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)





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





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











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



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








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





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







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



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





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



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





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



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



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)(



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





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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2-Diffusivity Equation-linear.pdf