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

(15) |

(16) |

(17) |

where *V*_{ijk} is the grid block volume and

(18) |

(19) |

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

(20) |

(21) |

(22) |

The finite-difference operators in equations (15) to (17) are given by

(23) |

(24) |

(25) |

(26) |

where subscript *l* = *i*,*j*,*k* and

for equation (15),

for equation (16),

for
equation (15) and

for equation
(16).

(27) |

(28) |

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 *X*_{g,w} and *X*_{g,h}.
The absolute permeability as a function of *S*_{h} is determined
using harmonic weighting. Upstream weighting is used for the relative
relative permeabilities and viscosities. Defining the phase potential as

(29) |

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.

11/12/1997