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Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells Tom BLASINGAME | [email protected] | Texas A&M U. Decline-Curve Analysis for Gas Wells At the end of this module, you will: Be able to sketch the Arps exponential, hyperbolic, and harmonic decline relations. Be able to state and derive the exponential rate decline relation (Eq. 9.2). Be able to derive Eq. 9.10 (cumulative exponential) and explain its practical aspects. Be able to state the form of the harmonic rate decline relation (Eq. 9.11). Be able to derive Eq. 9.16 (cumulative harmonic) and explain its practical aspects. Be able to state the form of the hyperbolic rate decline relation (Eq. 9.1) Be able to derive Eq. 9.21 (cumulative hyperbolic) and explain its practical aspects. Be able to describe the concept of a "decline type curve". Be able to sketch the Fetkovich "rate/time decline type curve" (i.e., Fig. 9.10). Be able to explain the flow regimes seen on the Fetkovich decline type curve. Be able to apply the dimensionless variables given by Eqs. 9.22-9.24. Be able to state the procedure to apply the Fetkovich decline type curve. Be able to demonstrate the solution of Example 9.2 (reproduce all details). Be able to sketch the Carter "rate/time decline type curve" (i.e., Fig. 9.11). Be able to explain the flow regimes seen on the Carter decline type curve. Be able to state the procedure to apply the Carter decline type curve. Be able to demonstrate the solution of Example 9.3 (reproduce all details). Be able to explain the strengths/limitations of using "decline type curve" analysis. Be familiar with "decline type curve" analysis without boundary-dominated flow. Be familiar with "decline type curve" analysis for multi-well reservoirs. From: Lee, W.J. and Wattenbarger, R.A.: Gas Reservoir Engineering, SPE (1996). Slide — 1
Transcript

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Decline-Curve Analysis for Gas Wells

At the end of this module, you will:●Be able to sketch the Arps exponential, hyperbolic, and harmonic decline relations.●Be able to state and derive the exponential rate decline relation (Eq. 9.2).●Be able to derive Eq. 9.10 (cumulative exponential) and explain its practical aspects.●Be able to state the form of the harmonic rate decline relation (Eq. 9.11). ●Be able to derive Eq. 9.16 (cumulative harmonic) and explain its practical aspects.●Be able to state the form of the hyperbolic rate decline relation (Eq. 9.1)●Be able to derive Eq. 9.21 (cumulative hyperbolic) and explain its practical aspects.●Be able to describe the concept of a "decline type curve".●Be able to sketch the Fetkovich "rate/time decline type curve" (i.e., Fig. 9.10).●Be able to explain the flow regimes seen on the Fetkovich decline type curve.●Be able to apply the dimensionless variables given by Eqs. 9.22-9.24.●Be able to state the procedure to apply the Fetkovich decline type curve.●Be able to demonstrate the solution of Example 9.2 (reproduce all details).●Be able to sketch the Carter "rate/time decline type curve" (i.e., Fig. 9.11).●Be able to explain the flow regimes seen on the Carter decline type curve.●Be able to state the procedure to apply the Carter decline type curve.●Be able to demonstrate the solution of Example 9.3 (reproduce all details).●Be able to explain the strengths/limitations of using "decline type curve" analysis.●Be familiar with "decline type curve" analysis without boundary-dominated flow.●Be familiar with "decline type curve" analysis for multi-well reservoirs.

From: Lee, W.J. and Wattenbarger, R.A.: Gas Reservoir Engineering, SPE (1996).

Slide — 1

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Decline-Curve Analysis for Gas WellsSummary of Arps Time-Rate and Time-Cumulative Relations

Exponential: (b=0)

Hyperbolic: (0<b<1)

Harmonic: (b=1)

Exponential: (b=0)

Hyperbolic: (0<b<1)

Harmonic: (b=1)

)exp( tDqq ii

bi

itbD

qq /1)(1

)(1 tDqqi

i

Time-Cumulative Production

)]exp([1 tDDqN ii

ip

])(1[1)(1

/11 bi

i

ip tbD

DbqN

)ln(1 tDDqN ii

ip

Time-Flowrate

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Slide — 2

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Arps Exponential Decline Relation

)exp( tDqq ii

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Flowrate

Constant"Decline"Parameter

InitialRate

Parameter

Time

Slide — 3

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Derivation of Arps Exponential Decline Relation

Oil Material Balance Relation:

Steps:1. Differentiate both relations with respect to time.2. Assume pwf = constant (eliminates d(pwf)/dt term).3. Equate results, yields 1st order ordinary differential equation.4. Integrate.5. Exponentiate result.

oio

tpssoiii B

BNcb

DtDqq 11 exp,

poio

ti N

BB

Ncpp 1

s

r

ACekh

Bbqbpp

wAoo

pssoopssowf 2,,14ln

21 141.2

Derivation of the Exponential Rate Decline Relation

Slide — 4

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Derivation of Arps Exponential Decline Relation (SPE 98042)

From

:B

lasi

ngam

e, T

.A. a

nd R

ushi

ng, J

.A.:

"A P

rodu

ctio

n-B

ased

Met

hod

for D

irect

Est

imat

ion

of G

as-

in-P

lace

and

Res

erve

s,"

pape

r SPE

980

42 p

rese

nted

at t

he 2

005

SPE

East

ern

Reg

iona

l Mee

ting

held

in M

orga

ntow

n, W

.V.,

14–1

6 Se

ptem

ber 2

005.

Slide — 5

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Arps Hyperbolic Decline Relation

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Flowrate

Constant"Decline"Parameter

InitialRate

Parameter

Time

bi

itbD

qq /1)(1 Arps

"Hyperbolic"Parameter

Arps"Hyperbolic"

Parameter

Slide — 6

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

a. Hyperbolic flowrate relations for the case of constant pressure production from a solution gas drive reservoir (Camacho and Raghavan (1989)).

b. Hyperbolic decline type curve with data simulation performance data superimposed (Camacho and Raghavan (1989)).

(Details of derivation are omitted, see paper SPE 19009, Camacho and Raghavan (1989)).

Hyperbolic Decline — Orientation

Discussion: Arps' Hyperbolic Time-Rate Relation●Assumes decline parameter based on average mobility/compressibility.●Assumes hyperbolic parameter = f[d/dt(average mobility/compressibility)].

Slide — 7

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Derivation of Arps Hyperbolic Decline Relation (SPE 98042)

From

:B

lasi

ngam

e, T

.A. a

nd R

ushi

ng, J

.A.:

"A P

rodu

ctio

n-B

ased

Met

hod

for D

irect

Est

imat

ion

of G

as-

in-P

lace

and

Res

erve

s,"

pape

r SPE

980

42 p

rese

nted

at t

he 2

005

SPE

East

ern

Reg

iona

l Mee

ting

held

in M

orga

ntow

n, W

.V.,

14–1

6 Se

ptem

ber 2

005.

Slide — 8

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Derivation of Arps Hyperbolic Decline Relation (SPE 98042)

From

:B

lasi

ngam

e, T

.A. a

nd R

ushi

ng, J

.A.:

"A P

rodu

ctio

n-B

ased

Met

hod

for D

irect

Est

imat

ion

of G

as-

in-P

lace

and

Res

erve

s,"

pape

r SPE

980

42 p

rese

nted

at t

he 2

005

SPE

East

ern

Reg

iona

l Mee

ting

held

in M

orga

ntow

n, W

.V.,

14–1

6 Se

ptem

ber 2

005.

Slide — 9

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Arps Harmonic Decline Relation

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Constant"Decline"Parameter

InitialRate

Parameter

Time

)(1 tDq

qii

Flowrate

Slide — 10

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Decline-Curve Analysis for Gas WellsSummary of Arps Rate-Cumulative Relations

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Flowrate- Cumulative ProductionExponential: (b=0)

Hyperbolic: (0<b<1)

Harmonic: (b=1)

p

i

ii N

qDqq exp

pNiDiqq

b

i

bi

p qDb

qNN

1)1(

)(or

Plot of: q versus Np

Plot of: log(q) versus Np

Plot of:log(N-Np) versus log(q)

iibp

i DbqN

NN

qq)(1

1 )1(1 Plot of:

log(q/qi) versus log[1-(Np/N)]

Slide — 11

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

1:1 Slope

1:3 Slope

Slide — 12

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Exponential Trend

Slide — 13

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Slide — 14

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

1:1 Slope

Slide — 15

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Harmonic Trend

Slide — 16

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Exponential Trend

Slide — 17

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

(Production) Decline Type Curve Analysis

Slide — 18

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Fetkovich "Empirical" Decline Type Curve:●Log-log "type curve" for the Arps "decline curves" (Fetkovich, 1973).●Initially designed as a graphical solution of the Arps' relations.

Fetkovich Decline Type Curve — Empirical

Slide — 19

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

From: SPE 04629 — Fetkovich (1973).

From: SPE 04629 — Fetkovich (1973).

"Analytical" Rate Decline Curves: Data from van Everdingen and

Hurst (1949), re-plotted as a rate decline plot (Fetkovich, 1973).

This looks promising — but this is going to be one really big "type curve."

What can we do? Try to collapse all of the trends to a single trend during boundary-dominated flow (Fetkovich, 1973).

"Analytical" stems are another name for transient flow behavior, which can yield estimates of reservoir flow properties.

Van Everdingen-Hurst Type Curves — Radial Flow

Slide — 20

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Fetkovich "Analytical" Decline Type Curve: (constant pwf)●Log-log "type curve" for transient flow behavior (Fetkovich, 1973).●First "tie" between pressure transient and production data analysis.

Fetkovich Decline Type Curve — Analytical Transient Radial Flow Stems

Slide — 21

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Fetkovich "Composite" Decline Type Curve:●Assumes constant bottomhole pressure production.●Radial flow in a finite radial reservoir system (single well).

Fetkovich Decline Type Curve — Analytical and Empirical Stems

Slide — 22

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Fetkovich Example Match: SPE 04629 — (Fetkovich)●Lack of early time data is an omen of things to come.●Late time data follow an exponential trend (constant pwf).

Fetkovich Example Analysis — Decline Type Curve

Slide — 23

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

From: SPE 04629 — Fetkovich (1973).

From: SPE 04629 — Fetkovich (1973).

Discussion:●Assumptions:Constant bottomhole pressure. "Liquid" flow (not gas).

●"Empirical" Decline Type Curve: "Arps" empirical trends presented in

dimensionless "decline" format for "boundary-dominated flow behavior" (Fetkovich, 1973).

●"Analytical" Transient Type Curve:Collapses the transient flow trends

(dimensionless "decline" rates) into "stems" related to reservoir size and skin factor (Fetkovich, 1973).

●Comments (gas flow behavior): Fetkovich (and others) have noted

that most gas cases lie on or near the stems for 0.4<b<0.6.

No (direct) physical support for this "rule of thumb."

Fetkovich Decline Type Curves — General

Slide — 24

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Fetkovich "Composite" Decline Type Curve: b>1 Cases●b=1 is the constant rate case — no theory to support b>1 cases.●Rule: Transient flow — q concave UP, PSS flow — q concave DOWN.

Fetkovich Decline Type Curve — b>1

Slide — 25

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Type Curves for Gas Wells:Gas cases cannot be fully represented

using Arps' (hyperbolic) relations. However, the Arps' relations are often an acceptable approximation.

Constant pwf gas cases are dependent on the pwf/pi ratio (path-dependent non-linearity) — and cannot be extended to variable-rate, variable pressure drop.

Reconstruction of Fetkovich (SPE 04629 —1973) and Carter (SPE 12917 — 1985) type curves for the gas case (various pwf/pi).

(Zoom View) Reconstruction of Fetkovich (SPE 04629 — 1973) and Carter (SPE 12917 — 1985) type curves for the gas case (various pwf/pi).

Reconstruction of Fetkovich-Carter Type Curves

Slide — 26

From

:Fe

tkov

ich,

M.J

. : "

Dec

line

Cur

ve A

naly

sis

Usi

ng T

ype

Cur

ves,

" JP

T(J

une

1980

) 106

5-77

.C

arte

r, R

.D.:

"Typ

e C

urve

s fo

r Fin

ite R

l and

Lin

ear G

as-F

low

Sys

tem

s: C

onst

ant-T

erm

inal

Pr

essu

re C

ase,

" SP

EI(O

ct. 1

985)

719

-28.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Fetkovich-Carter Decline Type Curve

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Slide — 27

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Example 9.1 — Lee and Wattenbarger

From

:Le

e, W

.J. a

nd W

atte

nbar

ger,

R.A

.: G

as R

eser

voir

Engi

neer

ing,

SPE

(199

6).

Slide — 28

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Modern Time-Rate Analysis

Slide — 29

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Work Path ― Analysis of Well Performance

Completions Production Reservoir Fluids Geomodel

Time-Rate

Time-Rate-

PressureReservoir

Model

Model: Time-RateBasis: Proxy model●Predictions●EUR●CorrelationsTime: Minutes/well

Model: Time-Rate-PressureBasis: Analytical/Numerical●Predictions●EUR/SRV●Estimate PropertiesTime: ~1 hour/well

Pres

sure

Time

Pres

sure

Time

Rat

es

Time

Rat

es

Time

Rat

es

Time

Model: Time-Rate-PressureBasis: Full Numerical●Predictions●EUR/SRV●Flow MechanismsTime: Days to weeks/well

Slide — 30

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Typical Flow Regimes in Unconventional Reservoir Systems

Required Model Parameters: ●Permeability (k)●Fracture half-length (xf)●Fracture conductivity (Fc)●Drainage area (A)●Skin factor (s)●Well length (Lw)●Number of fractures (nf)

Linear Flow:(fracture flow does not

interfere)

"SRV" Flow: ("depletion")(fracture flow does interfere)

"Post-SRV" Flow:("Compound Linear Flow")

Slide — 31

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

(Formation) Linear Flow — Theory

ktcBwfpipC

txfACq

hfxxfAt

xfAktcBwfpipq

DxftDp

1)( 128494.8

1 1

2 11)( 128494.8

1

Solution for a Single Fracture: (transient linear flow)

+ + + → 1)(

1]...

[

tottot

,4,3,

2,1,tot

tACq

tAAA

AACq

xf

nxfxfxf

xfxf

Note:These solutions are only valid for transient linear flow [i.e., the case of non-interfering pressure distributions (due to the fractures)].

Slide — 32

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

(Formation) Linear Flow — Practice (Synthetic Example)

Formation Linear Flow●Log-log diagnostic plot: log[q(t)] versus log[t ] (slope = -1:2)●"qDb" (time-rate) plot: log[q(t)] log[D(t)] log[b(t)] versus log[t ]●"Traditional" plot: q(t) versus 1/SQRT[t ] (straight-line portion)●Extrapolation using a linear flow model will over-predict EUR…

Region ofover-

extrapolation…

Slide — 33

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Time-Rate Relations — Power Law Exponential Rate Model

Stretched Exponential: (SEM)— Observed Behavior of q(t):

— Differentiating to solve for D(t):

— Differentiating to solve for b(t):

Power-Law Exponential: (PLE)— Observed Behavior of D(t):

— Integrating to solve for q(t):

— Differentiating to solve for b(t):

ˆ)()(

1)( )1( nitDnD

dttdq

tqtD

] ˆ exp[ ˆ)( nii tDtDqtq

nn

i

i ttDDnnDntb

2)1( ] ˆ[

)1(ˆ)(

1)()(

1)( nntndttdq

tqtD

])/(exp[ ˆ)( ni tqtq

nntnntb

1)(

Discussion:●Models are the same when D∞ = 0.●The Power-Law Exponential model was derived from observations (Blasingame/Ilk).●The Stretched-Exponential model was taken from a statistics text (Valko).

Literature:●Kohlrausch (1854).●Phillips (1996).●Kisslinger (1993)●Decays in random, disordered,

chaotic, heterogeneous systems (e.g., relaxation, aftershock decay rates, etc.).

])/( exp[ˆ)( ni tqtq

Valkó (2009)

Jones (1942) and Arps (1945)

)1( 100

exp )(1

mtD

qtqm

oo

Slide — 34

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Time-Rate Relations — Power-Law Exponential Rate Relation

PLE Rate Relation:

Decline Function: D(t)

Hyperbolic Function: b(t)

nn

i

i ttDDn

nDn

tDdtdtb

2)1( ] ˆ[

)1(ˆ

)(1 )(

]ˆexp[ˆ)( nii tDtDqtq

)1(ˆ

1)(

nitDnD

dtdq

qtD

Ilk, D.: "Well Performance Analysis for Low to Ultra-Low Permeability Reservoir Systems," Ph.D. Dissertation, Texas A&M University, College Station, TX (Aug 2010).

Clean-up/ flowback effects

are not significant for

this case

Slide — 35

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U. Slide — 36

Introduction — Modern Time-Rate Relations

DCA Model Rate Relation

Power Law Exponential Model

Stretched Exponential Decline Model

Duong Model

Logistic Growth Model

Weibull Model

]ˆexp[ˆ)( nigi tDtDqtq

]]/[exp[ˆ)( ni tqtq

1at ,)1(1

exp)( 11

1

tqtmatqtq mm

ttMtq exp)(

1

)ˆ(

ˆˆ)(

n

n

ta

tnKatq

PLE: Derived by introducing terminal decline D∞ in D-parameter for matching BDF.SEDM: Linear superposition of simple exponential decays — analogous to PLE.Duong: Straight line behavior of q/Gp vs. Time (log-log) plot for linear/bilinear flow regime.LGM: Population growth models — modified form of hyperbolic logistic growth models.Weibull: Application of Weibull distribution for modeling time to failure.

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Flow Regimes: (Barnett Shale Example)●Schematic illustrates flow regimes exhibited by time-rate-pressure data.●Duration/existence of flow regimes is DIFFERENT for each play.

EURLF (VERY OPTIMISTIC)

EURDep (CONSERVATIVE ???)

Pseudo-elliptical flow regime (flow from matrix to collection of fractures) might exist after fracture interference.

Slide — 37

Time-Rate Behavior — Flow Regimes for a Multi-Fracture Horizontal Well

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Calibration — Linear Flow (Gas Shales)Data taken from publicly available sources — Horizontal Shale (Dry) Gas Wells ONLY

Heckman, T.L., et al (2013): Best Practices for Reserves Estimation in Unconventional Reservoirs — Present and Future Considerations, Keynote presentation presented at the 2013 SPE Unconventional Resources Conference, The Woodlands, TX (USA), 10-12 April 2013.

Discussion:●START of "Linear Flow" (~3-6 months).●END of "Linear Flow" (~9-36 months).● "Linear Flow" is represented by b = 2.●EUR requires at least 20+ months (except Haynesville ~1 year; and Barnett ~3 years).

Slide — 38

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

Discussion:●START of "Linear Flow" (~3-6 months).●END of "Linear Flow" (~9-36 months).● "Linear Flow" is represented by linear trends on these plots.●Square root time plot used to show linear portion of trend (Gp(t) vs. SQRT(t) is most clear).

Data taken from publicly available sources — Horizontal Shale (Dry) Gas Wells ONLY

Heckman, T.L., et al (2013): Best Practices for Reserves Estimation in Unconventional Reservoirs — Present and Future Considerations, Keynote presentation presented at the 2013 SPE Unconventional Resources Conference, The Woodlands, TX (USA), 10-12 April 2013.

Calibration — Linear Flow (Gas Shales)

Slide — 39

Petroleum Engineering 613 — Natural Gas Engineering Lecture 8 — Decline-Curve Analysis for Gas Wells

Tom BLASINGAME | [email protected] | Texas A&M U.

(Sort of) "Big Data" Analysis ― Barnett Shale Example (Data prior to Mar 2013)

Slide — 40

Correlation of Gp,1Yr vs. Initial Gas Production (Barnett Shale horizontal gas wells).

Correlation of Gp,1Yr using Initial Gas Production and various completion parameters (Barnett Shale horizontal gas wells).

Histogram of Gp,1Yr (Barnett Shale horizontal gas wells).

Histogram of EUR30Yr (Barnett Shale horizontal gas wells).

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