Page 60 - Principles of Applied Reservoir Simulation 2E

Part I: Reservoir Engineering Primer 45

vdC/dx. When the diffusion term is much larger than the convection term, the
C-D equation behaves like the heat conduction equation, which is a parabolic
partial differential equation (PDE). If the diffusion term is much smaller than
the convection term, the C-D equation behaves like a first-order hyperbolic PDE.
The C-D equation is especially valuable for studying numerical solutions
of fluid flow equations because the C-D equation can be solved analytically and
the C-D equation may be used to examine two important classes of PDEs
(parabolic and hyperbolic). To solve the C-D equation, we must specify two
boundary conditions and an initial condition. The two boundary conditions are
needed because the C-D equation is second-order in the space derivative. The
initial condition satisfies the need for a boundary condition in time associated
with the first-order derivative in time. The boundary conditions for the miscibie
displacement process are that the initial concentration of displacing fluid is equal
to one at the inlet (x = 0), and zero for all other values of x. The mathematical
expressions for these boundary conditions are concentration C(0, t) = I at the
inlet, concentration C(°°, f) = 0 at the edge of the linear system for all times t
greater than the initial time t = 0, and the initial condition C(x, 0) = 0 for all
values of x greater than 0.
The propagation of the miscibie displacement front is calculated by
solving the C-D equation. The analytical solution of the one-dimensional C-D
equation is [Peaceman, 1977]

JC - Vt X + Vt
C(jc, 0 = -U erfc (5.17)
2 2 ]/Dt
l
where the complementary error function erfc(y) is defined as

2 r z 2
erfc(j) = 1 - — je rfz (5,18)
V* o
Abramowitz and Stegun [ 1972] have presented an accurate numerical algorithm
for calculating the complementary error function erfcO/). A comparison of the
analytical solution of the C-D equation with numerical solutions is given in
Fanchi [2000].