Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 26

and p c ʱ A is the capillary pressure function

p c ʱ A = p ʱ − p A , (115)

which is used to evaluate all other phase pressures, we obtain the pressure equation

m

ˆC ∂

p A
∂ t −∇· (ʻ

k ·∇ p A ) =∇· ʻ ʱ k · ( ∇ p c ʱ A − ˁ ʱ g ∇ z ) + L k , (116)

ʱ

k = 1

where C is the total compressibility defined as

m

C = ˆ 0
ˆ

1 ˁ k 0 c k (C kA + C R ) .

(117)

k =

We note that relation ( 111 ) is actually a sum over all volume-occupying compo-
nents, with

m

ˁ A C A =

1 ˁ kA C kA .

(118)

k =

This problem involves P ( n + 3 ) + n + m + 1 independent variables: c k , c k , c k ʱ dž , v ,

ʱ

, and T , while Eqs. ( 103 ) and ( 116 ), Darcy's law for the phase velocity, and the
energy balance provide n

p , S

ʱ ʱ

+ P + 2 differencial equations. The other ( n + 2 ) P + m − 1

relations are provided by the constraints ( 6 ), ( 79 ), ( 101 ), ( 107 ), ( 115 ), ( 118 ), and

m

c k ʱ = 1 , c k =dž dž c k ( c 1 , c 2 ,..., c k ) , (119)

k = 1

for the phase concentration and adsorption constraints.

6 Compositional Flow in Fractured Porous Media

A fractured porous medium has throughout its extent a system of interconnected
fractures dividing the medium into a number of essentially disjoint blocks of porous
rocks, called the matrix blocks . The fractured medium has two main length scales of
interest: the microscopic scale of the fracture thickness ( 10 − 4 m) and the macro-

scopic scale of the average distance between fracture planes, which is the size of
the matrix blocks (from

∼

1m). Since the entire medium is about 10 3 -10 4
m across, the flow can be numerically simulated only in some average sense. The
analysis of fluid flow through fractures in rocks is a process that is relevant in many
areas of the geosciences, ranging from ground-water hydrology to oil production. For

∼ 0 . 1to ∼