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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
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
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