According to the poroelasticity theory of consolidation developed by Lo et al. (2014), the momentum balance equations for a deformable homogeneous porous medium bearing one compressible, viscous fluid, in the absence of body force and inertial force, can be expressed as

where the subscripts “s” and “f” represent the solid phase and fluid phase, respectively; ∅ represents the porosity; p_f indicates the gauge pressure (excess pressure); u_f and u_s designate the displacements of each phase; $R_{\mathrm{22}}(=-{\mathrm{\varnothing }}^{\mathrm{2}}{\mathit{\mu }}_{\mathrm{f}}/k_{\mathrm{s}})$ signifies the coefficient of viscous drag due to the velocity difference between solid phase and fluid phase, where μ_f is the dynamic shear viscosity of the fluid phase, k_s is the intrinsic permeability. $\stackrel{\mathrm{‾}}{\stackrel{\mathrm{‾}}{\mathit{\sigma }}}$ denotes the total stress tensor for the porous medium, using the sign convention that tension is positive. On the basis of Biot's theory (1962), the total stress components $\stackrel{\mathrm{‾}}{\stackrel{\mathrm{‾}}{\mathit{\sigma }}}$ of the bulk material is defined as

where ${\stackrel{\mathrm{‾}}{\stackrel{\mathrm{‾}}{t}}}_{\mathrm{s}}$ is the stress tensor for the solid phase, $\stackrel{\mathrm{‾}}{\stackrel{\mathrm{‾}}{\mathit{\delta }}}$ is the unit tensor.

2.2 Boundary and initial conditions

A stratum of clay sandwiched between sandy strata that are highly permeable and much stiffer than the clay is shown in Fig. 1, water level depression h₁ and h₂ due to pumping take place in the sandy strata above and below the clay. Due to significant differences of permeability and compressibility between sand and clay, excessive pore pressure only exists in the clay as consolidation proceeds. And nearly all of the consolidation happens due to the volume change within the clay, while the sandy strata may be considered rigid media as compared with the clay. Furthermore, the steady-state solutions, which are usually applied in engineering practice, will be considered.

Figure 1Sketches of soil consolidation due to water table depression.

Download

The initial conditions can be expressed as $p_{\mathrm{f}}(z,\mathrm{0})=\mathrm{0}$. With respect to the boundary conditions, the top and bottom surfaces of the porous layers are considered to be permeable so that the pore fluids can drain at both sides of the sample, which leads to $p_{\mathrm{f}}\left(\mathrm{0},t\right)=-{\mathit{\rho }}_{\mathrm{f}}g{h}_{\mathrm{2}}$ and $p_{\mathrm{f}}\left(L,t\right)=-{\mathit{\rho }}_{\mathrm{f}}g{h}_{\mathrm{1}}$. Based on Tsai et al. (2006), combined flow equation and Darcy's law and assuming an isotropic and homogeneous porous media and one-dimensional steady-state consolidation, the pore water pressure in the clay stratum can be represented as function of depth as $p_{\mathrm{f}}=-{\mathit{\rho }}_{\mathrm{f}}g[h_{\mathrm{2}}+z(h_{\mathrm{1}}-h_{\mathrm{2}})/L]$. By combining of initial and boundary equations into Eq. (10), we can get

$$\begin{array}{}\text{(12)}& {\displaystyle }\begin{array}{rl}{p}_{\mathrm{f}}\left(z,t\right)& ={\displaystyle \frac{{\mathit{\rho }}_{\mathrm{f}}g\left[-h_{\mathrm{2}}L+z\left(h_{\mathrm{2}}-h_{\mathrm{1}}\right)\right]}{L}}\\ & +\sum _{n=\mathrm{1}}^{\mathrm{\infty }}{\displaystyle \frac{\mathrm{2}{\mathit{\rho }}_{\mathrm{f}}g}{n\mathit{\pi }}}\left[h_{\mathrm{2}}-h_{\mathrm{1}}\cos \left(n\mathit{\pi }\right)\right]\sin \left({\displaystyle \frac{n\mathit{\pi }z}{L}}\right)\cdot e^{-c_v(\frac{n\mathit{\pi }}{L}{)}^{\mathrm{2}}t}.\end{array}\end{array}$$

The time-dependent total settling s(t) can be consequently deduced

$$\begin{array}{}\text{(13)}& {\displaystyle }\begin{array}{rl}s\left(t\right)& =-\underset{\mathrm{0}}{\overset{L}{\int }}{\displaystyle \frac{\partial w}{\partial z}}\mathrm{d}z={\displaystyle \frac{-\left[\left(Q+R\right)d_{\mathrm{2}}\right]{\mathit{\rho }}_{\mathrm{f}}g(h_{\mathrm{1}}+h_{\mathrm{2}})L}{\mathrm{2}{K}^{\left(u\right)}}}\\ & +{\displaystyle \frac{(Q+R)d_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{f}}g}{K^{\left(u\right)}}}\\ & \times \sum _{n=\mathrm{1}}^{\mathrm{\infty }}{\displaystyle \frac{\mathrm{2}L(\mathrm{1}-\cos \left(n\mathit{\pi }\right))(h_{\mathrm{2}}-h_{\mathrm{1}}\cos (n\mathit{\pi }\left)\right)}{(n\mathit{\pi }{)}^{\mathrm{2}}}}\cdot e^{-c_v(\frac{n\mathit{\pi }}{L}{)}^{\mathrm{2}}t}.\end{array}\end{array}$$

Equation (13) implies that, as for t→∞, the final total settlement Δs is equal to

$$\begin{array}{}\text{(14)}& \mathrm{\Delta }s=-(Q+R)d_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{f}}g\left[\right(h_{\mathrm{1}}+h_{\mathrm{2}}\left)L\right]/\left(\mathrm{2}{K}^{\left(u\right)}\right).\end{array}$$

In order to investigate quantitatively the deformation of porous media due to water table depression, the governing equations have been implemented in MATLAB and a textbook case is simulated. The ground parameters used in the study are summarized in Table 1, which also lists the numerical value of elastic coefficients and hydraulic parameters necessary for the numerical simulations performed in the present study. Before the consolidation of soil subjected to water level depression is examined, the implementation of the poroelastic consolidation is verified. Considering water level depressions $h_{\mathrm{1}}/L=h_{\mathrm{2}}/L=\mathrm{0.01}$ and elapsed dimensionless time $T=c_{v}t/L^{\mathrm{2}}$, the excess dimensionless pore water pressure $P^{\ast }(=p_{\mathrm{f}}(zt)/({\mathit{\rho }}_{\mathrm{f}}gL\left)\right)$ with respect to the dimensionless depth $Z^{\ast }(=z/L)$ are shown in Fig. 2. The pore water pressure increases with time at the inside of the clay stratum. Because the boundary of clay stratum is subjected to a constant water table depression, the dimensionless pore water pressure at top and bottom of clay stratum remains equal to the value caused by the water table depression. The dimensionless pore water pressure at different depth for various dimensionless elapsed times is also plotted in Fig. 3. The dimensionless pore water pressure near the boundary is greater than that near the center of stratum. The change of pore water pressure in the clay layer is directly related to pressures at the upper and lower boundaries resulting from pumping, and the change of pore water pressure is from the boundary to the medium of the layer.

Figure 2The dimensionless pore water pressure with respect to the dimensionless depth at different elapsed times.

Download

Figure 3The dimensionless pore water pressure at different depths versus time.

Download

Table 1Material parameters for saturated clay (Rawls et al., 1992; Lo et al., 2007).

Download Print Version | Download XLSX

Figure 4 shows the dimensionless settlement $s^{\ast }(=\mathrm{s}(t)/\mathrm{\Delta }s)$ with respect to the dimensionless elapsed time. We can easily calculate the quantity of settlement at any time from Fig. 4. The dissipation of pore water pressure is significantly influenced by permeability (Fig. 5). When the intrinsic permeability is higher, the pore water pressure reaching stabilize is faster.

Figure 4The dimensionless surface settlement versus time.

Download

Figure 5Effect of permeability on the dissipative pore water pressure as function of time.

Download

Considering the current situation, the groundwater table may be a time series change. The upper water level changes with time was assumed as $h_{\mathrm{1}}=\mathrm{0.1}\times \sin \left(\frac{t\mathit{\pi }}{\mathrm{12}}\right)m$ (the time interval is 1 h, so the water level changes for 12 h as a cycle), lower water level changes do not change with time $h_{\mathrm{2}}/L=\mathrm{0.02}$. It can be clearly seen from the results in Fig. 6 that because of the greater fluctuation of the fixed water level at the lower boundary, the trend of the consolidation with time is similar to the above-mentioned fixed water level fluctuation, but at the same time, it is affected by the fluctuation of the upper water level, and the periodic fluctuation appears. When the upper groundwater table raises, the settlement significantly decreases, which is regarded as the phenomenon of soil swelling. It can also be seen that the amount of rebound is less than that of compression. Therefore, increasing the groundwater replenishment from land subsidence prevention, the soil layer will partially rebound.

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

CLY designed a study concept and made a general text writing and editing, WCL help to develop the concept of the approach. CWL and CFD provided assistance for data analyses.

The authors declare that they have no conflict of interest.

This article is part of the special issue “TISOLS: the Tenth International Symposium On Land Subsidence – living with subsidence”. It is a result of the Tenth International Symposium on Land Subsidence, Delft, the Netherlands, 17–21 May 2021.