Consolidation around a cylindrical heat source

Only half of the problem is modeled, taking advantage of the symmetry in the vertical direction. This problem is solved using both three-dimensional and axisymmetric coupled temperature–pore pressure elements. For the purpose of presenting the results, three-dimensional element type C3D8RPT is chosen. Both the three-dimensional and axisymmetric analyses are carried out using different variants (such as reduced-integration and hybrid) of the basic three-dimensional 8-node or axisymmetric 4-node elements, as well as the modified tetrahedral elements.

The response of the soil is assumed to be linear elastic, with a Young's modulus of 60.0 MPa and a Poisson's ratio of 0.4. The specific weight of the pore fluid is assumed to be 9800.0 N/m³ (1 lb/in³). The permeability is assumed to be constant, with a value of 4.63 × 10⁻⁸ m/sec. The thermal expansion coefficients of the soil and pore fluid are assumed to be 0.3 × 10⁻⁶ per °C and 0.21 × 10⁻⁵ per °C, respectively. The density, specific heat, and conductivity of the soil and the pore fluid are assumed to be the same, with values of 1000 kg/m³, 40.0 cal/(kg°C), and 11.9 W/cal/(m°C), respectively. The void ratio is assumed to be 1.0 initially throughout the soil volume. The initial temperature and pore pressure are assumed to be zero everywhere. It is also assumed that the pores are fully saturated with pore fluid.

There are two distinct time scales associated with this problem: one for each of the two diffusive mechanisms that are operational. The first time scale is associated with diffusive heat transfer and is given by $T_1=r_0{}^2/\kappa$, where $r_0$ is the radius of the cylindrical heat source, while $\kappa$ represents the thermal diffusivity of the surrounding medium and is given by $\kappa =K/\rho C_p$. In the preceding expression, $\rho$, $C_p$, and $K$ represent the density, specific heat capacity, and conductivity, respectively, of the surrounding medium. In general, the thermal properties used in this expression need to be weighted average quantities based on the volume fraction of the pores. However, such averaging is not necessary in this example as the thermal properties of the soil and the pore fluid are assumed to be the same. The second time scale is associated with diffusive flow of the pore fluid and is given by $T_2=r_0{}^2/c$. In the preceding expression, the quantity $c$ represents the consolidation coefficient that is defined as $c=k\left(\lambda +2G\right)/{\gamma }_w$, where $k$ is the permeability of the porous medium, $\lambda$ and $G$ are elastic constants, and ${\gamma }_w$ is the specific weight of the pore fluid. The choice of the different parameters for the problem is such that the ratio $T_1/T_2=c/\kappa$ is approximately equal to 2.

Booker and Savvidou obtained an analytical solution for consolidation around a point heat source in an otherwise infinite medium and utilized this analytical solution to approximate the solution to the problem of consolidation around a cylindrical heat source. The latter was accomplished by simply integrating the point source solution throughout the cylindrical volume. The expressions for the temperature and pore pressure fields, as given in the above reference, are reproduced below. These expressions are used to obtain the analytical solutions for comparison with the numerical results. The value of the temperature at the point ($x_F$, $y_F$, $z_F$) and at time $t$ is given by

where $q_v$ is the strength of the heat source (heat energy radiated from the source per unit volume per unit time), $R^2={\left(x_F-x_S\right)}^2+{\left(y_F-y_S\right)}^2+{\left(z_F-z_S\right)}^2$, and ($x_S$, $y_S$, $z_S$) represent the coordinates of a point source within the cylindrical volume $V_s$. The function $f\left(\frac{\kappa t}{R^2}\right)$ is expressed in terms of the complementary error function $\mathrm{erfc}$ as

Likewise, under the assumption that $c/\kappa =2$, the pore pressure field can be expressed as

where the quantity $X$ depends upon the elastic properties of the soil skeleton and the (volumetric) thermal expansion coefficients of both the soil skeleton and the pore fluid and is given by $X=\left[\left(1-n\right){\beta }_s+n{\beta }_w\right]\left(\lambda +\mathrm{\hspace{0.17em}2}G\right)-\left(\lambda +\mathrm{\hspace{0.17em}2}G/3\right){\beta }_s$. Booker and Savvidou note that the temperature reaches a maximum value of ${\theta }_N=1.45\frac{q_{v}r_0{}^2}{K}$ at the midpoint of the cylindrical source. If the soil were to be impermeable ($c=\mathrm{\hspace{0.17em}0}$), the pore pressure would reach a maximum value of $p_N=X{\theta }_N$ at the same point.

The expression for pore pressure given in the above paragraph clearly suggests that the effect of pore fluid flow is to reduce (with time) the pore pressure at a given point. For the special case of an impermeable fluid, the pore pressure will simply build up over time and never reduce. On the other hand, if $c\approx \kappa$, the fluid diffuses at the same rate as the heat and, hence, the pore pressure never builds up.

The accuracy of the time integration for the transient soils consolidation procedure that also models heat transfer is controlled by the maximum allowable pore pressure and temperature changes per time step. Even in a linear problem these values control the accuracy of the solution because the time integration operators for the consolidation and heat transfer problems are not exact (the backward difference rule is used in both cases). In this case the allowable pore pressure change per time step is chosen as 0.5 Pa, which is a relatively large value compared to $p_N$. Simulations with smaller values, such as 0.1 Pa and 0.075 Pa, produce essentially the same results, although the analysis takes progressively more increments to complete with lower values. The allowable temperature change per time step is chosen as 0.003°C, which is approximately 0.1${\theta }_N$. A value of 0.0003°C (along with a value of 0.075 Pa) produced essentially the same results, although the analysis took a significantly larger number of increments to complete.

The analysis is continued for a time period of approximately 1000 $T_1$.

The simulations use solution controls to specify a nondefault initial value of the time average pore fluid flux. The default choice may not be appropriate in situations such as those encountered in the present problem, where the fluid velocities are, in relative terms, lower compared to typical flux values encountered for other fields (such as displacements or rotations). Without the above specification, the increments would be treated as linear from the viewpoint of the continuity equation. In other words, without using solution controls to specify a nondefault initial value of the time average pore fluid flux, the pore fluid part of the incrementation will be treated as linear. Consequently, the continuity equation would be assumed to have been satisfied at the first iteration itself, without performing any further iterations to compute corrections to pore pressure.

For the pore fluid flow equations, the simulations also use a nondefault and a relatively large value of the ratio of the largest residual flux to the average flux, which sets the convergence criterion for an increment. This setting is helpful as the fluid velocity in this problem is very small and ensures that the pore pressure increment is not considered converged without at least one correction (iteration). There is not much advantage in reducing the tolerance further as the flow residuals are already sufficiently small, and any further reduction in the residual does not make any difference to the overall solution.