Terzaghi one-dimensional consolidation
This benchmark validates the coupled solid–water response in numgeo against Terzaghi's analytical solution for one-dimensional consolidation. A 1 m high, fully saturated soil column initially carries a uniform excess pore-water pressure of 50 kPa. At the start of consolidation, the top boundary is drained by reducing its prescribed pore-water pressure to zero while the 50 kPa surface load is maintained. Water then leaves through the top boundary, the excess pore-water pressure dissipates, and the column settles.
The column is discretised with either 20 or 40 elements, depending on the interpolation order. The lateral boundaries prevent horizontal displacement, the base prevents vertical displacement, and all boundaries except the top are hydraulically closed.
The material parameters used in the supplied input files are:
| Parameter | Value |
|---|---|
| Young's modulus \(E\) | 500 kPa |
| Poisson's ratio \(\nu\) | 0.35 |
| Initial void ratio \(e_0\) | 1.0 |
| Solid density \(\rho^s\) | 2.7 t/m\(^3\) |
| Water density \(\rho^w\) | 1.0 t/m\(^3\) |
| Water bulk modulus \(K^w\) | \(2\times10^6\) kPa |
| Intrinsic permeability \(\bar{K}\) | \(1\times10^{-12}\) m\(^2\) |
| Dynamic water viscosity \(\mu^w\) | \(1\times10^{-6}\) kPa·s |
Using \(\gamma^w=9.81\) kN/m\(^3\), the prescribed intrinsic permeability and viscosity correspond to a saturated hydraulic conductivity of approximately \(9.81\times10^{-6}\) m/s. The distinction between intrinsic permeability and hydraulic conductivity is explained in the material reference.
Figure 1: Finite-element model, initial conditions, and boundary conditions for the one-dimensional consolidation problem.
Element formulations
The archive contains two- and three-dimensional versions of the three saturated two-phase formulations available in numgeo:
- \(u\)–\(p\) elements, including Taylor–Hood, reduced-integration, and MINI variants;
- \(u\)–\(U\) elements with solid and water displacement degrees of freedom;
- \(u\)–\(p\)–\(U\) elements combining solid displacement, pore-water pressure, and water displacement.
In an element label, the number following u is the number of solid-displacement nodes, the number following p is the number of pore-pressure nodes, and the number following the second u—representing \(\boldsymbol{U}\) in the formulation notation—is the number of water-displacement nodes. The suffix -sat denotes full saturation and -red denotes reduced integration. See the two-phase benchmark overview for links to the corresponding element-reference pages.
Input files
The input files, analytical reference data, and post-processing scripts can be downloaded here.
Simulation results
Figure 2 compares the excess pore-water-pressure profiles at \(t=0.1\), 1, 10, 100, 1000, and 10,000 s, together with the settlement history at the top of the column. The numerical solutions generally converge to Terzaghi's analytical solution. Differences are most pronounced at early times, when the steep drainage gradient makes the response particularly sensitive to the pressure interpolation, integration rule, mesh spacing, and time increment.
Figure 2: Excess pore-water-pressure profiles and top settlement for the investigated two-phase element formulations.