Page 47 - Principles of Applied Reservoir Simulation 2E

32 Principles of Applied Reservoir Simulation

mass entering the block - mass leaving the block
= accumulation of mass in the block.
If the block has length A*, width Ay, and depth Az, then we can write the mass
entering the block in a time interval A/ as

(
| (',/ ) AvAz+ (j } AxAz + j } AjcAv A/ = Mass in (4,1)
I '< x ' x \ y I v \ z I : " \
where we have generalized to allow flux in the y and z directions as well, The
denotes the x direction flux at location*, with analogous meanings
notation (J x) x
for the remaining terms,
Corresponding to mass entering is a term for mass exiting which has the
form
+
z
A A A
A A
lW*.*,*y* (^W * * + (^) A, * ^ '
2+
(4.2)
+ gAxAyAzA/ = Mass out
We have added a source/sink term q which represents mass flow into (source)
or out of (sink) a well. A producer is represented by q > 0, and an injector by
q<0.
Accumulation of mass in the block is the change in concentration of phase
<! (C 4) in the block over the time interval A?. If the concentration C t is defined
as the total mass of phase 0 (oil, water, or gas) in the entire reservoir block
divided by the block volume, then the accumulation term becomes

[(^X+A* ~ (C^JAjtAyAz = Mass accumulation (43)

Using Eqs. (4.1) through (4.2) in the mass conservation equality
Mass in - Mass out = Mass accumulation
gives
[(/^AyAz H- (J^AjcAz + 0/ Z ) Z A*A7]A?
A A +
- lW .***y* z + (^W * * W ^ &x&y]&t (4.4)
x
z z
- ^AxAyAzA/ = [(C ( )^ Ar - (C f ),]A*A;pAz

Dividing Eq. (4.4) by AxAyAzA/ and rearranging gives