A Unidirectional Flow Problem
The case presented here for single
phase incompressible flow tests the ability of the numerical model to obtain
the correct unidirectional flow rate for an anisotropic 2D rectangular
reservoir with constant pressure boundary conditions.
The horizontal reservoir is 10000
ft long (xdirection) and 2000 ft wide (ydirection). The grid is
described with Nx=20, Ny=10, Nz=1, dx=500 ft, dy=200 ft, dz=1000 ft, all
block depths (to top) at 8000 ft, kx=10 md, ky=2 md, kz=10 md. Initial
pressure is 5000 psia. The left (i=1) boundary is constrained to 5100
psia, and the right boundary (i=nx) is constrained to 4900 psia. PVT
and other data are based on SPE1. The correct steady state rate given
by Darcy's law and reproduced by
Sensor in the output file given below, independent of the entered
permeability in the direction perpendicular to flow (ky), is
(kA/L) * dp = (10(1000)(2000)/(19*500)) * 200 *.00633/5.6146 =
474.7 stb/d
kx Lz Ly
Lx dp conversion
Sensor can obtain the same result using as little as 2 total
grid cells. 4 cells are required to demonstrate independence of
results on entered ky.
Sensor data: uni.dat
Sensor output:
uni.out
This problem
is a test for korthogonality and will demonstrate:

An exact Finite Difference solution using a Cartesian
grid, independent of ky.

The requirement for use of a multipoint flux
approximation, and its accuracy, when solved with unstructured grids using, for example,
hexagonal prisms, or with any grid that is not orthogonal in the
direction of flow, for example, corner point grids differing from
Cartesian geometry.

Smooth boundary approximation issues with
unstructured grids not using rectangular cells or mesh.

The requirement for a Cartesian grid or equivalent
finite element rectangular mesh in order to prevent grid orientation
effects that otherwise cause transverse dispersion of a tracer injected
at a point on the left boundary. This illustrates that all fixed
grids can suffer from grid orientation effects, even if they are
korthogonal.
