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8/14/2019 2-Diffusivity Equation-linear.pdf
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UNIVERSITI TEKNOLOGI PETRONAS
PAB3053RESERVOIR MODELLING AND SIMULATION
SEPT 2013Dr. Mohammed Abdalla Ayoub
Diffusivity Equation-LinearPetroleum Engineering Department (GPED)
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Introduction
x
x=0 x x+ x x=L
flowrate, q in x=0
x
x+ x
x=L
x
area, A
porosity, f
X axis
X axis
flowrate, q out
flowrate, q outflowrate, q in
isometric view
plan view
The mass accumulation(increase or decrease)over time t
=
The mass thatflows IN overtime t
- The mass thatflows OUT over time t
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Fluid flows in at position x = 0,and flows out at x= L
Element from x to positionx+ x is examined
Bulk volume of the element isthe product of the area, A andlength, dx, i.e. bulk volume = A
x
The pore volume of the elementis the product of the bulkvolume and the porosity, Φ , i.e.pore volume = A x Φ x
x=0 x x+ x x=L
flowrate, q in x=0
x
x+ x
x=L
x
area, A
porosity, f
X axis
X axis
flowrate, qout
flowrate, qout
flowrate, qin
isometric view
plan view
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o Density can be related to pressure through the isothermal compressibility
o Density is equal to mass over volume, which is:
o Hence, the isothermal compressibility is (using quotient rule):
P V
V c
1
V m
P P
m
mc
1
2[g(x)](x)gf(x)-(x)f g(x)
g(x)f(x)
dxd
Derivation of the Diffusivity Equation
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o Since,
o Then,
o This is NON-LINEAR PARTIAL DIFFERENTIAL EQUATION, whichmeans some variables in the equation (inputs) depend on the quantity weare trying to find.
t P
ct
P P t
t P
c x P k
x
f
Derivation of the Diffusivity Equation
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o By dividing both sides by the area A:
o By dividing both side by x t
o By taking the limit for both sides
x x xt t t v At v At A x f f
x x xt t t vt vt x f f
x
vvt
x x xt t t f f
x
vvt
x x x x
t t t t
f f 00 limlim
f v xt
Derivation of the Diffusivity Equation
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Derivation of the Diffusivity Equation
If Darcy’s law is applicable:
But:
Therefore:
NON-LINEAR PARTIAL DIFFERENTIAL EQUATION
t
P P t f f
x P k
xt f
x P k
v
x P k
xt P
P f
t P
c x P k
x
f
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Simplifying Assumptions for the Non-Linear PDE1. Viscosity, μ, is constant (with x and P)
Quite reasonable since viscosity does not vary greatly for most oils (or
water) over small pressure ranges.
2. Permeability is constant with x and P, i.e. the system is homogeneous.Quite drastic since it says that permeability (k) is constant through thereservoir i.e. that the system is homogeneous in k. For a real system, this isa very simplifying assumption.
Justifications
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Justifications
Simplifying Assumptions for the Non-Linear PDE
3. The pressure gradient, , is very small such that:
it is odd but it is designed to get rid of “difficult” terms with terms like(∂P/∂x)2 in them
4. The fluid has a constant compressibility.is reasonable for most reservoir oils.
x P 0
2
x P
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Working assumptions
ρ is a function of pressure, therefore, the RHS of the equationcan be expressed as:
Assuming the pressure gradient is very small, the RHS of theequation can be simplified to:
The diffusivity equation becomes
2
2
x P
x P
x P
P x P
x
2
2
x P
x P
x
2
2
x P k
t P
P
f
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By expanding the LHS of the equation:
Formation compressibility can be expressed as:
Therefore the LHS of the equation becomes:
t
P P P t
P P
f
f f
P c f
f f 1
t
P c
P t P
P f f
f f
Working assumptions
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Final form
By equalising the expanded RHS and LHS:
By simplifying the equation:
And this is the diffusivity equation for Linear, Horizontal, Singlephase fluid. or
2
2
x P k
t P
ct
f
2
2
x P
ck
t P
t f
t
P k c
x P t f
2
2
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Fluid compressibility
• As previously quoted
• For slightly compressible fluids where c is small and constant
o P P cecP a
cdP d
0 ln
Pc
11
1
dxdP c
dxd c
dP d
dP d
dP d
c
dP mV d
V m
dP dV
V c
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For black oil (P >P b)
• Diffusivity Equations for a Black Oil:
• Slightly Compressible Liquid: (General Form)
• Slightly Compressible Liquid: (Small p and c form)
t p
k c
p t f 2
t p
k c
p pc t f 22)(
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Checking dimensions and conversions
t
P k
c x P t
00633.02
2 f
t
P k c
x P t f
2
2
sec
sec1
22
Pam
Pa Pa
m Pa
Quantity 1 = Quantity 2
12
00633.0
2211
number number
unit number unit number
daymd
cp psi
ft
11
100633.0
2
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Radial Diffusivity Equation
r P r
r r ck
t P
r P k
r r r t
P c
t
t
1
1
f
f
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