# Compression of a Poroelastic Sample – Mandel’s Problem (FLAC2D)

## Problem Statement

Note

The project file for this example is available to be viewed/run in FLAC2D. The main data files used are shown at the end of this example. The remaining data files can be found in the project.

During compression of a poroelastic specimen under constant boundary conditions, the pore pressure will display a non-monotonic variation with consolidation time. At initial consolidation times, an increase in the pore pressure will be induced near the center of the sample when it is subjected to a constant vertical load and drained laterally. Subsequently, the pore pressure falls. This effect pointed out by Mandel (1953). It was also predicted by Cryer (1963), and thus is also known as the Mandel-Cryer effect, and was demonstrated experimentally by Verruijt (1965).

In Mandel’s problem, a sample of saturated poroelastic material is loaded under plane-strain conditions by a constant compressive force applied on rigid impervious platens (see Figure 1). The width of the sample is 2a its height is 2b, and the force intensity is 2F. The application of the load is instantaneous, the platens are impervious and the sample is free to drain laterally.

The short-term response of the material corresponds to a uniform vertical stress across the sample. As lateral drainage takes place, the nonuniform dissipation of induced pore pressure causes an apparent softening of the material near the edges of the sample. The resulting stress concentration in the stiffer (still undrained) core is then responsible for an additional increase in pore pressure in the middle of the sample. As drainage proceeds and the pore pressure gradient decreases, the vertical load is again transmitted in a uniform manner. The non-monotonic variation of pore pressure with time observed in Mandel’s problem serves to illustrate the main difference in prediction between the Biot and Terzaghi theories.

## Analytical Solution

The solution to Mandel’s problem, generalized for the case of compressible constituents, is given by Cheng and Detournay (1988). The expression for the pore pressure is

$p = {{2FB(1+\nu_u)}\over{3a}} \sum_{i=1}^{\infty} {{\sin \alpha_i}\over{\alpha_i - \sin \alpha_i \cos \alpha_i}} \left[ \cos {{\alpha_i x}\over a} - \cos \alpha_i \right] \exp (- \alpha_i^2 ct/a^2)$

where $$B$$ is Skempton’s pore pressure coefficient (the ratio of induced pore pressure to variation of confining pressure under undrained conditions), $$ν$$ and $$ν_u$$ are drained and undrained Poisson’s ratio, $$c$$ is the true diffusivity (or generalized consolidation coefficient), $$t$$ is time and $$α_i, i = 1, ∞$$, are the roots of the equation

$\tan \alpha_i = {{1 - \nu}\over{\nu_u - \nu}} \alpha_i$

If the solid grains that form the material are assumed to be incompressible, as is the case in FLAC2D, the following relations between material constants apply.

\eqalignno{ B &= {{K_w} \over {K_w + n K}} &\cr K_u &= K + {K_w \over n} \cr c &= k \ / \left({n \over K_w} + {1 \over {K + 4 G /3}}\right) & \cr }

where $$K_w$$ is fluid bulk modulus, $$n$$ is porosity, $$K$$ and $$K_u$$ are drained and undrained bulk moduli, $$G$$ is shear modulus and $$k$$ is mobility coefficient. The drained and undrained Poisson’s ratios are related to the moduli $$G$$, $$K$$ and $$K_u$$ as

\eqalignno{ \nu &= {{3 K - 2 G} \over {6 K + 2 G}} &\cr \ \ \ & &\ \ \cr \nu_u &= {{3 K_u - 2 G} \over {6 K_u + 2 G}} & \cr }

The formulae for horizontal displacement at $$x = a$$ and vertical displacement at $$y = b$$ are

\eqalignno{ u_x(a,t) &= {{F \nu} \over {2G}} + {{F (1 - \nu_u)} \over {G}} \sum_{i=1}^{\infty} {{\sin \alpha_i \cos \alpha_i}\over{\alpha_i - \sin \alpha_i \cos \alpha_i}} \exp (- \alpha_i^2 ct/a^2) &\cr \ \ \ & &\ \ \cr u_y(b,t) &= - {{F(1- \nu)b} \over {2Ga}} + {{F (1 - \nu_u)b} \over {G a}} \sum_{i=1}^{\infty} {{\sin \alpha_i \cos \alpha_i}\over{\alpha_i - \sin \alpha_i \cos \alpha_i}} \exp (- \alpha_i^2 ct/a^2) & \cr }

According to the exact solution, the initial (instantaneous) and final vertical displacements of the upper platen are

\eqalignno{ u_y(b,0) &= -Fb {{1-\nu_u} \over{2Ga}} &\cr \ \ \ & &\ \ \cr u_y(b,\infty) &= -Fb {{1-\nu} \over{2Ga}} & \cr }

The lateral boundaries of the sample stay plane during drainage; they first move outward and then inward. The initial (instantaneous) and final lateral displacements of the right boundary are

\eqalignno{ u_x(a,0) &= F {{\nu_u} \over{2G}} &\cr \ \ \ & &\ \ \cr u_x(a,\infty) &= F {{\nu} \over{2G}} & \cr }

The degrees of consolidation in the horizontal and vertical directions, $$D_h$$ and $$D_v$$, are defined as

\eqalignno{ D_h &= {{u_x(a,t) - u_x(a,0)} \over{u_x(a,\infty) - u_x(a,0)}} &\cr \ \ \ & &\ \ \cr D_v &= {{u_y(b,t) - u_y(b,0)} \over{u_y(b,\infty) - u_y(b,0)}} & \cr }

These degrees are found to be identical, and the analytic expression for $$D = D_h = D_v$$ is

$D = 1 - {{4(1-\nu_u)} \over {1-2\nu}} \sum_{i=1}^{\infty} {{\sin \alpha_i \cos \alpha_i}\over{\alpha_i - \sin \alpha_i \cos \alpha_i}} \exp (- \alpha_i^2 ct/a^2)$

Because the discharge has only a horizontal component in Mandel’s problem, the pore pressure, stress and strain solutions are independent of the $$y$$-coordinate.

## FLAC Model

By symmetry, only a quarter of the sample is considered in the numerical model. The grid has 20 zones in the x-direction and 2 in the y-direction. Mandel’s problem is solved for the case

 drained Poisson’s ratio ($$ν$$) 0.2 Skempton coefficient ($$B$$) 0.9

The FLAC2D simulation is carried out to produce results in terms of normalized pore pressure, $$\hat{p}$$, distance, $$\hat{x}$$, and time, $$\hat{t}$$, defined as

\eqalignno{ \hat p &= {{ap} \over{F}} &\cr \hat x &= {{x} \over{a}} &\cr \hat t &= {{ct} \over{a^2}} & \cr }

The normalized results are not affected by the absolute magnitude of material properties used, as long as their combination yields the values specified above for $$ν_u$$ and $$B$$. The FLAC2D grid dimensions, applied force and property values used in the simulation may be viewed as scaled for the purpose of producing the normalized results. They are selected as follows.

 model width ($$a$$) 1 model height ($$b$$) 0.1 applied force ($$F$$) 1 drained bulk modulus ($$K$$) 1 shear modulus ($$G$$) 0.75 fluid bulk modulus ($$K_w$$) 9 porosity ($$n$$) 0.5

With the above values, the true diffusivity in the FLAC2D model is unity, and the model time is t ̂. The model mechanical boundary conditions correspond to roller boundaries along the x- and y-axes of symmetry. The sample is initially stress-free, and the pore pressure is equal to zero. Undrained conditions are established first by applying a constant unit mechanical pressure at the top boundary of the model. In the second part of the simulation, the pore pressure is fixed at zero on the right side of the model to allow drainage to occur. The rigid plate condition is enforced by applying a vertical velocity at the top of the model. The velocity magnitude is derived from the exact displacement solution . As a verification to the numerical solution, the reaction force on the top platen is monitored to check whether it remains constant and equal to one.

## Results

The numerical simulation is carried out for a total value of normalized time equal to 4, with intermediate results at $$\hat{t} = 0.01, 0.1, 0.5, 1$$ and $$2$$. (Results at $$\hat{t} = 0$$ correspond to the undrained response.) The pore pressure profiles at those times are checked against exact solutions in Figure 2. At early times, the pore pressure at $$x = 0$$ (center of sample) is seen to rise above the undrained value, before decreasing as drainage evolves. Figure 2 This is also shown in Figure 3, which compares FLAC2D to the analytical solution for the actual pore pressure versus consolidation time at the center of the sample. As may be seen in Figure 4, the reaction force stays equal to unity throughout the simulation. (The approach taken to apply the force boundary condition is thus equivalent to a servo-controlled velocity.) Numerical values for the horizontal and vertical degrees of consolidation are plotted versus the analytical solution values in Figure 5. Figure 2: Pore pressure profile comparison : $$\hat{p}$$ vs. $$\hat{x}$$. Figure 3: Pore pressure versus consolidation time at the center of the sample. Figure 4: History of y-reaction force on top platen. Figure 5: Degree of consolidation versus log time.