Retention and Transfer of Pollution by Hydrocarbons: Mechanisms and Numerical Modeling - Environmental Geomechanics - page 394

where:

n + 1

⎛ dS ⎞

S

'
w

= S

n
w

+ ∆ p

cw

⎜ ⎜
⎝

w

⎟ ⎟
⎠

cw

dp

n + 1

n + 1

⎛ dS ⎞

⎛ dS ⎞ g

'
o

n
o

cw

⎜ ⎜
⎝

w

⎟ ⎟
⎠

cg

⎜ ⎜
⎝

⎟ ⎟
⎠

S = S − ∆ p − ∆ p

cw cg

dp dp

n + 1

dS

⎛ ⎞ g

'
g

n
g

cg

⎜ ⎜
⎝

⎟ ⎟
⎠

S = S + ∆ p

cg

dp

cw cw , n + 1 cw , n

∆ p = p − p

cg cg , n + 1 cg , n

∆ p = p − p

These relations are written directly for an implicit procedure, which will be used
for the solution of the equation system. The superscripts n and n+1 indicate the time
steps.

The linear momentum balance equation of the system, neglecting convective and
inertial terms, reads:

⎛ − ⎞

w

∂ ε D m ∂ p

∫

T

∫

T '

w

δ ε D dΩ − δ ε S

⎜ ⎜
⎝

m

T

⎟ ⎟
⎠

dΩ

T

∂ t 3K ∂ t

Ω Ω

s

⎛ − D m ⎞

o

⎛ − D m ⎞

g

[14.8]

∂ p ∂ p

∫

T '

o

∫

T '

g

− δ ε S

⎜ ⎜
⎝

m

T

⎟ ⎟
⎠

dΩ − δ ε S

⎜ ⎜
⎝

m

T

⎟ ⎟
⎠

dΩ

3K ∂ t 3K ∂ t

Ω Ω

s s

ˆ

∂ ε ∂ f

∫

T

∫

T

− δ ε D c dΩ − δ ε D

o

dΩ − = 0

T T

∂ t ∂ t

Ω Ω

where ε is the deformation vector, D T the tangential stiffness matrix, K s the bulk
modulus of the grains, c the compliance function, ε o the strain vector not dependent
on stresses, and f the load vector. This equation can be used for the simulation of
reservoirs in different conditions; for example if S 0 is zero, i.e. there is no oil, the
equation refers to a gas reservoir, which usually contains water. This is the case in
the second example.