Numerical analysis of seepage–deformation in unsaturated soils

Abstract

A coupled elastic–plastic finite element analysis based on simplified consolidation theory for unsaturated soils is used to investigate the coupling processes of water infiltration and deformation. By introducing a reduced suction and an elastic–plastic constitutive equation for the soil skeleton, the simplified consolidation theory for unsaturated soils is incorporated into an in-house finite element code. Using the proposed numerical method, the generation of pore water pressure and development of deformation can be simulated under evaporation or rainfall infiltration conditions. Through a parametric study and comparison with the test results, the proposed method is found to describe well the characteristics during water evaporation/infiltration into unsaturated soils. Finally, an unsaturated soil slope with water infiltration is analyzed in detail to investigate the development of the displacement and generation of pore water pressure.

Appendices

Appendix 1

From the simplified phase diagrams (see Fig. 8) for an unsaturated soil, we have

$$ V_{\text{v}} = V_{\text{a}} + V_{\text{w}} ,\quad {\text{and}}\quad V = V_{\text{a}} + V_{\text{w}} + V_{\text{s}} = 1, $$

$$ S_{\text{r}} = V_{\text{w}} /V_{\text{v}} ,\quad n = V_{\text{v}} /V, $$

so we obtain V _a = V _v − V _w. However, for the unsaturated soils, some air is embedded in the water, which can be formulated by the simplified method as V _a′ = c _h V _w, so the total air content of a unit volume of unsaturated soil element can be obtained:

$$ V_{\text{a}} + \, V\prime_{\text{a}} = \, V_{\text{v}} - \, V_{\text{w}} + \, c_{\text{h}} V_{\text{w}} = nV\left[ {1 - \, S_{\text{r}} + c_{\text{h}} S_{\text{r}} } \right] = \left[ {1 - \, S_{\text{r}} + c_{\text{h}} S_{\text{r}} } \right]n. $$

We define n _a = [1 − S _r + c _h S _r]n = [1 − (1 − c _h)S _r]n, where n is the ratio of the pore air of a unit soil element, which signifies the air content in the voids of a soil element within unit volume.

Fig. 8

Simplified three phase diagram

Appendix 2

For a unit volume soil element, the pore air pressure u _a will be generated upon loading. Under the conditions that the air in the voids cannot be discharged, from Boyle’s law, we have

$$ \left( {p_{\text{a}} + u_{\text{a}} } \right)\left( {V_{\text{a}} + c_{\text{h}} V_{\text{w}} } \right) = \left( {p_{\text{a}} + u_{\text{a0}} } \right)\left( {V_{\text{a0}} + c_{\text{h}} V_{\text{w0}} } \right), $$

(34)

in which p _a is the atmospheric pressure, u _a is the pore air pressure, V _a0 is the initial content of pore air of a unit volume soil element (the corresponding pore air pressure u _a0 = 0), and V _a is the content of pore air of a unit volume soil element. For a unit volume soil element (see Fig. 8), we have V _w = nS _r and V _a = n(1 − S _r). Combining equation (4), n _a = [1 − (1 − c _h)S _r]n, and Eq. (34), we have

$$ \left( {p_{\text{a}} + u_{\text{a}} } \right)n_{\text{a}} = \left( {p_{\text{a}} + u_{\text{a0}} } \right)n_{\text{a0}} , $$

(35)

in which n _a0 is the initial ratio of pore air of a unit soil element, n _a0 = [1 − (1 − c _h)S _r0]n ₀, S _r0 is initial degree of saturation, and n ₀ is initial porosity.

Because the initial air pressure is equal to the atmospheric pressure (u _a0 = 0), from Eq. (35), we have:

$$ u_{\text{a}} = \left( {\frac{{n_{\text{a0}} }}{{n_{\text{a}} }} - 1} \right)p_{\text{a}} . $$

(36)

Appendix 3

In the following, we ignore the flow of the dissolved air in the pore water, the vapor in the pore air, and the influence of temperature. For a unit volume soil element, the mass of pore air discharged per unit time has the following equation (mass conservation equation):

$$ \frac{\partial }{\partial t}\left[ {\rho_{\text{a}} \left( {1 - S_{\text{r}} } \right)n + \rho_{\text{a}} c_{\text{h}} S_{\text{r}} n} \right] = \frac{\partial }{\partial t}\left[ {\rho_{\text{a}} n_{\text{a}} } \right] = \frac{{{\text{d}}q_{\text{a}} }}{{{\text{d}}t}}, $$

(37)

in which q _a is the mass of pore air discharged within a unit soil element. From Eq. (34), we can deduce the following equation in incremental form:

$$ \Delta \rho_{\text{a}} \cdot n_{\text{a}} + \rho_{\text{a}} \cdot\Delta n_{\text{a}} =\Delta q_{\text{a}} . $$

(38)

Differentiating equation (5), ρ _a = ρ _a0(1 + u _a/p _a),  we have

$$ {{\Delta }}\rho_{E} = \rho_{\text{a0}} \cdot {{\Delta }}u_{\text{a}} /p_{\text{a}} . $$

(39)

From Eq. (7), we have

$$ {{\Delta }}q_{\text{a}} = \xi \cdot \rho_{\text{a}} \cdot {{\Delta }}n_{\text{a}} , $$

(40)

Substituting Eqs. (39) and (40) into Eq. (38), we obtain

$$ n_{\text{a}} \cdot \rho_{\text{a0}} \cdot {{\Delta }}u_{\text{a}} /p_{\text{a}} + \rho_{\text{a}} \cdot {{\Delta }}n_{\text{a}} = \xi \cdot \rho_{\text{a}} \cdot {{\Delta }}n_{\text{a}} , $$

(41)

Substituting Eq. (5), ρ _a = ρ _a0(1 + u _a/p _a), into Eq. (41), we have

$$ {{\Delta }}u_{\text{a}} = - \frac{{p_{\text{a}} + u_{\text{a}} }}{{n_{\text{a}} }}\left( {1 - \xi } \right){{\Delta }}n_{\text{a}} . $$

(42)

Appendix 4

The Eq. (8) can be expressed as:

$$ \frac{{du_{\text{a}} }}{{p_{\text{a}} + u_{\text{a}} }} = - \frac{1 - \xi }{{n_{\text{a}} }}{\text{d}}n_{\text{a}} , $$

(43)

Integrating the above equation, we obtain the following:

$$ { \ln }\left( {p_{\text{a}} + u_{\text{a}} } \right) = - \left( {1 - \xi } \right){ \ln }\left( {n_{\text{a}} } \right) + {\text{const}} ., $$

(44)

Substituting the initial conditions n _a0 and u _a0 into Eq. (44), we have

$$ {\text{const}} . {\text{ = ln}}\left[ {\left( {p_{a} + u_{a0} } \right)n_{a0}^{{\left( {1 - \xi } \right)}} } \right], $$

(45)

Combining Eqs. (44) and (45), we have

$$ u_{\text{a}} = \left[ {\left( {\frac{{n_{\text{a0}} }}{{n_{\text{a}} }}} \right)^{{\left( {1 - \xi } \right)}} - 1} \right]p_{\text{a}} . $$

(46)

Appendix 5

The elements of the coefficient matrix in the finite element Eq. (33) are as follows:

$$ k_{ij}^{11} = \int {\left[ {\left( {d_{11} + A_{2} } \right)\frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + d_{44} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + d_{14} \left( {\frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z} + \frac{{\partial N_{j} }}{\partial x}\frac{{\partial N_{i} }}{\partial z}} \right)} \right]} \;{\text{d}}x{\text{d}}z, $$

(47)

$$ k_{ij}^{12} = \int {\left[ {\left( {d_{12} + A_{2} } \right)\frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z} + d_{14} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + d_{24} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + d_{44} \frac{{\partial N_{j} }}{\partial x}\frac{{\partial N_{i} }}{\partial z}} \right]\;} {\text{d}}x{\text{d}}z, $$

(48)

$$ k_{ij}^{21} = \int {\left[ {\left( {d_{12} + A_{2} } \right)\frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial x} + d_{14} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + d_{24} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + d_{44} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z}} \right]} \;{\text{d}}x{\text{d}}z, $$

(49)

$$ k_{ij}^{22} = \int {\left[ {\left( {d_{22} + A_{2} } \right)\frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + d_{44} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + d_{24} \left( {\frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial x} + \frac{{\partial N_{j} }}{\partial z}\frac{{\partial N_{i} }}{\partial x}} \right)} \right]} {\text{d}}x{\text{d}}z, $$

(50)

$$ k_{ij}^{33} = - \int {\left[ {k_{x} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + k_{z} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z}} \right]} \;{\text{d}}x{\text{d}}z, $$

(51)

$$ k_{ij}^{13} = - \rho_{\text{w}} g\int {A_{1} } \frac{{\partial \bar{N}_{i} }}{\partial r}\bar{N}_{j} d{\text{d}}z, $$

(52)

$$ k_{ij}^{23} = - \rho_{\text{w}} g\int {A_{1} \frac{{\partial \bar{N}_{i} }}{\partial z}\bar{N}_{j} {\text{d}}x{\text{d}}z} , $$

(53)

$$ k_{ij}^{31} = - \rho_{\text{w}} g\int {S_{\text{r}} } \frac{{\partial \bar{N}_{j} }}{\partial x}\bar{N}_{i} {\text{d}}x{\text{d}}z, $$

(54)

$$ k_{ij}^{32} = - \rho_{\text{w}} g\int {S_{\text{r}} } \frac{{\partial \bar{N}_{j} }}{\partial z}\bar{N}_{i} {\text{d}}x{\text{d}}z, $$

(55)

$$ s_{ij}^{{}} = - \rho_{\text{w}} g\int {c_{\text{s}} N_{i} N_{j} } {\text{d}}x{\text{d}}z, $$

(56)

where c _s = μn/S _r