Effect of transient flow and pseudo-static forces on the lateral earth pressures developed in retained unsaturated backfills

Abstract

This article primarily investigates the effect of transient flow and seismic loadings on the active and passive earth pressures developed in retained unsaturated backfills. The classical Rankine’s earth pressure theory for saturated soils is duly modified to account for the dynamic loadings and the partial saturation state of the soil under transient infiltration. The influence of the matric suction on the water content and flow rate is apprehended by incorporating a suitable soil water characteristics curve and a hydraulic conductivity function. The suction stress-based effective stress formulation is adopted for the purpose of analysis. The one-dimensional Richards' equation models the transient water flow in the vadose zone. The seismic loading is idealized as a statically applied inertial force by using the pseudo-static method. A comprehensive parametric study is carried out to investigate the impact of soil strength properties, hydromechanical parameters, seismic loading conditions, and flow behaviour (rate and duration) of the transient infiltration on the two failure states -- active and passive. The tension crack depths in the vadose zone under transient flow conditions are also calculated.

Appendices

Appendix A

1.1 A.1 Deriving the governing differential equation

$$ {\text{Richards}}^{\prime}\;{\text{ equation}}:\;\frac{\partial }{\partial z}\left[ {k\left( {h_{m} } \right)\left( {\frac{{\partial h_{m} }}{\partial z} + 1} \right)} \right] = \frac{\partial \theta }{{\partial t}} $$

(A.1)

$$ {\text{Gardener}}^{\prime}{\text{s }}\;{\text{one}} - {\text{parameter}}\;{\text{ HCF }}\;{\text{model}}:k = k_{s} \exp \left( {\alpha h_{m} } \right) $$

(A.2)

$$ {\text{Gardener}}^{\prime}{\text{s}}\;{\text{SWCC}}\;{\text{model}}:\Theta_{n} = S_{e} = \frac{{\theta - \theta_{r} }}{{\theta_{s} - \theta_{r} }} = \exp \left( {\alpha h_{m} } \right) $$

(A.3)

Eq. (A.3) can be further simplified as follows:

$$ \theta = \theta_{r} + \left( {\theta_{s} - \theta_{r} } \right)\exp \left( {\alpha h_{m} } \right) $$

(A.4)

Performing partial differentiation of Eq. (A.4) with respect to time (t), the following expression can be obtained:

$$ \frac{\partial \theta }{{\partial t}} = \alpha \left( {\theta_{s} - \theta_{r} } \right)\exp \left( {\alpha h_{m} } \right)\frac{{\partial h{}_{m}}}{\partial t} $$

(A.5)

Substituting Eq. (A.2) and Eq. (A.5) into Eq. (A.1), the following differential equation can be obtained:

$$ k_{s} \left[ {\exp \left( {\alpha h_{m} } \right)\frac{{\partial^{2} h_{m} }}{{\partial z^{2} }} + \alpha \exp \left( {\alpha h_{m} } \right)\left( {\frac{{\partial h_{m} }}{\partial z}} \right)^{2} } \right] + \alpha k_{s} \exp \left( {\alpha h_{m} } \right)\frac{{\partial h_{m} }}{\partial z} = \alpha \left( {\theta_{s} - \theta_{r} } \right)\exp \left( {\alpha h_{m} } \right)\frac{{\partial h{}_{m}}}{\partial t} $$

(A.6)

$$ \Rightarrow k_{s} \exp \left( {\alpha h_{m} } \right)\left[ {\left\{ {\frac{{\partial^{2} h_{m} }}{{\partial z^{2} }} + \alpha \left( {\frac{{\partial h_{m} }}{\partial z}} \right)^{2} } \right\} + \alpha \frac{{\partial h_{m} }}{\partial z}} \right] = \alpha \left( {\theta_{s} - \theta_{r} } \right)\exp \left( {\alpha h_{m} } \right)\frac{{\partial h{}_{m}}}{\partial t} $$

(A.7)

$$ \Rightarrow \left[ {\frac{{\partial^{2} h_{m} }}{{\partial z^{2} }} + \alpha \left( {\frac{{\partial h_{m} }}{\partial z}} \right)^{2} } \right] + \alpha \frac{{\partial h_{m} }}{\partial z} = \frac{{\alpha \left( {\theta_{s} - \theta_{r} } \right)}}{{k_{s} }}\frac{{\partial h_{m} }}{\partial t} $$

(A.8)

A.2 Numerical Scheme for solving the IVP

$$ {\text{GD}}\;{\text{ SWCC}}:\left[ {\frac{{\partial^{2} h_{m} }}{{\partial z^{2} }} + \alpha \left( {\frac{{\partial h_{m} }}{\partial z}} \right)^{2} } \right] + \alpha \frac{{\partial h_{m} }}{\partial z} = \lambda \frac{{\partial h_{m} }}{\partial t};\;\lambda = \frac{{\alpha \left( {\theta_{s} - \theta_{r} } \right)}}{{k_{s} }} $$

$$ \left[ {(1 - \theta )\left. {\frac{{\partial^{2} h_{m} }}{{\partial z^{2} }}} \right|_{t + \Delta t} + \left. {\theta \frac{{\partial^{2} h_{m} }}{{\partial z^{2} }}} \right|_{t} + \alpha \left( {\left. {\frac{{\partial h_{m} }}{\partial z}} \right|_{t} } \right)^{2} } \right] + \alpha \left. {\frac{{\partial h_{m} }}{\partial z}} \right|_{t} = \lambda \frac{{\partial h_{m} }}{\partial t} $$

(A.9)

In terms of Crank-Nicolson finite difference terms (θ =0.5), the generalized equations for any arbitrary i^th node can be written as:

$$ \left[ {\frac{{h_{m,i - 1}^{t + \Delta t} - 2h_{m,i}^{t + \Delta t} + h_{m,i + 1}^{t + \Delta t} }}{{2\Delta z^{2} }} + \frac{{h_{m,i - 1}^{t} - 2h_{m,i}^{t} + h_{m,i + 1}^{t} }}{{2\Delta z^{2} }} + \alpha \left[ {\frac{{h_{m,i + 1}^{t} - h_{m,i}^{t} }}{\Delta z}} \right]^{2} } \right] + \alpha \left[ {\frac{{h_{m,i + 1}^{t + \Delta t} - h_{m,i}^{t + \Delta t} }}{\Delta z}} \right] = \lambda \frac{{h_{m,i}^{t + \Delta t} - h_{m,i}^{t} }}{\Delta t} $$

(A.10)

$$ \begin{aligned} \Rightarrow & - m_{1} h_{m,i - 1}^{t + \Delta t} + \left( {1 + 2m_{1} + m_{2} } \right)h_{m,i}^{t + \Delta t} - \left( {m_{1} + m_{2} } \right)h_{m,i + 1}^{t + \Delta t} = h_{m,i}^{t} + m_{1} \left( {h_{m,i - 1}^{t} - 2h_{m,i}^{t} + h_{m,i + 1}^{t} } \right) \\ & + \alpha m_{1} \left( {h_{m,i + 1}^{t} - h_{m,i}^{t} } \right)^{2} \\ \end{aligned} $$

Here,$h_{m,i + 1}^{t} ,h_{m,i}^{t}$, and $h_{m,i - 1}^{t}$ are the matric suction head at time t corresponding to i+1, i, and i-1 grid-points.

$$ m_{1} = \frac{\Delta t}{{2\lambda \Delta z^{2} }}, \, m_{2} = \frac{\alpha \Delta t}{{\lambda \Delta z}};L_{1} = \frac{{k_{s} }}{{\alpha \left( {\theta_{s} - \theta_{r} } \right)}} $$

Top boundary constraint yields the following algebraic relation:

$$ \left. {k_{s} \exp \left( {\alpha h_{m} } \right)\left( {\frac{{\partial h_{m} }}{\partial z} + 1} \right)} \right|_{{@(n + 1)^{{{\text{th}}}} {\text{node}}}} = q $$

$$ \Rightarrow \left( {\left. {\frac{{\partial h_{m} }}{\partial z}} \right|_{n + 1} + 1} \right) = \frac{q}{{k_{s} }}\exp \left( { - \alpha h_{m,n + 1} } \right)\; \Rightarrow \left. {\frac{{\partial h_{m} }}{\partial z}} \right|_{n + 1} = Q\exp \left( { - \alpha h_{m,n + 1} } \right) - 1 $$

$$ \Rightarrow \frac{{h_{m,n + 2}^{t + \Delta t} - h_{m,n}^{t + \Delta t} }}{2\Delta z} = Q\exp \left( { - \alpha h_{m,n + 1}^{t} } \right) - 1;\;Q\left( {\text{flow ratio}} \right) = q/k_{s} ; $$

$$ \Rightarrow h_{m,n + 2}^{t + \Delta t} = h_{m,n}^{t + \Delta t} + 2\Delta zQ\exp \left( { - \alpha h_{m,n + 1}^{t} } \right) - 2\Delta z $$

(A.11)

$$ {\text{The }}\;{\text{simultaneous}}\;{\text{ set }}\;{\text{of }}\;{\text{linear}}\;{\text{ equation}}\;{\text{ can }}\;{\text{be }}\;{\text{rewritten }}\;{\text{as}}:{\varvec{AH}}_{m}^{t + \Delta t} = {\varvec{B}}^{t} $$

(A.12)

$$ {\varvec{H}}_{m}^{t + \Delta t} = \left[ {\begin{array}{*{20}c} {h_{m,1}^{t + \Delta t} } & {h_{m,2}^{t + \Delta t} } & {...} & {...} & {h_{m,i - 1}^{t + \Delta t} } & {h_{m,i}^{t + \Delta t} } & {h_{m,i + 1}^{t + \Delta t} } & {...} & {...} & {\begin{array}{*{20}c} {h_{m,n}^{t + \Delta t} } & {h_{m,n + 1}^{t + \Delta t} } \\ \end{array} } \\ \end{array} } \right]_{1 \times (n + 1)}^{T} $$

(A.13a)

$$ {\varvec{B}}^{{\varvec{t}}} = \left[ {\begin{array}{*{20}c} {B_{1} } & {B_{2} } & {...} & {...} & {B_{i - 1} } & {B_{i} } & {B_{i + 1} } & {...} & {...} & {B_{n} } & {B_{n + 1} } \\ \end{array} } \right]_{1 \times (n + 1)}^{T} $$

(A.13b)

$$ \begin{gathered} B_{1} = 0;\;\,B_{i} = h_{m,i}^{t} + m_{1} \left( {h_{m,i - 1}^{t} - 2h_{m,i}^{t} + h_{m,i + 1}^{t} } \right) + \alpha m_{1} \left( {h_{m,i + 1}^{t} - h_{m,i}^{t} } \right)^{2} ; \hfill \\ B_{n} = h_{m,n}^{t} + m_{1} \left( {h_{m,n - 1}^{t} - 2h_{m,n}^{t} + h_{m,n + 1}^{t} } \right) + \alpha m_{1} \left( {h_{m,n + 1}^{t} - h_{m,n}^{t} } \right)^{2} \hfill \\ \end{gathered} $$

$$ B_{n + 1} = h_{m,n + 1}^{t} + p\left( {m_{1} + m_{2} } \right) + m_{1} \left( {2h_{m,n}^{t} - 2h_{m,n + 1}^{t} + p} \right) + \alpha m_{1} \left[ {h_{m,n}^{t} + p - h_{m,n + 1}^{t} } \right]^{ \, 2} $$

$p = 2\Delta z\left( {Q\exp \left( { - \alpha h_{m,n + 1}^{t} } \right) - 1} \right)$

$$ {\varvec{A}} = \left[ {\begin{array}{*{20}c} {1} & {0} & {0} & 0 & 0 & 0 & 0 \\ { - m_{1} } & {\left( {1 + 2m_{1} + m_{2} } \right)} & { - \left( {m_{1} + m_{2} } \right)} & 0 & 0 & 0 & 0 \\ {...} & {...} & {...} & {...} & {...} & {...} & {...} \\ 0 & 0 & { - m_{1} } & {\left( {1 + 2m_{1} + m_{2} } \right)} & { - \left( {m_{1} + m_{2} } \right)} & {...} & 0 \\ {...} & {...} & {...} & {...} & {...} & {...} & {...} \\ {...} & {...} & {...} & {...} & {...} & {...} & {...} \\ {...} & {...} & {...} & {...} & {...} & {...} & {...} \\ 0 & 0 & 0 & 0 & { - m_{1} } & {\left( {1 + 2m_{1} + m_{2} } \right)} & { - \left( {m_{1} + m_{2} } \right)} \\ 0 & 0 & 0 & 0 & 0 & { - \left( {2m_{1} + m_{2} } \right)} & {\left( {1 + 2m_{1} + m_{2} } \right)} \\ \end{array} } \right]_{{\left( \begin{subarray}{l} n + 1 \\ \times n + 1 \end{subarray} \right)}} \, $$

(A.14)

Equation (A.12) is further solved by employing Gauss Elimination scheme.

(A.15)