WELL TEST INTERPRETATION
Instructor:
Louis Mattar, B.Sc., M.Sc., P.Eng
Course Outline
WELL TEST INTERPRETATION
This course is intended for engineers and specialist who want to learn the reasons for
well testing, and the information that can be derived from it. The procedures and
principles for analyzing vertical well tests will be extended to apply to horizontal wells.
The course will deal with both oil and gas well test interpretation, drillstem tests, wireline
formation tests and production tests, interference tests, detection of boundaries,
estimation of stabilized flow rates from short tests, etc.
The Practice of well test interpretation will be emphasized over the Theory. To this end,
Data Validation and the PPD (Primary Pressure Derivative) will be used to illustrate
wellbore dynamics, and extricate these effects from the apparent reservoir response.
Throughout the course, the theme will be:
W.T.I >> P.T.A.
WELL TEST INTERPRETATION (W.T.I.)
involves a lot more than simply
PRESSURE TRANSIENT ANALYSIS (P.T.A.)
This course is aimed at obtaining an understanding of the concepts. These will be
presented graphically (using a computer), thus keeping equations to a minimum. The
Practical aspects of the interpretation process will be highlighted.
WELL TEST INTERPRETATION
Detailed Course Contents
1. Introduction
a. Why test
b. Course Philosophy
c. Types of tests
d. Types of fluids
e. Types of reservoirs
f. Approaches to well test interpretation
2. Basic Concepts
a. Simplifying assumptions – reservoir
b. Drawdown test – oil
c. Type curves (Dimensionless)
d. Skin effect
e. Wellbore storage/Bourdet et al type curves
3. Gas Flow Considerations
a. Turbulence
b. Pseudo-Pressure
c. Pseudo-Time
4. Flow Regimes – Vertical Wells
a. Segmented approach
b. Early Time – Wellbore Storage
- Linear – fracture Storage
- Bilinear
- Spherical
c. Transient Flow – Radial
d. Late Time – Transition
- Linear – channel
- Stabilized – steady state
- pseudo-steady state
5. Flow Regimes – Horizontal Wells
6. Useful Concepts
a. Radius of investigation
b. Time to stabilization
c. Superposition
7. Drawdown Analysis (or Injection)
a. Procedure
b. Specialized Analyses
c. Horizontal Wells
8. Buildup Analysis
a. Horner Plot
b. Equivalent Time
c. M.D.H. Plot
d. Average Reservoir Pressure
e. Detection of boundaries
f. Other Buildup Curves
g. D.S.T.
h. Horizontal Wells
9. Non-Reservoir Effects
a. Data Validation
b. Welbore Dynamics
c. Primary Pressure Derivative – PDD
10. Production Forecasting
a. Transient/Stabilized IPR
b. AOF – Sandface/Wellhead
11. Test Design
12. Complex Models
13. Pitfalls
14. References/Nomenclature
15. Miscellaneous
a. ERCB Chapter 3
b. Acoustic Well Sounders
c. EUB Guide 40
d. Partial Penetration
e. Practical Considerations
LOUIS MATTAR, M.Sc., P. Eng.
PRESIDENT
Fekete Associates Inc
B.Sc. Honours in Chemical Engineering, University of Wales in Swansea, 1965
M.Sc. in Chemical Engineering, University of Calgary, 1973
Membership: APEGGA; Petroleum Society of CIM; Society of Petroleum Engineers
Louis worked for the Alberta Energy Resources Conservation Board, where he was the
principal author of the world-renowned E.R.C.B. publication "Theory & Practice of the
Testing of Gas Wells, 1975", which is an authoritative text on the subject.
For several years, Louis was Associate Professor at the University of Calgary where he
taught courses in Reservoir Engineering and Advanced Well Testing, and conducted
research in tight gas reservoirs, and multi-phase flow.
Since 1981 he has been with Fekete Associates, a consulting company that specializes
in well testing and reservoir engineering. He has analyzed and supervised the
interpretation of thousands of well tests and specializes in the integration of practice
with theory. He has appeared as an expert witness in several Energy Board hearings.
He has conducted studies ranging from shallow gas reservoirs to deep sour wells, from
small pools to a 5000-well reservoir/completion/production study, and from waterfloods
to gas storage.
Louis teaches the CIM course in “Gas Well Testing, Theory and Practice”, as well as
“Modern Production Decline Analysis” to the SPE and to several companies. He has
authored 43 technical publications. He is an adjunct professor at the University of
Calgary.
AWARDS:
Louis was the SPE Distinguished Lecturer in Well Testing for 2002-2003. He is a
Distinguished Member of the Petroleum Society of CIM. In 1995, he received the CIM
Distinguished Author award and the Outstanding Service award. In 1987, he received
the CIM District 5 Technical Proficiency Award.
TECHNICAL PUBLICATIONS
BY
LOUIS MATTAR
43.
MATTAR, L.: “Analytical Solutions in Well Testing”, Invited Panelist, CIPC Panel
Discussion at the Canadian International Petroleum Conference, Calgary,
Alberta, June, 2003.
42.
MATTAR, L. and ANDERSON, D.M.: “A Systematic and Comprehensive
Methodology for Advanced Analysis of Production Data”, SPE 84472, presented
at the SPE Annual Technical Conference and Exhibition, Denver, Colorado,
October, 2003.
41.
RAHMAN, A.N.M., MILLER, M.D., MATTAR, L: “Analytical Solution to the
Transient-Flow Problems for a Well Located near a Finite-Conductivity Fault in
Composite Reservoirs”, SPE 84295, presented at the SPE Annual Technical
Conference and Exhibition, Denver, Colorado, October, 2003.
40.
ANDERSON, D.M. and MATTAR, L.: “Material–Balance–Time During Linear
and Radial Flow”, CIPC 2003-201, presented at the Canadian International
Petroleum Conference, Calgary, Alberta, June, 2003.
39.
ANDERSON, D.M., JORDAN, C.L., MATTAR, L.: “Why Plot the Equivalent
Time Derivative on Shut-in Time Coordinates?”, presented at the SPE Gas
Technology Symposium, May 2002, Paper number 75703.
38.
POOLADI-DARVISH, M. and MATTAR, L.: “SAGD Operations in the Presence
of Overlying Gas Cap and Water Layer-Effect of Shale Layers, CIM 2001-178
37.
THOMPSON, T. W. and MATTAR, L.: “Gas Rate Forecasting During BoundaryDominated Flow”, CIM 2000-46, Canadian International Petroleum Conference
2000, Calgary, Alberta, June 2000.
36.
JORDAN, C. L. and MATTAR, L.: “Comparison of Pressure Transient Behaviour
of Composite and Multi-layered Reservoirs,” presented at the Canadian
International Petroleum Conference, Calgary, Alberta, June, 2000.
35.
MATTAR, L.: “DISCUSSION OF A Practical Method for Improving the Accuracy
of Well Test Analyses through Analytical Convergence”, Journal of Canadian
Petroleum Technology, May 1999.
34.
STANISLAV, J., JIANG, Q. and MATTAR, L.: “Effects of Some Simplifying
Assumptions on Interpretation of Transient Data”, CIM 96-51, 47th Annual
Technical Meeting of the Petroleum Society of CIM, Calgary, Alberta, June 1998.
33.
MATTAR, L. and McNEIL, R. “The Flowing Gas Material Balance”, Journal of
Canadian Petroleum Technology (February, 1998), 52, 55
32.
MATTAR, L.: “Derivative Analysis Without Type Curves,” presented at the 48th
Annual Technical Meeting of the Petroleum Society of CIM, Calgary, Alberta,
June 8-11, 1997
31.
MATTAR, L.: “Computers - Black Box or Tool Box?” Guest Editorial, Journal of
Canadian Petroleum Technology, (March, 1997), 8
30.
MATTAR, L.: “How Useful are Drawdown Type Curves in Buildup Analysis?”,
CIM 96-49, 47th Annual Technical Meeting of the Petroleum Society of CIM,
Calgary, Alberta, June 1996.
29.
MATTAR, L. and SANTO, M.S.: “A Practical and Systematic Approach to
Horizontal Welltest Analysis”, The Journal of Canadian Petroleum Technology,
(November, 1995), 42-46
28.
MATTAR, L.: “Optimize Your Gas Deliverability With F.A.S.T. PIPERTM,
American Pipeline Magazine, August, 1995, 16-17.
27.
MATTAR, L.: “Commingling”, Internal Report
26.
MATTAR, L.: “Reservoir Pressure Analysis: Art or Science?”, Distinguished
Authors Series, The Journal of Canadian Petroleum Technology, (March, 1995),
13-16
25.
MATTAR, L.: “Practical Well Test Interpretation”, SPE 27975, University of Tulsa
Centennial Petroleum Engineering Symposium, Tulsa, OK, U.S.A., Aug., 1994
24.
MATTAR, L., HAWKES, R.V., SANTO, M.S. and ZAORAL, K.: "Prediction of
Long Term Deliverability in Tight Formations", SPE 26178, SPE Gas Technology
Symposium, Calgary, Alberta, June, 1993
23.
MATTAR, L.: "Critical Evaluation and Processing of Data Prior to Pressure
Transient Analysis," presented at the 67th Annual Technical Conference and
Exhibition of the Society of Petroleum Engineers, Washington, D.C., October 4-7,
1992
22.
MATTAR, L. and SANTO, M.S.: "How Wellbore Dynamics Affect Pressure
Transient Analysis," The Journal of Canadian Petroleum Technology, Vol. 31,
No. 2, February, 1992
21.
MATTAR, L. and ZAORAL, K.: "The Primary Pressure Derivative (PPD) - A New
Diagnostic Tool in Well Test Interpretation," The Journal of Canadian Petroleum
Technology, Vol. 31, No. 4, April, 1992
20.
ABOU-KASSEM, J.H., MATTAR, L. and DRANCHUK, P.M.: "Computer
Calculations of Compressibility of Natural Gas", Journal of Canadian Petroleum
Technology, Calgary, Alberta, Sep.-Oct. 1990, Vol. 29 No. 5 p. 105
19.
MATTAR, L.: "IPR's and All That - The Direct and Inverse Problem", Preprint
Paper No. 87-38-13, 38th Annual Technical Meeting of the Petroleum Society of
CIM, Calgary, Alberta, June 1987
18.
BRAR, G.S. and MATTAR, L.: "Reply to Discussion of: The Analysis of Modified
Isochronal Tests to predict the Stabilized Deliverability of Gas Wells without
Using Stabilized Flow Data", The Journal of Petroleum Technology, AIME
(January, 1987), 89-92
17.
LAIRD, A.D. and MATTAR, L.: "Practical Well Test Design to Evaluate Hydraulic
Fractures in Low Permeability Wells", Preprint Paper No. 85-36-8, 36th Annual
Technical Meeting of the Petroleum Society of CIM, Edmonton, Alberta, June
1985
16.
MATTAR, L. and ZAORAL, K.: "Gas Pipeline Efficiencies and Pressure Gradient
Curves", Preprint Paper No. 84-35-93, 35th Annual Technical Meeting of the
Petroleum Society of CIM, Calgary, Alberta, June 1984
15.
MATTAR, L. and HAWKES, R.V.: "Start of the Semi-Log Straight Line in Buildup
Analysis", Preprint Paper No. 84-35-92, 35th Annual Technical Meeting of the
Petroleum Society of CIM, Calgary, Alberta, June 1984
14.
WASSON, J. and MATTAR, L.: "Problem Gas Well Build-Up Tests - A Field
Case Illustration of Solution Through the Use of Combined Techniques", The
Journal of Canadian Petroleum Technology (March - April, 1983), 36-54
13.
NUTAKKI, R. and MATTAR, L.: "Pressure Transient Analysis of Wells in Very
Long Narrow Reservoirs", Preprint Paper No. SPE 1121, 57th Annual Fall
Technical Conference and Exhibition of the Society of Petroleum Engineers of
AIME, New Orleans, LA, September 1982
12.
LIN, C. and MATTAR, L.: "Determination of Stabilization Factor and Skin Factor
from Isochronal and Modified Isochronal Tests", The Journal of Canadian
Petroleum Technology (March - April, 1982), 89-94
11.
MATTAR, L. and LIN, C.: "Validity of Isochronal and Modified Isochronal Testing
of Gas Wells", Preprint Paper SPE 10126, 56th Annual Fall Technical
Conference of AIME, San Antonio, TX, October 1981
10.
KALE, D. and MATTAR, L.: "Solution of a Non-Linear Gas Flow Equation by the
Perturbation Technique", The Journal of Canadian Petroleum Technology
(October-December, 1980), 63-67
9.
ADEGBESAN, K.O. and MATTAR, L.: "Prediction of Pressure Drawdown in Gas
Reservoirs Using a Semi-Analytical Solution of the Non-Linear Gas Flow
Equation", Preprint Paper No. 80-31-39, 31st Annual Technical Meeting of the
Society of CIM, Calgary, Alberta, May 198077. MATTAR, L.:
“Variation of
Viscosity-Compressibility Product With Pressure of Natural Gas", Internal Report,
1980
8.
MATTAR, L.: “Variation of Viscosity-Compressibility Product With Pressure of
Natural Gas", Internal Report, 1980
7.
MATTAR, L., NICHOLSON, M., AZIZ, K. and GREGORY, G.: "Orifice Metering
of Two-Phase Flow", The Journal of Petroleum Technology, AIME (August,
1979), 955-961
6.
AZIZ, K., MATTAR, L., KO, S. and BRAR, G.: "Use of Pressure, Pressure
Squared or Pseudo-Pressure in the Analysis of Transient Pressure Drawdown
Data from Gas Wells", The Journal of Canadian Petroleum Technology, (April June, 1976), 1-8
5.
MATTAR, L., BRAR, G.S. and AZIZ, M.: "Compressibility of Natural Gases", The
Journal of Canadian Petroleum Technology, (October-December, 1975), 77-80
4.
E.R.C.B. (1975), "Theory and Practice of the Testing of Gas Wells, Third Edition"
(co-authored by L. MATTAR) Alberta Energy Resources Conservation Board,
Calgary
3.
MATTAR, L. and GREGORY, G.: "Air-Oil Slug Flow in An Upward-Inclined Pipe
- 1: Slug Velocity, Holdup and Pressure Gradient", The Journal of Canadian
Petroleum Technology, (January - March, 1974), 1-8
2.
GREGORY, G. and MATTAR, L.: "An In-Situ Volume Fraction Sensor for Two
Phase Flows of Non-Electrolytes", The Journal of Canadian Petroleum
Technology, (April - June, 1973), 1-5
1.
MATTAR, L.: "Slug Flow Uphill In an Inclined Pipe", M.Sc. Thesis, University of
Calgary, Alberta, 1973
EXPERT WITNESS TESTIMONY
LOUIS MATTAR, P.Eng
Appeared before National Energy Board / Alberta Energy Utilities Board to give
evidence and testimony relating to oil and gas issues on several occasions to represent:
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
xi)
NOVA Corporation of Alberta
Merland Exploration Limited
GasCan Resources Ltd.
Bralorne Resources Limited
Encor Inc.
Norcen Energy Resources Ltd.
Gulf Canada Resources Ltd.
Paramount Resources
Devon Canada Inc
Rio Alto
Alberta Energy Company
Appeared before the Alberta Court of Queens Bench, as an expert, to represent:
i)
Novalta Resources Ltd.
5.
1.
Traditional (Arps)
2.
Fetkovich
3.
Blasingame
4.
Agarwal-Gardner
NPI - Normalized Pressure Integral
6.
Modeling
1
Traditional Decline Analysis
(ARPS)
•Empirical
•Boundary Dominated Flow
Exponential, Hyperbolic and
Harmonic Equations
q = qie − Dit
exponential
hyperbolic
q
harmonic
q=
qi
(1 + bDit )1/ b
q=
t
qi
1 + Dit
b = 0 …Exponential
0 < b < 1…Hyperbolic
b = 1 …Harmonic
2
D is Constant
Rate
Slope
The graph on
the right is a replot of the one
on the left, but
the vertical
scale has been
changed to
Log flow rate.
This converts
the red curve
on the left into
a straight line
D = 2.303*Slope
Log Flow Rate
Flow Rate
D = Slope
Rate
D is Constant
Slope
Time
Time
Exponential Decline - D is Constant
D is Constant
Rate
Slope
Time
The graph on the
right is a re-plot of
the one on the left,
but the horizontal
scale has been
changed to
Cumulative
Production
instead of Time.
This converts the
red curve on the
left into a straight
line.
D = Slope
Flow Rate
Flow Rate
D = Slope
Rate
D is Constant
EUR
Slope
Cumulative Production
Exponential Decline - D is Constant
3
dq
− Dt = ln
q
qi
dq
dq
D = K * q = − dt
q
t
q dq
∫ 0 Ddt = −∫qi q
b
K=
∫
K=
Di
qb
∫
qt dq
Di
* dt = − ∫ b +1
0qb
qi q
i
t
t
0
t
t
0
0
Q = ∫ q * dt = ∫ q i* e − Dt * dt
Di t 1 1
= −
qi
qi qt
Q=
qi − qi * e − Dt
D
qi * e − Dt = q
Q=
qi − q
D
t
t
0
−1
Q = ∫ q * dt = ∫ q i (1 + bDi t ) b * dt
Q=
qi
(1 + bDi t )
(1 − b) Di
b
q
(1 + bDi t ) = i
q
Q=
EXPONENTIAL
q = q i (1 + Di t )
−1
q = qi (1 + bDi t ) b
0
qi
b
(1 − b )Di
(q
i
1− b
Di
qi
q dq
Di
dt = − ∫ 2
qi q
qi
bDi t
−b
= q − b − qi
b
qi
q = qi * e − Dt
dt
q
D = K * q1 = −
D = K * q = − dt
q
0
b −1
b
t
0
Q = ∫ q * dt = ∫ qi (1 + Di t ) −1 * dt
− 1
− q1−b
t
0
−1
Q=
qi
[ln(1 + Dit ]
Di
(1 + Dit ) = qi
q
)
HYPERBOLIC
Q=
qi qi
ln
Di q
HARMONIC
Log Flow Rate
Harmonic Decline
Abandonment Rate
Cumulative Production
Harmonic decline will become a straight line if plotted as log-Rate
versus Cumulative Production.
THE RATE WILL NOT REACH ZERO, and thus the ultimate
recoverable reserves (at zero rate) cannot be quantified, unless a (nonzero) abandonment rate is specified.
4
Fetkovich
Late Time
Boundary-Dominated
Early Time
Transient
•Constant Operating Conditions
Fetkovich Theory
-Developed because traditional decline curve
analysis is only applicable when well is in
boundary dominated flow
- Fetkovich used analytical flow equations to
generate typecurves for transient flow, and
combined them with emprical decline curve
equations from Arps
-Resulting typecurves encompass whole
production life of well
5
Fetkovich Theory – Empirical Portion
exponential
hyperbolic
harmonic
q
log(q)
t
log(t)
Theoretical Meaning of Exponential Stem
Boundary-Dominated Flow
Start of Boundary-Dominated Flow
pi
Boundary-Dominated
Pressure ( p )
Transient Flow
re
Distance ( r )
6
Other Type of Boundary Dominated Flow - Constant Rate
Pseudo-Steady State Flow
Start of Pseudo-Steady
pi
Transient Flow
Pressure ( p )
=
=
Pseudo-Steady State
Flow
Time
re
Distance ( r )
TRANSIENT
FLOW
7
Fetkovich Theory – Analytical Portion
Analytical solution for constant flowing pressure
Single
Curve
forisAll
re/rwa’s
Transient
Flow
a single
curve;
Different stems for
Transient
Flow
Boundary-Dominated
Flow
is a family
of curves
Same
SameBoundary
Transient for
for all
all re/rwa
re/rwa 's's
Curves separate during
boundary dominated flow
Different re/rw 's
Different re/rw 's
Curves separate
during transient flow
Fetkovich Theory – Analytical Portion
Analytical solution for constant flowing pressure
Single
Curve
forisAll
re/rwa’s
Transient
Flow
a single
curve;
Different stems for
Transient
Flow
Boundary-Dominated
Flow
is a family
of curves
qDd
Same Boundary for all re/rwa 's
Different re/rw 's
Curves separate
during transient flow
tDd
8
Fetkovich Theory – Analytical Portion (b)
Rate Decline Curves for "Constant Wellbore Flowing Pressure"
Transient Flow
Transient rFlow
Different
e/rw
10
Boundary
Boundary
Dominated
Flow
becomes Flow
Dominated
Exponential Decline
is Exponential
Decline
qDd
1
Analytical solution for
constant flowing
pressure
0.1
Matching will give reservoir parameters
0.01
0.0001
0.001
re/rw=10
re/rw=200
0.01
tDd
re/rw=20
re/rw=1000
0.1
1
re/rw=50
re/rw=10000
10
re/rw=100
exponential
Fetkovich Theory – Boundary Dominated and Transient
Fetkovich Decline Type Curves
10
Boundary
DominatedEmpirical
Stems
qDd
1
0.1
TransientAnalytical
Stems
0.01
0.001
0.0001
0.001
re/rw=10
re/rw=1000
b=0.6
0.01
re/rw=20
re/rw=10000
b=0.8
0.1
tDd
re/rw=50
b=0
b=1.0
1
re/rw=100
b=0.2
10
100
re/rw=200
b=0.4
9
Type Curve Matching
• The rate and transient stem matches are
used for kh calculations
141.2 Bo µ re
1
ln
−
qDd = q
kh ( p − p ) r
i
wf
wa match 2
k=
141.2 µB q
h ( pi − pwf ) qDd
re
1
−
ln
match rwa match 2
Type Curve Matching cont.
• The time and transient stem matches are
used for skin calculations
tDd =
t
rwa =
t Dd
0.00634kt
2
r
1
1
2 re
− 1 ln e
−
φµct rwa
2
2
r
r
wa match wa match
match
0.00634k
r
1
1 r
φµct e
− 1 ln e
−
2 rwa match rwa match 2
2
r
S = ln w
rwa
10
CUMULATIVE PRODUCTION TYPE CURVES
10
QDd
1
0.1
0.01
0.001
0.0001
0.001
0.01
0.1
1
10
100
tDd
re/rw=10
b=0
re/rw=20
b=0.2
re/rw=50
b=0.4
re/rw=100
b=0.6
re/rw=200
b=0.8
re/rw=1000
b=1
re/rw=10000
Fetkovich / Cumulative Type Curves
10
Fetkovich Type Curves
1
qDd,QDd
Cumulative Type Curves
0.1
0.01
0.001
0.0001
0.001
0.01
0.1
1
tDd
10
100
re/rw=10
re/rw=20
re/rw=50
re/rw=100
re/rw=200
re/rw=1000
re/rw=10000
re/rw=1000
re/rw=10
re/rw=10000
re/rw=20
b=0
re/rw=50
b=0.2
re/rw=100
b=0.4
re/rw=200
b=0.6
b=0.8
b=1.0
b=0
b=0.2
b=0.4
b=0.6
b=0.8
b=1
11
Constant Pressure
and
Constant Rate Solutions
The Two Solutions – Boundary Dominated
12
The Two Solutions – Boundary Dominated
Advanced Decline Analysis
(Blasingame et al)
13
P.54
Transient Flow is a family of curves;
Boundary-Dominated Flow is a single curve
10
Transient Flow
Boundary Dominated
Flow becomes
Exponential Decline
qDd
1
0.1
0.01
0.0001
0.001
re/rw=10
re/rw=1000
0.01
re/rw=20
re/rw=10000
tDd
0.1
re/rw=50
exponential
1
re/rw=100
10
re/rw=200
Concept of Material Balance Time
Actual Rate Decline
Equivalent Constant Rate
q
Q
Q
actual
time (t)
material
balance = Q/q
time (tc)
14
P.92
Decline Based on Time or Material-Balance-Time
10
1
qDd
Material-Balance-Time
0.1
Exponential Decline
becomes Harmonic
Decline when plotted using
Material-Balance-Time
0.01
Time
0.001
0.0001
0.001
0.01
0.1
1
10
100
tDd, tcDd
re/rw=10
re/rw=20
re/rw=50
re/rw=100
re/rw=200
re/rw=1000
re/rw=10000
Exp ---t
Exp --- tc
Concept of Rate Integral
rate integral
= Q/t
actual rate
Q
Q
actual
time
actual
time
15
B lasingame Typecurves (Vertical W ell - R adial Flow Model)
P.95
1.00E+03
1.00E+02
1.00E+01
qDd, qDdi, qDdid
q Ddi
1.00E+00
1.00E-01
q D did
1.00E-02
q Dd
1.00E-03
1.00E-04
1.00E-05
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05
tDd
q
∆p
i
d
q
∆p
i
q
∆pp
i
d
q
=−
=tc
∆pid dln(tc)
dtc
(oilwells
d
q
∆pp
i
d
q
=−
=tca (gaswells
∆pp i d
d ln(tca)
dtca
Concept of Pseudo-Time
1
b
compressibility
q
=
αpi
∆p
tc + 1
µctGib
reservoir pressure
For gas wells, compressibility (and viscosity)can
not be considered constants at low reservoir
pressure
Equation becomes non-linear for gas wells no longer follows the harmonic decline
16
17
Exponential Decline
Constant pressure
(varying rate)
q
OGIP
EUR
Cumulative Production
Flowing Material Balance
Initial pressure
p/z
Constant rate
(varying pressure)
flo
wi
ng
pr
es
su
re
s
OGIP
Cumulative Production
18
Agarwal-Gardner Flowing Material Balance
q/∆p
variable rate and
variable pressure
OGIP
Cumulative/(C*∆p)
Modern Production Decline
Analysis
Review
19
Traditional
• Empirical
– Single-phase and two-phase (0 < b < 1)
• Boundary-dominated regime
• Data q vs. t
• Constant operating conditions
Fetkovich
• Empirical and theoretical
– Single-phase and two-phase (0 < b < 1)
• Analytical solution for single-phase flow
– Exponential decline (b = 0)
– Introduction of transient stems (k and S)
• Boundary-dominated regime and transient
• Constant bottomhole pressure
• Data q vs. t
20
Blasingame
• Theoretical – Analytical solution
– Single-phase
• Accounts for variable BHP
– Introduction of MB Time
– Exponential decline turned to Harmonic (b = 1)
• Boundary-dominated regime and transient
• Data q/∆p vs. tc (makes use of pressure data)
• Be careful of sparse data points on the bottom
of Harmonic stem
Blasingame (cont.)
• Application to gas reservoirs
– The vertical axis is changed to q/∆m(p)
– The horizontal axis is changed to MB pseudotime, tca
– Requires iteration for correct determination of
MB pseudo-time
• Analytical solution for single-phase flow
– Accounts for variable BHP
– Harmonic (b = 1)
• Data q vs. MB time, tc
21
Agarwal Gardner
• Uses the same data as Blasingame
– The same analysis techniques and plotting
apply
• The flowing material balance plot allows an
alternative representation of data
– Very advantageous for determination of OGIP
22
DELIVERABILITY
CHAPTER 3
1
TESTS
INTRODUCTION
Deliverability
tests have conventionally
been called "back
pressure" tests because they make possible
the prediction
of well flow
rates against any particular
pipeline
"back pressure."
Since most
flowing well tesks are performed to determine the deliverability
of a
well, the term "deliverability
tests" is used in this publication
rather than "back pressure tests."
The purpose of these tests is to
predict
the manner in which the flow rate will decline with reservoir
depletion.
The Absolute Open Flow (AOF) potential
of a ~~11 is defined
as khe rate at which the well would produce against a zero sandface
back pressure.
It cannot be measured directly
but may be obtained from
deliverability
kcsts.
It is often used by regulatory
authorities
as a
guide
in setting
maximum allowable
1.1
producing
rates.
History
It iu interesting
to note the historical
development of
In the early days, a well was tested by opening
deliverability
tests.
it fully
CO the atmosphere and measuring the gas flow rate, which was
This method ~8s recognized as
termed the practical
open flow pokential.
undesirable
because khe pokential
thus obtained depended on khe size of
the well tubing, and apart from the serious waskage of gas resulting
from such practices,
wells were ofken damaged through water
attrition
by sand particles.
The basic work towards development of a practical
coning
test
and
was
carried out by Pierce and Rawlins (1929) ,of the U.S. Bureau of Mines
and culminaked wikh the publication
of the wel.l-known and widely used
Monograph 7 of Rzwlinu and Schellhardt
(1936).
Their kesk,
known as the
3-1
3-2
"conventional
back pressure test,"
several different
flow rates with
consisted of flowing
each flow rate being
the well
continued
at
to
pressure stabilization.
They observed that a plot of the difference
between the square of the static reservoir
pressure and the square of
the flowing
would yield
showed that
sandface pressure versus the corresponding
rate of flow
a straight
line on a logarithmic
coordinate
plot.
They
this stabilized
deliverability
plot could be empLoyed to
determine the well capacity at any flowing
sandface presaute,
including
zero, corresponding
to absolute open flow conditions,
and also showed
that it could be used to predict
the behaviour
of a well with reservoir
depletion.
The critical
aspect of the Rawlins and Schellhardt
conventional
deliverability
test is that each separate flow rate must be continued
to stabilized
conditions.
In Low permeability
reservoirs,
the time
required
to achieve pressure stabilization
can be very large.
As a
consequence the actual duration
of flow while conducting
conventional
tests on such reservoirs
is sometimes not lengthy enough, and the
resulting
data can be misleading.
Cullender
(1955) described the
"isochronal
test" method which involves
flowing
the well at several
different
flow rates for periods of equal duration,
normally much less
than the time required
for stabilization,
with each flow period
commencing from essentially
static conditions.
A plot of such pressure
and flow rate data, as is described above for the conventional
test,
gives a straight
line or a transient
deliverability
plot.
One flow rate
is extended to stabilization
and a stabilized
pressure-flow
rate point
is plotted.
A line through this stabilized
point parallel
to that
established
by the isochronal
points gives the desired stabilized
deliverability
plot.
This stabilized
deliverability
line is essentially
the same as that obtained by the conventional
test.
Another type of isochtonal
test was presented by Katz et al.
(1959, p. 448).
This "modified
iaochroiial
test" has been used
extensively
in industry.
The modification
requires
that each shut-in
period between flow periods,
rather than being long enough to attain
essentially
static conditions,
should be of the same duration
as the
3-3
flow period.
calculating
point.
test.
The actual
the difference
Otherwise,
1.2
unstabilized
shut-in
pressure is used for
in pressure squared for the nexr flow
the data plot
New Approach
is identical
to Interpreting
to that
for
Gas Well Flow
an isochronal
Tests
It is observed that there has been a progressively
greater
saving of time, and a reduction
in flared gas with the evolution
of
various deliverability
tests.
Application
of the theory of flow of
fluids
through porous media, as developed in Chapter 2, results
in a
greater understanding
of the phenomena involved.
Accordingly
more
inFormation,
and greater accuracy, can result
from the proper conduct
and analysis of tests.
It will be shown in a later chapter that the analysis of data
from an isochronal
type test, using the laminar-inertial-turbulent
(LIT) flow equation will yield considerable
reservoir
in addition
to providing
reliable
information
deliverability
concerning the
data.
This
may be achieved even without conducting the extended flow test which
is normally associated with the isochronal
tests,
thus saving still
more time and gas. For these reasons, the approach utilizing
the LIT
flow analysis is introduced
and its use in determining
deliverability
is illustrated
in this chapter.
This will set the stage for subsequent
chapters where the LIT flow equation will be used fo determine certain
reservoir
parameters.
2 FUNDAMENTALEQUATIONS
The relevant
theoretical
considerations
of Chapter 2 are
developed further
in the Notes to this chapter to obtain the equations
applicable
to deliverability
tests.
Two separate treatments with
varying degrees of approximation
may be used to interpret
the tests.
These will be called the "Simplified
analysis"
and the "LIT flow
analysis. "
3-4
2.1
(Rmulins
This approach
and Schellhardt,
of empirical
the form
SC
Analysis
is based on the well-known
Monograph 7
1936) which was the result of a Large number
observations.
q
Simplified
- c (p; - p$
The relationship
5
is co~~+~~nlyexpressed
c(Ap')n
in
(3-U
where
9BC -.
fl.ow rate
at standard
conditions,
=
(14.65 psia, 60oF)
average reservoir
pressure
3
=
=
of the well to complete stabilization,
flowing
sandface pressure,
psia
(pi - p:f)
c
=
G
obtained
MMscfd
by shut-in
psia
a coefficient
which describes
the position
stabilized
deliverability
line
of the
n
stabilized
= an exponent which describes the inverse of the slope
of the stabilized
deliverability
line.
It should be noted that pwf in the above equation is the
flowing
sandface pressure resulting
from the constant flow
If the pressure is not srabilized,
C decreases with
rat=, q,,.
duration
of flaw but eventually
becomes a fixed comcam at
stabilization.
Time to stabilization
and related matters is
discussed in detail
in Section 7.1. In the Note$ to this chapter,
it
is shown that n may vary from 1.0 for completely
laminar flow in the
formation
to 0.5 for fully turbulent
flow, and it may thus be considered
to be a measure of the degree of turbulence.
1.0 and 0.5.
Usual.ly n will
be between
A plot of Ap* (= pi - pif) versus q,, on logarithmic
coordinates
is a straight
Line of slope i a6 shown in Figure 3-l.
Such a plot is used to obtain the deliverability
potential
of the well
against
any sandface
pressure,
including
the AOF, which is the
3-5
deliverability
against
considered
to be constant
expected
that
this
only
the
range
for
beyond
the
a zero
sandface
pressure.
for
a limited
range
form
of
tested
of
the
flow
flow
of
deliverability
rates
raee~
C ad
used
to
flow
rates
reLationship
during
can lead
n may be
the
will
is
be used
Extrapolation
test.
erroneous
and, it
results
(Govier,
1961).
100
I
IO
q,JAMscfd
FIGURE 3-1. DELIVERABILITY
To
C and n,
to the
the
more
relationships
(3N-7)
and
obta-ln
a greater
empirically
of
understanding
analysis
interest
representative
of
the
gas properties
such
in
are
These
(3N-8).
TEST PLOT-SIMPLIFIED
derived
rigorous
rauge
100
Equation
the
to
equations
as viscosity,
the
equation,
given
of
of
flow
Notes
this
by Equations
temperature
Factors
(3-l)
for
tested,
that
is
a flow
affect
compared
The
chapter.
(3N-3),
show that,
rates
FLOW ANALYSIS
(3N-4),
rate
C and n depend
and compressibility
on
3-6
factor,
and reservoir
properties
such as permeability,
net pay thickness,
external boundary radius, wellbore
radius and well damage. As long as
these factors do not change appreciably,
the same stabilized
deliverability
plot should apply throughout
the life of the well.
In practice,
the viscosity,
the compressibility
of the well may change during
advisable
to check the values
2.2
Pressure-squared
factor
of the gas and the condition
the producing Life of the well,
of C and n occasionally.
and it
is
LIT Flow Analysis
Approach
The utility
The theory
of Equation (3-l),
is Limited by its approximate
of flow developed in Chapter 2 and in the Notes to
narure,
this chapter confirms that the straight
really only an approximation
applicable
rates tested.
The true relationship
if
line plot of Figure 3-l is
to the limited
range of flow
plotted
on logarithmic
slope of i = 1.0 at very low
coordinates
is a curve with an initial
values of q,,, and an ultimate
slope of i = 2.0 at very
high values
of cl,,.
Outside North America, there has been in general use a
quadratic
form of the flow equation often called the Forchheimer or the
Houpeurt equation or sometimes called the turbulent
flow equation.
It
is actually
the laminar-inertial-turbulent
(LIT) flow equation of
Chapter 2, developed
by Equation (3N-2)as
further
AP2 E ;2R - pif
in the Notes to this
chapter,
and is given
= a' qac + b' q&
(3-2)
where
alqsc=
b'q;c
Equation
pressure-squared
drop due to laminar
flow
and wellbore
effects
= pressure-squared
drop due to intertial-turbulent
flow
(3-2)
effects.
applies
for
all
values
of q,,.
It
is shown in
3-7
the Notes to this chapter that Equation (3-l) is only an approximation
of Equation (3-Z) for limited
ranges of p,,.
In the derivation
of Equation (3-21, an idealized
situation
was assumed for the well and for the reservoir.
It is important
to
know the extent and the applicability
of the assumptions,made
when
test results are being interpreted.
Sometimes anomalous results may be
explainable
in terms of deviations
from the idealized
situations.
Accordingly,
the assumptions which are clearly
defined in Chapter
Section 5.1 are summarized below:
1.
2.
3.
4.
5.
6.
7.
Isothermal
conditions
prevail
throughout
Gravitational
effects
are negligible.
The flowing fluid
is single phase.
The medium Is homogeneous and isotropic,
2,
the reservoir.
and the
porosity
is constant.
Permeability
is independent
of pressure.
Fluid viscosity
and compressibility
factor are constant.
Compressibility
and pressure gradients
are small.
The radial-cylindrical
flow model is applicable.
Pressure
Approach
Since this approach is seldom used for the analysis of
deliverability
tests, relevant
equations have not been derived in the
Notes as was done for the pressure-squared
approach.
However, it can
be shown, by procedures similar, to those for the pressure-squared
approach, that
Ap Z sR - P,f = a I 1 qsc+b"
4zc
(3-3)
where
a' 'qsc = pressure drop due to laminar flow and well effects
flow
b"q' SC = pressure drop due to inertial-turbulent
effects
The application
of Equation (3-3) is also restricted
by the
assumptions listed
for the pressure-squared
approach.
3-8
Pseudo-Pressure
Approach
Assumption
enors,
particularly
pressure
the
(6)
in
gradient
is
pseudo-pressure
or pressure
the
the
(3-3)
for
equation
is
Equation
(3N-9)
of
gas from
is
used,
instead
need
for
assumption
the
is more
all
is
rigorous
ranges
in
It
tight
approach
developed
4
flow
can be a cause
small.
equation
Equation
above
seldom
approaches,
resulting
mentioned
shown
the
to
Notes
where
Chapter
the
2 that
is
if
eliminated
Equation
rigorous
chapter
the
pressure-squared
(6)
The
this
in
either
of pressure.
serious
reservoirs
of
than
of
(3-2)
LIT
and is
and
or
flow
given
by
as
‘$
q
- qwf = a qs,
+ b q2SC
where
$R
Ilrwf
a 4sc
=
pseudo-pressure
corresponding
to
sR
=
pseudo-pressure
corresponding
to
pwf
=
pseudo-pressure
drop
due to
leminar
drop
due
to
inertial-turbulent
is
more
well
=
b q2SC
Since
either
the
the
is
used
effects.
pseudo-pressure
pressure
incorporating
approach,
the
or the
pressure-squared
this
is
its
constructed
Example
versa,
for
2-l).
as easy
of
referred
is
treated
reviewed
here.
gas at
is
using
then
used
in
LIT
approach
to
as the
LIT(q)
greater
detail
in
A curve
reservoir
for
p or p2 as the
has been
When Qwf reflects
q,,.
a no Longer
at
the
than
I/J versus
temperature
converting
working
this
constructed,
of
p to
q, and vice
variable,
approach
a stabilized
the
stabilized
increases
value.
pressure
with
A plot
duration
9 is
becomes
used.
just
of
A@ versus
due to a constant
of
flow
q,,
p
(see
p2 approach.
rate
constant
Is
curve
$ - p curve
as the
concept
a particular
This
analyses,
henceforth
application
and instead
Once the
flow
2 but
rigorous
manuel.
The pseudo-pressure
Chapter
analysis
pseudo-pressure,
in
and
conditions
pseudo-pressure
flow
flow
but
stays
on arithmetic
3-9
coordinates
0rigii-l.
would give a curve, concave upwards, passing through the
This CUFV~ has an initial
slope of 1, cor,resposding
to laminar
flow, whereas at the higher
reflecting
turbulent
flow.
fl.ow rates the slope increases to 2,
Consequently,
for large extrapolations,
a
considerable
difference
would be obsened in the AOF values obtained from
this curve and from the straight
line plot of the Simplified
analysis.
In order to obtain a plot that ia consistent
with Figure 3-1,
the arithmetic
coordinate
plot is discarded
in favour of a logarithmic
plot of Equation (3-4).
A straight
line may be obtained by plotting
This particular
method is
(A$ - bq;,) Y~TSUG g,, as shown in Figure 3-2.
chosen since the ordinate
then represents
the pseudo-pressure
drop due
to laminar flow effects,
a concept which iu consistent
with the Simplified
q,,, MMscfd
FIGURE 3-2. DELlVERABlLlTY TEST PLOT-LIT(q)
FLOW ANALYSIS
3-K
The deliverability
pressure
particular
may be obtained
value of A9
q SC =
potential
by solving
-a + J(a2
the quadratic
against
any sandface
Equation
(3-4)
for
-c 4 b A$)
2b
a and b in the LIT($)
reservoir
viscosity
of a well
flow
the
(3-5)
analysis
depend on the same gas 'and
properties
as do C and n in the Simplified
analysis except for
and compressibility
factor.
These two variables
have been
taken into account in the conversion
of p to @, and consequently,
will
not affect
the deliverability
relationship
constants a and b.. It
EOllOWS, therefore,
that the stabilized
deliverability
Equation (3-41,
or its graphical
representation,
is more likely
to be applicable
throughout
the life
of a reservoir
than Equations
(3-l),
(3-2)
or (3-3).
3 DETERMINATION OF STABILIZED FLOW CONSTANTS
Deliverability
tests
have to be conducted on wells to
the values of the stabilized
flow
determine,
among other things,
constants.
Several techniques are available
to evaluate
the Simplified
analysis,
and a and b, of the LIT($)
flow
from deLiverability
data.
3.1
Simplified
A logarithmic
coordinate
pl.ot
a straight
line over the range of flow
stabilized
deliverability
line gives $
The coefficient
C in Equation (3-l) is
C and n, of
analysis,
Analysis
of Ap' venus qs, should yield
rates tested.
The slope of this
from which n can be calculated.
then obtained
from
(3-6)
3-11
3.2
LIT($)
Flow Analysis
Least Squares Method
A plot of (A$-b&)
versus q,,, on logarithmic
coordinates,
should give the stabilized
deliverability
line.
a and b may be obtained
from the equations given below (Kulczycki,
1955) which are derived by
the curve fitting
method of least squares
(3-7)
(3-8)
where
N
=
number of data points
Graphical
Method
This method utilizes
the "general curve,"
developed by Willis
Before discussion
on the use of the
(1965), shown in Figure 3-3.
general curve method, the details
of its development
should be clearly
understood.
Equation
(3-4),
with
a = b = 1 can be written
as
O-9)
A$ = qsc + 4’9c
The straight
plot
11neu in Figure
of A$ versus
3-3, which is a logarithmic
q, are represented
A+ = qSC
coordinate
by the equations
(3-10)
(3-11)
3-12
If the plots of Equations
(3-10) and (3-11) are added for the same
value of q SC' the resulting
plot is the general curve.
To distinguish
Figure 3-3 from a data plot, the latter
will
be referred
to as the deliverability
plot.
To determine a and b, actual data are plotted
on logarithmic
coordinates
of the same size as Figure 3-3.
This stabilized
deliverability
data plot is laid upon the general curva plot, and
keeping the axes of the two plots parallel,
a position
is found where
the general curve best fits the points on the data plot.
The stabilized
deliverability
curve is now a trace of the general curve.
The value of
a is read directly
as A$ for the point on the deliverability
plot where
the line given by Equation (3-10) intersects
the line qac = 1 of the
dellverability
plot.
The value of b is read directly
as A$ for the
point on the deliverability
plot where the line given by Equation (3-11)
plot,
intersects
the line pSC = 1 of the deliverability
If the point at which'a*is
to be read does not intersect
the
plot, 'a"may instead be read where
PSC = 1 line of the deliverability
by 10 or 100, respectively,
qsc equals 10 or 100 and must then be divided
Similarly,
b may be read where q,, equals 10
to get the correct
value.
or 100 and must then be divided by 10' or loo*, respectively.
The advantage of this method is the speed with which
deliverability
data can be analyzed.
However, it should be used only
when reliable
data are available.
The above procedure may be applied to data from a conventional
test to yield a stabilized
deliverability
curve.
With isochronal
data,
however, it will yield a transient
deliverability
curve.
To obtain the
stabilized
deliverability
curve, it should be remembered that the value
of b is independent
of duration
of flow and must be the same for the
Accordingly,
stabilized
and the transient
deliverability
relationships.
the general curve is positioned
so that it passes through the stabilized
flow point and maintains
the value of b obtained from the transient
deliverability
is
illustrated
curve.
The application
of this
graphical
by Example 3-4 in Section
method to calculate
4.3.
a and b
3-13
101
LEQUATION
(3-11)
/
100
g,,,
FIGURE
3-3.
GENERAL
MMscfd
CURVE FOR THE ANALYSIS
From R. 8. Willil
(19451
OF DELIVERABILITY
DATA
3-14
The general
curve
of Figure
3-3 may also be used with
LIT(p')
approach.
The method is the same as described
Equation (3-2) is now fit instead of Equation (3-4).
the
above except
4 TESTS INVOLVING STABILIZED FLOW
In the preceding analyses, C or a are constant only when
stabilization
has been reached.
Before stabilization
is achieved,
the
Tests to determine the stabilized
flow is said to be transient.
deliverability
of a well may combine both transient
and stabilized
conditions.
Various tests that may be used directly
to obtain the
deliverability
or the AOF of a well are described in this section along
with examples of their Interpretation
by both the Simplified
and the
General guidelines
for the field
conduct and
LIT($)
flow analyses.
All the
reporting
of these tests are discussed in a later chapter.
tests treated in this section have at least one, and sometimes all, of
the flow rates run until pressure stabilization
is achieved.
This is
the deliverability
obtained will not
very important
as, otherwise,
Tests in
reflect
stabilized
conditions
and will thus be incorrect.
which no one flow race is extended
discussed in Section 5.
4.1
to stabilized
Conventional
conditions
will
be
Test
As mentioned in Section 1, Pierce and Rawlins (1929) were the
first
to propose and set out a method for testing gas wells by gauging
This
the ability
of the well to flow against various back pressures.
type of flow test has usually
been designated
the "conventional"
TO perform a conventionaL
test, the stabilized
deliverability
test.
shut-in reservoir
pressure,
p,, is determined.
A flow rate, qsc, is
The stabilized
then selected and the well is flowed to stabilization.
The flow rate is changed three or
flowing pressure,
p,f, is recorded.
four times and every time the well is flowed to pressure stabilization.
The flow-rate
and pressure histories
for such a test are depicted in
3-15
Figure 3-4.
Interpretation
below will give the desired
of the pressures
deliverability
----7.------
and flow
relationship.
rates
as shown
7-----l-.--
P
t-
FIGURE 3-4. CONVENTIONAL
TEST- FLOW RATE AND
PRESSURE DIAGRAMS
Simplified
Analysis
A graph of bp* (= ;; - p;f) versus qsc, on logarithmic
This gives a
coordinates,
is constructed
a~ shown in Figure 3-1.
straight
line of slope i or reciprocal
slope, n, known as the "back
From this straight
pressure line" or the deliverability
relationship.
line and Equation (3-l) the AOF or the deliverability
against any sandface back pressure may be obtained.
of the well
LIT($)
Flow Analysis
The values of pwf are converted to Q,, using the applicable
The values of a and b are
$ - p curve, similar
to Figure 2-4.
calculated
by the methods of Section 3 and the deliverability
relationship is expressed in form of Equation (3-4).
The deliverability
for any known A$ may then be obtained from Equation (3-S).
It is recommended that even though the deliverability
q,,
3-16
relationship
is derived by computation,
the equation obtained should be
plotted
on logarithmic
coordinates
along with the data points.
Data
which contain significant
errors will then show up easily.
ErrOIleOUS
data points must be discarded and the deliverability
relationship
then
recalculated.
A sample
both the Simplified
(for gss composition
Although
flow analyses will
analysis beyond the
deliverability
calculation
for a conventional
test by
and the LIT($) flow analyses is shown In Example 3-l
see Example A-l; for the Q - p curve see Figure 2-4).
in many instances,
both the Simplified
and LIT(@)
give the same reuult,
extrapolation
by the Simplified
range of flow rates tested can cause significant
errors.
Such il situation
is well
conventional
Cest (Example 3-l).
illustrated
The LIT($)
by the calculations
flow analysis gives
for a
an AOF
of 37.8 MMscfd while the Simplified
analysis yields an AOF of 44.0 MMscfd.
This method of testing and the interpretation
of the data iu
and the method has been considered
the basic
relatively
simple,
acceptable
standard fur
In a reservoir
testing
gas wells for many years.
of very high permeability,
the time required
to obtain
stabilized
this type
test may
hand, in
stabilized
flow fates and flowing
pressures,
as well as a
shut-in
formation
pressure is usually not excessive.
In
of reservoir
a properly
stabilized
conventional
deliverability
be conducted in a reasonable period of time.
On the other
low permeability
reservoirs
the time required
to even
In this
approximate
stabilized
flow conditions
may be very long.
situation,
It
IS not practical
to conduct a completely
stabilized
test,
and since
the results
of an unstabilized
test can be very misleading,
other methods of testing
should be used to predict
well behaviour.
4.2
Isochronal
Test
The conventional
delivetsbilLcy
test carried out under
qualifies
as an acceptabLe approach to attslning
stabilized
conditions,
the relationship
which is essential
to the proper interpretation
of
tests,
because it extends each flow rate over a period of time
3-17
sufficient
to permit
edge of the reservoir
the radius of investigation
or the point of interference
wells.
This ensures that the effective
The effective
dralnage radius concept is
If each fl.ow rate of a multi-point
test
time insufficient
for stabilization,
the
to reach the outer
between neighbouring
drainage radius is constant.
discussed in Section 7.1.
extends for a fixed periad of
effective
drainage radius,
td,
which is a function
of the duration
of flow, is the same for each point.
The isochronal
flow test which was proposed by Cullender
(1955), is
based on the principle
that the effective
draInage radius in a given
reservoir
is a function
only of dimensionless
time, and is independent
of the flow i-ate.
He suggested that a series of flow tests at different
rates for equal periods of time would result in a straight
line on
logarithmic
coordinates
and demonstrated
that such a performance curve
would have a value of the exponent n essentially
the same as that
LIT($)
flow theory
established
under stabilized
flow conditions.
confirms that b too is independent
and may, therefore,
be determined
flow rates,
c and a stay constaflt
of the duration
of flow (Section 3N.3)
from short flow tests.
For different
provided the duration
of each flow is
constant 1 Whereas n or b may be obtained from short (transient)
from stabilized
isochronal
flow tests, C or a can only be derived
conditions.
The isochronal
flow data may thus be used in conjunction
with only one stabilized
flow point to replace a fully
stabilized
the isochronal
test consists
conventional
deliverability
test.
Briefly,
of alternately
closing in the well until
a stabilized,
or very nearly
the well at different
stabilized
pressure,
&, is reached and flowing
rates for a set period of time t, the flowing
sandface pressure,
pwf,
at time c being recorded.
One flow test is conducted for a time period
long enough to atrain
stabilized
conditions
and is usually referred
CO
The flow rate and pressure sequence are
as the extended flow period.
depicted in Figure 3-7.
A brief
discussion
of the theoretical
validity
of isochronal
tests is given in Section 3N.5 of the Notes to this chapter.
3-18
EXAMPLE
3-1
ILLUSTRATING
TEST.
SEE
(A*-
AND
_.,-.^
DELlVERAi3lllTY
FIGURES
bqtt)
CALCULATIONS
3-5
AND
3-6
VERSUS
q,,,
RESPECTIVELY.
FOR
FOR
PLOTS
A
OF
CONVENTIONAL
Ap’
(NOTE:
q
VERSUS
IMPLIES
FLOW
4
SC
q,,)
“.
,..,-
SH”f-lN
^. --,_--
q
-----------------------
0.00229
?
4
190
x
36.1
4.3
5.50
AOF l~tkcfd~
RESULTS
DISCARDED
POINT
FLow
’
TRANSIENT
FLOW!
I.C.
-kt
STABILIZED
i.e.
FLOW:
3’56
- $Irf
ii
-
k‘
z
= a+q + bq2
q*
4 +
GR -
qw+
: 0.0625
: aq
+ b$
q + 0.00084 q~
DELIVERABILITY:
q = ib[-a
FOR vJ*‘ zD,
qEAOF
+ /++4b
I
37.8
(Ir,
-J;,)]
MMrcfd
3-19
q,,,MM,cfd
FIGURE 3-5. PLOT OF Ap2 VERSUS q,, - CONVENTIONAL
FIGURE 3-6. PLOT OF (A*-bq:)
VERSUS q,c- CONVENTIONAL
TESIT
TEST
3-20
7
EXTENDED FLOW RATE
I
1
t-
.
FIGURE 3-7. ISOCHRONAL
TEST- FLOW RATE AND PRESSURE DIAGRAMS
Simplified
Analysis
The best straight
line is drawn through the isochronal
points
plotted on logarithmic
coordinates.
This is the transient
deliverability
line.
A straight
line parallel
to the transient
deliverability
line
drawn through the stabilized
point is the stabilized
deliverability
line
from which the AOF or flow against any sandface back pressure can be
read.
LIT($)
Flow
Analysis
From the isochronal
flow rates
and the corresponding
pseudo-
pressures at and b can be obtained from Equations (3-7) and (3-8); at
refers to the value of a at the isochronal
time t. A logarithmic
plot of
data are also plotted.
(A$ - bq;J versus qgc is made and the isochronal
This plot is used as before to identify
erroneous data which must be
rejected
and a t and b recalculated,
if necessary.
The data obtained from the extended flow rate,
used with the value of b already determined in Equation
the stabilized
value of a. This is given by
4$ and qsc are
(3-4) to obtain
3-21
(3-12)
a and b are now known and the stabilized
deliverability
relationship
may
be evaluated
from Equation (3-4) and plotted
on the deliverability
plot.
A sample calculation
of stabilized
deliverability
from an
isochronal
fest is shown in Example 3-2 (for gas composition
see
Example A-l; for the $ - p curve see Figure 2-4).
The values of AOF
calculated
by rhe twcl methods are not too different
since only a small
extrapolation
is required.
However, the LIT($)
a more correct value and should be used instead
analysis.
4.3
Modified
Isochronal
In very tight reservoirs,
it
attain a completely stabilized
reservoir
flow period,
nor is it always practical
flow analysis does give
of the Simplified
Test
is not always practical
to
pressure before the initial
during the test to shut-in
the
reservoir
until the original
pressure is attained.
Aa a result,
the
true isochronal
test proves impractical
as a means of testing many
wells.
Katz et al, (1959, p. 448) suggested that a modified
isochronal
test conducted with a shut-in
period equal to the flow period
may give satisfactory
results
provided
the associated
unstabilized
shut-in pressure is used instead of pR in calculating
the difference
of
pseudo-pressure
or pressure-squared
for the next flow rate.
This method
has been used for testing
many wells, and indeed has given results which
As in the isochrdnal
test, two lines are
appear quite satisfactory.
obtained,
one for the isochronal
data and one through the stabilized
point.
This latter
line 1s the desired stabilized
deliverability
curve.
This method, referred
to as the modified
isochronal
test, does not yield
The
a true isochronal
curve but closely approximates
the true curve.
pressure and flow rate
are depicted in Figure
sequence of the modified
3-10.
isochronal
flow
test
3-22
EXAMPLE
ILLUSTRATING
3 -2
TEST.
SEE FIGURES
I&SIMPLIFIED
DELIVERABILITY
baf,)
VERSUS
3-8
Q...
AND
CALCULATION5
3,-9
FOR
RESPECTIVELY.
FOR
PLOTS
(NOTE:
OF
AN
ISOCHRONAL
Apz VERSUS
q IMPLIES
clsc AND
q,,)
ANALY
RESULTS
q _ c
(
p @z_
k*
1952
x
3810
I.320
x
1742
p,;
)”
*I I
i 0.000017
RESULTS
DISCARDED
POINT
Flow
2
TRANSIENT
1.e.
316
STABILIZED
I.e.
316
FLOW!
- $w.r
FLOW;
4,
-
Q
zL5.182
qR
-
A'# -
bq*
9
:
22.28
:
uuqwt z22.28
FOR $w‘ =0, q = AOF :
+g
* bq2
q .+ 1.870
qwf
DEL'VERAB'L'~~~+b Cm0 +&
0 =
:
eq
qz
+ bq2
q + 1.870
qz
+ *b ('JR - ew, ) ]
8.3
MMrcfd
3-x
I
!
loo
1
FIGURE 3-8.
I
I
AOF: 9.0 MM,cfd
1 II/l
10
q,<,MMscfd
100
PLOT OF Ap2 VERSUS q,, - ISOCHRONAL
q=, MMscfd
GURE 3-9.
I
I
PLOT OF (At/t-bq,:)
VERSUS q,,-ISOCHRON
3-24
lsochronal
A brief
tests
discussion
of the theoretical
validity
of modified
is given in Section 3N.5 of the Notes to this chapter.
92
EXTENDED FLOW RATE
t-
P
t---w
FIGURE 3-10. MODIFIED ISOCHRONAL
TEST-FLOW
AND PRESSURE DIAGRAMS
RATE
Analysis
The method of analysis of the modified
isochronal
test data
is the came es that of the preceding
isochronal
method except that
instead of &, the preceding shut-in
pressure is used In bbtainfng
ap2
or A$. The shut-in pressure to be used for the stabilized
point is p,,
the true stabilized
shut-in
pressure.
A sample calculation
of stabilized
deliverability
from a
modified
isochronal
test is shown in Example 3-3 (for gas composition
see Example A-l; for the I) - p curve eee Figure 2-4).
The values for
ilDF obtained by the different
methods are very nearly the eeme because
of the small extrapolation.
analyzed by the graphical
example, Example 3-4.
The test of Example 3-3 may also be
method of Section 3.2 as shown in the following
3-25
EXAMPLE
3-3
ILLUSTRATING
TEST.
qsc
SIMPLIFIED
LIT ($)
DELlVERAElllTY
SEE
AND
FIGURES
IA9
3-11
-h,c)
VERSUS
CALCULATIONS
AND
q,,.
3 - 12
FOR
FOR
A MODIFIED
PLOTS
RESPECTIVELY.
OF
(NOTE:~
ISOCHRONAL
Ap’
VERSUS
IMPLIES
4,<)
ANAlYSIS
ANALYSIS
RESULTS
DISCARDED
N=
POINT
TRANSIENT
<,
4
=
315
MMpri&p
i.e.
FLOW:
315
STA81tlZED
1.d.
315
&
-J;{
lr;,
= a,q + bqZ
z 3.273
- h
FLOW:
-
$
-
q + -LAG_
= oq
$v;r
: 9.747
qz
+ bqz
q +
1.641
qz
DELIVERABIIITY:
(EXTENDED
FLOWI
q'
A+;~, 183
0 -
A'# -
bq2
0
8.00
b:
=
9.747
q = ib[-”
1.641
FOR qwf -0,
q :AOF
t /a2
-
+4b
11.2
($
- VJ”,)
MMrcfd
1
3-26
+,MMscfd
FIGURE 3-11. PLOT OF &I’
VERSUS q,,-
MODIFIED
ISOCHRONAL
q,,, MMscfd
FIGURE 3-12. PLOT OF (A$-bq,:)
VERSUi q,, -MODlF
IED ISOCHRONAL~~TEST
3-27
EXAMPLE 3-4
Introduction
This,example
method of Section
Problem
Plot
logarithmic
So,lution
Figure
3-2 to the analysis
the application
of modified
of the graphical
isochronal
Calculate
the values of a, b and AOF for
test data of Example 3-3.
isochronal
3x3
illustrates
test data.
the modified
A$ versus qsc (transient,
modified
isochronal
coordinates
of the same size as the general
data) on
curve of
3-3.
This deliverability
data plot is shown in Figure 3-13:
The transient
deliverability
curve is drawn from the best
match of the deliverability
data plot and the general curve.
The values
of a and b are obtained from the intersections
of the straight
lines,
repr:sented
by Equations
(3-10) and (3-U),
the deliverability
data plot.
This gives
at
=
b
= 1.6
with
the q
SC
= 1 line
of
3.3
Plot the stabilized
flow point and maintaining
the value of
b = 1.6 draw the stabilized
deliverability
curve.
The intersection
of
the straight
line, represented
by Equation (3-lo),
with the q,, = 1 line
of the deliverability
data plot gives
a =
and the resulting
9.75
deliverability
AOF = 11.7
curve
shows an
MMscfd
Figure 3-3 may be used to obtain good approximations
for
a, b, and AOF, but it is recommended that the calculation
methods of
DiSCUSSiOIl
Examples 3-1, 3-2 and 3-3 using
better results.
the LIT($,)
flow
analysis
be used Ear
3-28
3-29
4.4
Single-Point
Test
If
from previous
tests conducted on the well,,the
reciprocal
slope n or the inertial-turbulent
(IT) flow effect
constant,
b, is howa,
only one stabilized
flow point is required
CO give the deliverability
relatXonship.
This is done by selecting
one flow rate and flowing
the
well at that tate to stabilized
conditions.
Often this fest is
conducted as part of a pressure survey 1n a reservoir
on production.
The gas in this test is usually flowed into a pipeline
and not wasted.
Care is taken to ensure that the well is producing at a constant rate
and has stabilized.
This rate and the flowing pressure
The well is then shut-in
long enough that the stabilized
are recorded.
shut-in
pressure
GR can be determined.
Knowing the static pressure p,, the stabilized
flowing
sandface pressure,
pwf, and the rate q,,, either the Simplified
or the
LIT($)
analysis may be used to obtain
the srabilized
deliverability
of
the well.
For the Simplified
analysis the stabilized
point is
on the usual logarithmic
coordinates
and through it a straight
inverse slope, n, is drawn.
In the LIT($)
flow analysis,
the
data, AIJJand q
are inserted with the previously
known value
SC
into Equation (3-12) to yield a value for a. The stabilized
plotted
line of
stabilized
of b
deliverability
is then given by Equation (3-4).
A sample calculation
of stabilized
deliverability
from a
single-point
test is shown in Example 3-5.
n and b are known from
previous
tests; n = 0.60, h = 1.641 (for gas composition
see Example
A-l; for the IJ - p c"r"e see Figure 2-4).
5 TESTS NOT INVOLVING STABILIZED FLOW
In the previous
sections,
tests
which would yield
the
deliverability
of a well, directly,
we're described.
Each of those tests
included at least one flow rate being rm to pressure stabilization.
In
the case of tight reservoirs,
stabilization
could take months or even
ye&Y.
This is obviously
a great inconvenience
and alternative
methods
3-30
must
to
be used
conduct
to determine
stabilized
flow
teats,
used
to obtain
the
In
and that
the
of
(3N-10)
from
the
Notes
of
the
Sections
isochronal
same value
is
to
placed
rate
before
analysis
chapter
is
single-point
deliverability.
the
and modified
it
the
calculated
and 4.3
may be
by calculation.
and using
4.2
transient
volume,
on production,
b is
having
of
drainage
that
applicable
this
flow
well’s
flow
accuracy
stated
conditions.
be obtained
the
without
relationship
has been
an extended
has been
stabilized
of
deliverability
well
confirm
deliverability
The LIT($)
a knowledge
to monitor
It
stabilized
tests.
a stabilized
analysis
given
with
when the
desirable
could
flow
along
Subsequently,
test
the
same for
it
transient
was shown
isochronal
that
flow
to
stabilized
flow.
the
stabilized
value
or
b
data,
From Equation
for
a is
by
x lo6
a = 3.263
0.472
T
n
re
rw
(3-13)
+*
I
where
k
=
effective
h
=
net
T
=
temperature
-
external
rw
=
well
radius,
ft
s
=
skin
factor,
dimensionless
r
usu;llly
re,
before
it
is
stabilized
shown
to note
of
the
radius
of
T are
value
know0
of
or build-up
that
data.
reliable
reservoir,
the
OR
drainage
and onSy
area,
k and
of
the
by the
present
ft
s need
a can be calculated.
For
values
md
ft
how k and a may be obtained
In
it
be determined
Chapters
analysis
purpose
to
of
is
k and s may be obtained
the
only
from
4 and
transient
necessary
transient
alone.
Thus
is
to gas,
pay thickness,
11, and
rw,
the
drawdown
tests
e
permeability
sufficient
Sections
isochronal
to
to
4.2
obtain
the
stabilized
deliverability
conduct
the
isochxonal
part
and 4.3.
data
are
The
used
extended
to obtain
flow
the
of
points
value
of
relationship,
the
tests
are
not
b from
it
described
required.
Equation
in
The
(3-8).
5
EXAMPLE
3-s
ILLUSTRATING
TEST.
b-b-
SEE
DELIVERABILITY
FIGURES
ha:,)
VERSUS
3-14
qsc,
AND
CALCULATIONS
3-15
F,OR
RESPECTIVELY.
FOR
PLOTS
(NOTE:
Of
A SINGLE
Ap*
POINT
VERSUS
q IMPLIES
q,,
AND
q,,)
RFSIJITS
i
0.00108
AOF (MMrcfd)
=
9.5
RESULTS
DISCARDED
POINT
TRANSIENT
FLOW!
1.e.
-hYt
STABILIZED
b=
[EXTENDED
FLOW1
NIXA'!-ZqZ
'+'
N Es2 - Eq Zq
A+:
0 z
q'
183
A'k 9
bql
I.#.
_
JR -
309
FLOW:
qwf
=
9
T@ -
-e,,
= +q
qwc;,
113.601
+ bq2
+
qz
: aq
+ bq*
q +
1.641
qz
DELIVERABILITV:
7.2
br
i
13.601
q : tb[-O
1.641
FOR $*# -4,
q’AoF
+ b2+4b
?
10.2
(qn
-J;,)]
MMscfd
3-32
10000
FIGURE 3-14. PLOT OF Ap2 VERSUS q,< - SINGLE
FIGURE 3-15. PLOT OF (A+bq$
VERSUS &-SINGLE
POINT TEST
POIN T TEST
3-33
The value
of
a i$
k and s from
the
calculated
from
dtawdown
or build-up
6
is,
in
all
the
practice
it
rate
the
and is
the
thus
shown
obtained
temperature
of
a straight
the
size
which
may cause
(Wentink
et
of
the
MOEOVer,
the
of
pipe,
of
a Eunctim
not
only
throughout
represent
3-17.
of
the
the
is
to
different
average
At any
condition
of
but
the
of
that
1967);
the
flowing
instead
in
which
well
drop
also
of
gas
does
in
reservoir
is
on the
itself
is
level.
relationship
to obtain
has
throughout
wellbore
is
curves
pressures,
it
flowing.
apply
pressure
represented
and may be used
back
depends
not
the
different
relates
However,
as it
the
it
it
pipeline
reservoir.
the
a well,
depletion
slope
to
because
gathering
deliverability
of
the
be a curve
useful
for
pressure
rate
wellhead
life
plot
or 3-2.
deliverability
and Cleland,
relationship
fl.ow
the
is valid
than
unique
the
deliverability
the
or annulus,
since
3-1
equal
of
flow
1971).
example,
being
HOWeVer,
Figures
coordinates
the
the
by
versus
wellhead
variations
plot
sandface
well
Because
constant
not
the
the
al.
accessible
tubing
unlike
life
for
mote
as the
at
conditions
as before.
of
In
B, and the
necessarily
made,
that
pressures
sandface
curses
(Edgington
are
the
may be plotted
not
deliverability
is
disadvantage
is
tests
sandface.
in Appendix
known
wellbore
situation,
to
On logarithmic
3-16.
A wellhrad
pressure,
Is
corrections
line
to a surface
sandface
the
ae the
be obtained
to
pressures
the
determined
conditions,
to measure
pressures
sandface
in
first
by the
sandface
measured
wellhead
plot
unless
to
detail
may then
deLiverability
using
moreover,
in
obtained
in Figure
wellhead
to are
given
similar
The relationship
refer
may be converted
relationship
a manner
obtained
more convenient
procedure
instances,
in
referred
pressures
calculation
some
sections
sometimes
deliverability
in
previous
These
having
analyses.
relationships
pressures
is
wellhead.
the
the
(3-13)
WELLHEAD DELIVERABILITY
The deliverability
described
Equation
are
not
needed
as shown in
by p,,
the
the
wrllhead
to
Figure
sandface
3-34
deliverability
by converting
the sandface pfess~res to wellhead
conditions
using the method of Appendix B, in reverse.
7 IMPORTANT CONSIDERATIONS PERTAINING
TO DELIVERABILITY TESTS
In all
of the tests described so far, the time to stabilization
is an important
factor,
and is discussed in detail
below.
Moreover, the
flow rate is assumed to be constant throughout
each flow period.
This
condition
is not always easy to achieve,in
ptac'cice.
The effect on
test results of a non-constant
flow rate is considered
In this section.
The choice of a sequence of increaslng
or decreasing
flow rates is also
discussed.
7.1
Time to Stabilization
and Related
Matters
Stabilization
originated
as a practical
consideration
and
reflected
the time when the pressure no longer changed significantly
with time; that is, it had stabilized.
With high permeability
reservoirs
this
point
was not too hard to observe.
However,
with tight
formations,
the
pressure
does
not
stabilize
for
a very
long
time,
except where there
mechanism acting on the pool, true steady-state
the pressure never becomes constant.
sometimes
years.
months
and
is a pressure maintenance
is never achieved and
MOreOVer,
Stabilization
is more properly
defined in terms of a radius of
investigation.
This is treated,
in detail
in Chapter 2, but will be
reviewed here.
When a disturbance
is initiated
at the well, it will
have an immediate effect,
however minimal, at all points in the
reservoir.
At a certain
distance
from the well,
however,
the effect
of
the disturbance
will be so small as to be unmeasurable.
This distance,
at which the effect
is barely detectable
is called the radius of
investigation,
the formation
the
no-flow
rinv.
until
boundary
As time increases,
it reaches the outer
between
adjacent
this
radius
boundary
flowing
wells.
moves
of
the
outwards
reservoir
From
then
into
OF
on,
It
3-35
100
1
10
100
q,,, MMscfd
FIGURE 3-16.
WELLHEAD DELIVERABILITY
PLOT
3000
0
0
2
4
b
8
IO
12
14
lb
18
~7 MMscfd
FIGURE 3-1Z WELLHEAD DELIVERABILITY
VERSUS FLOWING WELLHEAD
PRESSURE, AT VARIOUS STABILIZED SHUT-IN PRESSURES
3-36
stays constant,
that is, r inv = re* and stabilization
Is said co have
been attained.
This condition
is also called pseudo-steady
state.
The pressure does not become constant but the rate of pressure decline
does.
The time to stabilization
and is given by Equation (3N-15)
can only be determined
approximately
as
(3-14)
where
ts, = time to stabilization,
hr
r
= outer radius of the drainage
e
= gas viscosity
at p,, cp
i;
porosity,
fraction
$ = gas-fllled
area,
ft
k
= effective
permeability
to gas, md
There exist various rule-of-thumb
methods for determining
when
stabilization
is reached.
These are usually based on a rate of pressure
decline.
When the specified
rate, for example, a 0.1 psi drop in 15
minutes, is reached, the well is sard to be stabilized.
Such oversimplified
criteria
can be misleading.
It is shown in the Notes to this
chapter that at stabilization,
the race of pressure decline at the well
is given by Equation (3N-19) as
(3-15)
This shows that the pressure decline in a given time varies
from well to well, and even for a particular
well, it varies with the
flow rate.
For these reasons, methods of defining
stabilization
which
make use of a specified
rate of pressure decline may not always be
reliable.
The radius of investigation,
rinvf
after t hours of flow is
given by Equation (3N-21).
This equation is portrayed
graphically
in
Figure 3-18.
3-37
for
rinv
< re
(3-M)
As long as the radius of investigation
is less than the
exterior
radius of the reservoir,
stabilization
has not been reached
and the flow is said to be transient.
Since gas well tests often
involve
interpretation
of data obtained in the transient
flow regime,
For transient
flow,
a review of transient
flow seems appropriate.
Equations (3-l) and (3-4) still
apply but neither
C nor a IS constant.
Both C and a will change with time until stabilization
is reached.
From this time on, C and a will stay constant.
Effective
Drainage
Radius
A concept which relates
transient
and stabilized
flow
equations is that of effective
drainage radius,
rd, which is discussed
in detail
in Chapter 2. It is defined 8s that radius which a
hypothetical
steady-state
circular
reservoir
would have if the pressure
at that radius were s R and the drawdown at the well at the given flow
rate were equal to the
at the well increases
investigation
reaches
the effective
drainage
rd
= 0.472
The above
radius of drainage
be emphasized that
reservoir
and that
unsteady-state
flow
distinction
between
of investigation
Section 6.4.
actual drawdown.
Initially,
the pressure drop
and so does rd.
Ultimately,
when the radius of
the exterior
boundary, re, of a closed reservoir,
radius is given by Equation (Z-101)
re
(3-17)
equation is the source of the popular idea that the
It should
only moves half-way
into the reservoir.
at all times, drainage takes place from the entire
r d is only an equivalent
radius which converts an
the
equation to a steady-state
one. Furthermore,
the concepts of effective
drainage radius and radius
should
be understood
as,described
in Chapter
2,
3-38
-
3-39
7.2
The usual
to use,
where
test,
decreasing
practice
possible,
conventional
hold-up
in
there
is
ig
the
wellbore
of
Rates
deliverability
increasing
a likelihood
to
a problem,
in
form
rates.
hydrates,
results
In
a
forming,
higher
a
wellbore
Where
hydrates.
a decreasing
is
tests
flow
of
as it
tendency
is
Flow
conducting
advisable
and a decreased
in
of
a sequence
if
sequence
‘temperatures
Sequence
sequence
liquid
may be
preferred.
If
the
conducted,
rate
conventional
that
Is
the
or a decreasing
Ilowever,
should
rate
sequence
for
the
be used,
,or the
stabilization
is,
selected,
fesf
of
modified
pressure
sequence
is
give
the
will
isochronal
otherwise
the
isochronal
true
are
observed
relationship.
an increasing
loses
the
isochronal
a new
an increasing
deliverability
method
properly
before
Either
imaterial.
test,
test
is
test
rate
accuracy,
sequence
and may not
be acceptable.
The extended
isochronal
test
already
at
beginning,
conditions,
Often,
In
a shut-in
as the
at
must
so far,
In
practice
measured
the
is
to
tare
this
situation
it
in
the
to
the
it
isochronal
to
is
need
is
conducted
stabilized
flow
periods.
stabilization,
(isochronal)
this
last
well
essentially
may be chosen
and the
time
of
necessarily
not
with
isochronal
with
or without
rate,
as long
stabilization.
theory
each
of
Flow
Rate
applicable
flow
rarely
flow
If
test.
appropriate
Constancy
is
if
extended
rate
pressure
within
a critical
of
simply
flow
the
flow
the
be shut
at the
or modified
beginning,
However,
between
incerpretlng
through
then
taken
any suitable
extended
the
end of
rate
7.3
In
the
stabilization.
intervening
is
at
commencement
being
fact,
flow
at
well
the
of
either
isochronal
reading
and later
be so.
to
last
a pressure
or
the
prior
the
rate
may be run
on production,
the
flow,
flow
period
achieved.
prover,
the
‘co the
is
tests
assumed
If
upstream
the
described
to be constant.
flow
pressure
is
being
declines
3-40
continuously
If
with
an orifice
prac’cice~is
time,
and hence
meter
is
set
the
to
being
This
declining
wellhead
is
changed
not
pressure
downstream
pressure
regulator,
often
the
short
rarely
constant,
the
well.
All
factors
flow
these
rate
for
approach.
The
%amnarized
in
the
results
are
the
values
should
In
but
as is
in
the
rate
case
coupled
due
involve
a
the
back
flow
the
gas flowing
temperature
CO a gradual
for
A
with
from
wellhead
a fixed
period.
declining
rates
the
waraing
is
up of
the
an absolutely
Constant
a method
of analysis
the
information
in
flow
such
the
rate
rate
of
their
drawdown
findings
flow
are
and the
to vary
this
testing
are
applicable
to
provers
need
not
be kept
in
order
has been
made
to run
to
well
and
and utilized
may include
absolutely
with
Since
orifice
to
sudden
plates
adhere
to a
has commenced.
DELIVERABILITY
the
entire
continuously
IXI change
period
the
meters.
in
rapid,
flowing
over
and
not even
FOR DESIGNING
excessively
averaged
or orifice
approach,
a flow
not
corresponding
smoothly
reason,
be collected
information
flow
rates
pertaining
should
and McCain
below.
flow
decision
to
some of
this,
once
Harrell
validity
related
values
whatever
Lee,
the
than
with
GUIDELINES
rate.
rather
of
schedule,
developed
study
given
invalidate
Once the
procedure.
of
view
for
8
investigation
flow
usual
at
flow
resulting
of
the
meter,
the
a continuously
but
may be allowed
permissible
the
flow,
orifice
in
flow
their
changes
be used
period.
prespecified
all
choke,
simulation,
chapter,
tests
constant,
is
gas
choke
difficult
in
of
a later
instantaneous
changes
the
(1965)
variations
Provided
time,
it
by numerical
deliverability
flow
the
periods,
and Colpitts
confirmed,
pressure
of
being
make
the
correspondingly.
to be maintained.
account
(1972)
flow
variation
Winestock
to
results
decreases
throughout
the
calculations
During
temperature.
of
rate
of
upstream
pressure
Moreover,
flow
to measure
upstream
constant
rate.
used
choke,
setting
setting.
the
logs,
TESTS
a deliverability
to
the
in
specifying
drill-stem
test,
reservoir
under
the
teats,
test
3-41
previous
delrverabiliey
history,
fluid
studies.
In
wells
the
composition
the
should
of
is
the
of
some of
same
formation
first-hand
certainly
conducted
field
have
on that
and temperature,
absence
completed
value
testa
a major
well,
cores
these
production
and geological
derails,
data
from
may be substituted.
experience
must
not
influence
on the
neighbouring
At
all
times,
be underestimated
design
and
and conduct
of
tests.
8.1
A knowledge
lmporrant
the
factor
in
such
or from
the
not
of
data
are
order
Otherwise
is
more
used
if
the
An important
only
approximate
the
the
that
the
in
the
is
Co be used
for
directly
tests,
accurate
than
greater
conducted
well.
If
well
previous
on the
such
will
a very
determining
from
for
well
information
behave
same pool,
of
should
point
rather
than
being
tested
is
the
type
the
in
which
a
the
flowing
tied
of
flowperiodswhen
A single-point
into
test,
the
is not
time
test
is
modified
It
‘co stabilization
is
may be conducted.
preferable.
isochronal
known,
The isochronal
test
and
should
be
is warranted.
a test
is
is
that
wells,
deliverability
If
‘ce*Es
discussed
if
isochtonal
test,
and calculating
to
a pipellne,
but
stabilized
flow
is
in
to be flared,
type
the
the
must
‘be taken
points
appropriate
rather
are
is
to
duration
than
ensure
the
flow
the
well
available
sufficiently
to be obtained.
when the
1,
stabilized
Where
flexibility
Chapter
by testing
teats
stabilization.
more
care
fully
may be accomplishad
using
a well
test
more
gas is
This
be minimized.
new exploratory
stabilization
a conventional
accuracy
choice
to
(3-14).
isochronal
conventional
long
test
of
time
Equation
consideration
test
choosing
stabilization
may be known
wells
of a few hours,
The
the
of
This
neighbouring
from
one of
test
of
type
for
available.
may be estimated
the
the
may be assumed
When the
of
Test
required
characteristics
it
to
of
or deliverability
production
similar
time
a well.
as drill-stem
available,
manner
the
deciding
deliverabiliey
tests,
is
of
Choice
deliverability
in
3-42
relationship
of
this
test
is
of
well
relationship
is
prior
is
stabilized
rate
is
known
for
flowing
mentioned
in
Chapter
equipment
are
the
of
the
gas and
possibility
This
the
of
to prevent
complete
at
plugging
in
rate
in
the
six
to eight
test
equipment
be measured
with
Calculations
inaccurate
the
of
the
Extremely
hydrate
because
will
of
liquid,
from
be at
the
temperature
least
well.
It
above
the
such a
of
well
the
the
to
hydrate
or
the
Failure
due to
flow
must
hole
Choice
of Flow
Rates
the
before
flow
to lift
the
to maintain
these
it
can
provers.
become
practically
mandatory.
equipment
sulphur
minimum
gas
be included
and wherever
bottom
times,
to
liquid
becomes
Where
affect
causes
liquid
bombs
be sufficient
The
data
pressures
wellbore
required
analysis
test.
flow
critical
or
point,
the
Long
of
wellhead
test,
test.
surface
corrosion
that
also
the
with
a multi-point
the
of
equipment.
before
meters
use of
the
during
be free
pressure
and
or condensate,
at
the
choice
A and wiJ.1
measurements.
from
in
hole
Appendix
the
needed
are
be investigated.
of
orifice
liquid
during
sections
water
testing
the
enomaLous
gas must
problems
should
gas well
in
are
the
may make
equal
updating
as the
in
result
be it
8.3
should
pool
a measurement
must
in
and pressure
bottom
1n conducting
the
is
to be used
pressures
is
of
conduct
pressures,
formation
outlined
hours,
sandface
gases
to
affecting
to be expected
standard
use of
sour
impossible
and
since
when there
possible
rates
One or two separators
stabilizes.
the
used
flow
equipment
or partial
of
of
and only
Equipment
factors
heating
Production
of
the
methods
formation
least
ratio
the
is needed
Some of
of
by the
time
survey
equipment
effluent
hydrate
fluctuations
of
6.
and location
choice
that
of
expected
may be done
a pressure
Choice
types
liquid
tests,
pressure.
8.2
The various
previous
A convenient
and all
and the
from
desired.
to a shut-in
probably
flow
the
deposition.
flow
liquids,
rate
if
used
any,
a wellhead
considerations
do not
3-43
apply, the minimum and maximum flow rates are chosen, whenever practical,
such that the pressure drops they cause at the well are approximately
5
par cent and 25 per cent, respectively,
of the shut-in pressure.
Alternatively,
they may be taken to be about 10 per cent and 75 per cent,
respectively,
of the AOF. High drawdown rates that may cause well
damage by sloughing of'the
formation
or by unnecessarily
coning water
into the wellbore must be avoided.
Care must also be taken to avoid
retrograde
condensation
within the reservoir
in the vicinity
of the
well or in the well itself.
In the isochronal
and modified
isochronal
tests, the extended flow rate is often taken to be approximately
equal
If flaring
is taking place, flow should
to the expected production
rate.
be at the mlnLmum rate consistent
with obtaining
useful information.
Some idea of the flow rates at which a we13 is capable of
flowing may be obtained from the drill-stem
test or from the preliminary
In the absence of any data whatsoever,
the AOF may
well clean-up flows.
be estimated from Equation (3N-12)
by assuming stabilized,
purely
lamisar
flow
in the reservoir.
k h qR
AOF =
(3-18)
3.263
s may be estimated
approximately
Chapter 7.
similar
x lo6 T[l,,
from similar
wells
8.4
(0.472
$)
stimulation
in the formation,
Duration
+ &]
treatments
performed
or from Table
7-l
on
in
of FLOW Rates
In conducting tests
which involve
stabilized
conditions,
the
conventional
test, a single-point
test and the extended rate of the
isochronal
and modified
isochrcnal
tests, the duration
of flow must be
at least equal to the approximate time to stabilization
as calculated
from Equation (3-14).
The duration
of the isochronal
periods is determined by two
considerations,
namely, (a) wellbore
storage time and (b) the radius of
3-44
investigation.
a. The wellbore
storage time, tws, it the approximate
time
required
for the wellbore
storage effects
to become negligible.
This
can be calxulated
from Equation (3N-24) which is developed in the Notes
to this chapter:
t
"*
=
36177 ii vws cws
kh
O-19)
where
vws = volume of the wellbare
tubing (and aanulus, if
is no packer)
= compressibility
c
of the wellbore
fluid evaluated
ws
the mean wellbore
pressure and temperature
there
at
Equation (3-19) is presented graphically
in Figure 3-19 for the case
of a three-inch
internal
diameter tubing string in a six-inch
internal
diameter casing, with and without an annulus packer.
b. The radius of investigation
has been discussed in Section
7.1.
Rarely does wellbore
damage or stimulation
extend beyond 100 feet.
In order to obtain data that are representacive.of
the formation,
the
flow period must last
feet.
For wells wlrh
investigate
100 feet
3-18.
From Equation
t
longer than the time to investigate
the first
100
no damage or improvement an approximate
time to
is obtained
(3-14)
loo = 1000 $
from Equation
(3-X)
loo2 = 1.0 x lo7 $
R
or from Figure
(3-20)
R
of flow that will
The greater of tws and tlOO is the minimum duration
yield data representative
of the bulk formation
rather than the wellbore
area.
A duration
equal to about four times this value is recommended
for
the isochronal
periods.
3-45
----
-
-
Z
Y
-
tus
NOT
PACKED
\
7
-----s
\
\\
*
\t\
-
-
kh, md-ft
FIGURE
3-19.
TIME
REQUIRED
FOR WELLBORE
TO BECOME
NEGLIGIBLE
STORAGE
EFFECTS
3-46
EXAMPLE 3-6
Introduction
to the design
This example illustrates
of a deliverability
test.
calculations
that
are essential
Problem
A well was completed in a dry, sweet gas pool which is being
developed with a one-section
spacing between wells.
It has been cored,
logged and drill-stem
tested, acidized and cleaned but no deliverability
tests have,
deliverability
so far, been performed
test.
on it.
Design
a suitable
Solution
Choice of Test
Before the choice of a suitable
test can be made, the
t
approximate time to stabilization,
S' must be known. This being the
first
well in the pool, and the drill-stem
test flow rate not being
stabilized,
the time to stabilization
is not known and should be
estimated from Equation (3-,14).
This requires
a knowledge of the
Following
factors:
re' P,, $9 k, i,
= 2640 ft, equivalent
obtained
= 2000 psia,
to a one-section
spacing;
from the drill-stem
test;
= 0.15, the gas filled
porosity
is obtained by
multiplying
the formation
porosity
by the gas
saturation,
from logs:
=
both quantities
temperature
Equation
(3-14)
deducible
120 md, the build-up
period of the drill-stem
test was analyzed by methods described
in Chapter
From logs,
to give an effective
kh = 1200 md-ft.
h = 10 ft;
= 0.0158 cp, the gas composition
same as that of Example A-l.
From
being
is
580%.
is known and is the
The reservoir
5
3-47
= (looo)(o.15)(o.o158)(2640~2
(120) (2000)
_ 69 hours
This time to stabilization
four rates of a conventional
is considered to be too long to conduct the
test.
The isoehronal
procedures will be
considered
instead.
The permeability
and the build-up
characteristics
experienced
during drill-stem
testing
suggest that if P modified
isochronal
test were to be used, the shut-in
pressures between flows
would build up sufficiently
to make the modified
isochronal
test's
validity
comparable to char of an isochronal
test.
Therefore,
a
modified
isochronal
test is chosen to determine the deliverability
relationship.
Flow Periods
The time necessary to investigate
is obtained from Equation (3-20)
t
100
the reservoir
R
from Figure
k PR
@D
into
= 1.0 x lo7 *
se (1.0 x 10’)(0.15)(0.0158)
(120) (2000)
alternatively,
100 feet
1.01 x lOa,
= o 1o hours
3-18 with
t lpo = 0.10 hours
The time required
for wellbore
storage effects
to become
negligible
is obtained from Equation
(3-19) or Flgure 3-19.
SiIlC@
there is a bottom hole packer, the wellbore
volume is that of the tubing
alone (diameter of tubing = 0.50 feet, length of tubing = 5000 feet).
The average compressibility
of the gas in the wellbore,
knowing the gas
composition
and an assumed average pressure in the tubing of about
1800 psia, is 0.00060 psi-l.
3-46
EXAMPLE 3-6
Introduction
This
to the design
Problem
example illustrates
of a deliverability
A well
was
completed
calculations
that
are essential
test.
in a dry,
sweet gas pool which is being
spacing between wells.
It has been cored,
acldized and cleaned but no deliverability
developed with a one-section
logged and drill-stem
tested,
tests have, so far, been performed
deliverability
test.
on it.
Design
a suitable
Solution
Choice of Test
Before the choice of a suitable
test can be made, the
t
approximate time to stabilization,
6' must be known. This being the
first
well in the pool, and the drill-stem
test flow rate not being
stabilized,
the time to stabilization
is not known and should be
estimated from Equation ,(3-14).
This requrres a knowledge of the
following
factors:
re'
a.
b.
C.
c
;;
9
=
=
=
d.
k
=
e.
From Equation
;
-
(3-14)
$9
$7
k,
L,
2640 ft,
equivalent
to a one-section
spacing;
obtained from the drill-stem
test;
2000 psia,
0.15, the gas filled
porosity
is obtained by
multiplying
the formation
porosity
by the gas
saturation,
both quantities
being deducible
from
logs;
120 md, the build- up period of the drill-stem
test was analyzed by methods described
in Chapter
From logs,
to give an effective
kh = 1200 md-ft.
h = 10 ft;
0.0158 cp, the gas composition
same as that of Example A-l.
temperature
is 580'R.
is known. and is the
The reservoir
5
3-47
= (1000)(0.~5)(0.015E)(2640)2
(120)(2000)
_ 6g hours
This time to stabilization
is considered to be too long to conduct the
four rates of a conventional
test.
The isochronal
procedures will be
considered
instead.
The permeability
and the build-up
characteristics
experienced
during drill-stem
testing
suggest that If a modified
isochronal
test were to be used, the shut-in pressures between flows
would build up sufficiently
to make the modified
isochronal
test's
validity
comparable to that of an isochronal
test.
Therefore,
a
modified isochronal
relationship.
test
is chosen to determine
Flow Periods
The time necessasy
is obtained
from Equation
to investigate
alternatively,
the reservoir
= o 1o ho"rs
3-18 with
1.01 x loa,
The time required
into
R
x 10')(0.15)(0.015s)
(120)(2000)
ftom Figure
k iR
m=
100 feet
(3-20)
c 100 = 1.0 x 10's
= (1.0
the deliverability
t loo = 0.10 hours
for
wellbore
storage
effects
to become
since
3-19.
negligible
is obtained from Equation (3-19) or Figure
volume is that of the tubing
there is a bottom hole packer, the wellbore
alone (diameter
of tubing = 0.50 feet, length of tubing = 5000 feet).
The average compressibility
of the gas in the wellbore,
knowing the gas
composition
1800 psia,
and an assumed average
is 0.00060 psi-l.
pressure
in the tubing
of
about
3-48
From Equation
(3-19)
36177 u U", cws
kh
t ws
= (36177)(0.015a)(n*
(120)
alternatively,
from
Figure
3-19
0.25"
(10)
5000)(0.00060)
= o.28 hours
with
; cw* Lt = 4.7 x 1o-2,
t "S = 0.28 hours
Since
the duration
of the isochronal. periods
=4t ws = 1.12 hours = 1.5 hours
(say)
of the extended flow period
= 69 hours = 72 hours
s
(say)
the duration
zt
Flow Rates
Because of a mal.function
in the flow metering recorder,
flow
Accordingly
an estimate
rates during well clean-up are not available.
of the AOF will be made from Equation (3-W.
This requires
a knowledge
uf the following
factors:
il.
b.
C.
d.
From Equation
r
= 0.25 fr
TW = 580°R, obtained during drill-stem
from the Q-p
TR = 330x10" psi'/cp,
= 0.0, no data available
for this
s
(3-18)
AOF z:
k h Ji,
3.263 x lo6 T [log
(0.47,
;)
+ &]
testing
curve of Figure
new pool
2-4
3-49
(120)(10)(330
=e
r105)
(3.263x106)(580)
A suitable
10% of
AOF
=
6 MMscfd
75% of
AOP
=
45 Mi%cfd
range
first
rate
=
6 MMscfd,
for
1.5
hr
=
12 MNscfd,
for
1.5
hr
-
24 MMscfd,
for
1.5
hr
=
48 MMscfd,
for
1.5
hr
rate
fourth
An extended
flow
Since
is
flow
rate
of
the
this
wastage
deliverability
is
would
25 MMscfd
is
for
to
of
flaring
connected.
well,
some
is
of
by deferring
from
the
recommended.
and since
75 MMscf
Meanwhile,
Section
be
72 hour8
the
be avoided
be calculated
described
would
rates
connected
involve
a pipeline
method
about
no pipeline
that
until
the
rate
would
recommended
test
rate
flow
= 57 MMscfd
approximate
rate
third
‘“.4:;‘;:;40’]
of
second
there
log
the
gas,
this
the
stabilized
isochronal
test
extended
it
is
part
of
data,
the
using
5.
Equipment
pressure
in
From
a knowledge
and the
reservoir
Appendix
anywhere
A it
in
necessary
choke
the
test
be ample
pressures
involved,
Because
water
orifice
measuring
of
that
heater
to handle
all
oE the
pressures.
run.
composition,
are
unforeseen
equipment
A bottom
of
single
hole
reservoir
the
not
method
Likely
heating
preceding
presence
the
and by using
No special
condensation,.a
meter
gas
hydrates
equipment.
standard
should
mostly
the
the
the
temperature
can be seen
and the
0peratioIl.
of
CO form
equipment
and following
hydrate
be rated
small
quantities
separator
pressure
is
the
problems.
should
will
gauge
for
adjustable
Because
of
high-pressure
of
liquids,
suffice
is
outlined
desirable
prior
to
for
9
CALCULATING ANTJPLOTTING TEST RESULTS
Earlier
sectione
and their application.
describe the various types of deliverability
tests
The calculation
of the flow rates and the
conversion
of surface measured pressures CD sub-surface
pressures are
discussed in Chapter 6 and Appendix B, respectively.
Familiarity
with
these will be assumed. The methods for calculating
and plotting
test
results are outlined
fn this sectidn.
The calculations
for determining
the deliverability
ship mey be carried out as shown in Examples 3-1 to 3-5.
examples both the Simplified
and the LIT($) flow analyses
the purpose of illustration,
preferably
the more rigorous
relationIn these
were used for
but only one of these interpretations,
LIT($) flow analysis,
is needed.
If
approxWate calculations
need to be done in the field,
the Simplified
analyeis
inay prove to be conventient.
The pressures used is the calculations
are those at the
sandface and may be obtained by direct measurement or by conversion
of
the wellhead pressures.
In obtaining
the differences
in pressuresquared or pseudo-pressure,
the pairs of pressures involved
in the
subtraction
vary for the different
tests.
They are summarized in
Figure 3-20 which shows the appropriate
pressures connected by a
vertical
link.
The conventional
test will be used to explain the
application
of Figure 3-20.
The initial
shut-ln
pressure and the pressure at the end of
Flow 1 are converted
to p', for the Stmplified
analysis,
or to $, by
using the appropriate
$ - p curve, for the LIT($) flow analysie.
The
difference
in these two pressure-squared
or pseudo-pressure
terms,
AP' Of 4, correspond to the flow rate, q,, of Flow 1. The came
procedure is carried
out for Flow 2, Flow 3 and Flow 4. For the other
tests, Ap' or A$ values are obtained from the pressures linked together
in Figure 3-20.
The points
plotted as detalled
below.
(Ap2,q,,)
or (AIJJ- bq&,q,,)
are then
3-51
:ONVENTlONAL
I
I
’ :7”’
INITIAL SHUT-IN
FLOW
MODIFIED
ISOCHRONAL
ISOCHRONAL
1
SHUT-IN
FLOW
.--J
:I
:I
2
SHUT-IN
FLOW 3
SHUT-IN
FLOW 4
SHUT-IN
EXTENDED
FLOW
STABILIZED
SHUT-IN
‘I
I
I
:Il
(I) In tha modified irochronol test, the initial shut-in preraure may not bs fully stabilized.
FIGURE 3-20.
SANDFACE PRESSURES USED IN COMPUTING
FOR DELIVERABILITY TEST ANALYSES
9.1
Simplified
Ap2 OR A$
Analysis
plot of Ap' versus q,, should be made on logarithmic
coordinates
and a straighr
line should be drawn Khrough a minimum
of
three points.
If a straight
line is not Indicated
by at least three
The
and
consideration
slope
of the
1.0 or less
well, unless
points,
different
also if the LIT($)
flow analysis is not meaningful,
The reciprocal.
should be given to retesting
the well.
line is the exponent n. If the value of n is greater Khan
should be given to retesting
the
than 0.5, consideration
experience
with wells in that pool indicates
Khat a
n value would not be obtained.
If a well has been retesred,
and the test ls still
unsatisfactory,
the best fit line may be drawn through the points of
If the resulting
value
the test which appear to be the most acceptable.
an n of 1.0 shall be dram
of n is greater than 1.0, a line reflecting
3-52
through
the
0.5,
a line
flow
rate
highest
reflecting
rate
point.
an n of
0.5
the
case
of
be positioned
plotting
the
reciprocal
illustrated
in
reflect
slope
Figure
n Is
through
less
the
line
a,
points.
If
at
the
consideration
than
lowest
in
points
line
This
1~ done
appropriate
the
point,
scatter
of
a and b from
p,,
flow
by
rate.
a8 is
and the
points
to
is
that
entire
with
least
excessive
has been
retested,
well,
from
the
procedure
three
or if
a different
and
deliverability
and the
the
(3-7)
deviation
at
(3-4)
coordinates
calculated
retesting
indicates
Equations
ari excessive
be repeated
data
by Equation
on logarithmic
be rejected,
be given
pool
represented
showing
or b should
that
deliverability
FLOW Analysis
points
should
should
wells
the
through
versus
data
data
plotted
calculating
the
conditions.
relationship
the
Any
tests
versus
drawn
LIT($)
(AIJJ - bqic)
be made with
straight
Ap’
by calculating
of
relationship.
with
of n Is
3-8.
be determined
A plot
type
of
The deliverability
should
value
be drawn
stabilized
value
9.2
(3-E).
the
shall
isochronal
to
stabilized
A line,of
should
If
point.
In
should
flow
line
data
b is
unless
of
negative,
experience
would
not
be
obtained.
If
the
unsatisfactory,
a least
acceptable
then
(for
should
a value
These
well
of
be made.
zero
In
in
involve
satisfactory
pressure
to
Simplified
test
only
a retest
alterations
measurements
be used
to
n = 1.0
of
the
(for
still
that
out
resulting
an estimated
procedure
This
if
place
is
points
turn
in
a one-year
test
relationship.
data
or b still
relationship
wrthin
in
the
test
appear
most
to be negative,
negative
number.
b = 0) and n = 0.5
analysis.
the
is
at
of
equivalent
any case,
unsatisfactory
be given
the
are
fit
If
should
two condi,tions
a = 0)
squares
and the
two-phase
one,
period.
in
change
flow
from
and
may involve
second
consideration
The fetest
an attempt
appears
this
to obtarn
direct
should
should
a
sandface
to be a possibility,
or
3-53
it may involve
another type of flow test.
In the case of isochronal
type rests, the deliverability
line
should be positioned
to reflect
stabilized
conditions.
This is done by
calculating
a from Equation (3-12) if a stabilized
flow was conducted,
and plotting
the resulting
stabilized
deliverability
line a6 shown in
Figure 3-9.
In the absence of stabilized
from Eqvation
(3-13).
flow
data a may be calculated
NOTES TO CHAPTER 3
3N.1
Pressure-Squared
LIT Flow Analysis
Relationship
Equation (3-l),
deliverability
equation,
to a kheoretically
the LIT(p2)
flow
for
the commonly used Rawlins and Schellhardt
was obtained empirically
but may be related
Equation (3-2), also called
derived relationship,
equation.
Combining Equations (2-101) and (2-102).
and substituting
various dimensionless
variables
from Tables 2-3 and 2-4 gives,
stabilized
flow
(pseudo-steady
1.417
2
PR - P,f =
-2
%
for
state)
x lo6 qSC u z T
kh
3.263 x lo6 qsc p z T
kh
The above equation assumes laminar flow in the reservoir.
The skis factor,
s, and inertial-turbulent
flow effects,
DqsC,
discussed in Chapter 2, Section 9, may be introduced
to give, from
Equation (2-143)
(3N-1)
3-'54
-2
2
PR - P,f = 3.263XX1~'YZT[log(o.4::re)+~]q*c
+
a
1.417 x 106 II Z T
D q:c
kh
,
(3N-2)
q,, + b' qic,
Therefore
(3N-3)
bl = 1.417 x IO6 v 2 T D
kh
(3-l)
Miller
(3N-4)
The interrelationship
of a' and b' to C and n of Equation
has been given in various
and Riley (1963), Willis
forms by Houpeurt (1959), Carter,
(1965)
and Cornelson (1974).
Tek,
Grove and Poetrman (1957) gave similar
relationships,
in graphical
One form of the interrelationship,
for various ranges of flow rates.
as expressed by Cornelson
(1974) assumes
a.
b.
c.
d.
form,
Equation (3-l)
is valid for qmin 2 qsc C q,,,.
This
defines the range of flowrates
within which the Fi - p$
Versus'q,,
plot is a straight
line on a Log-log Plot;
Equation (3-2) &valid
for 0 2 q,, 5 AOP;
and (3-Z) is
The function
;;; - p& from Equations (3-l)
equal with the range qmin to qmax;
The rate of change of the above functions
is
the geometric mean of qmin and q,,,,
to give
equal at
(3N-5)
3-55
(3N-6)
and
4*c
c=
ON-l)
a’ + b’ qsc
+ b’ qic
a’ + 2b’ q,,
ar + b’ qsc
*=
In addition
(3-l)
and (3-2),
for Equation
seen that:
(3N-8)
a’ t 2b’ qs,
it
(3-2)
to the above interrelationship
can be shown chat Equation
for
various
ranges
of flow
between Equations
(3-l)
rates.
is an approximation
It
IS
readily
and
for very low flow rates
a’qsc .> bq;, , Ap’ = a’q,,
Conversely,
f ram Equation UN-5)
n of Equation (3-l) = 1.0.
for n = 1, a’ = I3 and Equation (3-2) reduces to
Equation (3-l) :
and
for high flow rates
a’qsc << bq& , Apz = b’q;,
Conversely,
from Equation (3~-6)
n of Equation (3-Z) = !.5*
and Equation (3-2) reduces to
for n = 0.5,
b’ = ($)
Equation
(3-l).
Hence n may vary
turbulent
from 1.0 for
fully
laminar
flow
to 0,5 for
flow.
Pseudo-Pressure
Relationship
Equation (3-4).
the rigorous
form of the LIT($)
flow equation,
can be related to Equation (3-l) in a manner similar
to that of the
previous
section.
Equations (3N-5) to (3N-8) are applicable
with a’
and b’ replaced by a and b.
An equivalent
form of Equation (3N-2) in terms of pseudopressure
is obtaiaed
by combining
Equations
(2-101)
and (2-103)
with
3-56
appropriate
substitutions
from Tables
2-3 and 2-4,
and from Equation
(2-143)
f
1.417
T
x lo6
D Gc
kb
= a q*, + b qic
(38-9)
Therefore
a -
3-263;;@
b
1.417 x lo6 T D
kh
=
T [10g(o’4;;
‘@)
+ h]
(3N-10)
(3N-11)
The interrelationship
of a and b to C and n of Equation (3-l)
can be obtained from Equations
(3N-5)
to (3N-8) simply by replacing
a'
and b' by a and b.
An approximate
idea of the absolute open flow potential
of a
well may be obtarned from Equation (3N-9) by neglecting
the Dq' term
and estimating
7-1. Hence
the skin
factor,
%-"
Equations
3.263
x IO6 T [h+.472
Time to Stabilization
(3N-2)
only; that is, for t > ts,
with appropriate
substitution
7, Table
k h 5,
E
AOF = qsc
3N.2
s, by the methods of Chapter
;)+
and Related
*]
(3N-12)
Matters
and (3N-9) apply to stabilized
conditions
the time to stabilization.
Equation (2-104),
for
dimensionless
quantities
from
3-57
Table 2-3,
can be written
aa
ON-131
,Substituting
for
x from Table
2-4 gives
(3N-14)
Approximate
compressibility
ae reciprocal
pressure
gives
$P r;
2
1000
ts
(3N-15)
k iR
Stabilization
is often, in practice,
defined in terms of a
specified
rate of pressure decline.
Such an approach is theoretically
inconsistent
as shown below.
At stabilization,
the applicable
flow equation (excluding
skin and IT flow effects)
in Equation (2-83) which can be written,
with appropriate
substitutions
for dImensionless
quantities
in terms
of pressure from Table 2-3 as
PR - P”f
The rate bf pressure
(3N-16) with respect
=
decline
to
is obtained
by differentiating
Equation
time
(3N-17)
Substituting
for
y and h from Table
2-4 gives
3-58
2 (7.385x105)(2.637X10-')
apwf
-=at
2 T qsc
F$h?r;
(3N-18)
Approximating
compressibility
as reciprocal
pressure
gives
(3N-19)
Equati.on
(3N-19)
shows that
at stabilization
the rate
of
pressure decline depends upon the flow rate and reservoir
characteristics
pressure decline rate that does
such as T, $, h and re. Any specified
not take all of these factors
into account is obviously
unacceptable
as
a definition
of stabilization.
Before stabilization
is achieved,
the radius of investigation,
r.XIV
given
as .dcfined
by
by Equation
(2-105),
is a function
of time and is
(3N-20)
Substituting
for
X from Table
approximated
by the rcciprocnl
2-4 and assuming
compressibility
may be
pressure
(3N-21)
r.1,n-9 = 0.032
3N.3
(3N-1)
The deliverability
and (3N-9), apply
Transient
Relationshlp
relationships,
represented
at stabilized
conditions,
that
by Equations
is, for rinv=re.
When rinv c re, the flow conditions
are said to be transient.
transient
flow, combining Equations (2-72) and (2-143) with
substitutions
from Tables 2-3 and 2-4 gives
For
appropriate
% - $"f
-
3.263X106
k h
T
log
I’
+k
'SC +
2.637X1O-4
k t + 0.809
2.303
+ !Ji ci r;
1.417 x lo6 T
k h
D q:c
= at 4,. + b qic
(3N-22)
Therefore
at is obviously
equal durations
of flow,
therefore
isochronal
value for
and (3N-22).
a function
of the duration
of flow.
For
as in an isochronal
test, t is a constant and
at is a constant.
This
tests.
b is initially
transient
and stabilized
3N.4
Table
forms the theoretical
basis for
Independent of time and has the same
flow as shown by Equations (3N-9)
Wellbore
Storage
Time
Equation (2-154) with appropriate
substitutions
2-4 and 0 from Equation (2-150) becomes
for
h from
(38-24)
3N.5
Laochronal
Type Tests
Aziz (1967b)established
the theoretical
and modified
isochronal
tests using the Simplified
validity
of isochronal
flow equation,
3-60
Equafion (3-l),
radial
unsteady-state
laminar flow equations and several
simplifying
assumptions.
Noting that in the publication
by Aziz (L967),
Modified Isochronal
Testing and,Another Modification
of the Isochronal
Test should be reversed since the latter
is actually
the proper modified
isochronal
test, the theoretical
justification
may be extended quite
simply to include the LIT(e)
flow equation,
Equation (3-4), skin and IT
flow effects.
Such an analysis would, however, assume that the
principle
of superposition
may be applied CO the unsteady-state
LIT flow
equation.
Quiz – Section 1
Name……………………….
1. Give FIVE reasons for testing:
a.
b.
c.
d.
e.
2.
Which tests - Transient (T) or Stabilized (S) – have the primary objective
of obtaining the following information:
Permeability ( ), Damage ( ), Stimulation ( ), Hydrocarbons in place ( ),
Reservoir heterogeneities ( ), Deliverability ( )
3.
Give approximate time duration (hours) for the following tests:
Drawdown-----, Buildup------, DST (on-land)------,
DST (offshore)------, RFT------, Interference-------, Deliverability-------,
4.
Which is the better type of test : Pulse or interference. Why?
5.
Name the two tests that can be used to determine porosity. ----------------, -----------------
6.
If you had very limited funding and could only conduct one test, which test
would you choose, and why?
7.
During the first week of production, Well A produces @1000 bopd
whereas Well B produces @ 2000 bopd. Give three reasons for the
difference:
a.
b.
c.
Name……………………………………………………
Exercise 2-3 DP(skin)
Calculate Pressure drop caused by skin DP(skin),
for two wells in different reservoirs but with the same skin:
Case 1:
K = 1, h = 141.2, q = 500, B = 1, Mu = 1
DP(skin) =
s=5
(141.2*q*B*Mu / (k*h)) * s
= ……… psi Problem? or No Problem?
Case 2:
K = 100, h = 141.2, q = 500, B = 1, Mu = 1
DP(skin) =
s=5
(141.2*q*B*Mu / (k*h)) * s
= ……… psi Problem? Or No Problem?
Conclusion:………………………
Name:……………………………………………
Quiz Section - 2
1.) Name 5 assumptions basic to all well testing equations.
a.)
b.)
c.)
d.)
e.)
2.) Give the equation for emptying a tank.
3.) Name 2 advantages of dimensionless variables.
4.) What is a type curve?
5.) Define pressure drop due to skin.
6.) Define skin.
7.) State the relationship between skin and “Delta P” skin.
8.) a. – What is the theoretical limit for positive skin ?
b. – What is a realistic range of skin for positive skin ?
c. – What is a realistic range of skin for negative skin ?
9.) A well has a skin of 50; Is this a problem ? ( yes, no, don’t know ).
Explain why?
Name………………………………………………..
Exercise 3-1 Turbulence a,b,c
Determine the cause of skin ..Damage? or Turbulence?
s’ = s + D*q
a)
q2 = 6
s2’ = 14
q1 = 1
s1’ = 4
S= ??
D = ??
b)
q2 = 6
s2’ = 26
q1 = 1
s1’ = 1
S= -??
D = ??
c)
q2 = 6
s2’ = 2
q1 = 1
s1’ = 7
S= ??
D = ??
Name:……………………………………...
Quiz (Sections 4 – 5)
For the given Flow Regimes identify the slope of the derivative and the
functional form of ∆P = f(t), based on the following:
4
f(t) = t , t , t , log t ,
Vertical Well
1
t
Slope of Derivative
∆P = f(t)
Wellbore Storage
Linear Flow
Bi-linear Flow
Spherical
Radial
Channel
Pseudosteady State
Steady State
Horizontal Well
Vertical Radial Flow
Linear Flow
Horizontal Radial Flow
½
t
Name:……………………………………...
Quiz (Section 4 – 5) contd
Sketch the flow paths for:
a.) radial flow
b.) linear flow in a fracture
c.) linear flow in a channel
Define Derivative:
Sketch the derivative for:
1. WBS, radial flow, BDF
2. WBS, fracture, (no radial flow)
3. Fracture, radial flow, channel
4. Horizontal well
Name: …………………………………………………..
Section 6 - Quiz
Write, in dimensionless form, the radius of investigation relationship
Re-write it in field units (state units of each variable)
Is concept of radius of investigation exact or approximate?
Time to stabilization is time to reach : 1st , 2nd , 3rd or ALL boundaries?
In theory, stabilization means the same as:
Pseudosteady state
yes
Boundary Dominated Flow yes
Tank Type Behavior
yes
Steady state
yes
or
or
or
or
no
no
no
no
In the field, stabilization means:
Time to stabilization depends on: reservoir size (yes or no),
permeability (yes or no),
shape (yes or no),
fluid properties (yes or no), flow rate (yes or no),
Superposition means:
There are 2 wells at the same location (Yes or No).
2 wells cannot be side by side (Yes or No).
If the rate changes the pressure must change too (Yes or No).
Superposition in space deals with boundaries, changing rates, and
multiple wells. (Circle any that apply)
Pressure at the well is: ∆P = A (log t) + B
Pressure in reservoir is: ∆P = B Ei (r2/t)
Write an equation for the pressure drop at well A in presence of well B
which is located 200 ft away .
Name:…………………………………………..
Quiz - Section 7
Drawdown Analysis Procedure:
Log Log plot of Derivative
Linear flow - fracture
(slope =
)
Radial flow Analysis
(slope =
)
Wellbore Storage
(slope =
)
Linear Flow - Channel
(slope =
)
Pseudosteady State
(slope =
)
Vertical Well Specialized Plots (straight line analysis)
Semilog ……Vs……….
slope gives……….
Linear plot ……Vs……...
slope gives ………..or……….
Storage Plot ……Vs………
slope gives ………or……….
Horizontal Well Derivative plot:
E
B
A
1
0
D
C
2 1
1
1
0
1
you get vertical permeability from?
you get skin from?
you get horizontal permeability from?
Where do you get the negative skin that makes a horizontal well
equivalent to a vertical well?
What information can you get from C?
Name:…………………………………………………
Section 8 – Quiz
Given the Flow Equation:
pi − p wf = 162 .4
qBµ
k
′
−
+
s
3
.
23
0
.
869
log t + log
kh
φµct rw2
Derive the equation for a Build-Up following a single constant rate.
Define: Effective Producing time (tp or tc):
Define: Horner time:
Define: Equivalent time:
Explain Superposition time:
Define: p ∗ :
pR
:
pi
:
When are they equal?
Name:…………………………………………………
Section 8 – Quiz contd.
How are buildup and fall-off analysis related?
Which part of buildup test data is used to calculate skin?
What is a Horner plot?
What is an M.D.H. plot?
When is it appropriate to use one or the other?
What are M.B.H. plots?
Name:……………………………………………
Quiz Section - 9
Define the PPD.
What is it used for?
How does it differ from derivative?
Why are static gradients conducted?
When should they be conducted?
Why do we use bottomhole pressure recorders instead of wellhead
recorders?
Name two things that can cause an increase in the PPD.
If a Horner plot has the shape shown above, how can you tell whether
this is a multilayered reservoir effect or a wellbore dynamics effect?
If a Derivative plot has the shape shown above, how can you tell
whether this is a dual porosity reservoir or a wellbore dynamics effect?
Name:…………………………………………………..
Quiz Section 10
Write down the simplified AOF equation:
What are the limits on n, and what do they represent?
A modified Isochronal test should have 3 sets of information shown on
the plot. Name them:
If it takes too long to reach stabilization, what are the various options?
Discuss whether an AOF test is valid throughout the life of a well:
Write the LIT equation:
If there are no wellbore problems, what fraction of sandface AOF is the
wellhead AOF?
Why do we measure both wellhead and sandface pressures during a test?
The field tester says “the rate stabilized after 6 hours”. The well test
analyst says, “It takes 6 months to stabilize”. Discuss the above
statements: