10_spe_084287_(pratikno)_decline_curve_analysis_type_curves_frac_wells_(wpres).pdf

Decline Curve Analysis Using Type Curves — Fractured Wells Slide — 1

SPE 84287Decline Curve Analysis Using Type

Curves — Fractured Wells

H. Pratikno, ConocoPhillips (Indonesia)J. A. Rushing, Anadarko Petroleum Corp.

T.A. Blasingame, Texas A&M U.Contact: Department of Petroleum Engineering

Texas A&M UniversityCollege Station, TX 77843-3116

(979) 845-2292 — [email protected]








Executive Summary


Executive Summary — Introduction (1/5)This work is motivated by the need for a decline type curve for the case of a well with a finite conductivity vertical fracture in a bounded reservoir.

Top View of Well ConfigurationFinite Conductivity Vertical Fracture.No Flow Outer Boundary.

Side View of Well ConfigurationFixed width, planar fracture.After Cinco-Ley, et al.


Executive Summary — Solution Mechanism (2/5)Using "desuperposition," we have created a solution to represent the flow behavior of a well with a finite con-ductivity vertical fracture in a bounded circular reservoir:

Desuperposition Solution:Developed to provide a convenient mechanism to correlate performance.Validated as very accurate approximation.Recognize that other solutions are available, again — this solution was used due to its convenience and accuracy.

[ ]∫ ++==≤1

0 002101 ')()(),'(),,(, dxux'xKu-x'xKsxquyxp DDfDDDinfCMD

),('"),"(0

'

0sxqsf

sCdxdxsxq

C DfDfD

x x

fDfD

D++− ∫ ∫ ππ

⎥⎥⎦

⎤

⎢⎢⎣

⎡++ ∫∫

+− )1(

0 0)1(

0 01

1 )( )()()(1

21 DD xuxu

eD

eD dzzIdzzIruIruK

uu


Executive Summary — Solution Check (3/5)

Solution Check (FcD=2π):Comparison against numerical simulation (various "penetration" cases)."Desuperposition" model is consistent throughout range of data.Should be considered accurate enough for analytical developments.


c. Pratikno (2002): "Fetkovich-McCray" format —FINITE conductivity vertical fracture (FcD=0.5).

a. Doublet, et al (1996): "Fetkovich-McCray" format— INFINITE conductivity vertical fracture (FcD=∞).

b. Pratikno (2002): "Fetkovich-McCray" format —FINITE conductivity vertical fracture (FcD=10).

Decline Type Curves: Fractured WellsInfinite fracture conductivity:– Less complex solution, but

somewhat ideal for use in practice.

Finite fracture conductivity:– FcD=10: Moderate to high fracture

conductivity case.– FcD=0.5: Low fracture conducti-

vity case.

Executive Summary — Decline Curves (4/5)


Example 1: East Texas Gas WellGood record of production history:– Rate profile is consistent.– Pressure profile appears accurate.

Data editing used to filter poor or "extra-transient" data.Type Curve Match: (FcD=5)– Very good match of data functions.–Match of transient data is excellent,

good data acquisition (pressure).

Executive Summary — Example Analysis (5/5)

c. Example 1 — Data match on "Fetkovich-McCray" format type curve (FcD=5).

a. qg and pwf vs production time — Example 1 (East Texas "tight gas" case).

b. Δpp/qg vs material balance time (ta,bar) —Example 1 (East Texas "tight gas" case).








Outline of Presentation


Executive SummaryObjectives and Rationale

Need for a suite of decline type curves for the case of a well with a finite conductivity vertical fracture.

Desuperposition SolutionProposal and validation of the solution.

Development of the Decline Type CurvesType Curve Suite — Various cases

Library of decline type curves.Validation — Field Examples

Example 1 — East Texas (tight gas).Example 2 — East Texas (tight gas).

Summary

Presentation Outline








Objectives and Rationale


Objectives and RationaleObjectives:

Develop a rigorous suite of decline type curves for the case of a well with a finite-conductivity vertical fracture (previous efforts were numerical and did not consider auxiliary data functions).Provide a systematic procedure for analysis based on multiple data plots — ultimately culminating in the "summary plot" — the decline type curve. Provide illustrative applications of the new type curves using field data.

Rationale: The decline type curve model (in "Fetkovich-McCray" format) for the case of a well with a finite-conductivity vertical fracture is missing from the literature. The motivation for this work is based on the observation that many cases can not beresolved with the existing infinite-conductivity fracture model (presented in 1996).








Desuperposition Solution


Desuperposition SolutionSchematic Model:

Well with a finite-conductivity vertical fracture pro-duced at a constant flowrate (we will deal with the variable-rate issue as part of the data analysis).Closed outer boundary (used a circle for simplicity).

Desuperposition: (Approximate Solution)Start with transient flow solution for the case of a well with a finite-conductivity vertical fracture (Cinco-Meng).Add this solution to the boundary-dominated flow relation for a well with an infinite-conductivity vertical fracture (Ozkan-Raghavan).


Desuperposition — Schematic Model (1/4)The schematic model for the case of a well with a finite conductivity vertical fracture in a bounded reservoir is:

Top View of Well ConfigurationFinite Conductivity Vertical Fracture.No Flow Outer Boundary.

Side View of Well ConfigurationFixed width, planar fracture.After Cinco-Ley, et al.


Desuperposition — Approximate Solution (2/4)Using "desuperposition," we have created a solution to represent the flow behavior of a well with a finite con-ductivity vertical fracture in a bounded circular reservoir:

Desuperposition Solution:Developed to provide a convenient mechanism to correlate performance.Validated as very accurate approximation.Recognize that other solutions are available, again — this solution was used due to its convenience and accuracy.

[ ]∫ ++==≤1

0 002101 ')()(),'(),,(, dxux'xKu-x'xKsxquyxp DDfDDDinfCMD

),('"),"(0

'

0sxqsf

sCdxdxsxq

C DfDfD

x x

fDfD

D++− ∫ ∫ ππ

⎥⎥⎦

⎤

⎢⎢⎣

⎡++ ∫∫

+− )1(

0 0)1(

0 01

1 )( )()()(1

21 DD xuxu

eD

eD dzzIdzzIruIruK

uu


Desuperposition — Validation (3/4)

a. Comparison of desuperposition solution with numerical simulation results (FcD=0.2π).

b. Comparison of desuperposition solution with numerical simulation results (FcD=π).

c. Comparison of desuperposition solution with numerical simulation results (FcD=2π).

d. Comparison of desuperposition solution with numerical simulation results (FcD=10π).


Desuperposition — (FcD=2π) (4/4)

Solution Check (FcD=2π):Comparison against numerical simulation (various "penetration" cases)."Desuperposition" model is consistent throughout range of data.Should be considered accurate enough for analytical developments.








Development of the Decline Type Curves


Development of the Decline Type CurvesDecline Type Curve Variables

Definition of tDd and qDdAuxiliary functions (qDdi, qDdid, NpDd and )

Correlation of the bDpss parameter:Use the desuperposition solution to establish the bDpss parameter for each case (considers fully developed pseudosteady-state flow behavior).Development of a correlating relation.

Ddt


Decline Type Curve Variables (1/2)Dimensionless Decline Time: tDd

Dimensionless Decline Rate: qDd

DpssDDd bqq =

DADpss

Dd tb

t π2=


Decline Type Curve Variables (2/2)

Dimensionless Cumulative Production: NpDd

Dimensionless Decline Rate Integral: qDdi

Dimensionless Decline Rate Integral-Derivative: qDdid

ττ dqt

ttN

q DdDd

DdDd

pDdDdi )(

0

1 ∫==

DdDdiDdDdi

DdDdDdi

Ddid qqdtdq

ttd

dqq −=−=−=

)ln(

ττ dqt

N DdDd

pDd )(0∫=

Dd

pDdDd

Dd

DdDd q

Ndq

t

qt == ∫ ττ )(

0 1

Dimensionless Material Balance Time: Ddt


bDpss Parameter — Definition (1/4)

Definition of the bDpss parameter:

DADpssDApss,D tbtp π2)( +=

DAds,DDpss tpb π2-=

General result for pseudosteady-state flow behavior:

bDpss is defined by combination of the pseudosteady-state relation and the new desuperposition solution:

A single value of bDpss (at pseudosteady-state) is esta-blished (for a given reD value) by plotting the bDpss para-meter versus tDA for a single reD case and selecting the asymptotic (or constant) value which occurs during fully developed pseudosteady-state flow behavior.


Correlation of bDpss versus tDA: Various reD values (FcD =1)Note asymptotic behavior as tDA grows large (i.e., pseudosteady-state flow).Obtain a database of bDpss—reD pairs for a particular FcD value.

bDpss Parameter — Definition (2/4)


Correlation for the bDpss parameter:

bDpss Parameter — Correlation (3/4)


Correlation of bDpss versus reD: Various FcD valuesSimilarity of bDpss vs. reD indicates consistency in bDpss definition.Correlation is more than sufficient for analytical and practical applications.

bDpss Parameter — Correlation Plot (4/4)








Library of Decline Type Curves


Library of Decline Type Curves:Decline type curve models are available for:

Unfractured Well: Production time format (1973)Unfractured Well: Material balance time format (1994)Fractured Wells: Material balance time format─ Infinite Conductivity Vertical Fracture (1996)─ Finite Conductivity Vertical Fracture (Present Work)

The present work (i.e., the case of a well with a finite-conductivity vertical fracture) evolved because of a need for a better physical representation of the frac-tured well case (compared to the infinite conductivity fracture solution). This is particularly true for the case of wells in low permeability (or tight) gas reservoirs.


Library of Decline Type Curves: Summary (1/7)

a. INFINITE conductivity vertical fracture (FcD=∞) (1996).

d. FINITE conductivity vertical fracture (FcD=10).

b. FINITE conductivity vertical fracture (FcD=0.1).

e. FINITE conductivity vertical fracture (FcD=100).

f. FINITE conductivity vertical fracture (FcD=1000).

c. FINITE conductivity vertical fracture (FcD=1).


Library of Decline Type Curves: Fetkovich (2/7)

"Fetkovich-McCray" Type Curve for an Unfractured Well:Production time format.Analogous to original Fetkovich type curve (note Arps' solutions).


Library of Decline Type Curves: Doublet, et al. (3/7)

"Fetkovich-McCray" Type Curve for an Unfractured Well:"Material balance time" format.Note the unified behavior for all trends during boundary dominated flow.


"Fetkovich-McCray" Type Curve for a Fractured Well:Well with an INIFINITE conductivity vertical fracture in circular reservoir.Material balance time format (unified pseudosteady-state behavior).

Library of Decline Type Curves: Doublet, et al. (4/7)


Library of Decline Type Curves: Present Work (5/7)

"Fetkovich-McCray" Type Curve for a Fractured Well:Well with a FINITE conductivity vertical fracture in a circular reservoir (FcD=10)Material balance time format (unified pseudosteady-state behavior).



"Fetkovich-McCray" Format Type Curve for a Fractured Well:Well with a FINITE conductivity vertical fracture in a circular reservoir (FcD=1)Material balance time format (unified pseudosteady-state behavior).



"Fetkovich-McCray" Type Curve for a Fractured Well:Well with a FINITE conductivity vertical fracture in a circular reservoir (FcD=0.1)Material balance time format (unified pseudosteady-state behavior).








Examples


Example 1: East Texas Gas Well (1/2)

a. qg and pwf vs. t: Example 1(East Texas Gas Well).

d. Δpp/qg vs. ta,bar: (data plot) Exam-ple 1 (East Texas Gas Well).

b. Δpp/qg vs. Gp: Example 1 (East Texas Gas Well).

e. Data match on type curve (FcD=5). Example 1 (East Texas Gas Well).

c. Δpp/qg vs. ta,bar: (edit plot) Exam-ple 1 (East Texas Gas Well).

Example 1:Good production history. Good trend of Δpp/qg vs. Gp(initial estimate for G)Good data edit and data plots.Type curve match is excellent— confirms reservoir model and yields extraordinary match for all data functions (this is likely due to the high-quality, high frequency pressure data).


Example 1: East Texas Gas Well (tight gas reservoir)Model: FINITE conductivity vertical fracture in a circular reservoir (FcD=5)Excellent match for all data functions.




a. qg and pwf vs. t: Example 2(East Texas Gas Well).

d. Δpp/qg vs. ta,bar: (data plot) Exam-ple 2 (East Texas Gas Well).

b. Δpp/qg vs. Gp: Example 2 (East Texas Gas Well).

e. Data match on type curve (FcD=2). Example 2 (East Texas Gas Well).

c. Δpp/qg vs. ta,bar: (edit plot) Exam-ple 2 (East Texas Gas Well).

Example 2:Very consistent production history. Acceptable trend of Δpp/qg vs. Gp (initial estimate for G)Good data edit and data plots.Type curve match (FcD=2) is very good to excellent. Reser-voir model is confirmed, and the characteristic behavior of a finite conductivity vertical frac-ture is observed.


Example 2: East Texas Gas Well (tight gas reservoir)Model: FINITE conductivity vertical fracture in a circular reservoir (FcD=2)Very good match for all data functions.









Summary


Summary: (1/2)

We have successfully constructed, validated, and applied a new set of unified decline type curve solu-tions for the behavior of a well with a finite conduc-tivity vertical fracture producing from a closed homogeneous reservoir.The decline type curves developed in this work can be applied to analyze and interpret production data from a fractured well, and are used to estimate the following parameters:

Formation permeability, Fracture half-length, Fracture conductivity, Volume of in-place fluids, and Reservoir drainage area.


Summary: (2/2)

The proposed method is error tolerant and generally robust — and should be the preferred method of analysis/interpretation for production data acquired from low permeability gas reservoirs.As always, we encourage vigilance in the acquisition of production data — particularly wellbore pressures as a function of time.








End of Presentation

10_spe_084287_(pratikno)_decline_curve_analysis_type_curves_frac_wells_(wpres).pdf

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