linear flow unconventional reservoirs
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SPE 39972
Analysis of Linear Flow in Gas Well Production
Ahmed H. E1-Banbi, and Robert A. Wattenbarger / SPE, Texas A&M University
Copyr ight 1995, ~ ie fy of Petro leum Enginee rs , Inc .
7h is paper was preparsd for p resen ta tion a t the 199S SPE Gas Technology Symposium held
inCalgary, Alber ta , Canada, 1S-18 March 199S
This paper was selec ted for p resen ta tion byan SPE Program Commdtee fol lowing rwew of
informat ion contahed in an
bs tract submitted by the au thor(s) . Con ten ts cd the paper, as
-fed. * n~ ~0 M* by Me .%cw@ d Petroleum Engineam and are subject to
corr~on by the atithor(s) lle material, as presantad, does not necessarily reflect any
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SPE meet ings are subj ect t o publi cat ion review by Edi torial Commt iees of the Societ y of
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Librarian, SPE. PO, 130xS3S93%, Richardson. TX 7S0S3-3.536, U S A , fax 01-972.9S2-943S
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 600A,The paper also
shows the application of these techniques in analyzing field
data.
Introduction
Many wells have been observed to show long-term linear
flowl-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 eases, lnear flow was present for
years before any boundary effects were reached.
Miller’” presented constant rate and constant pressure
solutions for linear aquifers at a variety of boundary
conditions. Nabor and Barham” wrote Miller’s solutions in
dimensionless form and added solutions for constant pressure
outer boundary case. The= authors considered a linear
Soeiefyof PetroleumEngin
reservoir model similar to the one in
Fig. 1.
They presen
the solutions in terms of the difference in pressure
constant rate inner boundary condition and in terms
cumulative production for constant
p.f
inner bound
condition. Their solutions were suitable for studying lin
aquifers.
In this paper, we present linear reservoir’ o-” solution
a form that can be used for a variety of models. We also s
how to use these solutions in analyzing pressure
production rate data.
Models and Solutions
Linear flow solutions can be adapted to yield the differe
in pressure for constant rate case and the production rate
constant pressure case. The solutions can be also adapted
use with a variety of models, In the following we show
different models and we also show how to use linear f
solutions to analyze pressure and rate data for these mod
Fig. 2 shows schematic drawings for these six models.
The first model (model
a)
is the original linear aqu
model’ 0“’. The second model (model b) is an infi
conductivity hydraulic fracture in a linear slab reservoir.
fracture extends all the way to the reservoir boundaries. T
model will show linear flow from the start of production u
the pressure transient reaches the outer reservoir boundarie
The third model (model c) is a general hydraulic fract
in a linear reservoir. It is expected for this model, initia
that linear or bilinear’2’13
flow will develop depending on
fracture conductivity. At a later time, linear flow will deve
because of the shape of the linear resemoir. When this I
linear flow develops, we can use the general linear f
solutions to analyze pressure and production data. T
model has been studied recently by Villegas’4. He develo
skin factors to account for fracture conductivity and late
penetration of the fracture. These skin factors are used in
linear flow equation for closed reservoirs.
The fourth model (model r.fj is a radial well in a lin
resetvoir. Radial flow will develop at early time followed
linear flow due to the shape of the reservoir. This linear f
will continue until other boundaries of the reservoir
reached. The simple relations presented in the paper can
used to analyze pressure and production data of this mo
after early radial flow effects are over.
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2
AHMED H. EL-BANBI,AND ROBERTA. WATTENBARGER SPE 3
The fifth and sixth models (models e and fl 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. LQw 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 p.f production,
respectively.
Where m(p) is the real gas pseudo-pressure’5 defined by:
tiP) = 2PJIP
...................................................(l)
The dimensionless pressure function at the flow face,
PWDL,nd the dimensionless rate function at the flow face,
9DL, 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 fimctions are based on general cross-
sectional area to flow, The cross-sectional area is different
and distance to bounda~ 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
p.f
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 boundaty, 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 cuwes 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 dmvn for
L-
several —
F
ratios.
Figs.
5 and 6 are type curves for
c
constant pressure outer boundary linear reservoirs producing
at constant rate and constant
p.y
respectively.
A useful way of plotting these type curves is by redefining
the dimensionless time fimction. The new definition uses the
length of the reservoir instead of the flow area. This
definition will collapse the type curves for closed reserv
for each case to just one type cume.
0.00633kt
t=
‘L ~ /JC,L2
....................................................(
We notice the relation between the dimensionless
defined by Eq. 2 and the usual dimensionless time define
both Tables 1 and 2 to be:
[)
_
2
tDL= 2 tDA
Lc
.................................................(
If the relation, given by Eq. 3, used in the solution
Tables 3 and 4, the solutions can be simplified to the f
given in Tables 6 and 7. The normalized dimension
functions given by the left-hand-side of the solutions
Tables 6 and 7 are plotted versus tD~in
Figs.
7 and 8. Fi
shows both constant rate and constant
p.f
solutions for cl
reservoirs. Fig. 8, on the other hand, shows the solutions
constant pressure outer boundary case.
Plotting both constant rate and constant p.f solut
together reveals that they are different even in the early
(infinite acting period). The difference between the
solutions is 7r/2.
Figs. 7 and 8 also show that the behavior of Ii
reservoirs deviates from the intlnite reservoir solution
dimensionless time (defined by Eq. 2) of 0.25 for constan
case and 0.5 for constant rate case. These values w
selected based on when the deviation becomes easily vis
These observations will be used to deduce information a
the reservoir.
Analysis of Pressure or Production Data With the
Hand Calculation Technique
In this section, we present an analysis technique that
simple plots and equations.
Log-Log Plot. The first step in analyzing pressure
production rate data is to identifi the linear flow from a
log plot of
Ap
or
q
versus time. If either of these plots sh
a half-slope line, this will be an indication of linear fl
This plot may also reveal when the data quit linear f
behavior. Pressure or production rate data may dev
upward or downward from the linear flow trend. The upw
bending could be due to a closed boundary reservoir.
downward bending could be due to a constant press
boundary or change from linear flow to pseudo-radial flow
in high conductivity fractured wells’2“3.
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SPE39972
ANALYSISOF LINEARFLOW IN GASWELL PRODUCTION
Square Root of Time PIOLAnother useful plot is the square
root of time plot in which Ap or
I/q
data is plotted
F
ersus tune. 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 bounckuy is reached,
teh
The
slope of the straight line will be different whether production
is at constant rate or constant p.f, Short-term drawdown
tests are usually performed at constant rate and long-term
production is usually assumed to be at constant
p.y
especially
in gas production. We use the slope of the line and end of
half-slope time, t,k, in the equations of Table 8 to calculate
fi~c 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 pseudo-
pressure, MscfD 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,
A,,
cannot be separated without
independent knowledge of one of the two.
Since constant rate and constant
p.f
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, teh,, and the
calculated pore volume will be a minimum (proven) volume.
Calculation of
O IP For gas wells, OGIP can be easily
calculated once the pore volume, VP, is determined. This is
done using the following equation:
~Glp = l“, (1- Sw)
Bm
...............................................(4)
This equation requires that average water saturation, Sw,
be known. However, if gas compressibility, c~, dominates the
total compressibility, c~,the equations used to calculate pore
volume (Table 8) would directly give
OGIP.
This way we
eliminate the problem of not knowing Swand 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 Vkual Basic for Excel. We used Excel
Solver to minimize the difference between the ac
recorded data and the model calculated results. This is d
by changing the model parameters (e.g.
k, A=, and L) u
the best match with the data is obtained.
We define an objective fi.mction to be minimized.
objective function for matching pressure data or produc
data is given by the following equation:
;J=X1OO ......
.
rror = — ~
We see that the objective finction is normalized tw
The first normalization is for the value of the data point.
norrnaliyation is required to give each data point use
calibration an equal weight (i.e. high values of data po
will not have higher effect than low values of data poi
The second normalization is for the number of data po
N, used in calibration. This normalization is usefhl to
the error on per point basis. The multiplication by 100 al
the calculation of the
error
to be on percent per point bas
In calibrating any of the models to match ac
production data, we do not have to use all the points in
calibration. This is easily done in Excel by selecting spe
points to be used in calibration. Consequently, we can a
using bad (inaccurate) data points and points affected
severe changes in operating conditions (large variation
rate or
pwf).
We have to notice that this procedure can give us
two independent parameters. In other words, we ca
separate
k
and
A.
exactly as in the hand calcula
approach. However, the calculated drainage area, A=L,
consequently, the pore volume,
VP,
are uniquely determin
Field Application
Wc chose to use the solutions presented in this pape
analyze production data from a tight gas well in a ticl
South Texas. The welt was hydraulically fractured and
been producing for almost 23 years. Monthly produc
rates were the only data available among some fluid
reservoir properties such as specific gravity of the gas
reservoir temperature, T, average porosity, +, and ave
water saturation, Sw. The fluctuations in the produc
history were caused by shut-ins. Unfortunately, we do
have much information about those shut-in periods.
Fig. 9 is a log-log plot of cumulative gas produc
versus time. The figure shows that a half-slope line (li
flow) exists for long time especially for production data
3 years. We choose to make two specialized plots that
identi& the linear flow behavior, These plots are log-log
of production rate versus time (Fig. 10 and reciproca
production rate versus square-root of time (Fig.
11 .
Fig. 10 shows negative half-slope line for almost 15y
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SF%39972 ANALYSISOF LINEAR FLOW IN GAS WELL PRODUCTION
because the effect of anisotropy due to natural fractures.
2. the drainage area is linear in shape,
i.e.,
the resewoir
is box-shaped and production is through an in.tlnite
conductivity fracture exlending 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 ~rosity linear resetvoir.
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 0 3Sor transient).
6. the reservoir is a transient dual porosity radial
resewoir where the boundary of the fracture system has been
reached and the boundaty of the matrix blocks has not
affected the product ion behavior yet.
7. the well is a fractured well, Linear flow may be
observed for any fractured WC1lif the fracture is of high
conductivity, This includes vertical wells with ~ertical,
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 cotilgurations 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.
J_
, We can calculate
k AC
from transient pressure or
production data. Howe\’er, separation of k from ACrequires
external information.
4, Pore volume and OGIP can be directly determined if
the outer boundary effect has been observed. (If the reservoir
is still intinitc acting, these would be
minimum
values).
Knowledge of k, +, and JC is not required.
5. If gas compressibility dominates c,, the calculation of
OGIP becomes insensitive to the value used for SW.
Knowledge of
k, +, A., and SWis not required.
6, Determination of pore
volume,
OG
and fi~C does not depend on which linear reservoir mo
we have. Correspondingly, we cannot distinguish wh
reservoir model is responsible for the linear flow from
pressure or production data. External information is requi
to select the appropriate reservoir model.
Nomenclature
A, =
cross-sectional area to flow, L2, ft2,
B =oil FVF, dimensionless, RB/STEt
Bgj =
gas FVF at initial pressure, dimensionless,
rcflscf
Ct= total compressibility, Lt2/m, psa-’
cti = total compressibility at initial pressure, Lt2/m
-1
h = formation thickness, L, ft.
GP =cumulative gas produced, L3, scf.
Jg = gas productivity index, L4t2/m,Mscf.cp/D/psi2
k =perrneability, L2, md
L =distance to boundary, L, R
&
~,/2
mcp = slope of llq~ vs. ,
/Mscf
mcR= slope of Am vs.&, psi2kp D’/2
mLJ= dimensionless real gas pseudo pressure
m(p) = real gas pseudopressure, m/Lt3, psiazlcp
m(~) = m(p) at average resexvoir pressure, rn/Lt3,
psia2/cp
m(p.j) =ndp) at flowing wellbore pressure, m/Lt3, psi2
OGIP =
Original Gas in Place, L3, scf
p =
absolute pressure, mfLt2, psia
~ =average reservoir pressure, rn/Lt2, psia
PDL = dimensionless pressure for linear reservoirs
/h/2L = dimensionless
pressure at wellbore
PwDLN = normalized dimensionless pressure
p.
=arbitrary lower limit of
m(p)
integration, m/L
psia
pwf =bottom-hole flowing pressure, rn/Lt2, psia
qL)I.= dimensionless flow rate for linear reservoirs
qDfJJ= norrnahmd dimensionless flOW rate
qg =gas flow rate, L3/t, MscWD
q = oil flow rate, L3/t, STB/D
SW= water saturation, fraction
t=
producing time, days
tD 4
dimensionless time based on
AC
tDL = dimensionless time based on L
T = reservoir temperature, T, “R
VP= pore volume, L3, ft3
w =fracture width, L. ft.
xf = fracture half-length, L, ft.
x. = reservoir half-width, L, ft.
y = distance in ydirection, L, ft.
~D= dimensionless distance in ydirection
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AHMED H. EL-BANBI,AND ROBERTA. WATTENBARGER
SPE3
y. = distance from fracture to outer boundary, L, ft.
z =gas deviation factor, dimensionless
=porosity, fraction
p =viscosity, miLt, ep
Subscripts
elm =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 projeet. We
also thank Coastal Oil & Gas Corp. and Tai Pham for
providing field data.
References
1.
2.
3.
4.
5.
6.
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the 1981 SPE/DOE Low Permeability
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Analysis of Tight Gas Well Production
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SPE 39972
ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION
Table 1- Dimensionless Variables for Con: ant Rate Production for Linear Reservoirs
I
Oil
~DL = k@’pi - p(y)]
141.2qBp
‘W(P,-
Pwf )
P .DL =
141.2qBp
0.00633kt
t–
“c – ( pc,AC
Y
yD=~
J_
c
G a s
kJ rJ ?z pi - t.?z(p(y))]
‘DL =
1424qg T
it~~(pi
)-
Tn(pw]
m=
wDL
1424q, T
..
0.00633kt
r
‘AC = (@/.J, )iAC
yD .-2-
lr
AC
Table 2- Dimensionless Variables for Constant
pti
Production for Linear Reservoirs
Oil
Gas
i - P Y))
‘DL ‘7GZJ
1
‘ Pi - Pwf )
1
k~[m(pi )-m(p@ )]
=
=
qDL
14
1.2qBp
qDL
1424q, T
0.00633kt
0.00633kf
t
W = PC,AC
t
‘“=
/ Mt)iAc
Y
Y
“D=—
F
yD. —
c
r
Ac
Table 3
Case
Constant Rate
Infinite Reservoir
Constant Rate
Closed Reservoir
Constant Rate
Constant Pressure
Outer Boundary
Reservoir
Linear Reservoirs Solutions for Constant Rate Production
Solution
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AHMED H. EL-BANBI,AND ROBERTA. WATTENBARGER
SPE
Table 4- Linear Reservoirs Solutions for Constant pti Production
Case
Solution
Constant pti
Infinite Reservoir
1
— = 2nJ<
qDL
Constant pti
Closed Reservoir
.=x
‘DL ~exp[ n~n
Constant pti
Constant Pressure
Outer Boundary
Reservoir
.=~
‘D’
{’ 2~exp[ n
Table 5- Solution Parameters for Linear Reservoirs Models
Model
A. L
a - linear slab
wh L
b - hydraulic fracture
4xfh
Ye
c - hydraulic fracture
4xeh
Ye
d- well in a slab reservoir
4xeh
Ye
e -
high permeability streak, single linear flow n
r~
h
~- high permeability streak, double linear flow
2 7cr.z
h/2
Table 6- Linear Reservoirs Solutions With Dimensionless Time Based on Length of the
Reservoir, t~~,
(Constant Rate Inner Boundary)
Case
Solution
Constant Rate
Closed Reservoir
~WDM=( }WDL=2n{[;+tDL]-;:(;~~,-n2m2tDL~
Constant Rate
Constant Pressure
Outer Boundary
Reservoir
~wDM. ~..L=2z{I- n~ ~~P[~rDL]}
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SPE 39972
ANALYSISOF LINEAR FLOW IN GAS WELL PRODUCTION
Table 7- Linear Reservoirs Solutions With Dimensionless Time Based on Length of the
Case
Constant pti
Closed Reservoir
Constant pti
Constant Pressure
Outer Boundary
Reservoir
Reservoir,
tDL
(Constant PM Inner Boundary)
Solution
1
[1
r
Ac 1
n
—= —— =
qDLN
L qDL m ~x ‘n2~2 ~
x P[
4
DL
nd,i
1
[
= 5
={1+2 Lz2.2,DL1}
DLN
Tahln fI Intmmretatirm Emlations
for Linear low
----
“ . . . ~..r..-.. . . . -~
-------- --- —------ - -— --
Case
GA,
Vp
Constant Rate
(Oil Production)
&A = 79.65 qBp
c ~~mCRL
V, =8.962@~
c; ‘CRL
Constant pti
(Oil Production)
diAC =
125.1 Bp
r
P
i
““1991*5
Constant Rate
(Gas Production)
Ac =
803.2qg T qgT ~
~mmCRL
V, = 90.36 —
@cf ), mcRL -“
Constant pti
(Gas Production)
&AC =
1262T
[m(Pi)-m(pWf)lJ(@ Pct)imCF’L
“ ‘20081.(Pi)-m~wf)~
r
t
ehs
—..
CI )i ‘CPL
153
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1
AHMED H. EL-BANBI,AND ROBERTA. WATTENBARGER
SPE 3997
Table 9- Data for Example Well
Initialpressure,pi
8800 psia
bottom-holeflowingpressure,PM 1600 psia
pseudo-pressureat PI,m(pi)
2.67
X
109
psizlcp
pseudo-pressureat PM,m(pti)
1,69x108
psizfcp
gas specificgravity,7g
0.717
reservoir temperature, T 290 ‘F
formation net pay thickness, h
92
ft,
formation porosity, 1
0.15
average water saturation, Sw
0.47
totalcompressibilityat Pi, cti
3.53 x 10-5
psi-’
Table 10- Estimated and Calculated Parameters
for Exan
Estimated Parameters by
Regression
Ac
XL
Calculated Parameters
O P
de Well
10,423
273,678,121
6.93
L
4
+
f
w
h
Fig. 1- Linear reservoir model.
2V
f
w
h
model a (linear slab)
2Y,
4
w
q
2X= 2xf
h
model c (hydraulic frecture)
?+
5
xf = 2X
h
model b (hydraulic fracture extends to boundaries)
md’n.~
ft3.
Bscf
154
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SPE 39972
ANALYSIS OF LINEAR FLOW IN GAS WELL PRODUCTION
model d (wel l in a slab reservoir)
Y’
odel e (high permeability streak, single l inear f low)
Fig.
2- Different
.,,
,., -
.,, . ..
, , ,,
.,
.,,
,,
.,,
,,,
,., .
,,,
,,.
,,.
,,
,,.
I
I
I
,,,
I
.,
.,, .
,. ,,.
.,
model f (high permeabil ity streak, double linear flow)
linear flow models.
1
r
,,,,
,,,
,,,
I m
i
P,,,,,,,.,,
r
,,, ,,.
,,, ,,
E=
,,
,, >,,
,,, ,,
,,, ,,
,,, ,,
,,, ,
1
,,,
,,,,,
,., ,,
,,.
,.,
,, .
,,
”,,.
J
,,
:,, .
r
,.,,,
.4
.,, ,’
,,, ,, ,,
,,, ,, .,,
1
.
,,,
,,,
I
L L——————
1 E.03
1.E-02
1.E.01 1.E
3
1.E.02
1 E.ot i E+IXI
l,E+o1
1,E+02
tDAC
t~Ac
Fig. 3- Constant rate type curves for closed IInesr reservoirs.
Fig. 4- Constant pt i type curves for closed linear reservoirs.
I w
1
:
1
01
t)
,, .,
,,,
,,
,,,
,,
,,, ~,
,,
,,
,,
,,,
,,
,,
,,
—..
,> ,4,,,
,1,
,..
1
,,
,,.
,.,
,,,
,,,
,,,
— —- ----
—— r- -——
,,,
,,
,,,
,,
,,,
,,
,,
,,,
,,
,,,
,, .,,
,,,
)3 1 E-02
1.E.ot
I,E+w
l,E+o1
1.E+02
t~A
Fig. 5 - Constent rate type curve for constant pressure outer
boundary linear reservoirs. -
155
3 1,E.02
1.E.01
1 E+30
1.EtOi
t ~Ac
Fig. 6 -
Constant pti type curve for constant pressure o
boundary linear reservoirs.
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12
AHMED H. EL-BANBI, AND ROBERTA. WATTENBARGER
SPE 3
J’” “<4.1
,, .,.
,.,
.,,
.,
/
,.,
. .
1
.
, .,
1E.03 t .E.02 1.E.01 1 E+OO 1,E+Q1
1.E+02
t ~A=
Fig. 7- Constant rate and constant
pti
type curves for closed
linear reservoir.
‘“’mm
,Mo
G=
h.
.-
. -
8
E
~
.-
.-
.-
Q’
tm
to
I
1
I
to Ica t
10,000
Time (days)
Fig. 9- Log-log plot of cumulative gas production versus time for
example well.
5
0,03
0
0
10=2040
60 70 20
&Td0v5Y”
001
ca
P
,,
I—
1-
rcd -“ -
,Y-
Y
.2X”;
1,E.03 1 E.o2
l,E.01 1.E+w
1.E+Ol
t~“c
Fig. 8- Constant rate and constant pti type curves for cons
pressure outer boundary linear reservoirs.
To,coa
1,Seu
g
g
:
100
10
lW
1
,m
Time (days)
Fig. 10- Log-log plot of production rate versus time for exam
well
Fig. 11- Reciprocal of production rate versus square-root of time for example well.
156