7/24/2014
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)] Rate changing smoothly and slowly q , e t a R
Slight modification of PDD constant rate model time
q6
q2
q , e t a R
qn
Rate changing abruptly (Multi-Rate Tests)
q4 q1
qn-
q3
Needs superposition technique to model
1
q5 t1
t2 t3
t4
t5
time
tn-1
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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. P i P wf q
162.6 B kt log 3 . 23 0 . 869 s 2 kh c r t w
Which suggests a plot of
with a slope of
P i P wf
versus
q
162.6
t
on semi-log graph paper.
B
Notice the similarity & differences with constant rate model
kh
Analysis of Smoothly Changing Rates
P i P wf
P i P wf P i P wf q q 2 1
Slope m '
q P i P wf q 2 P i P wf q 1
One log cycle
100
101
102
103
t
104
105
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Analysis of Smoothly Changing Rates From this plot the slope is
slope m' 162.6
B kh
The effective permeability to the fluid flowing in the drainage area of the well is estimated as;
k 162.6
B m' h
And the skin factor is
1 P i P wf k 3.23 log ' 2 ct r w m q 1hr
s 1.151
Where
P i P wf is the value of this quantity on the straight line or its q 1hr
extrapolation at a flowing time of 1 hr.
Analysis of Smoothly Changing Rates :Example t, hr 0.105 0.151 0.217 0.313 0.45 0.648 0.934 1.34 1.94 2.79 4.01 5.78 8.32 9.99 14.4 20.7 29.8 43
q (stb/d) pwf, psig (pi-pwf) (pi-pwf)/q 180 4332 80 0.4444 177 4302 110 0.6215 174 4264 148 0.8506 172 4216 196 1.1395 169 4160 252 1.4911 166 4099 313 1.8855 163 4039 373 2.2883 161 3987 425 2.6398 158 3952 460 2.9114 155 3933 479 3.0903 152 3926 486 3.1974 150 3926 486 3.24 147 3927 485 3.2993 145 3928 484 3.3379 143 3931 481 3.3636 140 3934 478 3.4143 137 3937 475 3.4672 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
Smoothly changing Rate PDD
q is constant, while only pwf is changing with time
Both q and pwf are changing with time
Semi-log plot: pwf-vs-t
Semi-log plot: (pi – pwf)/q –vs-t
Slope is
slope is
pwf@1hr for skin equation
[(pi-pwf)/q]@1hr for skin equation
162.6
qB kh
162.6
B kh
P i P wf Pwf
1 00
q
1 01
1 02
1 03
1 04
1 05
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Multi-rate Flow Tests Assume that the rate is changing as shown in the figure during a flow test
q , e t a R
q6
q2
qn
q4 q1
qn-1
q3 q5 t1
t2
t3
t4
t5
time
tn-1
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
Multi-rate Flow Tests Remember the equation for drawdown test:
k 162.6qB 3.23 0.869s P w P i log t log 2 kh c r t w To simplify the algebra, let us write the solution given by the above equation as:
P i P wf m' qlog t s where
and
m' 162.6
B kh
k 3.23 0.869s 2 ct r w
s log
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Multi-rate Flow Tests With this nomenclature for n different rates and for t>t n-1;
P i P wf m' q1 log t s m' q2 q1 log t t 1 s m' q3 q2 log t t 2 s m' q4 q3 log t t 3 s ... m' qn qn1 log t t n1 s Which can be written more conveniently as:
P i P wf qn
k q q 3 . 23 0 . 869 s m j j 1 logt t j 1 m' log 2 qn j 1 ct r w n
'
For qn 0. Most of the time for practical purposes q0 and t 0 are set equal to 0
Two-Rate Flow Tests For a two rate flow test whose production history given in the figure
q1 q , e t a R
q2
0
P i P wf
t1
time
162.6q2 B q1 kh
k q2 q1 logt t 1 log 3 . 23 0 . 869 s logt 2 q2 ct r w q2
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Two-Rate Flow Tests If we rearrange and define t 1=tp1 and t-tp1=t’, then equation becomes
P wf P i
162.6q2 B k log 3 . 23 0 . 869 s 2 kh c r t w
162.6q1 B t p1 t q2 ' ......... log ' q log t kh t 1 '
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
t p1 t ' q2 ' log t log ' t q1 on Cartesian graph paper.
2-Determine the slope, m, from the straight line on the plot and use it to calculate permeability, k , from k
162.6q1 B mh
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Analysis of Two-Rate Flow Tests 3.
Calculate the skin factor, s, from
k P 1hr P wf 1 log 3 . 23 2 ct r w q1 q2 m
s 1.151
q1
where P 1hr is the flowing pressure at
t’ =1
hr on the
straight line or its extrapolation, and P wf1 is the flowing pressure at the time the rate is changed (i.,e. t’ =0 ). 4.
The initial reservoir pressure P i is obtained by solving the drawdown equation for P i .
kt p1 3.23 0.869s 2 ct r w
P i P wf 1 mlog
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 tp1 q2 Pwf1 h
= 50 STB/D = 72 hours = 25 STB/D = 1142.24 psia = 43 ft
rw B ct
= 8.2 % = 0.45 ft = 1.143 RB/STB = 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 0.149 0.201 0.245 0.299 0.401 0.488 0.592
1354.12 1371.25 1382.65 1387.69 1391.29 1395.01 1396.87 1398.50
0.719 0.872 1.057 1.552 2.070 3.035 4.043 5.384
1400.01 1401.46 1402.87 1405.61 1407.58 1410.12 1411.94 1413.68
5.923 7.170 7.887 9.546 10.502 15.502 19.502 24.000
1414.25 1415.34 1415.87 1416.90 1417.39 1419.25 1420.20 1420.96
2-Rate Test Example
Data with plotting function of
t p1 t ' q2 ' log t log ' t q1
Time (hours)
Pressure (psi)
Plotting Function
Time (hours)
Pressure (psi)
Plotting Function
0.109 0.149 0.201 0.245 0.299 0.401 0.488 0.592 0.719 0.872 1.057 1.552
1354.12 1371.25 1382.65 1387.69 1391.29 1395.01 1396.87 1398.50 1400.01 1401.46 1402.87 1405.61
2.34 2.27 2.21 2.16 2.12 2.06 2.02 1.97 1.93 1.89 1.85 1.77
2.070 3.035 4.043 5.384 5.923 7.170 7.887 9.546 10.502 15.502 19.502 24.000
1407.58 1410.12 1411.94 1413.68 1414.25 1415.34 1415.87 1416.90 1417.39 1419.25 1420.20 1420.96
1.71 1.63 1.58 1.52 1.51 1.47 1.45 1.42 1.41 1.35 1.32 1.29
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2-Rate Test Example 1500
m
1450
1397 1429 2 1
32
1429
P1hr=1401
i s p , f
1400 1397
w
P
1350
Plotting Function 1300 0
0.5
1
1.5
1.863
2
2.5
2-Rate Test Example
Formation permeability:
k 162.6
q1 B
k 162.6
501.1431.278 4332
mh
k 8.63md
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2-Rate Test Example Skin factor:
k P 1hr P wf 1 log 3 . 23 2 ct r w q1 q2 m
s 1.151
q1
72 1 25 t p1 t ' q2 ' log t log log 1 1.863 t ' q 1 50 1
PF t ' 1 log
From the straight line; P 1hr 1401 psi
P wf 1 1142.24 psi
and
8.63 1401 1142.24 3 . 23 log 2 6 32 0.0821.27810.5 x10 0.45 50 25
s 1.151
50
s 11.81
2-Rate Test Example Initial Pressure:
kt p1 3 . 23 0 . 869 s 2 ct r w
P i P wf 1 mlog
8.6372 3.23 0.86911.81 2 6 0.0821.27810.5 x10 0.45
P i 1142.24 32log
P i 1669.54 psi
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