Part 12 Superposition Single and Multi Well

8/13/2019 Part 12 Superposition Single and Multi Well

1/22

Well Test Analysis

Superposition

Superposition

It is a fundamental tool of pressuretransient test analysis and reservoirengineering.

ma es us o cons ruc reservo rresponse functions in complex situations(boundaries, variable rate, variablepressure producti on) using only simplebasic model solutions , namely constant-rate (or constant-pressure) solutions.

Superposition

It can be used to represent the response dueto several wells by adding up the individualwell responses (multi-well applications).

By appropriate choice of flow rate and welllocation, it is also possible to representvarious types of r eservoir boundaries (no-flow and cons tant-pressure).

It should be noted that principle ofsuperposition holds only for linear systems(in the mathematical sense).


2/22

Well Test Analysis

Superposition However, these include most o f the standard response

functions used i n pressure transient analysis, makingthe assumption of slightly compressible fluid of

constant viscosity and compressibility.

Infinite acting radial flow for homo geneousreservoirs

Double-porosity and double-permeability models

Fractured, horizontal wells

Bounded systems, etc.

It can be used for gas wells with t he appropriatetransformations (e.g., real-gas pseudo-pressure) andcorrections (e.g., material balance correction) as to bediscussed later in gas pressure transient testing.

Principle of Superposition

It states that the response of the systemto a number of perturbations is exactlyequa o e sum o e responses oeach of the perturbations as if they werepresent by themselves.


Well 2q2

Well 4q4

Well 1

q1 Well 3q3

p(M,t) = ?


3/22

Well Test Analysis


= r1

+

r2

q1

Mq2

q3

q4

q1

M Mq2

p(M,t) = p1(r1,q1,t) + p2(r2,q2,t) + p3(r3,q3,t) + p4(r4,q4,t)

++ r3 r4M

q3

Mq4

Superposition

Using the principle of superposition and theimage well concept (to be discussed), it isrelatively straightforw ard to account for theeffects of complex boundary shapes,

constant-pressure boundaries.

As said previous ly , can be used to combinea series of different constant-rate solutions(or response functions) to describe thepressure response in a variable-rate

pressure transient t est.

Homogeneous Infinite System:


= r1

+r2

q1

Mq2

q3

q4

q1

M Mq2

p(M,t) = p1(r1,q1,t) + p2(r2,q2,t) + p3(r3,q3,t) + p4(r4,q4,t)

++ r3 r4M

q3

Mq4


4/22

Well Test Analysis



Suppose that that four wells in the the previous

slide start to production at different rates, butconstant, at the same time.

How could we describe each w ells pressure

change pj(rj,qj,t) that will created at the point M

at time t in the reservoir i f each well were

producing alone with a constant-rate of qj?

==

kt

rc948.05Ei

kh

B70.6q)r,p(qpt),r,(qp

2

tj

jjijjj

j

for j = 1,2,3,4.



=

rc948.05B70.6q

kt

rc948.05Ei

kh

B70.6qt)p(M,p

2

2t2

2

t1i

1

kt

rc948.05Ei

kh

B70.6q

kt

rc948.05Ei

kh

B70.6q

ktkh

2

4t4

2

3t3



Now, suppose that we wish to compute the

pressure drop at Well 1 in our previous 4- well

example, i.e., the point M is at w ell.

Well2

q2

Well1Well3

Well4

M

q3

q4

q1

r2

r3

r4

How to take r1? We take as the wellbore radius o f Well 1, r1 = rw1

r1


5/22

Well Test Analysis



Then, the pressure drop at Well 1, with the

presence of other three production wells, canbe written as:

rc948.05B70.6q 2

wt1 1

=

kt

rc948.05Ei

kh

B70.6q

kt

rc948.05Ei

kh

B70.6q

kt

rc948.05Ei

kh

B70.6q

ktkh,

2

4t4

2

3t3

2

2t2

1w1i

Skin factor

at Well 1


Principle of Superposition: Example

500 ft350 ft

Well 3

(a) Compute pr essure and pressur e drop at Well 3 at t = 60 hr.

(b) Compute pressure and press ure drop at Well 1 at t = 100 hr.

650 ftWell 2Well 1


Principle of Superposition: Example

(a) Compute pressu re and pressure drop at Well 3at t = 60 hr.

= rc948.05EiB70.6qrc948.05EiB70.6qt),(r

2

2t2

2

1t1w3i

thth

=

605

3501.5x100.70.2.05948Ei

1005

0.71.27512070.6

605

5001.5x100.70.2948.05Ei

1005

0.71.27545070.6t),p(rp

25

25

w3i

[ ] [ ] psi9.725.244.480.74Ei15.11.66Ei56.7t),p(rp w3i =+==

p(rw3,t = 60 hr)=5000-9.72= 4990.3 psi


6/22


7/22

Well Test Analysis

Homogeneous Infinite System-

Classical Two Well Problem

For t t* at point M,p i-p(x,y,t)=p1(x,y,t)

For t t* at point M,p i-p(x,y,t)=p1(x,y,t)+ p2(x,y,t-t*)



For t t* at point M,p i-p(x,y,t)=p1(x,y,t) =

kt

rc948.05Ei

kh

B70.6q 2

1t1

For t t* at point M,p i-p(x,y,t)=p1(x,y,t)+ p2(x,y,t-t*)

=

)tk(t

rc948.05Ei

kh

B70.6q

kt

rc948.05Ei

kh

B70.6q

*

22t2

2

1t1



Now suppose that both wells start

production wit h the same rate q at t = 0,

then the pressure drop at point M for t 0is iven b

p i-p(x,y,t)=p1(x,y,t)+ p2(x,y,t)

=

kt

rc948.05Ei

kh

70.6qB

kt

rc948.05Ei

kh

70.6qB

2

2t

2

1t


8/22

Well Test Analysis



Then, we can rewrite by using:

2221 yx)(dr ++=

222

2 yx)(dr +=

[ ]

[ ]

+

++=

kt

yx)(dc948.05Ei

kh

70.6qB

kt

yx)(dc948.05Ei

kh

70.6qBt)y,p(x,p

22

t

22

ti

0x

p

y)0;(x

=

=From this equation, we obtain:



y

d d

xWell 2, q

No-flow boundary

Well 1,q


Single No-Flow (Fault ) Problem

dd d

Sealing Fault

Well, q

Well, q Image

Well, q

No-flow boundary

+

=

t4

(2d)Ei

kh

70.6qB2s

t4

rEi-

kh

70.6qB(t)pp

22

wwfi c

k2.637x10

t

4

=

(prod.)

Actual System


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Well Test Analysis



2D Pressure

Profile, hereZ-axis represents

pressure

3D Pressure

profile

No-flow boundary



3D Pressure profiles

No-flow boundary

Image well

Numerical Simulation

with a single no-flow boundaryCondition in a semi-infini te sys.

Numerical Simulation

With two wells producing at thesame constant rate q , separated

by a distance 2d, in an infinite sys.



Now suppose that Well 1 start produc tion

with rate q at t = 0, but Well 2 start injection

with rate q at t = 0, the pressure drop at

oint M for t 0 is iven bp i-p(x,y,t)=p1(x,y,t)+ p2(x,y,t)

( )

=

kt

rc948.05Ei

kh

Bq-70.6

kt

rc948.05Ei

kh

70.6qB

2

2t

2

1t


10/22

Well Test Analysis



Then, we can rewrite by using:

2221 yx)(dr ++=

222

2 yx)(dr +=

[ ]

( ) [ ]

+

++=

kt

yx)(dc948.05Ei

kh

Bq-70.6

kt

yx)(dc948.05Ei

kh

70.6qBt)y,p(x,p

22

t

22

ti

tallforpt)y,0,p(x0t)y,0,p(x-p ii ====

From this equation, we obtain:



y

d d

Well 1,q

x

Well 2, -q

constant-pressureboundary (C-P Boundary)


Single C-P Boundary Problem

dd d

C-P boundary

Well, q

Well, q Image

Well, -q

C-P boundary

+

=

t4

(2d)Ei

kh

70.6(-q)B2s

t4

rEi-

kh

70.6qB(t)pp

22

wwfi c

k2.637x10

t

4

=

(Inj.)


11/22

Well Test Analysis

Examples: Imaging Based on

Superposition

dary

N-F boundary

d2

Image

Well, q(prod.) Image

Well, q

(prod.)

d2

N-F

bou

Well, q

1

Well, q

d1d2

Image

Well, q(prod.)

Actual System

d1

Write down an expression for computin g pressure drop at the well.


Superposition

dary

N-F boundary

d2

Image

Well, q(prod.) Image

Well, -q

(Inj.)

d2

C-P

bou

Well, q

1

Well, q

d1d2

Image

Well, -q(Inj.)

Actual System

d1



Superposition

d

ary

nd

ary

N-F

bou

Well, q

1

Well, qN-F

bou 2

Image

Well, q

Actual System

2

Image

Well, q

1 2 1 1 2.. ..

Infinitely many image wells (all prod .)



12/22

Well Test Analysis


Superposition

dary

ndary

C-P

bou

Well, q

1

Well, qN-F

bou 2

Image

Well, q

Actual System

2

Image

Well, -q

1 2 1 1 2.. ..

Infinitely many image wells (prod + inj .)

Write down an expression for computing pressure drop at the well.


Superposition

Infinitely many array of image wells (all prod .)


Superposition

Infinitely many array of image wells (both prod. and inj .)


13/22

Well Test Analysis

Superposition in Time



r2 = 2d

y-axis2

w

2

w1

2

1 rrr ==22

2 (2d)r =

Well 1, q Well 2, -q

x,y)r1 = rw1=rw

d d

0,0x-axis

Suppose Well 1 starts production w ith q from t = 0

Suppose Well 2 start producti on with -q from t = t*, t* >0

How can we compute pressure drop at Well 1 for agiven time t>0?



For t t* at point M, i.e., at Well 1p i-p(rw,t)=p1(rw,t)

For t t* at Well 1,p i-p(rw,t)=p1(rw,t)+ p2(r2,t-t*)


14/22

Well Test Analysis



For t t* at Well 1,p i-p(rw,t)=p1(rw,t) =

+

12s

kt

rc948.05Ei-

kh

B70.6q 2

1t1

For t t* at Well 1,p i-p(rw,t)=p1(rw,t)+ p2(r2,t-t*)

( ) ( )

+

=

)tk(t

2dc948.05Ei

kh

Bq-70.6

2skt

rc948.05Ei

kh

70.6qB

*

t

1

21t

2

Rate Superposition

Now, we bring Well 2 to the location of Well1 so that two well s are operating at thesame location of Well 1:

One production well started productionwith q at t = 0, and

an injection well started injection with qat t = t *, t* > 0.

Rate Superposition

Rate histories of the wells are:

Rate

q > 0 Well 1

Well 1

Rate

0

Time

Time- q < 0

t*Well 2

0

Rate

0Time

q = 0

t*

Buildup Test

+


15/22

Well Test Analysis

Rate Superposi tion Equation

For t t* at Well 1,p i-p(rw,t)=p1(rw,t) =

+

12s

kt

rc948.05Ei-

kh

70.6qB2

wt

For t t* at Well 1,p i-p(rw,t)=p1(rw,t)+ p2(rw,t-t*)

( )

+

+

+

=

1*

2

wt

1

2

1t

2s)tk(t

rc948.05Ei-

kh

Bq-70.6

2skt

rc948.05Ei-

kh

70.6qB

Buildup Test Pressure/Rate History

ressure

P

Time Time

Examples: Two Step-Rate Changes:

Single WellRate Histories

+


16/22

Well Test Analysis

Example: Two Step-Rate Changes:

Single Well

+

p

Timet10

p

pi

pi

+

p

Timet10

pp

pi

pi

Pressure Histories

p

0Time

t2Timet1

0t2

pip

0Time

t2Timet1

0t2

Timet10

t2

pi

Write down the pressure equation usin g superposition:

Assume infi nit e-acting homogeneous reservoi r,

fully penetrating line-source well.


Single WellRate Histories

q1 > 0qq

q1

q1 > 0qq

q1

q

0

Time

Time

q2-q1

t1

t2

Timet10

t2

2

q

0

Time

Time

q2-q1> 0

t1

t2

Timet10

t2

2


Single Well

+

p

Timet10

p

p i

p i

+

p

Timet10

p

p i

p i

Pressure Histories

p

0Time

t2Timet1

0t2

pip

0Time

t2Timet1

0t2

pi

Write down the pressure equation usin g superposition:

Assume infi nit e-acting homogeneous reservoi r,

fully penetrating line-source well.


17/22

Well Test Analysis

Multirate Superposition:Illustration

Multirate: illustration

q1

q2

q3 q5q6

q8

q1q3 q5

q6

q8

t1 t2 t3 t4 t5 t6 t7q4 q7

t1 t2 t3 t4 t5 t6 t7

q2

q4 q7

Multirate Superposition Equation

Although , so far, we have consideredinfinite acting homogeneous reservoirsolution, superposition is quite general canbe used for all models assuming slightl y

compressible fluid of constant viscosityan compress y, e.g., ou e poros y,layered systems, etc.

All we need is to have the cons tant -rateresponse of the system considered togenerate the solution for the multirate case.

Next, I develop the general equation ofmultirate for a single-well case.


18/22

Well Test Analysis


Let pu

(r,t) be the rate-normalized pressurechange in psi /(B/D) created at location rinthe reservoir at a time t from a well

- .

It includes the skin factor if it is evaluatedat the production/injection w ell.

In well testing literature, it is also referredto as the constant uni t-rate pressurechange.


For example, for a fully-penetrating lin e-sourcewell with no w ellbore storage effects in aninfinite-homogeneous reservoir, pu(r,t) would begiven by:

pu(r,t) =

=

+

>

=

w

2

wt

w

2

t

i

rr2skt

rc948.05Ei-

kh

70.6B

rr,kt

rc948.05Ei

kh

70.6B

q

t)p(r,p

,


Now, lets consid er the following rate historyat the producing well:

Time

q1q2

q3

qi

qn

t0=0 t1 tn-1tit2 ti-1t3


19/22

Well Test Analysis


Now, we apply superposition to ob tain thesolution:

( ) ( ) )t(tpqq)t(tpqq(t)pqt)p(r,p 2u231u12u1i ++=( ) ( ) )t(tpqq)t(tpqq 1nu1nn1iu1ii ++++ LL

( ) ( )=

+ =n

0j

juj1ji ttr,pqqt)p(r,p

where q0 = 0, and t0 = 0.

( ) ( )=

=n

1j

j-1uj-1ji ttr,pqqt)p(r,p

OR:


For a line-source fully penetrating well in aninfinite acting reservoir, we can express thesuperposition equation as:

( ) w

n

1j

tj-1ji rr,

kt

rc948.05Eiqq

kh

70.6Bt)p(r,p >

= =

( ) w

n

1j

2

wtj-1ji rr,2s

kt

rc948.05Eiqq

kh

70.6Bt)p(r,p =

+

=

=

Single-Well Convolution Equation

The superposition equation can also bewritten in an m ore general or compactintegral form as:

( ) ( )

ddt

tr,dpqt)p(r,p u

t

0

i

=

For a line-source well:

( )

=

kt

rc948.05exp

kh

70.6B

dt

tr,dp 2tu

convolution equation


20/22

Well Test Analysis

Multiwell/MultirateSu er osition

Multiwell/Multirate Superposition Now, we extend to a more general case

where we can have more than one wellproducing wi th a variable history i n thereservoir.

Lets consider a two-well system, Well Aand Wel l B with the fol lowin rateh istor ies ,at each w ell.

100

700

qB, stb/D

t, hour5

How do we calculate pressure at Well A at t = 25 hr?

0 (buildup)

250

100

qA, stb/D

10 20t, hour

Multiwell/Multirate Superposition

Combining superpositi on in space andsuperposition in time (or ratesuperposition), we can wri te the

case:

( ) ( )= =

=w

kN

1k

n

1j

k

j-1

k

u

k

j-1

k

ji tt,rpqqt)p(r,p


21/22

Well Test Analysis

Multiwell/Multirate Superposition

Suppose, we wish to compute pressure at

Well A at t = 25 hr for the exampleconsiderd, then w e will apply thesuperposition equation as:

( ) ( )

( ) ( ) ( ) ( )

= =

= =

+==

=

3

1j

2

1j

B

j-1

BB

u

B

j-1

B

j

A

j-1

A

w

A

u

A

j-1

A

j

A

wi

2

1k

n

1j

k

j-1

k

u

k

j-1

k

j

k

wi

tt,rpqqtt,rpqq25)t,p(rp

tt,rpqqt),p(rp

k

Pressure change caused

at Well A by Well A itself.

Pressure change caused

at Well A by Well B

Multiwell Convolution Equation

The multiwell/multirate superpositionequation can also be written in an moregeneral or co mpact integral form as:

( ) ( )

=

=

wN

1j

kj

u

t

0

jkk

i ddt

tdpqt),p(rp

Exercise 1

700

qB, stb/D250

100

qA, stb/D

100

t, hour5

Calculate pressure at Well A at t = 25 hr.

0 (buildup)

10 20t, hour


22/22

Well Test Analysis

Exercise 2

qA, stb/D 450250

500

0 shut-in

qB, stb/D

Calculate pressure at Well A at t = 72 hr.

24 48 t, hour 80 t, hour

Part 12 Superposition Single and Multi Well

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Transcript of Part 12 Superposition Single and Multi Well