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Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017
Chapte
r 2 – T heory- p15/743
Derivation of the diffusivity equation:
The diffusion equation is derived from the combination of three elementary equations:
The law of
conservation of mass
: this is a ‘common sense’ law that states that nothing is
ever created or lost; things just move or transform themselves (Antoine Lavoisier, France,
1785). The formulation for an elementary piece of rock is that ‘mass in’ minus ‘mass out’ =
‘accumulation’.
We consider the flow in the x direction of a fluid through a (small) area A, between x and x+
x
and between time t and t+
t. Considering that 1 bbl/day = 0.23394 ft
3
/hr, we get the
following equation
Conservation of mass:
before
after
out
in
Mass
Mass
on
Accumulati
Mass
Mass
Conservation of mass:
t
t t
x x x
x x
xA
xA
tq
tq
23394 .0
23394 .0
Differential form:
t
A
x
q
x
23394 .0
The second equation we will use relates the rate in a direction to the pressure gradient in this
direction. The simplest among these equations is
Darcy’s law
(Henry Darcy, France, 1856),
which establishes a linear relation between the speed and the pressure gradient, the linear
factor being a function of one reservoir property (the permeability) and one fluid property (the
viscosity).
Darcy’s law in the x direction:
x
p Ak
q
x
x
2. 887
So we get:
t
A
x
p Ak
x
x
2. 887
23394 .0
This simplifies to:
x
p
x
k
t
x
0002637
.0
It is from the equation above that we will start when dealing with real gas. Now we are going
to focus on slightly compressible fluids. The first term can be developed as:
First term:
t
p
p
p
t
p
p p
t
p
p
t
1
1
New differential form:
t
p
p
p
x
p
x
k
x
1
1
0002637
.0
The two terms between brackets in the second member of the equation are the formation
compressibility and the fluid compressibility:
Formation compressibility:
p
c
f
1
Fluid compressibility:
p
c
fluid
1