Research Article  Open Access
Chuanxun Li, Jinyang Xiao, Yang Yang, Wenbing Wu, "OneDimensional LargeStrain Nonlinear Consolidation of Overconsolidated Clays with a Threshold Hydraulic Gradient", Advances in Civil Engineering, vol. 2018, Article ID 5918492, 15 pages, 2018. https://doi.org/10.1155/2018/5918492
OneDimensional LargeStrain Nonlinear Consolidation of Overconsolidated Clays with a Threshold Hydraulic Gradient
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 onedimensional largestrain 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 largestrain supposition is faster than that under smallstrain 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 largestrain supposition is the same as that under smallstrain supposition. For the existence of the threshold hydraulic gradient, the final settlement of the clay layer with largestrain supposition is greater than that with smallstrain supposition.
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 (m_{v}) of overconsolidated clays is smaller than that of normally consolidated clays corresponding to the same pressure, for instance, and the coefficient of consolidation (c_{v}) 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 finegrained 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 nonDarcian flow. The model of the water flow in clays with a threshold gradient [11] is the simplest one in these nonDarcian 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 nonDarcian flows may decrease rather abruptly under low gradient. Under this condition, an approximation to the effective flow velocity described by nonDarcian 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 smallstrain 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 onedimensional largestrain 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 largestrain consolidation in numerical and analytical methods [22–26]. However, all these studies on onedimensional largestrain consolidation essentially based on Darcy’s law may not be applicable to finegrained overconsolidated clays under low gradients. As stated above, for finegrained clays, there is a theoretical significance in largestrain consolidation theory with consideration of a threshold gradient.
The main objective of this paper is to develop a new model for onedimensional largestrain consolidation of overconsolidated clays with a threshold hydraulic gradient. Moreover, the influences of the threshold hydraulic gradient and overconsolidated state on largestrain consolidation behavior will be analyzed.
2. Effective Stress, Compressibility, and Permeability of Overconsolidated Clays
As shown in Figure 1, 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 selfweight stress of the clay layer is labeled as . The vertical preconsolidation stress, , 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 and in the Lagrangian coordinate is as follows:where 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.
The relationship between the void ratio and the vertical effective stress of overconsolidated clays is shown as a semilog plot in Figure 2. AP and PB in Figure 2 correspond to the normal compression line in the curve of , and the normal compression line can be expressed as follows:where is a given vertical effective stress on the normal compression line of clays; is the void ratio corresponding to the vertical effective stress on the normal compression line; and is called as the compression index, and it is the slope of the normal compression line. If the vertical effective stress reduces at point P to the current vertical stress 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, , and the corresponding void ratio, , can be expressed as follows:
The initial void ratio of the overconsolidated clay, , which corresponds to , is as follows:
The relationship between and for overconsolidated clays obeys the following equations:where is called as the recompression index, and it is equal to the slope of the recompression line of overconsolidated clay from to . If the influence of sedimentation on is considered, at Lagrangian coordinate can be calculated by the following integration [27]:where is the specific gravity of clay particles and is the unit weight of water. By substituting equation (2) into equation (7), can further be expressed as follows:
It can be found that the initial effective stress in clay does not increase linearly with depth when sedimentation of the deposit is considered. If is given, the preconsolidation stress 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:where is the velocity of water flow in clays; is the hydraulic gradient; is the coefficient of vertical permeability; and 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 but yields a form of nonDarcian flow when . The following wellknown logarithmic relation is used to describe the nonlinear variations of coefficient of permeability during consolidation [2]:where is the permeability index, and it is equal to the slope of an curve, and is the coefficient of permeability corresponding to the given void ratio .
3. Analysis of Moving Boundary
As shown in Figure 3, the clay layer subjects to a timedependent uniform load on the top surface. The ultimate magnitude and the time of establishment for the timedependent load are labeled as and , respectively. The initial value of timedependent load is labeled as . As indicated by the dotted line in Figure 4, a ramp load is a special case of timedependent load. If the timedependent load is applied at the top surface over a very large area, the excess pore water pressure in the clay layer, , will increase at all depths of the clay layer, and the increase in 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 will not dissipate for the region of the clay layer in which . Only for the region of the clay layer in which , there is water flow in the clay and 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 . According to the studies [16, 17], the gradient of excess pore water pressure and the excess pore water pressure at the moving boundary are as follows:
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 coordinate a is noted as . If , 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 can be expressed in the Lagrangian coordinate as follows:where is the void ratio which varies with Lagrangian coordinate and time . 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 onedimensional largestrain consolidation in the Lagrangian coordinate can be given as follows:where is the time. If the creep effect of the clay skeleton is ignored and the consolidation is assumed to be monotonic, the void ratio is solely dependent on the effective stress , and then equation (14) can be rewritten as follows:
According to the theory of effective stress, the effective stress at time can be expressed as follows:
Substituting equations (9), (13), and (16) into equation (15), the following equation can be obtained:
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:
Substituting equations (19) and (20) into equation (18), the governing equation for onedimensional largestrain consolidation of overconsolidated clays with a threshold gradient in the Lagrangian coordinate can be derived as follows:where is the coefficient of consolidation corresponding to the given effective stress , and it can be determined by . In form of , the governing equation for largestrain consolidation of overconsolidated clays with a threshold gradient can be rewritten as follows:
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:
The initial condition for this model is as follows:
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 largestrain consolidation of overconsolidated clays with a threshold gradient is as follows:
Under this condition, the top boundary condition is the same as equation (24), and the bottom boundary condition is as follows:
If smallstrain supposition is adopted, the governing equations for onedimensional consolidation with consideration of the threshold gradient and stress history can be easily obtained and change as follows:
The corresponding moving boundary condition under smallstrain supposition can be expressed as follows:
5. Numerical Solutions for the Model
To obtain the numerical solutions for the above model, the following dimensionless variables are firstly defined as follows:
In terms of these dimensionless variables, the dimensionless governing equations for onedimensional largestrain consolidation of overconsolidated clays described by equations (22) and (28) can be rearranged as follows:
In terms of dimensionless variables, the solution condition described by equations (24)–(27) can be rewritten as follows:
Under small strain assumption, the corresponding solution condition can be expressed as follows:
In order to obtain finite differential solutions for above models, a differential grid is placed in the 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 . The nodal point of the spatial domain is noted as , and . So the nodal point of the top surface is , and the nodal point of the bottom surface is . Meantime, the time domain is also divided into a number of small time intervals. If the time interval is noted as , the final time of the time interval, , is as follows:
The dimensional value of current effective stress at the nodal point, , can be obtained by the following equation:in which can be determined by equation (8). The dimensional value of preconsolidation stress at the nodal point, , can be calculated by the following equation:in which can be obtained by equations (1) and (8). The dimensionless value of excess pore water pressure when and is noted as , and is the dimensionless value of timedependent load when . The void ratio at the nodal point, and , can be determined by and 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 the jth thin layer at and the average hydraulic gradient in the thin layer is greater than the threshold gradient, therefore, the flow front is supposed to reach the bottom of the thin layer during the next time interval .
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 can be expressed as follows:where , , , and 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:
With equations (42) and (43), the following equation can be obtained:
Substituting equation (44) into equation (40), the time for the moving flow front to reach the second thin layer can be derived:
also is the function of , so can be obtained by the iteration method. When , 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 thin layer at time , the differential equation and the moving boundary condition during the kth interval are as follows:where , , , and are given in Appendix.
If the following equation can be satisfied at time , according to supposition (2), the flow front reaches the bottom of the thin layer during the next time interval :
On the contrary, if equation (49) cannot be satisfied, the moving flow front still stays in the bottom of the thin layer during the next time interval . Denote the time for the moving flow front to reach as . If equation (49) is satisfied at time , can be determined by the following equation:
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:
Excess pore water pressure may not thoroughly dissipate for the existence of the hydraulic gradient. The excess pore water pressure at time can be considered as the residual value of excess pore water pressure if the norm of the differences between and is small enough, that is,where is a small number to be specified according to the tolerable error. By a number of calculations, small distinctions come out between and . Therefore, is adopted in the following analysis to complete the process of calculation rapidly. If equation (52) is valid, the residual excess pore water pressure, , can be determined by . The final void ratio also can be determined by equation (5) or equation (6). According to the study on largestrain consolidation in the Lagrangian coordinate [26], the settlement of the clay layer at time , , can be derived:
The settlement of the clay layer can be obtained by integrating equation (53) with respect to from 0 to H:
The final settlement of the clay layer, , can be expressed as follows:
The average degree of consolidation in terms of deformation at , , is as follows:
The average degree of consolidation in terms of excess pore water pressure at , , can be written as follows:
6. Analysis of Consolidation Behavior
To study the influence of one parameter on consolidation behavior, other parameters may remain constant, and the parameters in Table 1 are adopted in the following analysis.

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 Figure 6, the greater the i_{0}, 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 i_{0} is great enough. If stress history and largestrain supposition were considered, this consolidation behavior about the location of the flow front would not change. In Figure 6, when i_{0} = 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 7 describes the influence of different geometric suppositions on the moving rate of the flow front. In general, the flow front under largestrain supposition moves faster than that under smallstrain 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)).
(a)
(b)
6.2. Influences on the Dissipation of Excess Pore Water Pressure
Figure 8(a) indicates that the residual excess pore water pressure in clays with largestrain supposition is smaller than that with smallstrain supposition when T_{v} = 0.5. At the same time, the residual excess pore water pressure with i_{0} = 0.1 is smaller than that with i_{0} = 0.5. It further indicates that the dissipation rate of excess pore water pressure decreases with increasing i_{0}. 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 largestrain supposition is faster than that with smallstrain 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 largestrain supposition is smaller than that with smallstrain supposition.
(a)
(b)
The preconsolidation pressure also has great influence on the dissipation of excess pore water pressure (in Figure 9). 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 largestrain supposition is faster than that with smallstrain supposition.
(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 (Figure 10). The consolidation rate with largestrain supposition is faster than that with smallstrain supposition, and the final average degree in terms of excess pore water pressure with largestrain supposition is also greater than that with smallstrain 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 i_{0}. If i_{0} is so small (for instance, i_{0} = 0.1), then this difference can be neglected.
Figure 11 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 largestrain supposition is faster than that with smallstrain 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.
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 Figure 12, 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 largestrain supposition is greater than that with smallstrain 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 i_{0} = 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 largestrain supposition with consideration of the threshold hydraulic gradient is smaller than that under smallstrain supposition (in Figure 8). Therefore, when the threshold hydraulic gradient in clays under a low hydraulic gradient is considered, the final settlement under largestrain assumption is greater than that under smallstrain assumption.
7. Conclusions
In case of onedimensional largestrain 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 largestrain supposition is faster than that with smallstrain 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 largestrain supposition is the same as that with smallstrain supposition. However, if a threshold hydraulic gradient exists in overconsolidated clays, the final settlement with largestrain supposition is greater than that with smallstrain supposition.
Appendix
The Expressions of , , and
The expressions of , , and in numerical solutions for the excess pore water pressure in equation (46) are as follows:
Notations
:  Coefficients in finite differential solutions which are determined by equations (A.2) and (A.3) 
:  Coefficient in finite differential solutions which is determined by equation (A.1) 
:  A small number to be specified 
:  Coefficient in finite differential solutions which is determined by 
:  The reduction in effective stress for an overconsolidated clay layer 
:  Current effective stress resulting from selfweight stress 
:  Preconsolidation stress 
:  Unit weight of water 
:  Lagrangian coordinate 
:  Compression index 
:  Permeability index 
:  Recompression index 
:  Coefficient of consolidation 
:  Void ratio 
:  Initial void ratio 
:  at the nodal point 
:  Void ratio corresponding to the vertical effective stress on the normal compression line 
:  Void ratio e at depth x_{j} and time T_{vk} 
:  The final void ratio at depth x_{j} 
:  Void ratio corresponding to preconsolidation vertical effective stress on the normal compression line 
:  at the nodal point 
:  Specific gravity of clay particles 
:  Initial thickness of the clay layer 
:  Lagrangian coordinate of the flow front at time t 
:  Hydraulic gradient 
:  Threshold hydraulic gradient 
:  Coefficient of volume compressibility 
:  Total number of equal thin layers divided in the spatial domain 
:  A dimensionless parameter, 
:  A dimensionless parameter, 
:  at time T_{vk} 
:  A dimensionless parameter, 
:  Initial magnitude of timedependent load 
:  Ultimate magnitude of timedependent load 
:  Timedependent load 
:  A dimensionless parameter, 
:  A dimensionless parameter, 
:  S at the nodal point 
:  A dimensionless parameter, 
:  at the nodal point 
:  The settlement of the clay layer at time t 
:  The final settlement of the clay layer 
:  Time for the moving flow front to reach 
:  Dimensionless time factor, 
:  Dimensionless time factor corresponding to , 
:  The final time of the time interval 
:  The time interval 
:  Time 
:  Time of establishment for the timedependent load 
:  A dimensionless parameter, 
:  U at depth and time 
:  Residual excess pore water pressure 
:  Average degree of consolidation in terms of excess pore water pressure 
:  Average degree of consolidation in terms of deformation 
:  Excess pore water pressure 
:  Velocity of water flow 
:  A dimensionless parameter, 
:  A dimensionless parameter, 
:  at the nodal point 
:  Dimensionless value of the thickness of each thin layer. 
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (No. 51878320). This support is gratefully acknowledged.
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Copyright © 2018 Chuanxun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.