Darcys Law into Continuity Equation In 1D and horizontal flow - - PowerPoint PPT Presentation

Darcy’s Law into Continuity Equation

In 1D and horizontal flow (gravity neglected) Replacing fluid velocity in the continuity equation Assuming homogenous and isotropic permeability and viscosity (NOT always the case!)

Formation Volume Factor (Bw)

We measure volume at surface but do the mass balance at the reservoir

SC RC w

B ρ ρ =

Formation volume factor and ρ depend on both pressure and temperature

• Bo = formation volume factor for oil (> 1.0)
• Bg = formation volume factor for gas (<< 1.0)
• Bw = formation volume factor for water (~1.0)

depth

Surface/Standard Conditions Reservoir Conditions

14.7 psi; 60 °F;

sc

P T V V = = = ; ;

R R R

P P T T V V = = =

Formation Volume Factor—Continuity Equation

At reservoir conditions, density is: Replacing density in continuity equation and divide through by ρSC (a constant) Using the product rule on the left-hand side of the equation

1

w w sc

k p m x B x t B φ µ ρ     ∂ ∂ ∂ = −     ∂ ∂ ∂     

2 2

1 1

w w w sc

k p p m B x x B x t B φ µ ρ       ∂ ∂ ∂ ∂ + = −       ∂ ∂ ∂ ∂       

2 2 2

1 1

w w w sc

k p p m B x p B x t B φ µ ρ       ∂ ∂ ∂ ∂   + = −         ∂ ∂ ∂ ∂           

…And chain rule on the left hand side,

Expansion of the Time Derivative

A few definitions:

1 1 (rock compressibility) 1 1 1 1 (fluid compressibility) (total compressibility)

P r P w f w w w T T t r f

V c V p p B V c B V p p p B B p c c c φ φ ρ ρ ∂ ∂ = = ∂ ∂   ∂ ∂ ∂ ∂ − = − = = =   ∂ ∂ ∂ ∂   = +

2 2 2

1 1 1 1

w w w w sc

k p p p m B x p B x p B B p t φ φ µ ρ         ∂ ∂ ∂ ∂ ∂ ∂   + = + −           ∂ ∂ ∂ ∂ ∂ ∂             



2 2 2

1 1 1 1 1

r f f

w w w w w w w sc c c c

k p P p m B B B x B p B x B p B p t φ φ µ φ ρ             ∂ ∂ ∂ ∂ ∂ ∂       + = + −       ∂ ∂ ∂ ∂ ∂ ∂                      Chain and product rule on time derivative (right hand side) With some manipulation:

2

fluid compressibility

f

LT c M   ≡    

2

rock compressibility

r

LT c M   ≡    

1D Diffusivity Equation

Diffusivity constant Mobility Source

t w SC SC

k c k B m q α µφ λ µ ρ ≡ = ≡ = ≡ =  

2 2 2

1

f t w w w sc

c c k p p p m B x B x B t φ µ ρ   ∂ ∂ ∂   + = −     ∂ ∂ ∂       

≈0 slightly compressible fluid

If the fluid is “slightly compressible” (liquid), the compressibility is small (< 10-5) and constant and terms involving can be ignored. 1D diffusivity (with homogenous fluid and reservoir properties) can be written: If no sources or sinks (wells) are present, we get the “heat equation”

2 2

1 p p x t α ∂ ∂ = ∂ ∂

Generalized Diffusivity (Heat) Equation

In 2D (x-y plane) In 3D and potential Φ accounting for gravity

Slightly Compressible Fluids: Liquids

1 1 1 1

w f w w w T T

B V c B V p p p B B p ρ ρ   ∂ ∂ ∂ ∂ − = − = = =   ∂ ∂ ∂ ∂  

1

p f p c dp

d

ρ ρ

ρ ρ =

∫ ∫

V1 V2 More Pressure P1 P2

Recall fluid compressibility factor (cf) at constant temperature Integrating from a reference point (p0, ρ0), to any other point

Taylor Series Expansion

2 3

1 1 ( ) ( ) ( ) ( ) "'( ) ... 2! 3! f x x f x f x x f x x f x x ′ ′′ + ∆ = + ∆ + ∆ + ∆ +

2 3

1 1 ( ) ( ) ( ) "( ) "( ) ... 2! 3! p p p p p p p p p p ρ ρ ρ ρ ρ ∂ + ∆ = + ∆ + ∆ + ∆ + ∂

( )

( )

f

c p p f f

p c e c p ρ ρ ρ

−

∂ = = ∂

2 2 3 3

1 1 ( ) ... 2! 3!

f f f

p p c p c p c p ρ ρ ρ ρ ρ   + ∆ = + ∆ + ∆ + ∆ +    

1 1 ( )

f w

c p p B ≈ + −

Using Taylor series to expand density around a reference density, Differentiate exponential equation for density: For slightly compressible (cf < 10-5 psi-1) liquids, higher order terms are small:

1 ( )

f

c p p ρ ρ   ≈ + −  

Negligible for small cf

Therefore,

Bw

0=1 (assume reference is

standard conditions)

Simple 1D Problem: Core Flooding

( ) ( ) ( )

2 2 1 2

1 ,0 0.0 psi 0, 1000.0 psi , 0.0 psi

init B B

p p x t p x p p t p p L t p α ∂ ∂ = ∂ ∂ = = = = = =

( )

2 2

(2 1) 4 1

4 ( 1) (2 1) , cos 2 1 2

n

n t n init L B n

p n x p x t p e n L

α π

π π

+ ∞ =

− + = − +

∑

Analytical Solution to PDE “Heat” Equation

x=0 x=L

PB1 PB2

p=pinit

Analytical Solution

Steady state solution Time increasing

( )

2 2

(2 1) 4 1

4 ( 1) (2 1) , cos 2 1 2

n

n t n init L B n

p n x p x t p e n L

α π

π π

+ ∞ =

− + = − +

∑

Real Reservoirs

That was the easy solution… Real reservoirs have:

• Spatially varying permeability, porosity, etc.
• Time-varying viscosity, formation volume factor
• Geometries that are 2D or 3D and are not on a regular grid
• Sources and sinks (called wells) spaced irregularly throughout the reservoir
• Complex boundary conditions

(and this is just for single phase flow…)

( )

t SC w w

c k p p g z q B B t φ ρ µ   ∂ ∇ ∇ + ∇ = −   ∂   

Solving “ugly” PDE

So how do we solve this complex, nonhomogeneous, 3D PDE?

• 1. Break the reservoir into manageable blocks that have contain reservoir and fluid

properties

• 2. Write algebraic equations for each block by “discretizing” PDE
• 3. Solve system of linear equations

1 2 1 1 2 3 2 2 3 4 3 1

3 2 2

N N N

TP TP Q TP TP TP Q TP TP TP Q TP TP Q

−

− = − + − = − + − = − + = 

1 1 n n

P P t t

+ +

  + = +   ∆ ∆   B B T Q