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Pressure Drawdown

(Variable/Multi-Rate)

Test

Lecture Outline

Introduction

Test Types

Information obtained

Mathematical Model

Interpretation

Semi-log analysis

Cartesian

Practice Problems

Common mistakes

Summary

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Lecture Outcomes

At the end of this class, a student should be able to do the following:

Handle the rate variation during a draw down test

Familiar with the general form of equation to describe ‘n’ number of flow rates

Model the Single Rate Drawdown, PBU, and 2-rate tests from the general equation

Synthesize the various data and information to interpret a pressure draw down test with

smoothly varying rates

2-rate test

Make qualitative judgment on the data and choose appropriate interpretation method

Isolate the correct data for interpretation

Draw conclusions from results

Make suggestions on the improvement of the test

Test Outcomes

Information gathered/required

Pressure versus time recording (pwf – vs – t)

Flow rate (q – vs - t)

Fluid properties- B, µ

Formation and well parameters –Ø, ct, h, rw

Test interpretation results

Formation permeability (k)

Initial pressure (pi)

Wellbore condition - damage or stimulation- skin (s)

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Types of Flow Tests

PDD is also known as Flow Test. Actually, Flow Test is a more generalized term.

There may me several types of Flow Tests, as follows:

Single or Constant Rate Test (q = constant)

variable Rate Test [q = f(t)]

Rate changing smoothly

Rate changing abruptly (Multi-Rate Tests)

Variable Rate Test [q = f(t)]

Ra

te,

q

time

q1

q2

q3

q4

q5

q6

qn-

1

qn

t1 t2 t3 t4 t5 tn-1

Ra

te,

q

time

Rate changing smoothly

and slowly

Slight modification of

PDD constant rate

model

Rate changing

abruptly

(Multi-Rate Tests)

Needs superposition

technique to model

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Analysis of Smoothly Changing Rates

Difficult to maintain q = constant during flow tests

Winestock and Colpitts method:

Even when both Pwf and q vary with time, the following equation can be used to model variable-tests as long as the rate is changing slowly and smoothly.

s

rc

kt

kh

B

q

PP

wt

wfi869.023.3log

6.1622

Which suggests a plot of versus on semi-log graph paper.q

PP wfi t

with a slope of

kh

B6.162 Notice the similarity &

differences with constant rate

model

Analysis of Smoothly Changing Rates

100 101 102 103 104 105

One log cycle

q

PP wfi

t

2

q

PP wfi

1

q

PP wfi

12

'

q

PP

q

PPmSlope

wfiwfi

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Analysis of Smoothly Changing Rates

From this plot the slope is

kh

Bmslope

6.162'

The effective permeability to the fluid flowing in the drainage area of the well is estimated as;

hm

Bk

'6.162

23.3log

1151.1

2

1

'

wthr

wfi

rc

k

q

PP

ms

And the skin factor is

Where is the value of this quantity on the straight line or its

extrapolation at a flowing time of 1 hr.hr

wfi

q

PP

1

Analysis of Smoothly Changing Rates :Example

t, hr q (stb/d) pwf, psig (pi-pwf) (pi-pwf)/q

0.105 180 4332 80 0.4444

0.151 177 4302 110 0.6215

0.217 174 4264 148 0.8506

0.313 172 4216 196 1.1395

0.45 169 4160 252 1.4911

0.648 166 4099 313 1.8855

0.934 163 4039 373 2.2883

1.34 161 3987 425 2.6398

1.94 158 3952 460 2.9114

2.79 155 3933 479 3.0903

4.01 152 3926 486 3.1974

5.78 150 3926 486 3.24

8.32 147 3927 485 3.2993

9.99 145 3928 484 3.3379

14.4 143 3931 481 3.3636

20.7 140 3934 478 3.4143

29.8 137 3937 475 3.4672

43 134 3941 471 3.5149

q1 = table

Pi = 4412 psia

h = 69 ft

= 3.9%

rw= 0.198 ft

B = 1.136 RB/STB

ct = 17 x 10-6 psi-1

= 0.8 cp

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Analysis of Smoothly Changing Rates :Example

k = 7.65 mD

s = +6.4

Constant Rate versus Smoothly Changing Rate

Constant Rate PDD

q is constant, while only pwf is changing with time

Semi-log plot: pwf-vs-t

Slope is

pwf@1hr for skin equation

Smoothly changing Rate PDD

Both q and pwf are changing with time

Semi-log plot: (pi – pwf)/q –vs-t

slope is

[(pi-pwf)/q]@1hr for skin equation

kh

B6.162

kh

qB6.162

Pwf

1

00

1

01

1

02

1

03

1

04

1

05

q

PP wfi

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Multi-rate Flow Tests

Assume that the rate is changing as shown in the figure during a flow test

To model this:

• Apply superposition in time

• logarithmic approximation of Ei function

• Reservoir must be infinite acting for the total time elapsed (t) since beginning of production at q1

Ra

te,

q

time

q1

q2

q3

q4

q5

q6

qn-1

qn

t1 t2 t3 t4 t5 tn-1

Multi-rate Flow Tests

stqmPP wfi log'

kh

Bm

6.162'

To simplify the algebra, let us write the solution given by the above

equation as:

s

rc

kt

kh

qBPP

wt

iw 869.023.3loglog6.162

2

Remember the equation for drawdown test:

where

and src

ks

wt

869.023.3log2

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Multi-rate Flow Tests

With this nomenclature for n different rates and for t>tn-1;

sttqqmstqmPP wfi 112

'

1

' loglog

sttqqm

sttqqm

sttqqm

nnn

11

'

334

'

223

'

log

...log

log

Which can be written more conveniently as:

For qn 0. Most of the time for practical purposes

q0 and t0 are set equal to 0

s

rc

kmtt

q

qqm

q

PP

wt

n

j

j

n

jj

n

wfi869.023.3loglog

2

'

1

1

1'

Two-Rate Flow Tests

For a two rate flow test whose production history given in the figure

Ra

te,

q

time

q1

q2

t10

s

rc

ktt

q

qqt

q

q

kh

BqPP

wt

wfi 869.023.3logloglog6.162

21

2

12

2

12

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Two-Rate Flow Tests

If we rearrange and define t1=tp1 and t-tp1=t’,

then equation becomes

'

1

2

'

'

11

2

2

loglog6.162

.........

869.023.3log6.162

tq

q

t

tt

kh

Bq

src

k

kh

BqPP

p

wt

iwf

Analysis of Two-Rate Flow Tests

Below procedure is suggested to analyze Two-Rate Flow Tests:

1-Plot Pwf vs. the time plotting function of

on Cartesian graph paper.

2-Determine the slope, m, from the straight line on the plot and use it to calculate permeability, k, from

'

1

2

'

'

1loglog t

q

q

t

tt p

mh

Bqk

16.162

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Analysis of Two-Rate Flow Tests

23.3log151.1

2

11

21

1

wt

wfhr

rc

k

m

PP

qq

qs

3. Calculate the skin factor, s, from

where P1hr is the flowing pressure at t’ =1 hr on the

straight line or its extrapolation, and Pwf1 is the flowing

pressure at the time the rate is changed (i.,e. t’ =0).

4. The initial reservoir pressure Pi is obtained by solving the

drawdown equation for Pi.

s

rc

ktmPP

wt

p

wfi 869.023.3log2

1

1

2-Rate Test Example

Example:

A well is produced at 50 STB/D for 72 hours. The rate is then reduced to 25 STB/D for 24 hours. Estimate formation permeability, skin factor, and initial pressure from the two-rate drawdown test data given the following formation and fluid properties:

q1 = 50 STB/D

tp1 = 72 hours

q2 = 25 STB/D

Pwf1 = 1142.24 psia

h = 43 ft

= 8.2 %

rw = 0.45 ft

B = 1.143 RB/STB

ct = 10.5 x 10-6 psi-1

= 1.278 cp

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2-Rate Test Example

Drawdown Test Data:

Time

(hours)

Pressure

(psi)

Time

(hours)

Pressure

(psi)

Time

(hours)

Pressure

(psi)

0.109 1354.12 0.719 1400.01 5.923 1414.25

0.149 1371.25 0.872 1401.46 7.170 1415.34

0.201 1382.65 1.057 1402.87 7.887 1415.87

0.245 1387.69 1.552 1405.61 9.546 1416.90

0.299 1391.29 2.070 1407.58 10.502 1417.39

0.401 1395.01 3.035 1410.12 15.502 1419.25

0.488 1396.87 4.043 1411.94 19.502 1420.20

0.592 1398.50 5.384 1413.68 24.000 1420.96

2-Rate Test Example

Data with plotting function of

Time

(hours)

Pressure

(psi)

Plotting

Function

Time

(hours)

Pressure

(psi)

Plotting

Function

0.109 1354.12 2.34 2.070 1407.58 1.71

0.149 1371.25 2.27 3.035 1410.12 1.63

0.201 1382.65 2.21 4.043 1411.94 1.58

0.245 1387.69 2.16 5.384 1413.68 1.52

0.299 1391.29 2.12 5.923 1414.25 1.51

0.401 1395.01 2.06 7.170 1415.34 1.47

0.488 1396.87 2.02 7.887 1415.87 1.45

0.592 1398.50 1.97 9.546 1416.90 1.42

0.719 1400.01 1.93 10.502 1417.39 1.41

0.872 1401.46 1.89 15.502 1419.25 1.35

1.057 1402.87 1.85 19.502 1420.20 1.32

1.552 1405.61 1.77 24.000 1420.96 1.29

'

1

2

'

'

1loglog t

q

q

t

tt p

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1300

1350

1400

1450

1500

0 0.5 21 1.5 2.5

Pw

f, p

si

Plotting Function

1397

1400

1429

3212

14291397

m

1.863

P1hr=1401

2-Rate Test Example

2-Rate Test Example

mh

Bqk

16.162

3243

278.1143.1506.162k

mdk 63.8

Formation permeability:

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Skin factor:

2-Rate Test Example

81.11s

23.3log151.1

2

11

21

1

wt

wfhr

rc

k

m

PP

qq

qs

863.11log50

25

1

172logloglog1 '

1

2

'

'

1'

t

q

q

t

tttPF

p

From the straight line;

psiPhr 14011

23.3

45.0105.10278.1082.0

63.8log

32

24.11421401

2550

50151.1

26xs

psiPwf 24.11421 and

Initial Pressure:

s

rc

ktmPP

wt

p

wfi 869.023.3log2

1

1

81.11869.023.3

45.0105.10278.1082.0

7263.8log3224.1142

26xPi

psiPi 54.1669

2-Rate Test Example