20120926 Askabe MS Pres (Pptx)

Integration of Production Analysis and Rate-Time Analysis

via Parametric Correlations — Montney Shale Case Histories

Yohanes ASKABEDepartment of Petroleum Engineering

Texas A&M UniversityCollege Station, TX 77843-3116 (USA)

[email protected]

Status PresentationCollege Station, TX (USA) — 12 August 2012

Status Presentation — Yohanes ASKABE — Texas A&M UniversityCollege Station, TX (USA) — 12 August 2012

Integration of Production Analysis and Rate-Time Analysis via Parametric Correlations Montney Shale Case Histories Sl

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(Alt.) Rate-Decline Relations for Unconventional Reservoirs and

Development of Parametric Correlations for Estimation of Reservoir Properties

●Objectives●Introduction●Rate-Time Models:— PLE Model— Logistic Growth Model (LGM)— Duong Model

●Models performance analysis●Modified rate decline models●A Parametric correlation study●Methodology:— Analysis of time-rate model parameters— Correlation of time-rate model parameters with reservoir/well

parameters— Development of Parametric Correlations

●Conclusions and Recommendations

Outline:



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■Ilk et al., (2011) have demonstrated that rate-time parameters can be correlated with reservoir/well parameters using limited well data from unconventional reservoirs.

■Theoretical verification and analysis of large number of high quality field data is necessary to test and verify the parametric correlations that correlate reservoir/well parameters with time-rate model parameters.

■This study will provide the opportunity to investigate performance of modern time-rate models in matching and forecasting rate-time data from unconventional reservoirs. The models considered are:

— PLE Model— Logistic Growth Model (LGM)—Duong Model

Objectives/ Problem Statement:



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■Modern time-rate models (PLE) have been shown to provide accurate EUR estimates and forecast future production when bottomhole flowing pressure (pwf) is constant.

■Time-Rate model Constraints:—Constant Bottomhole Pressure (pwf)—Constant Completion Parameters (Well lateral length, xf....)

■ Time-Rate model parameters can be correlated with reservoir/well parameters (k, kxf, EUR)

■A diagnostic Approach—Diagnostic Plots—Data Driven matching process

Introduction



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'qdb' type diagnostic plot—discussed below

●Time-Rate Analysis: Base Definitions■Based on the "Loss Ratio" concept (Arps, 1945).■Loss Ratio:

■Loss Ratio Derivative:

●Approach■Continuous evaluation of D(t) and b(t) relations provide a

diagnostic method for matching time-rate data.■Diagnostic relations are used to derive empirical models.

dtdqq

D g

g

/1

dtdqq

dtd

Ddtdb

g

g

/1

Governing Relations: Time-Rate Definitions



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●History:■SPE 116731 (Ilk et al., 2008)■Derived from data (D(t) and b(t))■Analogous to Stretched-Exponential, but derived independently■Has a terminal term for boundary-dominated flow (D∞)

●Governing Relations:■Rate-Time relation:

■PLE Loss Ratio relation:

■PLE Loss Ratio Derivative relation:

]ˆexp[ˆ)( nigi tDtDqtgq

DtDntD n

i1ˆ)(

2)ˆ()1(ˆ)(

ni

ni

ntDtDntnDtb

Time-Rate Analysis: Power Law Exponential



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●History:■SPE 137748 (Duong, 2011)■Based on extended linear/bilinear flow regime■Derived from transient behavior of unconventional-fractured

reservoirs■Relation extracted from straight line behavior of q/Gp vs. Time

(Log-Log) plot

●Governing Relations:■Duong Rate-Time relation:

■Duong Loss Ratio relation:

■Duong Loss Ratio Derivative relation:

)1(

1exp)( 1 mm

gig tmatqtq

matmttD 1)(

2)()()( m

mm

mtatattmttb

Time-Rate Analysis: Duong Model



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●History■SPE 144790 (Clark et al., 2011)■Adopted from population growth models■Modified form of hyperbolic logistic growth models

●Governing Relations:■LGM Cumulative and Rate-Time relation:

■LGM Loss Ratio relation:

■LGM Loss Ratio Derivative relation:

2

)1(

)()( n

n

g taaKnttq

)()1()( n

n

tattnanatD

2

222

))1(()1()1(2)1()( n

nn

tnanatntnanatb

Time-Rate Analysis: Logistic Growth Model (LGM)

n

n

g taKttQ

)(



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● PLE Model■ Transient■ Transitional and■ boundary-dominated

flow regimes.

● LGM Model■ Transient and■ Transitional flow

regimes.

● Duong Model■ Transient flow

regimes.

●Well 1: k = 2000 nD



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Theoretical Consideration: Time-rate analysis

●Well 1: k = 2000 nD

● PLE■ Excellent time-rate

data match.■ Accurate estimate of

EUR is possible.● LGM and Duong Models

■ Excellent match during Transient flow regimes.

■ Lack boundary conditions.

■ EUR is overestimated.



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● PLE, LGM and Duong Models.■ All models match transient flow-regimes very well.■ In the absence of boundary-dominated flow, all models provide reliable

EUR estimate.

●Well 2: k = 50 nD



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●Well 2: k = 50nD

● In the absence of boundary-dominated flow, PLE, LGM and Duong Models can:■match transient flow

regimes very well and■provide good estimate

of EUR.



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Integration of Production Analysis and Rate-Time Analysis via Parametric Correlations Montney Shale Case Histories

●Modified Time-Rate Relations



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●Modified Duong Model ■With boundary parameter, DDNG

■Boundary-dominated flow can be modeled.■Derivation is based on loss-ratio definition. The modified form of

loss-ratio relation is given by:

■It is derived by assuming constant loss-ratio during boundary-dominated flow regimes.■New time-rate relation can be derived from the loss-ratio relation.

It is given by:

■Cumulative production relation can not be derived. Numerical methods are necessary.

tDt

matqtq DNG

mmg 1

1exp)( 1

1

mDNG at

tmDtD )(

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38



Modified Time-Rate Models: Duong Model – (MODEL 1)

●Modified Duong Model ■The loss-ratio derivative is given by:

●Modified Duong Model ■ Boundary-dominated flows can be modeled.■ EUR estimates are constrained.■ Exponential decline characterizes boundary-dominated flow.

2))(()()(tDmtat

tatmttbDNG

m

mm

DNGD

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38



(Cont.) Modified Time-Rate Models: Duong Model - (MODEL 1)



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Modified Duong Model: 'qdb' type diagnostic plot. (MODEL 1)

● Derived based on loss-ratio derivation of Duong Model.

● Modified Duong Model ■ Boundary-dominated flows

can be modeled.■ EUR estimates are

constrained.■ Exponential decline

characterizes boundary-dominated flow.

DNGm Dat

tmtD )(

Added Constant

●Modified Duong Model ■With boundary parameter DDNG

■Boundary-dominated flow can be modeled.■Based on q/Gp Vs. time diagnostic plot.■New q/Gp model-relation:

■New time-rate relation:

■New Cumulative production relation:

],1[],1[)1(exp)( 11 tDmDmaDtDtqtq DNGDNG

mDNGDNGmg

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38



Modified Time-Rate Models: Duong Model - (MODEL 2)

]exp[ tDatGq

DNGm

p

],1[],1[exp)( 11 tDmDmaDDaqtG DNGDNG

mDNGDNGp

●Cont. (Model parameters)■The loss-ratio relation is given by:

■The loss-ratio derivative is given by:

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38




]exp[)( tDattmDtD DNG

mDNG

2)(]exp[

)(]exp[]exp[)(tDmttDat

tDmatmttDttDtbLGM

mLGM

LGMm

LGMm

LGM

Slid

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19/

38



Modified Time-Rate Models: Duong Model – (MODEL 2)● q/Gp vs. Time — Diagnostic Plot

● On log-log plot of q/Gp vs. time:■ Transient flow can be

characterized by a power-law relation, and

■ Boundary-dominated flow can be characterized by an exponential decline relation.

■ q/Gp data can be matched with the following relation:

]exp[ tDatGq

DNGm

p





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●Duong Model

Slid

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38



Modified Time-Rate Models: Duong Model (Cont.)

mGq

dtd

qGt

p

p

●Modified Duong Model - (Model-2)

tDmGq

dtd

qGt DNG

p

p

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38



Duong Model: Diagnostic Plot (Cont.)- Montney Shale Wells

p

p

Gq

dtd

qGt vs. time Diagnostic Plot

● m – Duong parameter describes rock-types, stimulation practices and fracture properties.

Slid

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38



Modified Time-Rate Models: Duong Model - (MODEL 2)● Numerical Simulation Case, k=8µD.● Model shows excellent data match for all flow regimes.

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Modified Time-Rate Models: Models Comparison

● Model Comparison■ Duong Model■ Model 1 and■ Model 2

● Modified Duong Models provide a better match

● EUR is constrained.

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Modified Time-Rate Models: Models ComparisonNumerical Simulation Case (k = 8 µD)

● Model Comparison■ Duong Model■ Model 1 and■ Model 2

● Modified Duong Models provide excellent match to Transient, Transition and boundary-dominated flow regimes.

● Duong Model can also match observed early time Skin and production constraints.



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●Modified Logistic Growth Model ■With boundary parameter DLGM

■Boundary-dominated flow can be modeled.■Modified LGM time-rate relation: Assuming exponential decline

during boundary dominated flow regimes.

■Modified LGM Loss Ratio relation:

■Modified LGM Loss Ratio derivative relation:

]exp[)(

)( 2

)1(

tDta

aKnttq LGMn

n

g

)()1()1()( n

LGMn

LGM

tattDnttDnatD

2

222

))1()1(()1()1(2)1()(tDnttDnatntnanatb

LGMn

LGM

nn

Modified Time-Rate Models: Logistic Growth Model (MODEL 3)



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Modified Logistic Growth Model: MODEL 3-qdb' type diagnostic plot.

● Modified Logistic Growth Model: ■ Boundary-dominated

flows can be modeled accurately

■ EUR estimates are constrained.

■ Exponential decline characterizes boundary-dominated flow.

● Prior knowledge of gas in place (K) is required.

● Direct formulation of Gp is not possible. Numerical methods are necessary.



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Modified Logistic Growth Model: (MODEL 4)

n

g

n

g

n

n

g

n

n

g

attQK

attQK

tta

tQK

taKttQ

1)(

1)(

)(

)(

● Using Diagnostic plot of [K/Qg – 1] vs. t or tmb

From LGM Model we have ● The last relation suggests that a log-log

plot of [K/Qgt – 1] versus time shows a power-law relation for transient flow regimes.

● Now, we can suggest the following relation with modification for boundary dominated flow regimes.

WhereK = Initial Gas in Place.R = Remaining Gas Reserve at t∞.

RtDattQK

LGMn

g

]exp[1)(



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Modified Logistic Growth Model: (MODEL 4)

● Now, we can derive the associated modified relations.

RtDattQK

LGMn

g

]exp[1)(

nLGM

LGMn

g ttDRatDKttQ]exp[)1(]exp[)(

21

]exp[)1()(]exp[)(

nLGM

LGMn

LGMg ttDRa

tDnKttDatq

R = Remaining Gas Reserve at t∞

RtQK t

g

1)(

lim

● Cumulative Production [Gp(t)] relation can be derived for Model 4.



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Modified Logistic Growth Model: Diagnostic Plot Corrected K/Q-1 Relation (MODEL 4)

● If K is known, we can estimate parameters a and n from the transient flow regime.

● .DLGM can be modified based on boundary behaviors.

a = 161n = 0.79K = 20,219,576.75Dlgm = 0.00029R = 0.157



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Modified Logistic Growth Model: Comparison

● Modified LGM models can match transient and boundary-dominated flow regimes better than LGM model.

● EUR is constrained.

● MODEL 4 provides a better match.

● Gp relation can be derived for MODEL 4.

● Prior knowledge of gas in place (K) is required.

(MODEL and MODEL 4)

■A horizontal well with multiple transverse fractures is modeled.

Reservoir Properties: Net pay thickness, h = 39.624 m Formation permeability, k = 0.25 µD - 5µD Wellbore Radius, rw = 0.10668 m Formation compressibility, cf

= 4.35E-7 kPa-1

Porosity, 𝝓 = 0.09 (fraction) Initial reservoir pressure, pi = 34,473.8 kPa Gas saturation, sg = 1.0 (fraction) Skin factor, s = 0.01 (dimensionless) Reservoir Temperature, Tr = 100 °C

Fluid Properties: Gas specific gravity, γg = 0.6 (air=1)

Hydraulically Fractured Well Model Parameters: Fracture half-length, xf = 50 m Number of Fractures = 20 Horizontal well length, l = 1,500 mProduction parameters: Flowing pressure, pwf = 3447.4 kPa Producing time, t = 10,598 days (30 Years)

Theoretical Consideration: Synthetic Case Examples

■The model inputs are as follows:

Transverse Fractures

Horizontal well with multiple transversefractures



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●Synthetic Examples■ 14 Models with permeability (k)

ranging from 0.25 µD - 5µD.■ All other reservoir/well and

fluid parameters are identical.

●PLE model parameters are related to EUR estimates from PDA



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Parameter Analysis: PLE Time-Rate Model

●PLE model parameters are related to permeability



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Parameter Analysis: PLE Time-Rate Model

■ A parametric correlation that relates reservoir permeability with rate-time model parameters can be produced.



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Parameter Correlation: Permeability

■ A parametric correlation that relates EUR estimates with rate-time model parameters can be produced.

■ The parametric correlation may not be unique.



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Parameter Correlation: EUR

●Field data example: Montney Shale, (Brassey) Wells

■Careful analysis of pressure/production data is necessary to accurately estimate reservoir/well parameters (k, EUR, xf).

■Decline curve analysis is then carried out to estimate EUR.



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Field Data Example: Permeability

■ EUR is normalized by initial BHP (Pi), and number of effective fractures.



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Field Data Example: EUR●Field data example: Montney Shale, (Brassey) Wells

■It is possible to integrate time-rate model parameters with reservoir/well parameters using parametric correlations.

■Parametric correlations solve the uncertainty regarding the number of unknown parameters in model based production data analysis.

■Modern rate decline models are successful at modeling different flow regimes observed from unconventional reservoirs. In summary:

— PLE Model ›Transient, transition, and, boundary-dominated flow regimes are

successfully modeled.— Logistic Growth Model (LGM)

›Transient, and transition flow regimes are successfully modeled.—Duong Model

›Only transient flow regimes are matched.›EUR is overestimated.›Doesn’t conform to ‘qdb’ type diagnostic plot.



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Conclusion:



Extra Slides



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Summary: Model ComparisonRate Decline Models

Model Relations Diagnostic Plots Recommendation

Power-law exponential

● Use diagnostic relation

Duong Model ● Use diagnostic relation

● Do not match boundary flow regimes.

Modified Duong ModelLogistic Growth Model (LGM)

Modified LGM


DtDntD ni

1ˆ)(

)1(

1exp)( 1 mm

gig tmatqtq

m

p

g atGq

],1[],1[)1(

exp)(1

1

tDmDmaDtD

tqtq

DNGDNG

mDNGDNG

mg

]exp[ tDatGq

DNGm

p



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Modified Logistic Growth Model: Corrected K/Q-1 Relation

● LGM K (Carrying capacity) is equivalent to Gas in Place volumetric estimate.

● Gas in place estimate should be available to use this model

● K/Q(t)-1 vs. tmb diagnostic plot can be used.





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)1(

1exp)( 1 mm

gig tmatqtq

2

)1(

)()( n

n

g taaKnttq

tDt

matqtq DNG

mmg 1

1exp)( 1

1


mDNGDNGmg

]exp[)(

)( 2

)1(

tDta

aKnttq LGMn

n

g

21

]exp[)1()(]exp[)(

nLGM

LGMn

LGMg ttDRa

tDnKttDatq

PLE

Duong

LGM

Modified LGM MODEL 2


Modified Duong MODEL 2






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DtDntD n

i1ˆ)( 2)ˆ(

)1(ˆ)(n

i

ni

ntDtDntnDtb

matmttD 1)(2)()()( m

mm

mtatattmttb

)()1()( n

n

tattnanatD

2

222

))1(()1()1(2)1()( n

nn

tnanatntnanatb

mDNG at

tmDtD )( 2))((

)()(tDmtat

tatmttbDNG

m

mm


mDNG

2)(]exp[


tDmatmttDttDtbLGM

mLGM

LGMm

LGMm

LGM

)()1()1()( n

LGMn

LGM

tattDnttDnatD

2

222

))1()1(()1()1(2)1()(tDnttDnatntnanatb

LGMn

LGM

nn

PLE

Duong

LGM





mDNGDNGmg

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]exp[ tDatGq

DNGm

p


mDNGDNGp


mDNG

2)(]exp[


tDmatmttDttDtbLGM

mLGM

LGMm

LGMm

LGM





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Power Law Exponential (PLE) ModelLoss Ratio Relation

Basis for PLE Model

Rate-Time Relation

Loss-Ratio Derivative

DtDntD n

i1ˆ)(


2)ˆ()1(ˆ)(

ni

ni

ntDtDntnDtb





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Duong Time-Rate Relationq/Gp vs. Production Time Log-Log Plot

m

p

atGq Basis for Duong Model

Rate-Time Relation

)1(

1exp)( 1

1mm

g tmatqtq

Cumulative-Time Relation

)1(

1exp 11 m

p tma

aqG

Loss-RatiomatmttD 1)(

2)()()( m

mm

mtatattmttb






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Logistic Growth Model (LGM)K/Q(t) -1 vs. Production Time Log-Log Plot

ntatQK ˆ1)(

Basis for LGM

Rate-Time Relation


Loss-Ratio


2

)1(

)ˆ(ˆ

)( n

n

g taKntatq

n

n

p taKtG

ˆ

)()1()( n

n

tattnanatD

2

222

))1(ˆˆ()1()1(ˆ2)1(ˆ)( n

nn

tnnaatntnanatb





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Modified Duong Model (Model 1)Loss Ratio Relation

Basis for Modified Duong Model

Rate-Time Relation


mDNG atmtDtD 1)(

tDt

matqtq DNG

mmg 1

1exp)( 1

1

2))(()()(tDmtat

tatmttbDNG

m

mm





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Modified Duong Model (Model 2)q/Gp vs. Production Time Log-Log Plot

Basis for Modified Duong Model (Model 2)

Rate-Time Relation


Loss-Ratio


]exp[ tDatGq

DNGm

p


mDNGDNGmg


mDNGDNGp


mDNG

2)(]exp[


tDmatmttDttDtbLGM

mLGM

LGMm

LGMm

LGM





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Modified Logistic Growth Model (Model 1)Loss Ratio Relation

Basis for Modified Logistic Growth Model(Model 1)

Rate-Time Relation


LGMn

n

Dtat

tnnaatD

)ˆ()1(ˆˆ

)(

]exp[)ˆ(

ˆ)( 2

)1(

tDta

Kntatq LGMn

n

g

2

222

))1()1(ˆ()1()1(ˆ2)1(ˆ)(tDnttDnatntnanatb

LGMn

LGM

nn





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Modified Duong Model (Model 2)

Basis for Modified LGM (Model 2)

Rate-Time Relation


K/Q(t) -1 vs. Production Time Log-Log Plot

RtDtatQK

LGMn

g

]exp[ˆ1)(

21

]exp[)1(ˆ)(]exp[ˆ

)(n

LGM

LGMn

LGMg

ttDRatDnKttDatq

nLGM

LGMn

g ttDRatDKttQ]exp[)1(ˆ]exp[)(

Loss-Ratio

Loss-Ratio Derivativedtdq

qD g

g

/1

dtdqq

dtd

Ddtdb

g

g

/1

20120926 Askabe MS Pres (Pptx)

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