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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

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-fed. * n~ ~0 M* by Me .%cw@ d Petroleum Engineam and are subject to

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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.

145



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.

146



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

147





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

149



6

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.

7.

8.

9.

10

11

Bagnall, W.D. andRyan,W.M. : “The Geology,Reserves, and

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presented at

the 1981 SPE/DOE Low Permeability

Symposium, Denver, CO., May, 27-29, 1981.

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Conference and Exhibition, New Orleans, LA, Sept. 26-29,

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Analysis of Tight Gas Well Production

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SPE/DOE Symposium on Low Permeability, Denver, CO.,

March, 14-16, 1983.

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

Symposium on Low Permeability, Denver, CO., March, 14-16,

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Anuner, J.R., Sawyer, W.K., and Drophin, M.J. : “Practical

Methods for Detecting Production Mechanisms in Tight Gas

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SPWDOE/GRl Unconventional Gas Recovery Symposium,

Pittsburgh, PA., May, 13-15, 1984.

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presented at Lhe SPE/DOE 1985 Low Permeability Gas

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“Fracture Half-Length and Linear

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Linear Reservoir,” Joutnal of the Institute of Petroleum,

Volume 48, Number 467- Nov. 1962,365-79.

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JPT,

(May, 1964), 561-563.

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13.

14.

15.

16.

17.

18.

19.

20.

21.

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and Performance Prediction of Low-Permeability Gas W

Stimulated by Massive Hydraulic Fracturing:

JPT

(M

1979) 362-372; Trans. AIME, 267.

Villegaa, M.E.

: Pe ormance of Fmctured Vertical

Horizontal Wells, Ph.D. Dissertation, Texas A&M Unive

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JPT

(May 1

624-636.

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150



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



8


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]}



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



1


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



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.



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

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