Sequential Methods for Coupled Geomechanics and Multiphase Flow

A.3. NUMERICAL EXAMPLES 217
Figure A.1: Left: the Terzaghi problem. Right: the two dimensional plain strain strip
footing consolidation problem.
A.3.1 Case A.1—Terzaghi’s problem
We have a drainage boundary for flow on the top and at the bottom, where the boundary
fluid pressure is Pbc = 2.125 MPa. The overburden is ¯σ = 2 × 2.125 MPa on the top, and
a no-displacement boundary condition is applied to the bottom. The domain has 50 grid
blocks. The length of the domain is Lz = 100 m with grid spacing ∆z = 2 m. The bulk
density of the porous medium is ρb = 2400 kg m −1 . Initial fluid pressure is Pi = 2.125 MPa.
The fluid density and viscosity are ρf,0 = 1000 kg m −1 and µ = 1.0 cp, respectively. The
fluid compressibility is cf = 1.0 × 10 −10 Pa −1 . Permeability is kp = 100 md, porosity is
φ0 = 0.3, the constrained modulus is Kdr = 6 GPa, and the Biot coefficient is b = 1.0.
There is no fluid production or injection of fluid and there is no gravity. The values for the
input parameters are shown in Table A.1.
The top of Figure A.2 shows the pressure distributions at early and late time when
the nodal based finite element methods are used with linear interpolation for pressure and
displacement (Wan, 2002). At early time, this space discretization causes an unstable
pressure solution, even though pressure becomes stable later in time. From Figure A.2, it