The equations that govern the fluid flow through a porous medium can
be solved numerically using a finite-difference method. For three dimensions,
the finite difference approximations of equation (12) through (14)
can be written as
where Vijk is the grid block volume and
are the gas and water transmissibilities. The cross-sectional area normal to the direction of the flow at the block-boundary is denoted by A, and the distance between two connecting grid points by .
The accumulation terms in equations (15) to (17) are given as
The finite-difference operators in equations (15) to (17) are given by
where subscript l = i,j,k and
for equation (15),
for equation (16),
for equation (15) and
for equation (16).
Furthermore, it is:
where is the cross-sectional area normal to the direction of the flow at the block-boundary . The distance between grid points l and l+1 is given by . The absolute permeability is taken in the direction of the flow.
Equations (20) to (28) use coefficients that must be evaluated at intercell boundaries between two grid points. Simple arithmetic mean is used to approximate the densities , , and , and the mass fractions Xg,w and Xg,h. The absolute permeability as a function of Sh is determined using harmonic weighting. Upstream weighting is used for the relative relative permeabilities and viscosities. Defining the phase potential as
Equations (15) to (28) represent the finite-difference set-up for a three-dimensional fluid-flow simulator of a gas-water-hydrate system, which can be solved using the Newton-Raphson method.