Abstract

Analytical solutions are developed for one-dimensional consolidation of double-layered saturated soil subjected to groundwater fluctuations. The solutions are derived by an explicit mathematical procedure using Duhamel’s theorem in conjunction with a Fourier series, when groundwater fluctuation is described by a general time-dependent function and assumed to be the pore water pressure variations at the upper boundary. Taking as an example the harmonic groundwater fluctuation, the relevant response of the excess pore water pressure is discussed in detail, and the main influencing factors of the excess pore pressure distribution are analyzed. A dimensionless parameter has been introduced because it significantly affects the phase and the amplitude of excess pore pressures. The influences of the coefficients of permeability and compressibility of soil on the excess pore pressure distribution are different and cannot be incorporated into the coefficient of consolidation in double-layered soil. The relative permeability ratio of two clayey soils also plays an important role on the curves of the distributions of the excess pore pressures. The effects of the thickness of the soil layer on the excess pore pressure distribution should be considered together with the dimensionless parameter and the permeability and compressibility of the double-layered soil system.

1. Introduction

Excess pore water pressures are generated within saturated clayey soil due to groundwater fluctuations arising from sources such as river or ocean tides, seasonal or climatic water table changes, and human factors (e.g., pumping or recharging of water) [1]. The more common analysis models for the generation and propagation of excess pore water pressures can be generalized as double clayey soil layers instead of a single layer of soil with pervious top and impervious bottom boundary conditions (PTIB) (Figure 1(a)) or with both pervious top and bottom boundary conditions (PTPB) (Figure 1(b)) [2]. On account of the marginal difference between the natural unit weight and the saturated unit weight of a soil layer with high permeability, the total stress in the clay soils generally varies slightly and, accordingly, can be regarded as constant. Therefore, the groundwater table variation in the top sand layer with high permeability can reasonably be considered as the excess pore water pressure variation at the upper boundary of clayey soil layers. In addition, the evolution of excess pore pressures with time in clayey soil layers generally do not reflect the changes occurring at the boundary.

(a) PTIB

(b) PTPB

(a) PTIB
(b) PTPB

Figure 1 

One-dimensional consolidation model of two clayey soil layers due to groundwater fluctuations for (a) PTPB and (b) PTIB boundary conditions.

Conte and Troncone [1] achieved analytical solutions of the excess pore pressures in single clayey soil caused by pore pressure varying at the boundary, which was conducted based on the calculation procedure for the solution of a sinusoidal pore pressure variation at the boundary [3, 4]. Many forms of analytical procedures of one-dimensional consolidation for layered systems have been provided by other authors. Schiffman and Stein [5] developed a general solution for the one-dimensional consolidation problem that considered a variety of boundary conditions, initial conditions, loading histories, and types of multilayered soil; however, this solution is quite tedious to use. Lee et al. [2] derived a general analytical solution for the problem of layered soils with a time-dependent load, which was more explicit than the one given by Schiffman and Stein [5]. Luan and Qian [6] developed an analytical solution for the case of double-layered soil profile with a depth-dependent ramp load. Xie [7] demonstrated a mathematical procedure for an analytical solution of one-dimensional consolidation subject to the time-dependent load and depth-dependent initially nonlinear pore pressure and subsequently developed a procedure for a multilayered system [8]. Xu et al. [9] applied the Laplace transform and the Fourier technique to solve a one-dimensional consolidation problem for layered soil under arbitrary loading. An analytical study involving one-dimensional consolidation of double-layered soil with partially drained boundaries was provided by Xie et al. [10]. The recent research studies on analytical solutions for consolidation of porous media to predict the variations of the pore pressures and settlement with time were also proposed by Ho et al. [11, 12]. Hydrogeologists studied the groundwater table response to tidal fluctuation in a coastal aquifer [13, 14]. The studies of tide-induced pore pressure were conducted by van der Kamp and Gale for a homogeneous porous medium [15] and by Wang and Davis for a multilayer medium [16].

However, no analytical solution was found for one-dimensional consolidation with such a complex upper boundary condition that takes the layered characteristics of soil into account. Analytical solutions are proposed for one-dimensional consolidation of double-layered saturated soils subjected to groundwater fluctuations for the simplest multilayer system: the double-layered system. Both PTIB and PTPB boundary conditions are taken into consideration in this paper. Groundwater table variation is assumed to be the pore water pressure variation at the boundary described by a general time-dependent function. The solution is derived by a more complicated mathematical procedure using Duhamel integration in conjunction with a Fourier series. The response of excess pore pressure to groundwater fluctuation and its main influencing factors were analyzed using the proposed solution, which can be significantly different from the solution of a one-layered system.

2. Solution Method

2.1. Governing Equations

With all the assumptions of Terzaghi’s one-dimensional consolidation theory [17], except for the pore water pressure at the upper boundary, the equation governing the generation and propagation of excess pore water pressures in clayey soil layers of this analytical model (Figure 1(b)) iswhere is the spatial coordinate; is time; and are the excess pore water pressures in the upper layer (at ) and the lower layer (at ) of the double-layered clayey soil system, which depend on both and ; and are the coefficients of swelling or consolidation of the upper soil layer and the lower soil layer, respectively, which are formulated by the expression (); and are the coefficient of permeability in the vertical direction and the coefficient of volume change of the soil, respectively. In practice, as well as and are generally accepted to be constant during the process [4].

According to the analytical model assumptions, the excess pore water pressure on the upper boundary of the double-layered clayey soil system is first assumed to be harmonic with time, which is expressed as follows:where and determine the excess pore pressure amplitudes and is the angular frequency, whose functional relation with the period is .

The bottom boundary of the double-layered clayey soil system may be fully permeable or impermeable; thus,where is the total thickness of this double-layered soil model.

In addition, the seepage and excess pore water pressures of the double clayey soil layers are continuous:where is the thickness of the upper layer soil of double-layered soil model.

Moreover, the initial condition is

Overall, the solution of this analytical model is formulated by (1), which should satisfy the above boundary conditions (2) and (3a) or (3b), (4), and the initial conditions in (5); the solution follows the analytical procedure of solving the partial differential equation with complicated inhomogeneous boundary conditions.

2.2. Analytical Solution of Harmonic Groundwater Fluctuations

Equation (2) implies that the excess pore water pressure at the upper boundary of the model varies harmonically with time. Furthermore, the solution procedure described as Duhamel’s theorem [1, 3, 18, 19] is used to derive (1) with the boundary and initial conditions given by (2) and (3a) or (3b), (4), and (5), which is obtained by the following integration:where is determined by (2) and denotes the solution to (1) and (3a) or (3b), (4), and (5), when the excess pore water pressure at the upper boundary is kept at unity (as the value ).

First, the excess pore water pressure in double clayey soil layers with pervious top and impervious bottom boundary conditions (Figure 1(a)) is systematically obtained in this section. In particular, is reduced to a separable partial differential equation:where is a function of and and is a function only of ; and are for and and are for . Subject to all the assumptions identified in a previous section, and should satisfy the following equations:

The function is determined by (8), with the boundary conditions in (9)–(12); the result takes the following form:

A more complex mathematical procedure is used to derive the solution for (13), with the boundary conditions in (14), (15), (17), and (18) and the initial conditions in (16), in which can also be reduced to a separable partial differential equation, expressed as follows:

Substituting (20) into differential equation (13) and introducing a constant , which is defined as the eigenvalue, we have

Deriving (21) yields

 where , , , , and are undetermined coefficients.

Based on the boundary conditions in (14) and (15), coefficients and are given in the forms

Substituting (24) into the boundary conditions in (17) and (18) yields

To guarantee nontrivial solutions of and , the determinant of the second-order correlation matrix composed of (25) is identically vanishing. Consequently, the eigenequation is obtained as follows: in which = + , is the thickness of the lower layer soil of double-layered soil model; is defined as the interlaminar seepage-compression comprehensive parameter ratio [6]. The eigenvalue of (26), which is proved to be infinite, can be derived by an iteration process in general, except for some special conditions; for example, when and , (26) yields .

Considering (24) and (25), in conjunction with (22), (23a), and (23b), the general solution of (20) can also be expressed in series formwhere is defined as the modal participating factor; the eigenfunction is represented as follows:

The above eigenfunctions are proved to be orthogonal to the coefficient of the volume change of the soil by the following equation:whereand is defined as follows:

In particular, the orthogonal property demonstrated in (29) is familiar and useful in dealing with the mathematical problems of the double-layered system [6, 7, 10].

Moreover, an inspection of the initial condition in (16) combined with (29) reveals

Substituting (19) and (26) into (31a), after some algebraic manipulation, renders , such that

Consequently, the solution for is developed, and the final form for is represented as follows:where , , and are determined by (28a), (28b), and (31b), respectively. Meanwhile, the parametric coefficient is determined by (26).

Therefore, substituting for in (2) and for in (7), at the same time performing the partial derivative of with respect to , (6) is developed into the following simplified calculation formulas:whereand denotes the time factor of Terzaghi’s theory.

Furthermore, the excess pore water pressures in double clayey soil layers with both pervious top and bottom boundary conditions (Figure 1(b)) can be obtained using a similar method, when the corresponding boundary conditions are described in (2) and (3b). After performing the same steps previously described, the solution for , is

where the eigenvalues , , , , and are determined by the equations

Based on (35a) and (35b), the final solutions for (6) in this case are given bywhereand and are defined by (33f) and (33g), respectively.

Note that (33a) and (33b) or (36a) and (36b) are applicable only when ; for simplification, (2) is used to calculate the values of excess pore water pressure at [1].

2.3. Analytical Solution of General Groundwater Fluctuations

Based on the derived equations, an analytical solution to (1) is derived using the Fourier series, when a more general time-dependent excess pore pressure due to an arbitrary groundwater fluctuation is applied to the top surface of double-layered soil.

First, should be expanded in harmonic components using the Fourier series theorem, as outlined by Xu et al. [9] and Conte and Troncone [1, 18], such thatin which the series amplitudes and associated with the frequency (with , and is the period of ) are represented as follows, respectively:and can be obtained from (10) by setting ; thus,

Second, the corresponding excess pore water pressure for each of the harmonic components in (37) is calculated by (33a) and (33b) or (36a) and (36b) in which is indicated as , and are substituted by and , respectively, and is replaced by . Finally, the actual excess pore water pressure at any time and depth is formulated by using the principle of superposing all the components of the determined :in which is represented by (40) and is represented by (32a) and (32b) or (35a) and (35b).

The analytical solutions are so far available for any arbitrary groundwater fluctuation of double-layered system.

3. Discussion of the Analytical Solutions

3.1. Comparison with Existing Analytical Solutions

The solutions by Conte and Troncone [1] and Ying et al. [19] assumed that there is only a single clayey soil layer. In this section, these solutions are demonstrated to be special cases of the solution presented in this paper. Taking as an example the PTIB boundary condition, if there is a single clayey soil layer (, ), from (26), the expression for becomes

Substituting (42) back in to (31b) yields

The combination of (33a), (33b), (42), and (43) directly leads towhich is essentially the same as the equation derived by Conte and Troncone [1] and Ying et al. [19]. The same procedure can be conducted for the case with PTPB boundary condition; the resulting equation takes the formwhere

It may be meaningful to distinguish the difference between the behavior of the single system and the layered system. In comparison with the expressions for the single system, the ones of the double-layered system are obviously much more complicated. Equations (44) and (45) indicate that the dimensionless parameter may be the primary influencing factor of excess pore pressure distribution in single clayey soil layer, which has been verified by Ying et al. [19]. The expressions of excess pore pressures in (33a)–(33g) and (36a)–(36e) indicate that four important parameters, the dimensionless parameter , the permeability ratio of the soil, the compressibility ratio of the soil, and the thickness ratio of the soil layers, are involved in the model. Those parameters may have a significant influence on the excess pore pressure distribution of the double clayey soil layers. The discussion above can be the basis of the following further discussion.

3.2. Influence of Various Parameters on Excess Pore Water Pressures

In this section, we pay principal attention to the influence of various parameters on the distribution of excess pore water pressure subject to groundwater fluctuation, for the purpose of understanding the behavior of the layered system and being aware of the potential complications.

For simplicity, let in (2), with the assumption that the groundwater fluctuation is a cosinusoidal wave, and define the following dimensionless parameters:

Typical values of those four important parameters (, , , and ) were selected to study their influence on the distribution of excess pore water pressures. In addition, a computation program was developed, and the main results are shown in Figures 2 to 6. The distributions of excess pore water pressure with the PTIB boundary condition and with the PTPB boundary condition are obviously different. However, the influence laws of these important parameters on the cases with PTIB and PTPB boundary conditions are alike. For the sake of simplicity, only the cases with PTIB boundary condition are analyzed in Figures 3 to 6.

(a) PTIB

(b) PTPB

(a) PTIB
(b) PTPB

Figure 2 

Influence of the dimensionless parameter on the excess pore water pressures.

Figure 3 

Influence of the permeability on the excess pore water pressures (PTIB).

Figure 2 shows the variation of the excess pore water pressure with dimensionless parameter for both PTPB and PTIB boundary conditions in clayey soil layers at when other dimensionless parameters , , and remain constant (i.e., , , and ). As can be observed from Figure 2, the greater the dimensionless parameter , the greater the excess pore water pressures and the propagation depth of the excess pore water pressures. When the value of is comparatively low, the excess pore water pressures attenuate rapidly with depth, the propagation depth of excess pore water pressures is relatively small, and the hysteresis of the propagation of excess pore water pressures exists in the deeper depth. When the dimensionless parameter increases and becomes greater than a certain value (say in this case), the distribution of excess pore water pressures better fits the linear distribution. Obviously, the propagation of excess pore water pressures is related to the dimensionless parameter , which is similar to the characteristics of the propagation of excess pore water pressures in single clayey soil [19]. However, there exist the inflection points of the isochrones of the excess pore water pressures in a double-layered system other than a single-layered system.

Figures 3 and 4 show the influence of the permeability and the compressibility, respectively, of soil on the propagation of excess pore water pressures for PTIB boundary condition in clayey soil layers at when other dimensionless parameters , are fixed (i.e., and ). As shown in Figure 3, the influence of the dimensionless ratio on the excess pore water pressures in the upper and the lower layers is opposite when the compressibility of two layers is the same. The greater the ratio , the smaller the excess pore water pressure in upper layer; in contrast, the behavior is opposite for the lower layer. The inflection points of the excess pore water pressure distribution curves exist in the interface between two clayey soil layers transparently, except for , which implies that the permeability of the two layers is the same. Figure 4 indicates that the inflection points of the excess pore water pressures distribution curves disappear if the permeability of two layers is the same. As expected, the excess pore water pressure isochrone moves gradually to the left, indicating that higher values of the dimensionless parameter results in higher attenuation of the excess pore water pressures. The data in Figures 3 and 4 implies that the hysteresis of the propagation of excess pore water pressures exists when the second clayey soil layer is with lower permeability or higher compressibility.

Figure 4 

Influence of the compressibility on the excess pore water pressures (PTIB).

Figure 5 shows the influence of the dimensionless parameter on the propagation of the excess pore water pressures for the PTIB boundary condition in clayey soil layers at . Obviously, for the case where the permeability in the lower layer is higher than the permeability in the upper layer (e.g., in Figure 5(a)), the thickness of the lower layer with higher permeability is inversely correlated to the excess pore water pressure attenuation in the lower layer and directly correlated to the excess pore water pressure attenuation in the upper layer. When the permeability ratio of the double-layered system is less than 1 (e.g., in Figure 5(c)), the thickness of the lower layer with lower permeability is directly correlated to the excess pore water pressure attenuation in the lower layer and inversely correlated to the excess pore water pressure attenuation in the upper layer. When the lower layer has higher compressibility (e.g., in Figure 5(c)), the thickness of the lower layer is directly correlated to the excess pore water pressure attenuation, while the opposite results occur for the cases of the lower layer with a lower compressibility (e.g., in Figure 5(d)). The results in Figure 5 reveal that the thickness of the lower layer with greater permeability or smaller compressibility is directly correlated to the easy of the excess pore pressure propagation in the double-layered system.

(a) ,

(b) ,

(c) ,

(d) ,

(a) ,
(b) ,
(c) ,
(d) ,

Figure 5 

Influence of the dimensionless parameter on the excess pore water pressures (PTIB).