SPE 39972

Analysis of Linear Flow in Gas Well Production

Ahmed H. El-Banbi, and Robert A. Wattenbarger / SPE, Texas A&M University

Copyright 1998, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the 1998 SPE Gas Technology Symposium held
in Calgary, Alberta, Canada, 1518 March 1998.
This paper was selected for presentation by an SPE Program Committee following review of
information contained in an abstract submitted by the author(s). Contents of the paper, as
presented, have not been reviewed by the Society of Petroleum Engineers and are subject to
correction by the author(s). The material, as presented, does not necessarily reflect any
position of the Society of Petroleum Engineers, its officers, or members. Papers presented at
SPE meetings are subject to publication review by Editorial Committees of the Society of
Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this
paper for commercial purposes without the written consent of the Society of Petroleum
Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not
more than 300 words; illustrations may not be copied. The abstract must contain
conspicuous acknowledgment of where and by whom the paper was presented. Write
Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract
Linear flow may be a very important flow regime in fractured
gas wells. It is also important in some unfractured wells. This
paper presents practical approach to analyze both pressure
(well testing) and production rate (decline curve analysis)
data which is influenced by linear flow. The paper explains
two approaches to analyze the data (hand calculations and
curve fitting). It uses analytical solutions that are adapted to
different reservoir models. These models include fractured
wells and wells producing reservoirs with high permeability
streaks. Permeability, flow area, and pore volume may be
obtained from either pressure or production rate data. The
constant rate solutions are different from the constant
pressure solutions. The use of the wrong equations in the
analysis may result in errors as high as 60%. The paper also
shows the application of these techniques in analyzing field
data.
Introduction
Many wells have been observed to show long-term linear
flow1-9. Linear flow can be detected by slope line in log-log
plots of either pressure drop or reciprocal of production rate
versus time. This linear flow is sometimes observed even
when the wells have small hydraulic fractures or no fractures
at all. In many of the cited cases, lnear flow was present for
years before any boundary effects were reached.
Miller10 presented constant rate and constant pressure
solutions for linear aquifers at a variety of boundary
conditions. Nabor and Barham11 wrote Millers solutions in
dimensionless form and added solutions for constant pressure
outer boundary case. These authors considered a linear

reservoir model similar to the one in Fig. 1. They presented

the solutions in terms of the difference in pressure for
constant rate inner boundary condition and in terms of
cumulative production for constant pwf inner boundary
condition. Their solutions were suitable for studying linear
aquifers.
In this paper, we present linear reservoir10-11 solutions in
a form that can be used for a variety of models. We also show
how to use these solutions in analyzing pressure and
production rate data.
Models and Solutions
Linear flow solutions can be adapted to yield the difference
in pressure for constant rate case and the production rate for
constant pressure case. The solutions can be also adapted for
use with a variety of models. In the following we show six
different models and we also show how to use linear flow
solutions to analyze pressure and rate data for these models.
Fig. 2 shows schematic drawings for these six models.
The first model (model a) is the original linear aquifer
model10,11. The second model (model b) is an infinite
conductivity hydraulic fracture in a linear slab reservoir. The
fracture extends all the way to the reservoir boundaries. This
model will show linear flow from the start of production until
the pressure transient reaches the outer reservoir boundaries.
The third model (model c) is a general hydraulic fracture
in a linear reservoir. It is expected for this model, initially,
that linear or bilinear12,13 flow will develop depending on the
fracture conductivity. At a later time, linear flow will develop
because of the shape of the linear reservoir. When this later
linear flow develops, we can use the general linear flow
solutions to analyze pressure and production data. This
model has been studied recently by Villegas14. He developed
skin factors to account for fracture conductivity and lateral
penetration of the fracture. These skin factors are used in the
linear flow equation for closed reservoirs.
The fourth model (model d) is a radial well in a linear
reservoir. Radial flow will develop at early time followed by
linear flow due to the shape of the reservoir. This linear flow
will continue until other boundaries of the reservoir are
reached. The simple relations presented in the paper can be
used to analyze pressure and production data of this model
after early radial flow effects are over.

AHMED H. EL-BANBI, AND ROBERT A. WATTENBARGER

The fifth and sixth models (models e and f) are for high
permeability streaks. Typically, in a radial well producing
layered reservoir with high permeability contrast between
layers, high permeability streaks are depleted first until their
boundaries are reached. Low permeability layers will then
drain into the high permeability streaks. This type of flow
will be vertical linear flow. Again, when this linear flow is
seen in pressure or production data, analysis methods
presented in the paper can be used to obtain reservoir
properties.
We choose useful definitions for dimensionless pressure
and dimensionless time functions for both single-phase oil
and gas flow. These definitions are given in Tables 1 and 2
for constant rate production and constant pwf production,
respectively.
Where m(p) is the real gas pseudo-pressure15 defined by:

SPE 39972

length of the reservoir instead of the flow area. This new

definition will collapse the type curves for closed reservoirs
for each case to just one type curve.

t DL =

0.00633kt
.................................................... (2)
c t L2

We notice the relation between the dimensionless time

defined by Eq. 2 and the usual dimensionless time defined in
both Tables 1 and 2 to be:
2

t DL

Ac
t DA ................................................. (3)
=
c
L

p
dp ...................................................(1)
z
po

m( p ) = 2

The dimensionless pressure function at the flow face,

pwDL, and the dimensionless rate function at the flow face,
qDL, appear to be reciprocal of each other. However, they are
different functions. The first is used for constant rate
production and the second is used for constant pwf
production.
The dimensionless functions are based on general crosssectional area to flow. The cross-sectional area is different
and distance to boundary is also different for each model.
Table 3 shows the general linear flow solutions for constant
rate inner boundary and a variety of outer boundary
conditions. Table 4 is for constant pwf solutions.
Gas flow solutions are obtained by using the
dimensionless pseudo-pressure function in place of the
dimensionless pressure function.
These solutions can be used with any model of Fig.2 if we
use the appropriate definition of cross-sectional area, Ac, and
the appropriate definition of the distance to boundary, L. The
definitions for these cross-sectional areas and distances to
boundary are given in Table 5.
Type Curves
The solutions presented in Tables 3 and 4 can be used to
draw type curves for linear reservoirs. Figs. 3 and 4 are type
curves for closed linear reservoirs producing at constant rate
and constant pwf respectively. The curves are drawn for
several

L
ratios. Figs. 5 and 6 are type curves for
Ac

constant pressure outer boundary linear reservoirs producing

at constant rate and constant pwf respectively.
A useful way of plotting these type curves is by redefining
the dimensionless time function. The new definition uses the

If the relation, given by Eq. 3, used in the solutions of

Tables 3 and 4, the solutions can be simplified to the form
given in Tables 6 and 7. The normalized dimensionless
functions given by the left-hand-side of the solutions of
Tables 6 and 7 are plotted versus tDL in Figs. 7 and 8. Fig. 7
shows both constant rate and constant pwf solutions for closed
reservoirs. Fig. 8, on the other hand, shows the solutions for
constant pressure outer boundary case.
Plotting both constant rate and constant pwf solutions
together reveals that they are different even in the early time
(infinite acting period). The difference between the two
solutions is /2.
Figs. 7 and 8 also show that the behavior of linear
reservoirs deviates from the infinite reservoir solution at a
dimensionless time (defined by Eq. 2) of 0.25 for constant pwf
case and 0.5 for constant rate case. These values were
selected based on when the deviation becomes easily visible.
These observations will be used to deduce information about
the reservoir.
Analysis of Pressure or Production Data With the
Hand Calculation Technique
In this section, we present an analysis technique that uses
simple plots and equations.

SPE 39972

ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION

Log-Log Plot. The first step in analyzing pressure or

production rate data is to identify the linear flow from a loglog plot of p or q versus time. If either of these plots shows
a half-slope line, this will be an indication of linear flow.
This plot may also reveal when the data quit linear flow
behavior. Pressure or production rate data may deviate
upward or downward from the linear flow trend. The upward
bending could be due to a closed boundary reservoir. The
downward bending could be due to a constant pressure
boundary or change from linear flow to pseudo-radial flow as
in high conductivity fractured wells12,13.
Square Root of Time Plot. Another useful plot is the square
root of time plot in which p or 1/q data is plotted
versus time . The data should follow a straight line and
then bend upward or downward depending on the following
flow regime. We can then record the slope of the straight line
and the actual time when the boundary is reached, tehs. The
slope of the straight line will be different whether production
is at constant rate or constant pwf. Short-term draw-down
tests are usually performed at constant rate and long-term
production is usually assumed to be at constant pwf especially
in gas production. We use the slope of the line and end of
half-slope time, tehs, in the equations of Table 8 to calculate

k Ac and pore volume, Vp, of the reservoir. For oil wells,

the units are psi for pressure, STB for production rate, and
days for time. For gas wells, the units are psi2/cp for pseudopressure, Mscf/D for production rate, and days for time.
Note that the pore volume, Vp, calculation is independent
of either the reservoir permeability or the reservoir geometry.
This could be a very useful calculation if reservoir
parameters are uncertain. Note also that permeability, k, and
cross-sectional area to flow, Ac, cannot be separated without
independent knowledge of one of the two.
Since constant rate and constant pwf solutions are
different in the linear flow region, we expect that the analysis
equations would have different constants for the two different
cases. If no boundaries are reached (i.e., the data shows
strictly linear flow or half-slope line), we can use the last
production time as the end of half-slope time, tehs, and the
calculated pore volume will be a minimum (proven) volume.
Calculation of OGIP. For gas wells, OGIP can be easily
calculated once the pore volume, Vp, is determined. This is
done using the following equation:

OGIP =

V p (1 S w )
B gi

...............................................(4)

This equation requires that average water saturation, Sw,

be known. However, if gas compressibility, cg, dominates the

total compressibility, ct, the equations used to calculate pore
volume (Table 8) would directly give OGIP. This way we
eliminate the problem of not knowing Sw and consequently,
we can determine OGIP even without knowledge of Sw.
Analysis of Pressure or Production Data With the
Curve Fitting Technique
The analytical solutions presented in both Tables 3 and 4 can
be programmed and used to match pressure or production
data when linear flow is observed. We chose to program
these solutions in Visual Basic for Excel. We used Excel
Solver to minimize the difference between the actual
recorded data and the model calculated results. This is done
by changing the model parameters (e.g. k, Ac, and L) until
the best match with the data is obtained.
We define an objective function to be minimized. The
objective function for matching pressure data or production
data is given by the following equation:

1
error =
N

Data Model

100 .......... (5)

Data

i =1
N

We see that the objective function is normalized twice.

The first normalization is for the value of the data point. This
normalization is required to give each data point used in
calibration an equal weight (i.e. high values of data points
will not have higher effect than low values of data points).
The second normalization is for the number of data points,
N, used in calibration. This normalization is useful to base
the error on per point basis. The multiplication by 100 allows
the calculation of the error to be on percent per point basis.
In calibrating any of the models to match actual
production data, we do not have to use all the points in the
calibration. This is easily done in Excel by selecting specific
points to be used in calibration. Consequently, we can avoid
using bad (inaccurate) data points and points affected by
severe changes in operating conditions (large variations in
rate or pwf).
We have to notice that this procedure can give us only
two independent parameters. In other words, we cannot
separate k and Ac exactly as in the hand calculation
approach. However, the calculated drainage area, Ac L, and
consequently, the pore volume, Vp, are uniquely determined.
Field Application
We chose to use the solutions presented in this paper to
analyze production data from a tight gas well in a field in
South Texas. The well was hydraulically fractured and has
been producing for almost 23 years. Monthly production
rates were the only data available among some fluid and
reservoir properties such as specific gravity of the gas, g,

AHMED H. EL-BANBI, AND ROBERT A. WATTENBARGER

reservoir temperature, T, average porosity, , and average

water saturation, Sw. The fluctuations in the production
history were caused by shut-ins. Unfortunately, we do not
have much information about those shut-in periods.
Fig. 9 is a log-log plot of cumulative gas production
versus time. The figure shows that a half-slope line (linear
flow) exists for long time especially for production data after
3 years. We choose to make two specialized plots that help
identify the linear flow behavior. These plots are log-log plot
of production rate versus time (Fig. 10), and reciprocal of
production rate versus square-root of time (Fig. 11).
Fig. 10 shows negative half-slope line for almost 15 years
of production. This negative half-slope is a confirmation that
linear flow existed for long period of time in this well. The
fluctuations in production rate were caused by shut-ins. The
same figure also shows that production data bend downward
after approximately 5,000 days (14 years). This downward
bending may be due to boundary effects. The effect of the
boundary was not seen in the cumulative production plot
(Fig. 9), however.
Fig. 11 shows that the reciprocal of production rate
follows a straight line when plotted versus square-root of
time. This is another confirmation that linear flow dominates
the production behavior of the well. Again, the fluctuations
in production rate were caused by shut-ins.
Based on the above plots, it becomes evident that this
well has been producing under the influence of linear flow.
Next step is to match the production data of this well with
the linear reservoir model assuming that production was at
constant pwf. We used the curve fitting procedure with Excel
outlined above to calculate the best values of the models
parameters. We used the data of Table 9 to calculate pseudopressures15 and dimensionless variables. Other gas properties
required for calculations (g, cg, Bg, and z-factor) were
calculated using correlations16-18.
A good match between the actual production rate and the
calculated production rate was obtained and shown in Fig.12.
The models calculated parameters are given in Table 10.
Forecasting After Reservoir Boundary is Reached
After determining the reservoir parameters with either the
hand calculation approach or the curve fitting approach, we
can use the calculated model parameters in forecasting the
production of the well. If no reservoir boundaries were
reached during the production, we can use the equations to
give us minimum pore volume and use this value in our
forecast.

SPE 39972

Forecast for Oil Wells. If the reservoir is an oil reservoir

producing above the bubble point, we can use the equations
presented in Tables 3 and 4 directly to forecast production
rate for a specified draw-down. If the bubble point is reached
at some point in time, the forecast calculated by the above
equations will be a conservative forecast.
Forecast for Gas Wells. If the reservoir is a dry gas
reservoir, we can still use the equations of Tables 3 and 4 to
forecast the future gas production of the well. However, if the
reservoir boundary is reached, we need to correct the
solutions of Tables 3 and 4 with the normalized time
suggested by Fraim and Wattenbarger19.
An alternative method of calculating the forecast is based
on solving Material Balance equation for volumetric gas
reservoirs with the productivity index equation. This
approach is similar to the approach used for solving
commingled gas reservoirs problem20.
First, the reservoir average pressure, p , is calculated
from the Material Balance equation:

Gp
p p
......................................... (6)
= 1
z z i OGIP
Everything on the right-hand-side of Eq. 6 is known and
average reservoir pressure will be the pressure corresponding
to the calculated p/z. Then, knowing the last production rate
and the bottom-hole flowing pressure, we can solve the
productivity index equation (Eq. 7) for productivity index, Jg.

[ ()

q g = J g m p m ( p wf

) ] .................................... (7)

Time-steps are selected and the cumulative gas

production is updated every time-step. The material balance
equation is used to calculate a new average pressure and used
in Eq. 7 to calculate the production rate.
Discussion
The models and analysis techniques presented in this paper
are useful in analyzing short-term and long-term pressure
and production data affected by linear flow. These techniques
are based on analytical solutions that have assumptions. The
most limiting assumption may be production at either
constant rate or constant pwf.
In actual production of oil and gas wells, many variations
in production constraints (shut-ins, changes in pwf, liquid
loading, recompletions, etc.) may occur. These variations
may conceal the linear flow characteristics and complicate
the analysis of the data. In such situations, judgement has to

SPE 39972

ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION

be made to choose the prevailing production constraint

(constant rate or constant pwf) and approximate results could
be obtained from the analysis with the tools provided in the
paper. Numerical reservoir simulation should be used to
confirm the results if variations in production constraints are
suspected to affect the data.
In the methods presented in the paper, we can calculate
pore volume without knowing either permeability, k, or
cross-sectional area to flow, Ac. This may be a big help since
we often dont know permeability. We dont have to assume
a reservoir model, either. This is similar to the radial
reservoir case. However, for the radial reservoir case, we use
a semi-log plot instead of a square-root of time plot.
When linear flow becomes the main flow regime affecting
pressure or production data, the log-log plot of pressure drop,
reciprocal of production rate, or cumulative production
versus time shows a positive half-slope line. Linear flow
could be due to one or more of the following reasons21:
1. the drainage area behaves like a linear reservoir
because the effect of anisotropy due to natural fractures.
2. the drainage area is linear in shape, i.e., the reservoir
is box-shaped and production is through an infinite
conductivity fracture extending to the lateral boundaries.
3. the reservoir is layered and there exist high
permeability streaks which cause vertical linear flow.
4. the reservoir is a channel reservoir (well between two
no-flow boundaries) with production from a radial or a
fractured well. Typically, pressure transient tests would show
a period of radial or early-linear flow before long-term linear
flow, due to reservoir shape, is established.
5. the reservoir is a dual porosity linear reservoir.
Typically, data will show two parallel half-slope lines
between which there exists a transition period. The shape of
the transition period depends on the type of the dual porosity
model (PSS or transient).
6. the reservoir is a transient dual porosity radial
reservoir where the boundary of the fracture system has been
reached and the boundary of the matrix blocks has not
affected the production behavior yet.
7. the well is a fractured well. Linear flow may be
observed for any fractured well if the fracture is of high
conductivity. This includes vertical wells with vertical,
horizontal, and diagonal fractures; horizontal wells with
longitudinal and transverse fractures.
8. the well intersects natural fractures that are of high
conductivity.
9. the well is a horizontal well. Horizontal wells show
two periods of linear flow (early-linear and late-linear).
Conclusions
Many wells in tight gas reservoirs have long-term production
trends which exhibit only linear flow. Several reservoir
models and well configurations can give linear flow. Many of
these situations are summarized in the paper.

Based on the work done in this paper, we can draw the

following conclusions:
1. In this paper, linear flow solutions have been adapted
to a variety of models useful in the analysis of both pressure
and production data.
2. Unlike the familiar radial reservoirs, constant rate
solutions are quite different from constant pwf solutions for
linear reservoirs. Consequently, the analysis equations for
either case are different.
3.

We can calculate k Ac from transient pressure or

production data. However, separation of k from Ac requires

external information.
4. Pore volume and OGIP can be directly determined if
the outer boundary effect has been observed. (If the reservoir
is still infinite acting, these would be minimum values).
Knowledge of k, , and Ac is not required.
5. If gas compressibility dominates ct, the calculation of
OGIP becomes insensitive to the value used for Sw.
Knowledge of k, , Ac, and Sw is not required.
6. Determination
of
pore
volume,
OGIP,
and k Ac does not depend on which linear reservoir model
we have. Correspondingly, we cannot distinguish which
reservoir model is responsible for the linear flow from just
pressure or production data. External information is required
to select the appropriate reservoir model.
Nomenclature
Ac =cross-sectional area to flow, L2, ft2.
B =oil FVF, dimensionless, RB/STB
Bgi =gas FVF at initial pressure, dimensionless,
rcf/scf
ct = total compressibility, Lt2/m, psa-1
cti = total compressibility at initial pressure, Lt2/m,
psa-1
h =net formation thickness, L, ft.
Gp =cumulative gas produced, L3, scf.
Jg =gas productivity index, L4t2/m, Mscf.cp/D/psi2.
k =permeability, L2, md
L =distance to boundary, L, ft.
mCP = slope of 1/qg vs. t , D1/2/Mscf
mCR
mD
m(p)
m( p )
m(pwf)
OGIP
p

p
pDL
pwDL

= slope of m vs. t , psi2/cp D1/2

= dimensionless real gas pseudo pressure
=real gas pseudopressure, m/Lt3, psia2/cp
=m(p) at average reservoir pressure, m/Lt3,
psia2/cp
=m(p) at flowing wellbore pressure, m/Lt3, psi2/cp
= Original Gas in Place, L3, scf
=absolute pressure, m/Lt2, psia
=average reservoir pressure, m/Lt2, psia
=dimensionless pressure for linear reservoirs
=dimensionless pressure at wellbore

AHMED H. EL-BANBI, AND ROBERT A. WATTENBARGER

pwDLN =normalized dimensionless pressure

po =arbitrary lower limit of m(p) integration, m/Lt2,
psia
pwf =bottom-hole flowing pressure, m/Lt2, psia
qDL = dimensionless flow rate for linear reservoirs
qDLN = normalized dimensionless flow rate
qg =gas flow rate, L3/t, Mscf/D
q = oil flow rate, L3/t, STB/D
Sw = water saturation, fraction
t = producing time, days
tDAc =dimensionless time based on Ac
tDL =dimensionless time based on L
T =reservoir temperature, T, oR
Vp = pore volume, L3, ft3
w =fracture width, L, ft.
xf = fracture half-length, L, ft.
xe = reservoir half-width, L, ft.
y = distance in y-direction, L, ft.
yD = dimensionless distance in y-direction
ye = distance from fracture to outer boundary, L, ft.
z =gas deviation factor, dimensionless
=porosity, fraction
=viscosity, m/Lt, cp
Subscripts
ehs =end of half-slope period
i =initial conditions
Acknowledgments
We thank the Reservoir Modeling Consortium and Texas
A&M University for providing funding for this project. We
also thank Coastal Oil & Gas Corp. and Tai Pham for
providing field data.
References
1.

2.

3.
4.

5.

6.

Bagnall, W.D. and Ryan, W.M. : The Geology, Reserves, and

Production Characteristics of the Devonian Shale in
Southwestern West Virginia, Appalachian Petroleum Geology
Symposium, Morgantown, W. Va., March 1-4, 1976, p.41.
Boardman, C.R. and Knutson, C.F. : Unita Basin Lenticular
Sandstone Reservoir Characteristics, paper SPE/DOE 9849
presented at the 1981 SPE/DOE Low Permeability
Symposium, Denver, CO., May, 27-29, 1981.
Hale, B.W. and Evers, J.F. : Elliptical Flow Equations for
Vertically Fractured Gas Wells, JPT (Dec. 1981) 2489-97.
Kohlhaas, C.A. and Abbott, W.A. : Application of Linear and
Spherical Flow Analysis Techniques to Field Problems-Case
Studies, paper SPE 11088 presented at the 57th Annual Fall
Conference and Exhibition, New Orleans, LA, Sept. 26-29,
1982.
Hale, B.W. : Analysis of Tight Gas Well Production
Histories, paper SPE/DOE 11639 presented at the 1983
SPE/DOE Symposium on Low Permeability, Denver, CO.,
March, 14-16, 1983.
Stright, D.H., Jr. and Gordon, J.I. : Decline Curve Analysis in
Fractured Low Permeability Gas Wells in the Piceance Basin,

7.

8.

9.

10.

11.
12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

SPE 39972

paper SPE/DOE 11640 presented at the 1983 SPE/DOE

Symposium on Low Permeability, Denver, CO., March, 14-16,
1983.
Ammer, J.R., Sawyer, W.K., and Drophin, M.J. : Practical
Methods for Detecting Production Mechanisms in Tight Gas
Reservoirs, paper SPE/DOE/GRI 12864 presented at the 1984
SPE/DOE/GRI Unconventional Gas Recovery Symposium,
Pittsburgh, PA., May, 13-15, 1984.
Maley, S. : The Use of Conventional Decline Curve Analysis
in Tight Gas Well Applications, paper SPE/DOE 13898
presented at the SPE/DOE 1985 Low Permeability Gas
Reservoirs, Denver, CO., May, 19-22, 1985
Nott, D.C. and Hara, S.K. : Fracture Half-Length and Linear
Flow in the South Belridge Diatomite, paper SPE 21778
presented at the Western Regional Meeting, Long Beach,
California, March 20-22, 1991.
Miller, F. G.: Theory of Unsteady-State Influx of Water in
Linear Reservoir, Journal of the Institute of Petroleum,
Volume 48, Number 467 - Nov. 1962, 365-79.
Nabor, G. W., and Barham, R. H.: Linear Aquifer Behavior,
JPT, (May, 1964), 561-563.
Cinco, H., Samaniego, F. and Dominguez, N..: Transient
Pressure Behavior for a Well With a Finite-Conductivity
Vertical Fracture, SPEJ (Aug. 1978), 253-264.
Agarwal, R.G., Carter, R.D. and Pollock, C.B.: Evaluation
and Performance Prediction of Low-Permeability Gas Wells
Stimulated by Massive Hydraulic Fracturing, JPT (March,
1979) 362-372; Trans. AIME, 267.
Villegas, M.E. : Performance of Fractured Vertical and
Horizontal Wells, Ph.D. Dissertation, Texas A&M University,
May 1997.
Al-Hussainy, R., Ramey, H.J., Jr., and Crawford, P.B.: The
Flow of Real Gas Through Porous Media, JPT (May 1966)
624-636.
Sutton, R.P.: Compressibility Factors for High-MolecularWeight Reservoir Gases, paper SPE 14265 presented at the
1985 SPE Annual Technical Meeting and Exhibition, Las
Vegas, Sept. 22-25.
Dranchuk, P.M. and Abou-Kassem, J.H.: Calculation of Z
Factors for Natural Gases Using Equations of State, J. Cdn.
Pet. Tech. (July-Sept. 1975) 34-36.
Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: The Viscosity of
Natural Gases. JPT (Aug. 1966) 997-1000; Trans., AIME,
237.
Fraim, M.L. and Wattenbarger, R.A.: Gas Reservoir DeclineCurve Analysis Using Type Curves With Real Gas Pseudopressures and Normalized Time, SPEFE (Dec. 1987) 671682.
El-Banbi, Ahmed H. and Wattenbarger, R.A. : Analysis of
Commingled Gas Reservoirs With Variable Bottom-Hole
Flowing Pressure and Non-Darcy Flow, paper SPE 38866
presented at the 72nd SPE Annual Meeting and Exhibition,
San Antonio, TX. Oct. 5-8, 1997.
El-Banbi, Ahmed H.: Analysis of Tight Gas Well Performance,
Ph.D. Dissertation, Texas A&M University, May 1995.

SPE 39972

ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION

Table 1 - Dimensionless Variables for Constant Rate Production for Linear Reservoirs
Oil
Gas

p DL =

k Ac [ p i p ( y )]
141.2q B

k Ac ( p i p wf

p wDL =

m DL =

0.00633kt
c t Ac

yD =

1424q g T

m wDL =

141.2q B

t DAc =

k Ac [m ( p i ) m( p ( y ))]

k Ac m( p i ) m ( p wf

)]

1424q g T

t DAc =

0.00633kt
( ct )i Ac

yD =

Ac

y
Ac

Table 2 - Dimensionless Variables for Constant pwf Production for Linear Reservoirs
Oil
Gas

p DL =

( pi

p ( y ))
(pi p wf )

k Ac ( p i p wf
1
=
q DL
141.2q B
t DAc =

m DL =

k
1
=
q DL

0.00633kt
c t Ac

yD =

[m( pi ) m( p( y ))]
[m( pi ) m( p wf )]
Ac [m( p i ) m( p wf )]
1424q g T

t DAc =

0.00633kt
( ct )i Ac

yD =

Ac

y
Ac

Table 3 - Linear Reservoirs Solutions for Constant Rate Production

Case
Solution
Constant Rate
Infinite Reservoir

Constant Rate
Closed Reservoir

Constant Rate
Constant Pressure
Outer Boundary
Reservoir

pwDL = 4 tDAc

wDL

= 2

p wDL

L
Ac

1
+
3

Ac
L

t
DAc

2
2

1
2 2
2 exp n

n =1 n

Ac
L

DAc

2

L 8 1 n 2 2 Ac

= 2
t DAc
2 exp

A 1 2 n
4
L
odd n

AHMED H. EL-BANBI, AND ROBERT A. WATTENBARGER

Case

SPE 39972

Table 4 - Linear Reservoirs Solutions for Constant pwf Production

Solution

Constant pwf
Infinite Reservoir

1
= 2 tDAc
qDL

Constant pwf
Closed Reservoir

A
c

1
=
q DL

Constant pwf
Constant Pressure
Outer Boundary
Reservoir

n 2 2
exp

4
nodd

Ac

t DA
c
L

A
1
c

=
q DL

Ac

2 2
1 + 2 exp n

n =1

t DA
c

Table 5 - Solution Parameters for Linear Reservoirs Models

Model
Ac
a - linear slab
wh
b - hydraulic fracture
4 xf h
c - hydraulic fracture
4 xe h
d - well in a slab reservoir
4 xe h
e - high permeability streak, single linear flow
re2
f - high permeability streak, double linear flow
2 re2

L
L
ye
ye
ye
h
h/2

Table 6 - Linear Reservoirs Solutions With Dimensionless Time Based on Length of the
Reservoir, tDL, (Constant Rate Inner Boundary)
Case
Solution
Constant Rate
Closed Reservoir

Constant Rate
Constant Pressure
Outer Boundary
Reservoir

Ac

p wDL = 2 1 + t DL 22 12 exp n 2 2 t DL
n=1 n
L
3

p wDLN =

Ac

p wDL = 2 1 82 12
L
nodd n

p wDLN =

2 2

n
exp
t DL

4

SPE 39972

ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION

Table 7 - Linear Reservoirs Solutions With Dimensionless Time Based on Length of the
Reservoir, tDL, (Constant pwf Inner Boundary)
Case
Solution
Constant pwf
Closed Reservoir

1
q DLN

Constant pwf
Constant Pressure
Outer Boundary
Reservoir

1
q DLN

Ac 1

=
=
L q DL

n 2 2

exp
t DL

nodd
4

Ac 1
2

=
=

L q DL

2 2

1 + 2 exp n t DL

n =1

Table 8 - Interpretation Equations for Linear Flow

Case
Constant Rate
(Oil Production)

Constant pwf
(Oil Production)

k Ac =

k Ac =

Constant Rate
(Gas Production)

Constant pwf
(Gas Production)

Vp

k Ac

(p

V p = 8.962

c t m CRL

125.1 B
i

k Ac =

k Ac =

79.65 qB

p wf

V p = 19.91

c t m CPL

803.2 q g T

1262 T

[m( p ) m( p )] ( c ) m
i

wf

t ehs
B
(pi p wf )ct mCPL

V p = 90.36

( ct )i mCRL

t i

CPL

Vp = 200.8

qB t ehs
c t m CRL

qgT

( c t )i

t ehs
mCRL

t ehs
T
m( pi ) m( pwf ) ( ct )i mCPL

10

AHMED H. EL-BANBI, AND ROBERT A. WATTENBARGER

Table 10 - Estimated and Calculated Parameters

for Example Well

Table 9 - Data for Example Well

Initial pressure, pi
bottom-hole flowing pressure, pwf
pseudo-pressure at pi, m(pi)
pseudo-pressure at pwf, m(pwf)
gas specific gravity, g
reservoir temperature, T
formation net pay thickness, h
formation porosity,
average water saturation, Sw
total compressibility at pi, cti

8800
1600
9
2.67 10
8
1.69 10
0.717
290
92
0.15
0.47
-5
3.53 10

SPE 39972

psia
psia
2
psi /cp
2
psi /cp

Estimated Parameters by
Regression

k Ac
Ac L

10,423
273,678,121

md1/2.ft2
ft3.

Calculated Parameters
OGIP

6.93

Bscf

F
ft.
-1

psi

L
w
h
Fig. 1 - Linear reservoir model.

2ye

2xf = 2xe

model b (hydraulic fracture extends to boundaries)

model a (linear slab)

2ye
2xe

2ye

2xf

2xe
h

model c (hydraulic fracture)

model d (well in a slab reservoir)

SPE 39972

ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION

re

11

re
h/2
h
h/2

model f (high permeability streak, double linear flow)

model e (high permeability streak, single linear flow)

Fig. 2 - Different linear flow models.

1,000

100

L
=1
Ac

L
=1
Ac

100

p wDL

1/q DL

10

10

1
1

0
1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-03

1.E+02

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

t DAc

t DAc

Fig. 4 - Constant pwf type curves for closed linear reservoirs.

Fig. 3 - Constant rate type curves for closed linear reservoirs.

100

100

L
=8
Ac

L
=8
Ac

10

10
1

p wDL

1/q DL

0.1
1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

t DAc

Fig. 5 - Constant rate type curve for constant pressure outer

boundary linear reservoirs.

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

t DAc

Fig. 6 - Constant pwf type curve for constant pressure outer

boundary linear reservoirs.

12

AHMED H. EL-BANBI, AND ROBERT A. WATTENBARGER

1,000

10

constant rate

constant p wf

constant p wf
production

p wDLN and 1/q DLN

100

p wDLN and 1/q DLN

SPE 39972

10

1
constant rate
production

0
1.E-03

0
1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E-02

1.E-01

Fig. 7 - Constant rate and constant pwf type curves for closed
linear reservoirs.

10,000

1.E+01

1.E+02

Fig. 8 - Constant rate and constant pwf type curves for constant
pressure outer boundary linear reservoirs.

10,000

Linear Flow
(1/2 Slope)

Linear Flow
(-1/2 Slope)

1,000

q g (Mscf/D)

G p (MMscf)

1,000

100

100

10
10

100

1,000

10,000

10
10

Time (days)

0.03

0.025

0.02

0.015

0.01

0.005

0
10

20

30

40

50

60

1,000

10,000

Fig. 10 - Log-log plot of production rate versus time for example

well.

0.035

100

Time (days)

Fig. 9 - Log-log plot of cumulative gas production versus time for

example well.

1/q g (Mscf/D)

1.E+00

t DAc

t DAc

70

80

90

100

time(days)

1/2

Fig. 11 - Reciprocal of production rate versus square-root of time for example well.