Nomenclature
Ca,	interactive constant with respect to the air phase;	Cvw,	coefficient of the volume change with respect to the water phase;
Cw,	interactive constant with respect to the water phase;	Cσa,	consolidation coefficient for the air phase;
Cva,	coefficient of volume change with respect to the air phase;	Cσw,	consolidation coefficient for the water phase;
g,	gravitational acceleration;	Ra,	parameter of the semi-permeable drainage for the air at the top boundary;
h,	soil layer thickness;	Rw,	parameter of the semi-permeable drainage for the water at the top boundary;
h0,	top boundary thickness;	Sr0,	initial degree of the saturation;
ka,	coefficient of the air permeability;	T,	absolute temperature;
kw,	coefficient of the water permeability;	t,	time;
ka0,	coefficient of the air permeability at the top boundary;	ua,	pore-air pressure;
kw0,	coefficient of the water permeability at the top boundary;	uatm,	atmospheric pressure;
M,	molecular mass of air;	ua0,	absolute pore-air pressure;
m1ka,	coefficient of the air volume change with respect to a change in (σ -ua);	ua0,	initial excess pore-air pressure;
m2a,	coefficient of the air volume change with respect to a change in (ua -uw);	uw,	pore-water pressure;
m1kw,	coefficient of the water volume change with respect to a change in (σ -ua);	uw0,	initial excess pore-water pressure;
m2w,	coefficient of the water volume change with respect to a change in (ua -uw);	w,	settlement;
n0,	initial porosity;	w*,	normalized settlement;
q0,	initial surcharge;	z,	depth;
R,	universal gas constant;	γw,	unit weight of water;
Q(s),	result of the Laplace transform of ;	εv,	volumetric strain.

1 Introduction

The boundary condition for the soil layer consolidation is generally considered as a fully permeable or impermeable boundary. In fact, a cushion covered on the top surface of a soil layer to accelerate the drainage speed should be treated as a semi-permeable boundary, which is neither a fully permeable boundary nor an impermeable boundary^[1]. Great progresses on the study of consolidation for saturated soils under semi-permeable drainage boundary conditions have been achieved^[1-3]. The study on the consolidation of unsaturated soils is more general in engineering practice. However, the solution to the one-dimensional (1D) consolidation of unsaturated soils under the semi-permeable drainage boundary condition is rarely found in the literature.

The consolidation of unsaturated soils as a common issue in geotechnical engineering has attracted much attention of geotechnical engineering researchers. A considerable amount of research has been conducted, and great progresses have been achieved. Scott^[4] estimated the consolidation of unsaturated soils with occluded air bubbles. Biot^[5] proposed a general consolidation theory. Barden^[6] presented an analysis of 1D consolidation of compacted unsaturated clay. On the basis of the hypothesis that the air and water phases were continuous, Fredlund and Hasan^[7] proposed a 1D consolidation theory, in which two partial differential equations were used to describe the dissipation processes of the excess pore pressures in unsaturated soils. This theory was widely accepted, and was later extended to three-dimensional (3D) cases by Dakshanamurthy et al.^[8]. With the assumption that all the soil parameters remained to be constant during the consolidation, Fredlund et al.^[9] presented a simplified form of 1D consolidation equations for unsaturated soils.

Based on the consolidation theory of unsaturated soils proposed by Fredlund and Hasan^[7], Qin et al.^[10-11], Shan et al.^[12-13], Zhou et al.^[14], Zhou and Zhao^[15], Ho and Fatahi^[16-17], and Ho et al.^[18] obtained several analytical or semi-analytical solutions by different mathematical methods. Qin et al.^[10-11] gave a series of analytical and semi-analytical solutions with the Laplace transform and Cayley-Hamilton technique, but the processes of derivation and simplification of the top surface state variables were very complicated. Shan et al.^[12-13] presented the exact solutions of 1D consolidation for the unsaturated soils under single, double, and mixed drainage boundary conditions by the method of separation of variables. However, the final equations have been left undisclosed as a result of the cumbersome derivation, and the solutions were obtained from a complex mathematical process and are difficult to be used by engineers. Zhou et al.^[14] introduced the analytical solutions for single and double drainage boundary conditions by use of two alternative terms ϕ₁ and ϕ₂ to convert the nonlinear inhomogeneous partial differential equations (PDEs) into traditional homogeneous PDEs and the method of separation of variables to solve them. Zhou and Zhao^[15] obtained a numerical solution to the 1D consolidation of unsaturated soils by the differential quadrature method (DQM). Ho and Fatahi^[16-17] and Ho et al.^[18] discussed simple yet precise analytical solutions of the 1D and two-dimensional (2D) plane strain and axisymmetric consolidations of unsaturated soils under homogeneous boundary conditions by adopting the eigen-function expansion method and Laplace transform technique. However, the solution to the 1D consolidation of unsaturated soils under the semi-permeable drainage boundary condition has not been obtained.

This paper presents the semi-analytical solutions to the pore-air and pore-water pressures, and the settlement for the unsaturated soil deposit with the semi-permeable drainage boundary with the 1D consolidation theory proposed by Fredlund and Hasan^[7]. The previous solutions proposed in Refs. [10]-[16] are obtained under the Dirichlet boundary and Neumann boundary conditions. The present solution focuses on the 1D consolidation of the unsaturated soil under the third boundary condition, which can be used to simulate the sand cushion at the top surface. To obtain the final solutions, the inhomogeneous governing equations for unsaturated soils are first derived into the homogeneous equations, which are then solved by the Laplace transform method. It is found that the current solutions are more general and agree well with the existing solutions in the literature, and the consolidation behavior of the unsaturated soil layer with single, double, mixed, or semi-permeable drainage boundaries can all be investigated by changing the drainage boundary parameters. Finally, several examples are given to illustrate the consolidation behavior of unsaturated soils. This study sufficiently investigates the effects of the semi-permeable drainage boundary condition, the air to water permeability ratio, i.e., k_a/k_w, and the soil depth on the changes in the pore-air and pore-water pressures and settlement.

2 Mathematical model 2.1 Governing equations

Fredlund and Hasan^[7] proposed the 1D consolidation equations for unsaturated soils, in which an unsaturated soil layer is considered in infinite horizontal extent with the thickness h under the vertical loading q₀ (see Fig. 1). The water flow, air flow, and settlement only occur in the z-direction. k_a are k_w are the coefficients of the air and water permeabilities in the unsaturated soil layer, respectively. In this paper, we presume that there is a semi-permeable drainage boundary with the thickness of h₀ above the unsaturated soil layer, and k_a0 and k_w0 are the coefficients of the air and water permeabilities of the top boundary, respectively.

The main assumptions are as follows:

(ⅰ) The solid particles and water phase are incompressible.

(ⅱ) The air and water phases are assumed to be continuous and independent.

(ⅲ) The effects of the air diffusing through water and dissolving in water and the movement of the water vapor are ignored.

(ⅳ) The coefficients of permeability with respect to the air and water phases and the volume changes for the soil remain to be constant throughout the consolidation process.

(ⅴ) Loading and deformation take place only along a 1D vertical direction.

(ⅳ) The thin soil layer with the thickness h₀ is considered as the top boundary, which is semi-permeable to the air and water phases.

The governing equations for the water and air phase pressures are as follows^[8]:

(1)

(2)

where u_a and u_w represent the pore-air pressure and the pore-water pressure, respectively. C_a and C_w are the interactive constants with respect to the air and water phases, respectively. C_v^a and C_v^w are the consolidation coefficients for the air and water phases, respectively. The consolidation parameters can be expressed as follows:

(3a)

(3b)

(3c)

(3d)

where m_1k^w and m₂^w are the coefficients of the water volume change with respect to a change in the net normal stress (σ -u_a) and the matric suction (u_a-u_w), respectively. The subscript "k" stands for the K₀-loading. m_1k^a and m₂^a are the coefficients of the air volume change with respect to a change in (σ -u_a) and (u_a -u_w), respectively. γ_w is the unit weight of water, i.e., γ_w=9.8 kN·m^-3. g is the gravitational acceleration, i.e., g=9.8 m·s^-2. S_r0 and n₀ are the initial degrees of saturation and the initial porosity, respectively. u_a⁰ is the absolute pore-air pressure, i.e., u_a⁰ =u_a⁰ +u_atm, where u_atm is the atmospheric pressure. It is assumed in this paper that u_a⁰ =u_atm. This assumption has a little effect on the solution, since u_a⁰ is much smaller than u_atm and rapidly dissipates during the consolidation. M is the molecular mass of air, i.e., M=0.029 kg·mol^-1. R is the universal gas constant, i.e., R=8.314 J·mol^-1·K^-1. T is the absolute temperature.

2.2 Initial condition

(4)

where u_a⁰ and u_w⁰ are the values of the initial excess pore-air and pore-water pressures due to the application of loading q₀ at t=0 s, respectively.

2.3 Boundary conditions

Case Ⅰ The top boundary is considered to be semi-permeable to the air and water phases, and the bottom boundary is considered to be impermeable to the air and water phases.

The top boundary is

(5)

where

and they are the parameters of the semi-permeable drainage boundary for the air and water phases at the top surface, respectively.

The bottom boundary is

(6)

Case Ⅱ The top boundary is considered to be semi-permeable to the air and water phases, and the bottom boundary is considered to be permeable to the air and water phases.

The top boundary is

(7)

The bottom boundary is

(8)

Let us investigate the possible existing boundary conditions in Cases Ⅰ and Ⅱ. When R_a and R_w approach ∞ in Case Ⅰ, the boundary is a single drainage boundary, which is permeable to the air and water phases at the top surface and impermeable to the air and water phases at the bottom surface. When R_a approaches ∞ and R_w=0 in Case Ⅰ, the boundary is a mixed boundary, which is permeable to the air phase and impermeable to the water phase at the top surface, and is impermeable to the air and water phases at the bottom surface. When R_a=0, and R_w approaches ∞ in Case Ⅰ, the boundary is another mixed boundary, which is impermeable to the air phase and permeable to the water phase at the top surface, and is impermeable to the air and water phases at the bottom surface. When R_a and R_w approach ∞ in Case Ⅱ, the boundary is a double drainage boundary, which is permeable to the air and water phases at the top and bottom surfaces. When R_a approaches ∞ and R_w=0 in Case Ⅱ, the boundary is a mixed boundary, which is permeable to the air phase and impermeable to the water phase at the top surface, and is permeable to the air and water phases at the bottom surface. When R_a=0 and R_w approaches ∞ in Case Ⅱ, the boundary is a mixed boundary, which is impermeable to the air phase and permeable to the water phase at the top surface, and is permeable to the air and water phases at the bottom surface. Therefore, the single, double, and mixed drainage boundaries are special cases of the semi-permeable drainage boundary.

3 Derivation of semi-analytical solution

Equations (1) and (2) can be rewritten as follows:

(9)

(10)

where

Equations (9) and (10) can be transformed into the equivalent set partial differential equations of ϕ₁ and ϕ₂ as follows:

(11)

(12)

The details of the derivation and meanings of ϕ₁, ϕ₂, Q₁, and Q₂ can be found in Appendix A.

By applying the Laplace transform to Eqs. (11) and (12), respectively, we have

(13)

(14)

where

and c₂₁ and c₁₂ can be found in Appendix A.

The general solutions of Eqs. (13) and (14) are

(15)

(16)

where x₁² = s/Q₁, x₂² = s/Q₂, and C₁, C₂, D₁, and D₂ are the arbitrary functions of s, which can be determined from the boundary conditions.

By applying the Laplace transform to Eqs. (A10) and (A11) in Appendix A, respectively, we have

(17)

(18)

Combining Eqs. (15) and (16) with Eqs. (17) and (18) gives

(19)

(20)

Taking the derivative with respect to Eqs. (19) and (20) leads to

(21)

(22)

Applying the Laplace transform to Eqs. (5), (6), (7), and (8), respectively, leads to the following boundary conditions in the Laplace domain.

Case Ⅰ

The top boundary is

(23)

The bottom boundary is

(24)

Case Ⅱ

The top boundary is

(25)

The bottom boundary is

(26)

Substituting Eqs. (23) and (24) into Eqs. (19)-(22) gives

(27)

(28)

where

Substituting Eqs. (25) and (26) into Eqs. (19)-(22) gives

(29)

(30)

where

According to the approach of two stress-state variables for unsaturated soils^[7], we have

(31)

where

Applying the Laplace transform to Eq. (31) and integrating along the depth, we can obtain the soil layer settlement in the Laplace domain as follows:

(32)

For Case Ⅰ, substituting Eqs. (27) and (28) into Eq. (32) gives

(33)

For Case Ⅱ, substituting Eqs. (29) and (30) into Eq. (32) gives

(34)

In conclusion, the expressions of ũ_a, ũ_w, and are obtained for the 1D consolidation of unsaturated soils under the semi-permeable drainage boundary conditions.

By adopting the Crump method^[19] to perform the Laplace inversion, we can obtain the semi-analytical solutions of the pore-air and pore-water pressures and soil settlement in the time domain. The details of the Crump method can be found in Ref. ^[20].

4 Examples and discussion

In this study, the parameters are assumed to be as follows:

Based on the proposed solutions, the variations in the pore-air pressure, pore-water pressure, and normalized settlement (w* = w/(m_1k^sq₀h)) are investigated under the semi-permeable drainage boundary condition. The parametric studies are conducted on the effects of the semi-permeable drainage boundary parameters, the ratio of the air to water permeability coefficients, and the depth on the changes in the pore pressures and normalized settlement.

4.1 Verification

Compared with the results from Qin et al.^[10-11], the variations of the pore-water and pore-air pressures with time at z=8 m are calculated under the semi-permeable drainage boundary condition. In order to verify the generality of the present solutions, four kinds of traditional mixed drainage boundary conditions are defined. The solution derivation follows Qin et al.^[11]. Then, the variations of the pore-water and pore-air pressures with time at z=8 m are calculated for the single, double, and mixed drainage boundaries, which are abbreviated as SDB, DDB, and MDB, respectively. The four kinds of traditional mixed drainage boundary conditions are as follows:

(ⅰ) Mixed drainage boundary 1 (MDB1)

The top boundary is permeable to the air and impermeable to the water, and the bottom boundary is impermeable to the water and air, i.e.,

(35)

(ⅱ) Mixed drainage boundary 2 (MDB2)

The top boundary is impermeable to the air and permeable to the water, and the bottom boundary is impermeable to the water and air, i.e.,

(36)

(ⅲ) Mixed drainage boundary 3 (MDB3)

The top boundary is impermeable to the air and permeable to the water, and the bottom boundary is permeable to the water and air, i.e.,

(37)

(ⅳ) Mixed drainage boundary 4 (MDB4)

The top boundary is permeable to the air and impermeable to the water, and the bottom boundary is permeable to the water and air, i.e.,

(38)

Figures 2 and 3 present the calculated results for the single, double, and mixed drainage boundary conditions. It is found that when the parameters R_a and R_w approach infinity, the results of the two solutions obtained by Qin et al.^[10] and this paper for single and double drainage boundaries are the same. Meanwhile, when one of the parameters R_a and R_w approaches infinity and the other parameter is zero, the results of the two solutions for the mixed drainage boundary are also identical.

Therefore, the present solutions are more general and applicable to all kinds of traditional drainage boundary conditions. Besides, the consolidation behavior of unsaturated soils can be investigated by changing the parameters R_a and R_w under the semi-permeable drainage boundary condition.

4.2 Consolidation under the semi-permeable drainage boundary condition

Equations (5) and (7) can outline different semi-permeable drainage boundary conditions when the parameters R_a and R_w take different values. The variations of the pore-water and pore-air pressures and soil settlement with time at z=8 m are calculated for various semi-permeable drainage boundary conditions.

4.2.1 Consolidation with R_a=R_w

Figures 4, 5, and 6 present the results of the pore-air and pore-water pressures and normalized settlement for the semi-permeable drainage boundary condition under different values of parameters R_a=R_w. It can be observed that the parameters of the semi-permeable drainage have a noticeable effect on the dissipation processes of the pore-water and pore-air pressures. From Fig. 4, we can see that, the smaller the values of R_a and R_w are, the more slowly the pore-air pressure dissipates. Compared the results of Cases Ⅰ and Ⅱ, the main effects all begin since the time of 10⁵ s, but the parameters R_a and R_w have a greater effect on the result of Case Ⅰ than that of Case Ⅱ, and the dissipation of the pore-air pressure in Case Ⅰ needs more time.

It can also be found that there are two pore-water pressure variations due to the different values of the parameters R_a and R_w (see Fig. 5). The first pore-water variation is observed before the plateau period, and is very small, which thus is shown in a local enlarged diagram in Fig. 5(b). The second pore-water pressure variation begins at about 10⁸ s, and the dissipation process of the pore-water pressure in Case Ⅰ needs more time. Similarly, the bigger the values of R_a and R_w are, the more quickly the pore-water pressure dissipates. Therefore, the dissipation rate of the pore pressures can be controlled by changing the values of R_a and R_w, which can investigate the consolidation behavior of the unsaturated soils under the semi-permeable drainage boundary condition.

The variations of the normalized settlement are investigated by use of different values of the parameters R_a=R_w under the semi-permeable drainage boundary condition (see Fig. 6). The increases in R_a and R_w increase the normalized settlement rate. Compared with the result in Case Ⅱ, the parameters R_a and R_w have a greater effect on the normalized settlement in Case Ⅰ. This is because that the pore pressures in Case Ⅰ dissipate at only the top surface, which is semi-permeable, and it needs more time to reach the final settlement.

4.2.2 Consolidation with changing R_a and R_w

In this section, the 1D consolidation of unsaturated soils is investigated by changing R_a and fixing R_w under the semi-permeable drainage boundary condition. The results of the variations in the pore-air and pore-water pressures and normalized settlement are presented in Figs. 7, 8, and 9. It can be seen that changing R_a under fixed R_w only affects the dissipation process of the pore-air pressure after 10⁵ s (see Fig. 7), which is the same as that in Fig. 4. Moreover, there is a slight effect on the dissipation process of the pore-water pressure at the intermediate stage, which corresponds to the dissipation process of the pore-air pressure and before the plateau period (see Fig. 8). According to Eq. (5), it is known that the parameter R_a mainly affects the dissipation of the pore-air pressure, and the parameter R_w has a significant effect on the dissipation of the pore-water pressure. Therefore, if R_a changes while R_w is fixed, the effect on the dissipation of the pore-air pressure is noticeable, and the dissipation of the pore-water pressure has minor changes (see the local enlarged diagram of Fig. 8). Figure 9 depicts the variations in the normalized settlement under the semi-permeable drainage boundary condition with changing R_a and fixing R_w. It can be found that a bigger parameter R_a induces a quicker rate of settlement, which only occurs at the early stage. Since the bottom boundary of Case Ⅰ is impermeable and that of Case Ⅱ is permeable, the time reaching the final settlement is different, and there is a smaller effect on the settlement in Case Ⅱ (see the local enlarged diagram of Fig. 9(b)). It is well-known that the variations of settlement depends on the dissipation of the pore pressures. Therefore, the affected stages of the pore pressures and normalized settlement are the same when R_a changes while R_w is fixed.

4.2.3 Consolidation with changing R_w and fixing R_a

In this section, the 1D consolidation of unsaturated soils is investigated by changing R_w with R_a fixed under the semi-permeable drainage boundary condition. Figures 10, 11, and 12 show the variations in the pore-air and pore-water pressures and normalized settlement. Obviously, there is no difference in the dissipation process of the pore-air pressure when R_w changes while R_a is fixed (see Fig. 10). There is only one difference due to R_w (see Fig. 11), which appears at the later stage of the dissipation process of the pore-water pressure. The differences in the pore-water pressure all begin since the time of 10⁸ s and end at the completed dissipation of the pore-water pressure. Moreover, the difference in the normalized settlement has the same pattern with the dissipation of the pore-water pressure (see Fig. 12). It can be found that the bigger the parameter R_w is, the more quickly they change, and the difference in the normalized settlement only appears at the later stage.

On the basis of the results in Figs. 4-12, we can conclude that changing R_a affects the dissipation of the pore-air pressure and normalized settlement at the early stage and the dissipation of the pore-water pressure at the intermediate stage. Besides, there is a noticeable effect on the dissipation of the pore-water pressure and normalized settlement at the later stage due to different R_w, which does not affect the dissipation of the pore-air pressure. Moreover, when R_a or R_w increases, the dissipation rate of the pore pressures and the rate of the normalized settlement increase.

4.3 Consolidation under different k_a/k_w

To investigate the 1D consolidation behavior of unsaturated soils at different values of k_a/k_w under the semi-permeable drainage and traditional drainage boundaries (including SDB and DDB) with k_w=10^-10 m/s and R_a=R_w=5, the pore-air pressure, pore-water pressure, and normalized settlement are computed, and the results are shown in Figs. 13, 14, and 15.

From Figs. 13 and 14, we can see that the pore pressures under the semi-permeable drainage boundary condition dissipate more slowly than those under the SDB and DDB. When the value of k_a/k_w increases, the dissipation of the pore pressures accelerates, which is the same as those in Refs. [10], [12], [14], and [16]. Moreover, there is a more noticeable effect on the pore-air pressure due to the increase in k_a. Since the dissipation of the pore pressures under the DDB is quicker than that under the SDB, the semi-permeable drainage boundary condition has a greater effect on the dissipation of the pore pressures in Case Ⅰ than in Case Ⅱ. Especially, the difference in the pore-water pressure in Case Ⅱ is slight (see the local enlarged diagram of Fig. 14(b)).

Figure 15 shows the normalized settlement curves at different values of k_a/k_w under the semi-permeable drainage boundary condition. The normalized settlement is significantly affected at the early stage when the value of k_a/k_w increases, which corresponds to the dissipation of THE pore pressures (see Fig. 15). From the results of Cases Ⅰ and Ⅱ, we can see that the bigger the value of k_a/k_w is, the more quickly the normalized settlement changes at the first half stage. When the pore-water pressures at different values of k_a/k_w dissipate along the same route, the curves of the normalized settlement converge into a single one. Since the semi-permeable drainage boundary impedes the dissipation of the pore pressures significantly, it needs more time to reach the final settlement compared with the SDB and DDB.

4.4 Consolidation at different depths

Figure 16 depicts the dissipation process of the pore-air pressure at different depths when k_a/k_w=10. It is found that the pore-air pressure dissipates more slowly with depth, but they are prone to dissipate completely at almost the same time at different depths (see Fig. 16). Moreover, the dissipation of the pore-air pressure proceeds more slowly when the point of interest is further away from the top surface. Meanwhile, the dissipation of the pore-air pressure under the SDB or DDB has a similar process with that under the semi-permeable drainage boundary condition, while the pore-air pressure dissipates more quickly under the SDB or DDB. The reason is that the boundary is fully permeable and in favor of the air flow.

Figure 17 demonstrates the dissipation patterns of the pore-water pressure at different depths when k_a/k_w=10. A similar dissipation behavior can be observed for the pore-air pressure in Fig. 16 and the pore-water pressure in Fig. 17. The closer it is to the top boundary, the earlier the pore-water pressure dissipates. The dissipation patterns of the pore-water pressure converge into a single one at almost the same time at different depths. The plateau periods in the dissipation curves of the pore-water pressure become longer with depth. Besides, the dissipation rates of the pore-water pressure are different under different drainage boundary conditions, and there are intersection points on the pore-water pressure curves. The pore-water pressure at the depth of 1 m under the semi-permeable drainage boundary condition dissipates more slowly than that at the depths of 3 m and 5 m under the DDB at the later stage (see Fig. 17(b)).

5 Conclusions

In this paper, a semi-analytical solution to the consolidation equation for unsaturated soils with the semi-permeable drainage boundary condition is obtained by introducing two related variables and the Laplace transform technique. The main conclusions can be summarized as follows:

(ⅰ) Compared with the previous available solutions in the literature, the present semi-analytical solutions are more general. By changing the values of the semi-permeable drainage parameters R_a and R_w, the single, double, and mixed drainage boundary conditions can be realized.

(ⅱ) The values of R_a and R_w have a noticeable effect on the changes of the pore pressures and normalized settlement. R_a affects the dissipation of the pore pressures, but R_w only affects the dissipation of the pore-water pressure. The bigger the values of R_a and R_w are, the more quickly the pore pressures dissipate. When the values of R_a or R_w are greater than 100, the dissipation patterns of the pore pressures do not change, and the boundary can be assumed to be permeable to air or water.

(ⅲ) With the decrease in k_a/k_w, the pore-air pressure dissipates more slowly at the later stage, and the pore-water pressure dissipates more slowly at the intermediate stage. The pore pressures tend to dissipate faster at the depth close to the top surface. Compared with the results under single and double drainage boundary conditions, the semi-permeable drainage boundary condition impedes the dissipation of the pore pressures, and the impeded effect in Case Ⅰ is more obvious than that in Case Ⅱ.

Appendix A

Multiplying Eqs. (9) and (10) by the arbitrary constants c₁ and c₂, respectively, and adding these two equations together lead to

(A1)

which can be transformed into a conventional diffusion equation with the variable ϕ= u_ac₁ + u_wc₂ by introducing a constant Q that satisfies

(A2)

(A3)

Therefore, the constant Q must satisfy

(A4)

which is a quadratic equation of Q, and has the following two roots Q₁ and Q₂:

(A5)

When Q = Q₁, the solutions of Eqs. (A2) and (A3) are c₁₁ and c₂₁. When Q = Q₂, the solutions are c₁₂ and c₂₂.

Without loss of generality, it is possible to assume that c₁₁= c₂₂ = 1. Then, c₁₂ and c₂₁ can be expressed as follows:

(A6)

(A7)

Therefore, Eq. (A1) can be rewritten as follows:

(A8)

(A9)

where

(A10)

(A11)

The transformed initial condition ϕ₁(z, 0) and ϕ₂(z, 0) are as follows:

(A12)

(A13)

By applying the Laplace transform to Eqs. (A12) and (A13), respectively, we have

(A14)

(A15)