


## 2. Effective Stress, Compressibility, and Permeability of Overconsolidated Clays

As shown in Figure1, the initial thickness of the clay layer is H. The bottom of the clay layer is fixed and referenced to the Lagrangian coordinate system. The Lagrangian coordinate a is measured downwards in the direction of gravity. The top surface of the layer (a = 0) is pervious, and the bottom surface (a = H) is impervious. The current vertical effective stress at Lagrangian coordinate a resulting from the self-weight stress of the clay layer is labeled as σv0′a. The vertical preconsolidation stress, σvpc′a, is defined as the maximum vertical effective stress experienced by the overconsolidated clay layer. For overconsolidated clays, the current vertical effective stress must be less than the vertical preconsolidation stress, and the reduction in vertical effective stress could have resulted from melting of ice sheets, erosion of overburden pressure, or a rise in the water table. The relationship between σvpc′ and σv0′ in the Lagrangian coordinate is as follows:(1)σvpc′a=σv0′a+σpc,where σpc is the reduction in vertical effective stress for an overconsolidated clay layer at the same Lagrangian coordinate. It is usually supposed to be constant with depth.Figure 1
An overconsolidated clay layer.The relationship between the void ratioe and the vertical effective stress of overconsolidated clays σv′ is shown as a semi-log plot in Figure 2. AP and PB in Figure 2 correspond to the normal compression line in the curve of e−logσv′, and the normal compression line can be expressed as follows:(2)e−e1=Cclogσv1′σv′,where σv1′ is a given vertical effective stress on the normal compression line of clays; e1 is the void ratio corresponding to the vertical effective stress σv1′ on the normal compression line; and Cc is called as the compression index, and it is the slope of the normal compression line. If the vertical effective stress σv′ reduces at point P to the current vertical stress σv0′ for some reason, DP corresponds to the unloading and reloading line of the overconsolidated clay. Upon reloading beyond P, the overconsolidated clay continues along the path that it would have followed if loaded from A to B continuously. The vertical effective stress at point P of the normal compression line is called as preconsolidation stress. The preconsolidation stress, σvpc′, and the corresponding void ratio, epc, can be expressed as follows:(3)epc−e1=Cclogσv1′σvpc′.Figure 2
The relationship betweene and σv′ of overconsolidated clays.The initial void ratio of the overconsolidated clay,e0, which corresponds to σv0′, is as follows:(4)e0−epc=Crlogσvpc′σv0′.The relationship betweene and σ′ for overconsolidated clays obeys the following equations:(5)e−e1=Cclogσv1′σv′,σv′≥σvpc′,(6)e−epc=Crlogσvpc′σv′,σv0′≤σv′<σvpc′,where Cr is called as the recompression index, and it is equal to the slope of the recompression line e−logσ′ of overconsolidated clay from σv0′ to σvpc′. If the influence of sedimentation on σv0′a is considered, σv0′a at Lagrangian coordinate a can be calculated by the following integration [27]:(7)σv0′a=∫0aγwGs−11+e0da,where Gs is the specific gravity of clay particles and γw is the unit weight of water. By substituting equation (2) into equation (7), σv0′a can further be expressed as follows:(8)1+e1+0.4343Ccσv0′a−Ccσv0′alogσv0′aσv1′=γwGs−1a.It can be found that the initial effective stress in clay does not increase linearly with depth when sedimentation of the deposit is considered. Ifσpc is given, the preconsolidation stress σvpc′ of an overconsolidated clay can also be determined by equations (1) and (8).The water flow in the clay is assumed to obey the flow model with a threshold gradient, and this model can be expressed as follows:(9)v=0,i<i0,kvi1−i0i,i≥i0,where v is the velocity of water flow in clays; i is the hydraulic gradient; kv is the coefficient of vertical permeability; and i0 is the threshold hydraulic gradient, and the value is supposed to be constant during the consolidation. Equation (9) is reduced to Darcy’s law when i0=0 but yields a form of non-Darcian flow when i0≠0. The following well-known logarithmic relation e−logkv is used to describe the nonlinear variations of coefficient of permeability during consolidation [2]:(10)e=e1+Cklogkvkv1,where Ck is the permeability index, and it is equal to the slope of an e−logkv curve, and kv1 is the coefficient of permeability corresponding to the given void ratio e1.


## 3. Analysis of Moving Boundary

As shown in Figure3, the clay layer subjects to a time-dependent uniform load qt on the top surface. The ultimate magnitude and the time of establishment for the time-dependent load are labeled as qc and tc, respectively. The initial value of time-dependent load is labeled as q0. As indicated by the dotted line in Figure 4, a ramp load is a special case of time-dependent load. If the time-dependent load is applied at the top surface over a very large area, the excess pore water pressure in the clay layer, u, will increase at all depths of the clay layer, and the increase in u will be equal to the increase of the external load. In this way, the water in the void spaces of the clay layer will be squeezed out and flow toward the pervious surface. If a threshold hydraulic gradient in water flow exists, there will be no water flow and u will not dissipate for the region of the clay layer in which i≤i0. Only for the region of the clay layer in which i>i0, there is water flow in the clay and u gradually dissipates with time. An existing boundary between the two regions is named as the flow front. Furthermore, the flow front moves from the top of the clay down with the dissipation of excess pore water pressure during the consolidation. When the flow front does not reach the bottom of the clay layer, the problem of consolidation with the threshold gradient consequently becomes a moving boundary problem as illustrated in Figure 3. The Lagrangian coordinate of the flow front at time t is noted as ht. According to the studies [16, 17], the gradient of excess pore water pressure and the excess pore water pressure at the moving boundary a=ht are as follows:(11)uht,t=qt,(12)ia=ht=i0.Figure 3
Analysis of moving boundary in the Lagrangian coordinate.Figure 4
Time-dependent load.For the existence of the threshold gradient, the excess pore water pressure cannot dissipate completely at the end of consolidation. The final residual value of excess pore water pressure at Lagrangian coordinatea is noted as ua,∞. If uH,∞<qc, it means that the moving boundary can reach the bottom of the clay layer. If the excess pore water pressure at the impervious surface is constant during the whole progress of consolidation, on the contrary, the moving boundary cannot reach the bottom of the clay layer.


## 4. The Mathematical Model: Governing Equations and Solution Conditions

Based on the study by Xie and Leo [26], the hydraulic gradients i can be expressed in the Lagrangian coordinate as follows:(13)i=1γw1+e01+e∂u∂a,where e is the void ratio which varies with Lagrangian coordinate a and time t. With consideration of the incompressibility of clay particles and pore water, according to the study by Xie and Leo [26], the general continuity condition for one-dimensional large-strain consolidation in the Lagrangian coordinate can be given as follows:(14)∂v∂a=−11+e0∂e∂t,where t is the time. If the creep effect of the clay skeleton is ignored and the consolidation is assumed to be monotonic, the void ratio e is solely dependent on the effective stress σv′, and then equation (14) can be rewritten as follows:(15)∂v∂a=−11+e0∂e∂σv′∂σv′∂t.According to the theory of effective stress, the effective stressσv′ at time t can be expressed as follows:(16)σv′a,t=qt+σv0′a,t−ua,t.Substituting equations (9), (13), and (16) into equation (15), the following equation can be obtained:(17)u=qt,1+e01+e1γw∂u∂a<i0,(18)∂∂akvγw1+e01+e1−i01/γw1+e0/1+e∂u/∂a∂u∂a=11+e0∂e∂σ′dqdt−∂u∂t,1γw1+e01+e∂u∂a≥i0.With equations (5), (6), and (10), the coefficient of permeability and the coefficient of compressibility of overconsolidated clays change with the effective stress as follows:(19)dedσv′=−Crσv′ln10,σv0′≤σv′≤σvpc′,−Ccσv′ln10,σv′>σvpc′,(20)kv=kv1σv1′σvpc′Cc/Ckσvpc′σv′Cr/Ck,σv0′≤σv′≤σvpc′,kv1σv1′σv′Cc/Ck,σv′>σvpc′.Substituting equations (19) and (20) into equation (18), the governing equation for one-dimensional large-strain consolidation of overconsolidated clays with a threshold gradient in the Lagrangian coordinate can be derived as follows:(21)∂u∂t−dqdt=cv11+e01+e1CcCrσv′σv1′∂∂aσv1′σvpc′Cc/Ckσvpc′σv′Cr/Ck1+e01+e−i01/γw∂u/∂a∂u∂a,1γw1+e01+e∂u∂a≥i0,σv0′≤σv′≤σvpc′,cv11+e01+e1σv′σv1′∂∂aσv1′σv′Cc/Ck1+e01+e−i01/γw∂u/∂a∂u∂a,1γw1+e01+e∂u∂a≥i0,σv′>σvpc′,where cv1 is the coefficient of consolidation corresponding to the given effective stress σv1′, and it can be determined by cv1=σv1′1+e1kv1ln10/γwcc. In form of ht, the governing equation for large-strain consolidation of overconsolidated clays with a threshold gradient can be rewritten as follows:(22)∂u∂t−dqdt=cv11+e01+e1CcCrσv′σv1′∂∂aσv1′σvpc′Cc/Ckσvpc′σv′Cr/Ck1+e01+e−i01/γw∂u/∂a∂u∂a,0<a≤ht,σv0′≤σv′≤σvpc′,cv11+e01+e1σv′σv1′∂∂aσv1′σv′Cc/Ck1+e01+e−i01/γw∂u/∂a∂u∂a,0<a≤ht,σv′>σvpc′,(23)u=qt,a>ht.When the moving boundary does not reach the bottom of the clay layer, the top boundary condition and the moving boundary condition described by equations (11) and (12) can be expressed as follows:(24)u0,t=0,(25)uht,t=qt,(26)1γw1+e01+e∂u∂aa=ht=i0.The initial condition for this model is as follows:(27)ua,0=q0=q0.If the moving flow front reaches the bottom of the overconsolidated clay layer, the moving boundary becomes a fixed boundary, and the governing equation for large-strain consolidation of overconsolidated clays with a threshold gradient is as follows:(28)∂u∂t−dqdt=cv11+e01+e1CcCrσv′σv1′∂∂aσv1′σvpc′Cc/Ckσvpc′σv′Cr/Ck1+e01+e−i01/γw∂u/∂a∂u∂a,0<a≤H,σv0′≤σv′≤σvpc′,cv11+e01+e1σv′σv1′∂∂aσv1′σv′Cc/Ck1+e01+e−i01/γw∂u/∂a∂u∂a,0<a≤H,σv′>σvpc′.Under this condition, the top boundary condition is the same as equation (24), and the bottom boundary condition is as follows:(29)1γw1+e01+e∂u∂aa=H=i0.If small-strain supposition is adopted, the governing equations for one-dimensional consolidation with consideration of the threshold gradient and stress history can be easily obtained and change as follows:(30)∂u∂t−dqdt=cv11+e01+e1CcCrσv′σv1′∂∂aσv1′σvpc′Cc/Ckσvpc′σv′Cr/Ck1−i01/γw∂u/∂a∂u∂a,0<a≤ht,σv0′≤σv′≤σvpc′,cv11+e01+e1σv′σv1′∂∂aσv1′σv′Cc/Ck1−i01/γw∂u/∂a∂u∂a,0<a≤ht,σv′>σvpc′.The corresponding moving boundary condition under small-strain supposition can be expressed as follows:(31)1γw∂u∂aa=ht=i0.


## 5. Numerical Solutions for the Model

To obtain the numerical solutions for the above model, the following dimensionless variables are firstly defined as follows:(32)x=aH,U=uσv1′,Q0=q0σv1′,Q=qtσv1′,Qu=qcσv1′,R=i0γwHσv1′,S=σv0′σv1′,Spc=σvpc′σv1′,Tv=cv1tH2,Tvc=cv1tcH2,X=htH.In terms of these dimensionless variables, the dimensionless governing equations for one-dimensional large-strain consolidation of overconsolidated clays described by equations (22) and (28) can be rearranged as follows:(33)∂U∂Tv−dQdTv=S+Q−UCcCr1+e01+e1∂∂x1SpcCc/CkSpcS+Q−UCr/Ck1+e01+e−R∂U/∂x∂U∂x,0<x≤X,S+Q−U≤Spc,S+Q−U1+e01+e1∂∂x1S+Q−UCc/Ck1+e01+e−R∂U/∂x∂U∂x,0<x≤X,S+Q−U>Spc,(34)∂U∂Tv−dQdTv=S+Q−U1+e01+e1∂∂x1SpcCc/CkSpcS+Q−UCr/Ck1+e01+e−R∂U/∂x∂U∂x,0<x≤1,S≤S+Q−U≤Spc,S+Q−U1+e01+e1∂∂x1S+Q−UCc/Ck1+e01+e−R∂U/∂x∂U∂x,0<x≤1,S+Q−U>Spc.In terms of dimensionless variables, the solution condition described by equations (24)–(27) can be rewritten as follows:(35)U0,Tv=0,UXTv,Tv=Q,1+e01+e∂U∂xx=XTv=R,Ux,0=Q0.Under small strain assumption, the corresponding solution condition can be expressed as follows:(36)U0,Tv=0,UXTv,Tv=Q,∂U∂xx=XTv=R,Ux,0=Q0.In order to obtain finite differential solutions for above models, a differential grid is placed in thex−Tv plane with spatial isometry and nonisometry time steps. As shown in Figure 5, the spatial domain is divided into n equal thin layers, and the dimensionless value of the thickness of each thin layer is Δx. The jth nodal point of the spatial domain is noted as xj, and xj=jΔx,j=0,1,2,3,…,n. So the nodal point of the top surface is x0=0, and the nodal point of the bottom surface is xn=1. Meantime, the time domain Tv is also divided into a number of small time intervals. If the kth time interval is noted as ΔTvkk=1,2,3,4,…, the final time of the kth time interval, Tvk, is as follows:(37)Tvk=∑r=1kΔTvr.Figure 5
A differential grid in thex-Tv plane.The dimensional value of current effective stress at thejth nodal point, Sj, can be obtained by the following equation:(38)Sj=σv0′jΔxHσv1′,in which σv0′jΔxH can be determined by equation (8). The dimensional value of preconsolidation stress at the jth nodal point, Spcj, can be calculated by the following equation:(39)Spcj=σvpc′jΔxHσv1′,in which σvpc′jΔxH can be obtained by equations (1) and (8). The dimensionless value of excess pore water pressure when xj=jΔx and Tv=Tvk is noted as Ujk, and Qk is the dimensionless value of time-dependent load qt when Tv=Tvk. The void ratio at the jth nodal point, e0j and epcj, can be determined by Sj and Spcj according to equations (3) and (4). In order to solve the moving boundary problem, the following two assumptions are made:(1)
The water flow in the first thin layer occurs as long as the external load is applied to the surface of clays. The flow front reached the bottom of the second thin layer after the first interval.(2)
If the water flow occurs at some point of a thin clay layer, the water flow occurs during the whole thin layer at the same time, and the whole thin layer begins to consolidate at the same time. If the flow front reaches the bottom of thejth thin layer at Tv=Tvk−1 and the average hydraulic gradient in the j+1th thin layer is greater than the threshold gradient, therefore, the flow front is supposed to reach the bottom of the j+1th thin layer during the next time interval ΔTvk.If the clay layer is divided into many thin layers, the computational error caused by assumption (1) can be ignored. Moreover, assumption (1) has been already adopted by Pascal et al. [16] in the study of consolidation with consideration of a threshold gradient. Similar to the studies [28, 29], the differential equation corresponding to equation (33) during the first time interval ΔTv1 can be expressed as follows:(40)U11=U10+Q1−Q0+λ1β11α1+1/21U21−U11−α1−1/21U11−U01,where λ1=ΔTv1/Δx2, β11, α1−1/21, and α1+1/21 are given in Appendix. When the moving boundary reaches the second thin layer, the top boundary and moving boundary conditions can be expressed in terms of the discrete point as follows:(41)U01=0,(42)3U21−4U11+U01=2RΔx1+e211+e02,(43)U21=Q1.With equations (42) and (43), the following equation can be obtained:(44)U11=3Q1−2RΔx1+e21/1+e024.Substituting equation (44) into equation (40), the time for the moving flow front to reach the second thin layer T2 can be derived:(45)T2=Tv1=ΔTv1=U11−Q1β11α1+1/21Q1−U11−α1−1/21U11−U01.Q
1 also is the function of T2, so T2 can be obtained by the iteration method. When Tv>T2, the moving flow front still stays in the second thin layer or moves down to another thin layer. If the moving flow front stays in the jth2≤j<n thin layer at time Tvk−1, the differential equation and the moving boundary condition during the kth interval ΔTvk are as follows:(46)Ulk=Ulk−1+Qk−Qk−1+λkβlk−1αl+1/2k−1Ul+1k−Ulk−αl−1/2k−1Ulk−Ul−1k,k≥2,0<l≤j−1,(47)Ulk=Qk,k≥2,l≥j,(48)3Ujk−4Uj−1k+Uj−2k=2RΔx1+ejk−11+e0j,where λk=ΔTvk/Δx2, βlk, αl−1/2k, and αl+1/2k are given in Appendix.If the following equation can be satisfied at timeTvk, according to supposition (2), the flow front reaches the bottom of the j+1th thin layer during the next time interval ΔTvk+1:(49)1+e0j1+ejkUj+1k−UjkΔx≥R.On the contrary, if equation (49) cannot be satisfied, the moving flow front still stays in the bottom of the jth thin layer during the next time interval ΔTvk+1. Denote the time for the moving flow front to reach xj+1 as Tj+1. If equation (49) is satisfied at time Tvk, Tj+1 can be determined by the following equation:(50)Tj+1=Tvk=∑r=1kΔTvr,j=2,3,4,…,n−1.When the moving flow front reaches the bottom of the clay layer, the moving boundary turns into a fixed boundary. Under this case, the boundary condition can be expressed in terms of the nodal point as follows:(51)3Unk−4Un−1k+Un−2k=2RΔx1+enk−11+e0n.Excess pore water pressure may not thoroughly dissipate for the existence of the hydraulic gradient. The excess pore water pressure at timeTvk+1 can be considered as the residual value of excess pore water pressure if the norm of the differences between Ujk+1 and Ujk is small enough, that is,(52)maxUjk+1−UjkUjk+1≤ε,j=1,2,3,…,n,where ε is a small number to be specified according to the tolerable error. By a number of calculations, small distinctions come out between ε=10−4 and ε=10−5. Therefore, ε=10−4 is adopted in the following analysis to complete the process of calculation rapidly. If equation (52) is valid, the residual excess pore water pressure, Uj∞, can be determined by Ujk+1. The final void ratio also can be determined by equation (5) or equation (6). According to the study on large-strain consolidation in the Lagrangian coordinate [26], the settlement of the clay layer at time t, St, can be derived:(53)dSt=e0−e1+e0da.The settlement of the clay layerSt can be obtained by integrating equation (53) with respect to a from 0 to H:(54)St=∫0He0−e1+e0da=∑j=1ne0j−ejk1+e0j.The final settlement of the clay layer,S∞, can be expressed as follows:(55)S∞=∫0He0−ej∞1+e0da=Hn∑j=1ne0j−ej∞1+e0j.The average degree of consolidation in terms of deformation atTvk, Ust, is as follows:(56)Ust=StS∞.The average degree of consolidation in terms of excess pore water pressure atTvk, Upt, can be written as follows:(57)Upt=∫0Hqt−uqcda=QkQu−1nQu∑j=1nUj−1k+Ujk2,Tv≤Tvc,1−1nQu∑j=1nUj−1k+Ujk2,Tv>Tvc.


## 6. Analysis of Consolidation Behavior

To study the influence of one parameter on consolidation behavior, other parameters may remain constant, and the parameters in Table1 are adopted in the following analysis.Table 1
The parameters.
Parameters
H (m)
σ
v
1
′ (kPa)
e
1
k
v1 (10−8m/s)
σ
pc (kPa)
G
s
γ
w (kN/m3)
q
c (kPa)
T
vc
C
c
C
k
C
r
i
0
ε
Values
20
50
1.5
1.0
25
2.75
10
200
0.05
0.85
0.8
0.05
0.5
10−4


### 6.1. Influences on the Location of the Moving Flow Front

The threshold hydraulic gradient gives rise to the existence of the flow front and moving flow boundary. As shown in Figure6, the greater the i0, the longer the time needed by the flow front reaching the bottom of the clay layer. Moreover, according to the previous studies [16, 17], the flow front cannot reach the bottom of the clay layer when i0 is great enough. If stress history and large-strain supposition were considered, this consolidation behavior about the location of the flow front would not change. In Figure 6, when i0 = 1.5, the moving flow front cannot reach the bottom of the clay layer because the threshold hydraulic gradient is large enough compared with the external load and the thickness of the clay layer.Figure 6
Influence ofi0 on the moving flow boundary.Figure7 describes the influence of different geometric suppositions on the moving rate of the flow front. In general, the flow front under large-strain supposition moves faster than that under small-strain supposition. However, the difference in moving rate between large strain and small strain is not evident when the threshold hydraulic gradient is equal to 0.1 (in Figure 7(a)). It must be noted that the preconsolidation pressure also has influence on the moving rate of the flow front. The moving rate of the flow front increases with the increasing value of preconsolidation pressure. Furthermore, the difference in moving rate of the flow front between large strain and small strain decreases with the increasing value of preconsolidation pressure (in Figure 7(b)).Figure 7
Influences of geometric suppositions on the moving rate of the flow front: (a) under different threshold hydraulic gradienti0; (b) under different preconsolidation pressure σpc.
(a)
(b)


### 6.2. Influences on the Dissipation of Excess Pore Water Pressure

Figure8(a) indicates that the residual excess pore water pressure in clays with large-strain supposition is smaller than that with small-strain supposition when Tv = 0.5. At the same time, the residual excess pore water pressure with i0 = 0.1 is smaller than that with i0 = 0.5. It further indicates that the dissipation rate of excess pore water pressure decreases with increasing i0. Figure 8(b) describes the dissipation of excess pore water pressure with time and indicates that the dissipation rate of excess pore water pressure with large-strain supposition is faster than that with small-strain supposition during the whole consolidation process. The difference in the dissipation rate of excess pore water pressure between large strain and small strain becomes evident with an increase in the threshold hydraulic gradient. Moreover, the final residual excess pore water pressure with large-strain supposition is smaller than that with small-strain supposition.Figure 8
Influences of geometric suppositions on the dissipation of excess pore water pressure. (a) Curves of excess pore water pressure versus Lagrangian coordinatea when Tv = 0.5. (b) Curves of excess pore water pressure versus time factor Tv at z = 10 m.
(a)
(b)The preconsolidation pressure also has great influence on the dissipation of excess pore water pressure (in Figure9). The dissipation rate of excess pore water increases with the increasing preconsolidation pressure when the threshold hydraulic gradient is constant. However, the preconsolidation pressure has no influence on the value of the final residual excess pore water pressure. If the preconsolidation pressure is also the same, the dissipation rate of excess pore water pressure with large-strain supposition is faster than that with small-strain supposition.Figure 9
Influences of preconsolidation pressure on the dissipation of excess pore water pressure. (a) Curves of excess pore water pressure versus the Lagrangian coordinatea when Tv = 0.5. (b) Curves of excess pore water pressure versus time factor Tv at z = 10 m.
(a)
(b)


### 6.3. Influences on the Average Degree of Consolidation

Since the dissipation rate of excess pore water pressure decreases with an increase in the value of the threshold hydraulic gradient, the average degree in terms of excess pore water pressure should increase with the decreasing threshold hydraulic gradient at the same time (Figure10). The consolidation rate with large-strain supposition is faster than that with small-strain supposition, and the final average degree in terms of excess pore water pressure with large-strain supposition is also greater than that with small-strain supposition. For the existence of the threshold hydraulic gradient, the excess pore water pressure cannot thoroughly dissipate, and the final average degree in terms of excess pore water pressure cannot run to 1. The difference in the average degree between large strain and small strain may become evident with an increase in i0. If i0 is so small (for instance, i0 = 0.1), then this difference can be neglected.Figure 10
Influence of geometric suppositions onUpt under different i0.Figure11 indicates that the preconsolidation pressure of overconsolidated clays has influences on the consolidation rate. If the threshold hydraulic gradient is constant, the consolidation rate increases with the increasing preconsolidation pressure. If both the preconsolidation pressure and the threshold hydraulic are constant, the consolidation rate with large-strain supposition is faster than that with small-strain supposition. It should be noted that the preconsolidation pressure has no influence on the final average degree when the dissipation of excess pore water pressure is completed.Figure 11
Influence of geometric supposition onUpt under different preconsolidation pressure.


### 6.4. Influences on the Settlement of the Clay Layer

The threshold hydraulic gradient has great influence on the settlement of the clay layer. As shown in Figure12, the final settlement decreases with the increasing threshold hydraulic gradient. From Figure 13, it can be seen that the settlement of the clay layer decreases with the increasing preconsolidation pressure of overconsolidated clays. If the threshold gradient and the preconsolidation pressure remain constant, the settlement of the clay layer with large-strain supposition is greater than that with small-strain supposition. However, this difference in settlement between large strain and small strain increases with an increase in threshold hydraulic gradient and a decrease in preconsolidation pressure of overconsolidated clays. Especially, if i0 = 0.0001 which can be approximately considered as Darcy’s law, the settlement of the clay layer under different geometric suppositions is the same. The reason for this phenomenon is that the residual excess pore water pressure under large-strain supposition with consideration of the threshold hydraulic gradient is smaller than that under small-strain supposition (in Figure 8). Therefore, when the threshold hydraulic gradient in clays under a low hydraulic gradient is considered, the final settlement under large-strain assumption is greater than that under small-strain assumption.Figure 12
Influence of geometric supposition onSt under different i0.Figure 13
Influence of geometric supposition onSt under different preconsolidation pressure.


## 6.1. Influences on the Location of the Moving Flow Front

The threshold hydraulic gradient gives rise to the existence of the flow front and moving flow boundary. As shown in Figure6, the greater the i0, the longer the time needed by the flow front reaching the bottom of the clay layer. Moreover, according to the previous studies [16, 17], the flow front cannot reach the bottom of the clay layer when i0 is great enough. If stress history and large-strain supposition were considered, this consolidation behavior about the location of the flow front would not change. In Figure 6, when i0 = 1.5, the moving flow front cannot reach the bottom of the clay layer because the threshold hydraulic gradient is large enough compared with the external load and the thickness of the clay layer.Figure 6
Influence ofi0 on the moving flow boundary.Figure7 describes the influence of different geometric suppositions on the moving rate of the flow front. In general, the flow front under large-strain supposition moves faster than that under small-strain supposition. However, the difference in moving rate between large strain and small strain is not evident when the threshold hydraulic gradient is equal to 0.1 (in Figure 7(a)). It must be noted that the preconsolidation pressure also has influence on the moving rate of the flow front. The moving rate of the flow front increases with the increasing value of preconsolidation pressure. Furthermore, the difference in moving rate of the flow front between large strain and small strain decreases with the increasing value of preconsolidation pressure (in Figure 7(b)).Figure 7
Influences of geometric suppositions on the moving rate of the flow front: (a) under different threshold hydraulic gradienti0; (b) under different preconsolidation pressure σpc.
(a)
(b)


## 6.2. Influences on the Dissipation of Excess Pore Water Pressure

Figure8(a) indicates that the residual excess pore water pressure in clays with large-strain supposition is smaller than that with small-strain supposition when Tv = 0.5. At the same time, the residual excess pore water pressure with i0 = 0.1 is smaller than that with i0 = 0.5. It further indicates that the dissipation rate of excess pore water pressure decreases with increasing i0. Figure 8(b) describes the dissipation of excess pore water pressure with time and indicates that the dissipation rate of excess pore water pressure with large-strain supposition is faster than that with small-strain supposition during the whole consolidation process. The difference in the dissipation rate of excess pore water pressure between large strain and small strain becomes evident with an increase in the threshold hydraulic gradient. Moreover, the final residual excess pore water pressure with large-strain supposition is smaller than that with small-strain supposition.Figure 8
Influences of geometric suppositions on the dissipation of excess pore water pressure. (a) Curves of excess pore water pressure versus Lagrangian coordinatea when Tv = 0.5. (b) Curves of excess pore water pressure versus time factor Tv at z = 10 m.
(a)
(b)The preconsolidation pressure also has great influence on the dissipation of excess pore water pressure (in Figure9). The dissipation rate of excess pore water increases with the increasing preconsolidation pressure when the threshold hydraulic gradient is constant. However, the preconsolidation pressure has no influence on the value of the final residual excess pore water pressure. If the preconsolidation pressure is also the same, the dissipation rate of excess pore water pressure with large-strain supposition is faster than that with small-strain supposition.Figure 9
Influences of preconsolidation pressure on the dissipation of excess pore water pressure. (a) Curves of excess pore water pressure versus the Lagrangian coordinatea when Tv = 0.5. (b) Curves of excess pore water pressure versus time factor Tv at z = 10 m.
(a)
(b)


## 6.3. Influences on the Average Degree of Consolidation

Since the dissipation rate of excess pore water pressure decreases with an increase in the value of the threshold hydraulic gradient, the average degree in terms of excess pore water pressure should increase with the decreasing threshold hydraulic gradient at the same time (Figure10). The consolidation rate with large-strain supposition is faster than that with small-strain supposition, and the final average degree in terms of excess pore water pressure with large-strain supposition is also greater than that with small-strain supposition. For the existence of the threshold hydraulic gradient, the excess pore water pressure cannot thoroughly dissipate, and the final average degree in terms of excess pore water pressure cannot run to 1. The difference in the average degree between large strain and small strain may become evident with an increase in i0. If i0 is so small (for instance, i0 = 0.1), then this difference can be neglected.Figure 10
Influence of geometric suppositions onUpt under different i0.Figure11 indicates that the preconsolidation pressure of overconsolidated clays has influences on the consolidation rate. If the threshold hydraulic gradient is constant, the consolidation rate increases with the increasing preconsolidation pressure. If both the preconsolidation pressure and the threshold hydraulic are constant, the consolidation rate with large-strain supposition is faster than that with small-strain supposition. It should be noted that the preconsolidation pressure has no influence on the final average degree when the dissipation of excess pore water pressure is completed.Figure 11
Influence of geometric supposition onUpt under different preconsolidation pressure.


## 6.4. Influences on the Settlement of the Clay Layer

The threshold hydraulic gradient has great influence on the settlement of the clay layer. As shown in Figure12, the final settlement decreases with the increasing threshold hydraulic gradient. From Figure 13, it can be seen that the settlement of the clay layer decreases with the increasing preconsolidation pressure of overconsolidated clays. If the threshold gradient and the preconsolidation pressure remain constant, the settlement of the clay layer with large-strain supposition is greater than that with small-strain supposition. However, this difference in settlement between large strain and small strain increases with an increase in threshold hydraulic gradient and a decrease in preconsolidation pressure of overconsolidated clays. Especially, if i0 = 0.0001 which can be approximately considered as Darcy’s law, the settlement of the clay layer under different geometric suppositions is the same. The reason for this phenomenon is that the residual excess pore water pressure under large-strain supposition with consideration of the threshold hydraulic gradient is smaller than that under small-strain supposition (in Figure 8). Therefore, when the threshold hydraulic gradient in clays under a low hydraulic gradient is considered, the final settlement under large-strain assumption is greater than that under small-strain assumption.Figure 12
Influence of geometric supposition onSt under different i0.Figure 13
Influence of geometric supposition onSt under different preconsolidation pressure.


## 7. Conclusions

In case of one-dimensional large-strain consolidation of overconsolidated clays, the moving rate of the flow front resulted by the threshold hydraulic gradient and the dissipation rate of excess pore water pressure decrease with the increasing threshold gradient. The greater the threshold hydraulic gradient is, the greater the final residual excess pore water pressure is and the smaller the final settlement of the clay layer is. The consolidation rate of overconsolidated clays with the threshold hydraulic gradient is faster than that of normally consolidated clays with the same threshold gradient, and it increases with an increase in preconsolidation pressure. The dissipation rate of excess pore water pressure with large-strain supposition is faster than that with small-strain supposition, and this difference between them increases with an increase in threshold gradient and a decrease in preconsolidation pressure. If Darcy’s law is valid, the final settlement with large-strain supposition is the same as that with small-strain supposition. However, if a threshold hydraulic gradient exists in overconsolidated clays, the final settlement with large-strain supposition is greater than that with small-strain supposition.



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*Source: 5918492-2018-12-13.xml*