Sequential Methods for Coupled Geomechanics and Multiphase Flow

A.1. BACKGROUND 215
which yields
h 2
∆t ≥ 1
. (A.5)
6 αcv
The lower bound on the time step size implies that enough pressure diffusion is required
to obtain a pressure distribution that is smooth enough, such that it can be interpolated us-
ing piecewise polynomials. ˇ Zeniˇsek (1984) shows the error estimates for the two dimensional
consolidation problem with an incompressible fluid as
p − ph l2 + um − u m h 1
≤ C
h n ∆t −1/2 u 0 + h n+1 n
+ ∆t , (A.6)
where p ∈ H n , u ∈ [H n+1 ] 2 , C is a constant independent of ∆t and h, and u m = u(t = tm).
·l2 and ·1 are defined as
f 2
m
= ∆t f l2 i 2
, f2 0 k = D a f 2
L2, (A.7)
i=1
where D a f is the a th order spatial derivative of f. From the first term of the right hand side
in Equation A.6, we identify the lower bound on the time step size. The first term is only
present during the first time step. The error estimate by Murad and Loula (1992) does not
show a lower bound on the time step size, because Gradp L 2 is assumed to be bounded
at initial time. A small oscillation is observed in the results by Murad and Loula (1992),
even though the elements satisfy the LBB condition, and this is because Gradp L 2 is
unbounded at the drainage boundary at initial time.
Even though there are two factors that cause the instability at early time, we can
eliminate the instability completely by introducing a stabilization technique (Wan, 2002;
White and Borja, 2008). The technique honors consistency in space and time, but it leads
to a loss in local accuracy. Local accuracy is important in reservoir engineering applications.
For example, the ability to saturation fronts of water accurately is considered (Wan, 2002).
a=k