Finite Difference Approximation - Cairo...

Finite Difference Approximation

12/26/2017 1

Dr. Helmy Sayyouh

Petroleum Engineering

Cairo University

Introduction

• The most widely used numerical technique in reservoir simulation is the finite-difference approach.

• The governing equations, as well as the boundary conditions used for describing flow in porous media, have only first-order and second-order derivatives, and so we will limit our discussion to these.

12/26/2017 Dr. Helmy Sayyouh 2

2

P

x 2

1

P

t

First-order derivative

• First-order derivatives appear in the governing equations on the right-hand side in the form of the time derivative (the accumulation term).

• In addition, first-order derivatives appear when the gradient is specified across a given boundary.

• To approximate the first-order derivatives, we use truncated Taylor series expansion.


• The derivative is approximated at the point x (also designated as i) at which the value of pressure is Pi.

• The two neighboring points to the central point are x - Dx (also designated as i -1) and x + Dx (also designated as i - 1).

• Accordingly, at these two neighboring points, the pressure values are Pi - 1 and P i + 1, respectively.


• In general, a Taylor series expansion for evaluating a function f (x) at x + Dx can be written


f x x f x xf

x

x f

x

x f

xx x x

2 2

2

3 3

32 3! !...

P P x

P

x

x P

x

x P

xi i

i i i

1

2 2

2

3 3

32 3

! !...

Using the notation of the previous Figure we can write the equation

Forward-Difference Approximation:

• The forward-difference approximation to the first-order derivative at point uses the values of the function at points i and i +1.


P

x

P P

xO x

i

i

1

( )

The second term on the right-hand side of the Equation denotes the error in

the approximation and is read as “order of .” The magnitude of the error is

the same as the magnitude of . In practice, the error term is dropped in

writing the finite-difference analogues. Thus, we write:

P

x i

P

i 1 P

x


The exact derivative of P at point is the slope of the tangent AB to the curve at

point O. Equation uses the slope of the secant OC to approximate this derivative.

As we can conclude from the figure, the accuracy of this approximation depends on

the shape of the curve, as well as the length of the interval between and in the limit

as point is moved progressively closer to the secant OC tends to obtain the same

shape as the tangent AB.

• In reservoir simulation, we use the forward difference approximation.

• The two neighboring points in this case represent the old time level and the new time level .

• When this derivative is written at a point i in the spatial domain it represents the rate of change of pressure with time at point i. In other words,


P

t

i

P

i

n 1 P

i

n

t

Backward-Difference Approximation:

• With the same strategy, but now using the neighboring point , we can obtain the backward-difference approximation to the first-order derivative from Equation using the (-) sign.


Central-Difference Approximation:

The central-difference approximation to the first-order derivative at point i uses the two adjacent neighboring points.


P

x i

P

i 1 P

i 1

2 x

Note that the secant CE is almost parallel to the

desired tangent AB.

It is apparent from Figure that central-difference provides a more accurate

approximation to the first-order derivative than either the forward- or backward-

difference approximation does.

Second-order derivative

• The left-hand-side of the flow equation is composed of second-order derivatives representing the flux terms. To approximate these second-order derivatives, we use central-difference approximation.


2 P

x 2

i

P

i 1 2 P

i P

i 1

( x ) 2

O x 2

2 P

x 2

i

P

i 1 2 P

i P

i 1

( x ) 2

OR

Finite-difference schemes

• There are two principal groups of finite-difference schemes: explicit andimplicit.

• We can illustrate the governing concepts for these schemes using the classical diffusivity equation as it is written in one dimension:


2

P

x 2

1

P

t

The explicit finite-difference analogue to the diffusivity Equation

is


P i 1

n 2 P

i

n P

i 1

n

( x ) 2

1

P i

n 1 P

i

n

t

Since pressures at the old time level are known at all locations, the only

unknown in Equation is the pressure at the new time level

P i

n 1

t

( x ) 2

P i 1

n 2

( x ) 2

t

P

i

n P

i 1

n

The implicit finite difference analogue is:

collecting all the unknown terms on one side, we obtain the characteristic form of the implicit finite-difference scheme


P i 1

n 1 2 P

i

n 1 P

i 1

n 1

( x ) 2

1

P i

n 1 P

i

n

t

t

( x ) 2

P i 1

n 1 1

2 t

( x ) 2

P

i

n 1

t

( x ) 2

P i 1

n 1 P

i

n

Flow equations in the finite-difference form


A x

k x

x B x

i , j , k

P i 1 , j , k

n 1 P

i , j , k

n 1

A x k

x

B x i , j , k

n

P i , j , k

n 1 P

i 1 , j , k

n 1

A y

k y

y

B y

i , j , k

P i , j 1 , k

n 1 P

i , j , k

n 1

A y

k y

B y i , j , k

n

P i , j , k

n 1 P

i , j 1 , k

n 1

A

z k

z

z B z

i , j , k

P i , j , k 1

n 1 P

i , j , k

n 1

A z k

z

B z i , j , k

n

P i , j , k

n 1 P

i , j , k 1

n 1

q

i , j , k

V

b c

5 . 615 t

i , j , k

P i , j , k

n 1 P

i , j , k

n

The term Axkx/Dx is the constant part of the transmissibility group,

while the term μB could depend on the pressure. To calculate these

two terms at the interface, we must use averaging procedures. For

the constant part of the transmissibility, if we consider the two blocks

as being connected in series, it becomes necessary to use harmonic

averaging, such as


A x k

x

x i

1

2 , j , k

2 A x

A x

k x

k x

A x

k x

x i 1 , j , k A x

k x

x i , j , k

• The second term (μB) is considered a weak function of pressure; that is, it exhibits weak non-linearity. We calculate this term at the arithmetic average of pressures at the two neighboring blocks:


• Note that under multiphase flow conditions, a typical transmissibility term will include relative permeability to the fluid for which the flow equation is being written. In that case, the transmissibility group between the blocks i, j, k and i+1, j, k is


A x k

x k

rf

f B

f x

i 1 / 2 , j , k

• This procedure is known as single-point upstream weighting,and can be summarized as


k r f

k r f

S f

if P i + 1, j, k

P i, j, k

k r f

S f

if P

i + 1, j, k P

i, j, k

• A more accurate representation of the relative permeability at the interface is known as two-point upstream weighting,and is expressed as


k rf

1

2 3 k

rfS

f

- k

rfS

f

if P i + 1, j , k

P i, j , k

1

2 3 k

rfS

f

- k

rfS

f

if P i + 1, j , k

P i, j , k

• Although we have illustrated the transmissibility groups only for the i+1/2, j, k interface, similar analyses apply to the other interfaces.

• In summary, it is imperative to note that the transmissibility group consists of three components, and that each is calculated differently.

• The constant component is calculated using harmonic averaging

• The weakly non-linear component is estimated using arithmetic averaging

• The strongly non-linear component is obtained using upstream averaging


Incorporation of boundary conditions

• To obtain a particular numerical solution to a flow problem, we must impose specific boundary conditions.

• Figure shows a pure Dirichlet-type boundary condition with a mesh-centered grid system.


The boundary conditions imposed on the system translate to the following equations:


P i , NY

P A

P i , 1

P C

for i = 1, ...NX

P i , j

P D

P NX , j

P B

for j = 1, ...NY

• Figure shows a body-centered grid with Neumann-type boundary conditions. In this case, finite-difference representation of the boundary conditions is as follows:


P NX 1 , j

P NX, j

x C

1

P i , j

P 0 , j

x C

4

f or j = 1, ..., NY

P i , 1

P i , 0

y C

2

P 1 , NY 1

P i , NY

y C

3

for i = 1, ..., NX


Figure shows a similar treatment for the mesh-centered grid system.

The important difference is the node being reflected. The implementation of

the finite-difference approximation in this case will be as follows:

P NX 1 , j

P NX 1 , j

2 x C

1

P 2 , j

P i , 0

2 x C

4

f or j = 1, ..., NY

P i , 2

P i , 0

2 y C

2

P 2 , NY 1

P i , NY

2 y C

3

for i = 1, ..., NX

Shorthand notation for the finite-difference equations

• Strongly-Implicit-Procedure, or SIP (Stone, 1968). For example, we can write the SIP form as follows:


Z P B P D P E P F P

H P S P Q

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

, , , , , , , , , , , , , , , , ,

, , , , , , , , ,

1

1

1,

1

1,

1 1

1,

1

1,

1

1

1

Z, B, D, F, H and S are the transmissibility terms representing the

interaction of block i, j, k with the neighboring blocks.

Figure graphically shows the interaction of block i, j, k with the surrounding blocks


• The principal advantage of the SIP notation is its inherent flexibility.

• For instance, having written it for three-dimensional flow, for one- and two-dimensional flows some of the SIP coefficients simply drop out.


Table summarizes the possible combinations and their corresponding SIP coefficients.

Special considerations

• In writing finite-difference analogues for partial differential equations, we must always consider a number of important points, which principally pertain to the accuracy of the resulting solution.

• The specific points of consideration include solution stability, consistency and truncation error.

• Rarely does the user of a reservoir simulator need to worry about these problems, but one should be aware of them.


• Stability analysis is meant to ensure that the round-off error (which arises over time due to the computer’s finite word length as time evolves) does not magnify in such a way as to obscure the solution.

• Stability analyses focus on defining the stability criterion for a given finite-difference scheme.

• These types of analyses provide results such as conditional stability and unconditional stability or instability


• When we encounter a conditional stability, the criterion will bring in certain limitations on the block dimensions and time step sizes.

• Although an unconditional stability implies no restriction on the block and time step sizes, we should keep in mind that the physical meaning of the solution can be lost if we assign unrealistically large block and time step sizes.


• The next important question to answer is whether the proposed scheme is consistent (i.e., compatible) with the original partial differential equation.

• In other words, does the proposed finite-difference analogue collapse to the original partial differential equation in the limit as the block dimensions and time step size approach zero?


• The truncation error is the difference between the original partial differential equation and its finite-difference analogue.

• For a compatible scheme, we should expect that the truncation error will disappear as the block dimensions and the time step size become infinitesimally small.

• If this does not happen, we have a consistency problem. In other words, the proposed scheme produces a solution to a different problem (partial differential equation).


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