analysis of layered gas reservoir performance using a quasi-analytical solution for

Analysis of Layered Gas Reservoir Performance Using a Quasi-Analytical Solution for

Rate and Pressure Behavior

I Nengah Suabdi

Department of Petroleum EngineeringTexas A & M University

9 May 2001

OutlineOutline

Introduction Objectives Assumptions Semi-analytical solutions New Type Curves for Layered Gas Reservoirs Field Application Conclusions

IntroductionIntroduction

Depletion Performance Analysis:

Can single-layer model performance detect layering..? , layer volume..?, or effect of drawdown..?

Is a single layer model satisfactory..?

Fetkovich, M.J. et.al (1990)– Using numerical simulations

Layered-gas reservoir depletion study:

k2

k1 Layer-1

Layer-2

No-Crossflow

Single layer model..?where : k1>k2 ..?

3000

2500

2000

1500

1000

500

p/z

, ps

ia

25x106 20151050

Total Cumulative Gas Production (Gpt),MSCF

k1/k2 = 1

pi/zi

Single Layer or Equivalent Single

Layer Model

1.To provide a quasi-analitycal solution for the depletion performance of a well produced at a common production pressure in a layered gas reservoir.

2. To utilize this quasi-analytical gas flow solution as a mechanism for charac-terizing the performance of layered gas reservoirs.

ObjectivesObjectives

The proposed analysis techniques will be used to estimate the following properties for a layered gas reservoir system:

The permeability ratio (2-layer case).Layer productivity index (Jg)The total original gas-in-place (G).The total flow capacity (kh product).The moveable reserves in each layer.

ObjectivesObjectives

Schematic diagram of layered reservoir

Layer-1

Layer- 2

Layer- 3

Layer- n

Assumptions:

h1

Physical ModelPhysical Model

h2k2

k1

No-CrossflowProduction is commingled

Two-layer (dry) gas reservoir No crossflow in the reservoir Homogeneous (except klayer)

Bounded radial system (pseudosteady-state flow)

Production is commingled at a constant BHP

Layer-1

Layer-2

Gas Diffusivity Equation in terms of :pressure (Gas Diffusivity Equation in terms of :pressure (pp), ), pseudopressure (pseudopressure (pppp), and time :), and time :

is not constant because µ and ct are functions of pressure

tp

z

pk

rp

zp

r r

r1 ct

0.0002637

t

p

k

r

p r

r

r1 ptp c

0.0002637

surepseudopres p

:

dp z(p) )p(

p

: .al et Hussainy,-Al

p

b

p

pp

p

where

2

Plot of the Viscosity-Compressibility FunctionPlot of the Viscosity-Compressibility Function(Ansah (Ansah et.alet.al))

Dt

tii p CC

Semi-Analytical SolutionsSemi-Analytical Solutions

We can then develop the dimensionless decline rate (qDd), pressure (pD), and cumulative production (GpD).

We consider the "first-order polynomial model" for correlating the curves. This result is given by Ansah, et al. as:

The fundamental form of stabilized flow equation is given by


Where :

pwfp p - p J q gg

D

Dp

wDpt

tii

ti

gg dp

c c

c

J q

Scf/D/psiindex, typroductivi well Jg

c Sc Sc c fwcwggt

Gas MBE for moderate to low pressure reservoirs:


Where the dimensionless pressures are defined by:

or G

Gp -

zp

zp

i

i

1

GGp -

zp

zp

ii

1

z

p

p

zp

ii

D

Dimensionless Pressure (pD)


Where :

0 p ; t

p wDDd

D

0.5 1

1

0

wD

Dd wDwDwD

Dd wDwDwDwDD p;

tp - exp p - - p tp - exp p - p

pp1111

zp

zp

ii

wfwf

wD p

z

p

zp i

ijD

j

ii

D p zp or

zp

p

Dimensionless Decline Rate (qDd)


Where :

p ; t

q wD

n

1 j Ddj Dd 0

2 0.5 1

1

0exp

4

12

2

p;tp - p-p

tpexpp q wD

n

j

Ddj wDwDwD

)Ddj wD -wDDd

-11

(

layer of number total n index layer j

Dimensionless Cumulative Production (GpD)


Where :

0

exp11

exp1 1

1

2

p; tp - p - -p

tp - - p - wD

DdjwDwDwD

DdjwDwDpD

n

j

G

p ; t

wDDdj

DdjpD

n

j0

0.5 1

0.5

1

tG

ppD Recovery Fractional

G G

G

In field units, the dimensionless "decline" time is defined as:


Where :t = Time, dayskj = Permeability ( layer j), mdj = Porosity ( layer j), fractioncti = Total system compressibility, psia-1

re = Radius of the external boundary, ft

21

ln1 -21

1

- rr

rrrc

t t

wa

e2

wa

e2

wa tiiDdj

j

jk

0.00634

21

ln1 -21

1

- rr

rrr c

t k t

wa

e2

wa

e2

tiiDd

wa0.00634


Where :Cj = Stabilized flow coefficient layer-j, Mscf/D/psi2

kj = Permeability ( layer j ), mdj = Porosity ( layer j ), fractioncti = Total system compressibility, psi-1

pref = (pi + pwf)/2, psi

Gas rate production for each layer (qgj) in-term of (p/z)2 is defined as

zp

zp

C q2

wf

wf2

jg j

j

cp

z

rr

h k C

refpt

wa

e

jj

43

lnT1.4232

j

Pressure Depletion Decline Type Curve Pressure Depletion Decline Type Curve

3000

2500

2000

1500

1000

500

0

Av

era

ge

Re

se

rvo

ir P

res

su

re/z

-fa

cto

r (p

bar

,j /

z j )

, ps

ia

25x106 20151050


Legend: (Case: pwD = 0.1)

p/z (k1/k2) = 1

Case: (pwD = 0.1)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2zi,1 = zi,2 = 0.915, zwf = 0.978

pi,1 = pi,2 = 2500 psia

pwf = 265 psia (constant)

pwf/z = 271 psia (constant)

pi/zi = 2732 psia

k1/k2 = 1 k2

k1


3000

2500

2000

1500

1000

500

0

Avera

ge R

eserv

oir

Pre

ssu

re/z

-facto

r (

pb

ar,

j / z

j ),

psia

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

Case: (pwD = 0.1)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1/3


pi/zi = 2732 psia

k2

k1


3000

2500

2000

1500

1000

500

0

Avera

ge R

eserv

oir

Pre

ssu

re/z

-facto

r (

pb

ar,

j / z

j ),

psia

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

Case: (pwD = 0.1)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1/3

1x10-1


pi/zi = 2732 psia

k2

k1


3000

2500

2000

1500

1000

500

0

Avera

ge R

eserv

oir

Pre

ssu

re/z

-facto

r (

pb

ar,

j / z

j ),

psia

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

Case: (pwD = 0.1)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

1/3

1x10-1

1x10-2


pi/zi = 2732 psia

k2

k1


3000

2500

2000

1500

1000

500

0

Ave

rag

e R

eser

voir

Pre

ssu

re/z

-fac

tor

(pb

ar,j

/ z j

),

psi

a

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

Case: (pwD = 0.1)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3


pi/zi = 2732 psia

k2

k1

Vol Layer-1

Vol Layer-2

pwD = 0.1 G


3000

2500

2000

1500

1000

500

0

Ave

rag

e R

eser

voir

Pre

ssu

re/z

-fac

tor

(p

bar

,j /

z j )

, p

sia

25x106 20151050


Legend: (Case: pwD = 0.1 to 0.5)

p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2zi,1 = zi,2 = 0.915, zwf =(0.978 - 0.931)

pi,1 = pi,2 = 2500 psia

pwf = (265-1253) psia (constant)

k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3

pwf/zwf = 271 psia (constant)

pi/zi = 2732 psia


k2

k1

G


3000

2500

2000

1500

1000

500

0

Ave

rag

e R

eser

voir

Pre

ssu

re/z

-fac

tor

(p

bar

,j /

z j )

, p

sia

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3


pi/zi = 2732 psia



k2

k1

G


3000

2500

2000

1500

1000

500

0

Ave

rag

e R

eser

voir

Pre

ssu

re/z

-fac

tor

(pb

ar,j

/ z j

),

psi

a

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3


pi/zi = 2732 psia




k2

k1

G


3000

2500

2000

1500

1000

500

0

Ave

rag

e R

eser

voir

Pre

ssu

re/z

-fac

tor

(p

bar

,j /

z j )

, p

sia

25x106 20151050



p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3


pi/zi = 2732 psia





k2

k1

G

Depletion Decline Rate Type Curve Depletion Decline Rate Type Curve

10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t

Properties: (pwD = 0.1 - 0.5)

k1/k2 = varying from 1 to 1000

h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2zi,1 = zi,2 = 0.915, zwf = (0.978 - 0.931)

pi,1 = pi,2 = 2500 psia

pwf = (265 - 1253) psia (constant)

Legend: (Case : pwD = 0.1)

pwD = 0.1 (pwf = 265 psia)

1x103

1x102

1x101

k1/k21

Layer-1 (m ore perm eable layer)

Layer-2 (less perm eable layer)

k2

k1

Rate Depletion Decline Type Curve Rate Depletion Decline Type Curve

10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t



h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


Legend: (Case : pwD = 0.1 - 0.2)

pwD = 0.1 (pwf = 265 psia)

pwD = 0.2 (pwf = 510 psia)

1x103

1x102

1x101

k1/k21



k2

k1


10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t



h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia



pwD = 0.1 (pwf = 265 psia)

pwD = 0.2 (pwf = 510 psia)

pwD = 0.3 (pwf = 775 psia)

1x103

1x102

1x101

k1/k21



k2

k1


10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t



h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2

zi,1 = zi,2 = 0.915, zwf = (0.978 - 0.931)

pi,1 = pi,2 = 2500 psia



pwD = 0.1 (pwf = 265 psia)

pwD = 0.2 (pwf = 510 psia)

pwD = 0.3 (pwf = 775 psia)

pwD = 0.4 (pwf = 1010 psia)

1x103

1x102

1x101

k1/k21



k2

k1


10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t



h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia



pwD = 0.1 (pwf = 265 psia)

pwD = 0.2 (pwf = 510 psia)

pwD = 0.3 (pwf = 775 psia)

pwD = 0.4 (pwf = 1010 psia)

pwD = 0.5 (pwf = 1253 psia)

1x103

1x102

1x101

k1/k21



k2

k1

GGpDpD vs. Dimensionless Decline Time ( vs. Dimensionless Decline Time (ttDdDd) )

10-3

10-2

10-1

100

101

102

Dim

ensi

on

less

Cu

mu

lati

ve G

as P

rod

uct

ion

(G

pD )

10-2

10-1

100

101

102

103

104

105

Dimensionless Decline Time (t Dd,t )


p/z (k1/k2) = 1

p/z (k1/k2) = 1x101

p/z (k1/k2) = 3x101

p/z (k1/k2) = 1x102

p/z (k1/k2) = 3x102

p/z (k1/k2) = 1x103

Properties: (Case: pwD = 0.1)

k1/k2 = varying from 1 to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2

zi,1 = zi,2 = 0.915, zwf = 0.978

pi,1 = pi,2 = 2500 psia


k1/k2 = 1 1x10

1

3x101

1x102 3x10

2 1x10

3

Layer-1 (more permeable layer)

Layer-2 (less permeable layer)

k2

k1

Stabilized Gas Flow Coefficient (cStabilized Gas Flow Coefficient (cjj))

102

103

104

105

106

107

108

(p

bar,

j / z

j )2

- (

pw

f / z

wf )

2,

psia

2

101

102

103

104

105

106

107

108

109

qg,j , Mscf/D


( p/z )2 - ( pwf/zwf )

2 (k1/k2) = 1

( p/z )2 - ( pwf/zwf )

2 (k1/k2) = 3

( p/z )2 - ( pwf/zwf )

2 (k1/k2) = 1x10

1

( p/z )2 - ( pwf/ zwf )

2 (k1/k2) = 1x10

2

( p/z )2 - ( pwf/ zwf )

2 (k1/k2) (k1/k2) = 1x10

3

Case: (pwD = 0.1)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2

zi,1 = zi,2 = 0.915, zwf = 0.978

pi,1 = pi,2 = 2500 psia


Layer-2 Layer-1

k1/k2 = 1 3 1x101 1x10

2 1x10

3

k2

k1

p/z vs G vs GpD,t pD,t Function

1.0

0.8

0.6

0.4

0.2

0.0

p/z

fu

nc

= (

p/z

- p

wf/z

wf)

/ (p

i/zi -

pw

f /z w

f )

1.00.80.60.40.20.0 GpD ,t func = Gp,t /G

Case: (pwD = 0.1-0.5)

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2zi,1 = zi,2 = 0.915, zwf = (0.978-0.931)

pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3

Legend: (Case: pwD = 0.1-0.5)

p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

Layer-1( more permeable layer )

Layer-2( less permeable layer )

Field Application (Field Application (p/zp/z vs. vs. GGptpt Curve Example)Curve Example)

Well Beavers 1-11 (Hugoton Field, Kansas, USA)400

350

300

250

200

150

100

50

0

p/z

, p

sia

30,00025,00020,00015,00010,0005,0000

Total Cumulative Gas Production (Gpt), MMSCF

Well Beavers 1-11("parent well")

Kansas Hugoton Field

p/zp/z versus Gpt —Cartesian format. versus Gpt —Cartesian format.

Well Beavers 1-11 (Hugoton Field, Kansas, USA)400

350

300

250

200

150

100

50

0

p/z

, p

sia

30,00025,00020,00015,00010,0005,0000

Total Cumulative Gas Production (Gpt), MMSCF



k1/k2 = 68

k2/k1 = 1 x 68-1

pw f/zw f = 20

More Permeable Layer

Less Permeable Layer

G = 24.11 BSCF

qqgg versus prod time —semilog format. versus prod time —semilog format.

Well Beavers 1-11 (Hugoton Field, Kansas, USA)

10-1

100

101

102

103

104

Ga

s P

rod

uc

tio

n R

ate

, q

g, M

SC

F/D

70x103 605550454035302520151050

Total Production Time , Days



Legend: Data Model

qqgg versus prod time —log-log format. versus prod time —log-log format.


10-1

100

101

102

103

104

Ga

s P

rod

uc

tio

n R

ate

, q

g, M

SC

F/D

10-1

100

101

102

103

104

105




Legend: Data Model

GGptpt versus prod time —semilog format.versus prod time —semilog format.


10-1

100

101

102

103

104

105

To

tal C

um

ula

tiv

e G

as

Pro

du

cti

on

, G

p,t, M

MS

CF

20x103 1614121086420




Legend: Data Model

GGptpt versus prod time —log-log format.versus prod time —log-log format.


10-1

100

101

102

103

104

105

To

tal C

um

ula

tiv

e G

as

Pro

du

cti

on

, G

p,t, M

MS

CF

101

102

103

104

105

106




Legend: Data Model

Estimate properties of Well Beavers 1-11Estimate properties of Well Beavers 1-11

- Total original gas-in-place (G) = 24.11 BSCF

- The permeability ratio (k1/k2) = 68

- Total reservoir thickness, (htot) = 130 ft

- Average reservoir radius, (re) = 3,250 ft

- Average area each layer, (Aavg) = 761.76 Acres

- The total flow capacity, (kh product) = 482 md-ft

- The magnitude of wellbore F. Press (Pwf) = 20 psia

Field Application (Rate typeField Application (Rate type Curve Example)Curve Example)

Nelson well (Hugoton Field, Kansas (USA))

10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t



h1 = h2, s1 = s2 = 0, 1 = 2

re,1 = re,2, cti,1 = cti,2 , i,1 = i,2

zi,1 = zi,2 = 0.915, zwf = (0.978 - 0.931)

pi,1 = pi,2 = 2500 psia



pwD = 0.1 (pwf = 265 psia)

pwD = 0.2 (pwf = 510 psia)

pwD = 0.3 (pwf = 775 psia)

pwD = 0.4 (pwf = 1010 psia)

pwD = 0.5 (pwf = 1253 psia)1x103

1x102

1x101

k1/k21

101

2

4

6

102

2

4

6

103

2

4

6

104

q, M

SC

F/D

10-2

10-1

100

101

102

103

104

t, Days

Legend: Nelson Well (Hugoton Field)

Matching parameters (Nelson Well)

1.0 q MPDd 1.0 t MPDd

MScf/D500 q MPg Days100 t

MP

10k

k

MP2

1

0.2 p MPwD

2

1

k

k = 10

Jg = 51. 553 Scf/D/psi G = 103.106 x 106 Scf or 0.103 Bscf

Field Application (Rate typeField Application (Rate type Curve Example)Curve Example)

Gas Well- B

10-4

10-3

10-2

10-1

100

101

q D

d

10-2

10-1

100

101

102

103

104

105

t Dd,t



h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia



pwD = 0.1 (pwf = 265 psia)

pwD = 0.2 (pwf = 510 psia)

pwD = 0.3 (pwf = 775 psia)

pwD = 0.4 (pwf = 1010 psia)

pwD = 0.5 (pwf = 1253 psia)1x103

1x102

1x101

k1/k21

102

2

4

6

103

2

4

6

104

2

4

6

105

qg, M

SC

F/D

100

101

102

103

104

105

106

t , Days

Matching parameters:

1.0 qMPDd 1.0 tMPDd

MScf/D500 q MPg Days100 t

MP

10k

k

MP2

1

0.2 pMPwD

2

1

k

k = 10

Jg = 674.327 Scf/D/psi G = 4.495 x 109 Scf or 4.495 Bscf

1. We successfully demonstrated the use of a semi-analytical solution for a single-layer gas system for layered gas reservoir cases presented by Fetkovich, et.al (numerical simulations).

2. A two-layer type curve was developed for the analysis of production performance. The single- layer case can not be used to model the 2-layer case.

3. The sensitivity of individual layer properties was investigated, in particular — permeability ratio, layer volumes, and the effect of drawdown.

Conclusions

Analysis of Layered Gas ReservoirUsing Production Data

I Nengah Suabdi

Department of Petroleum EngineeringTexas A & M University

3 February 2001

Field Application (Example)Field Application (Example)

Curtis well (Hugoton Field, Kansas, USA)

1.0

0.8

0.6

0.4

0.2

0.0

p/z

fu

nct

ion

1.00.80.60.40.20.0

GpD,t function

Case: ( pwD = 0.1-0.5 )

k1/k2 = 1x10-3

to 1x103

h1 = h2, s1 = s2 = 0, 1 = 2


pi,1 = pi,2 = 2500 psia


k1/k2 = 1

3

1x101

1x102

k1/k2 = 1x103

1/3

1x10-1

1x10-2

k1/k2 = 1x10-3 Legend:

p/z (k1/k2) = 1

p/z (k1/k2) = 3 or 1/3

p/z (k1/k2) = 1x101 or 1x10

-1

p/z (k1/k2) = 1x102 or 1x10

-2

p/z (k1/k2) = 1x103 or 1x10

-3

1.0

0.8

0.6

0.4

0.2

0.0

1.00.80.60.40.20.0

Legend: HUGOTON_FIELD Data Well Curtis1_A

Less Permeable Layer

analysis of layered gas reservoir performance using a quasi-analytical solution for

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