pdf p324 07a (for class) lec mod2 01b poz for abnprs (gonzalez)

7/25/2019 PDF P324 07A (for Class) Lec Mod2 01b PoZ for AbnPrs (Gonzalez)

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A New p/z-Gp2 M a t er i a l B a l a n c e f o r A b n o r m a l l y -P r es s u r e d R e s er v o i r s Slide

1

A Q u ad r a t ic C u m u l at i v e Pr o d u c t i o n

M o d el f o r t h e M a ter i al B al an c e o f A b n o r m a l l y - P r e s s u r e d G a s R e s e r v o i r s

F.E. Gonzalez, M.S. Thesis Defense

17 October 2003Department of Petroleum Engineering

Texas A&M University

College Station, TX 77843-3116


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2

Executive Summary "p/z-Gp2" Relation (1/4)

The rigorous relation for the material balance of a drygas reservoir system is given by Fetkovich, et al. as:

)(61551

))((1

ewinjwpg

swpinj WBWBWB

.RWGpG

Giz

ip

iz

ippippecz

p

Eliminating the water influx, water production/injection,and gas injection terms; defining Gp=ce(p )(pi-p) andassuming that Gp


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3

Simulated Dry Gas Reservoir Case Abnormal Pressure: Volumetric, dry gas reservoir withcf(p) (from Fetkovich).

Note extrapolation to the "apparent" gas-in-place (previous approaches).

Note comparison of data and the new "Quadratic Cumulative Production" model.



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4

Anderson L Reservoir Case Abnormal Pressure: South Texas (USA) gas reservoir with abnormal pressure.

Benchmark literature case.

Note performance of the new "Quadratic Cumulative Production" model.



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5

Executive Summary

Objectives and RationaleRigorous technique for abnormal pressure analysis.Development of the p/z-Gp

2 modelDerivation from the rigorous material balance.

Validation Field ExamplesCase 1

Dry gas simulation (cf(p) from Fetkovich).

Case 3 Anderson L (South Texas, USA).

Demonstration (MS Excel Anderson L case)Summary

Recommendations for Future Work

Presentation Outline


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6

Objectives and Rationale

Objectives: Develop a rigorous functional form (i.e., a model) for

the p/zvs.Gpbehavior demonstrated by a typicalabnormally pressured gas reservoir.

Develop a sequence of plotting functions for theanalysis ofp/zGpdata (multiple plots).

Provide an exhaustive validation of this new modelusing field data.

Rationale: The analysis ofp/zGpdata for abnorm-ally pressured gas reservoirs has evolved from empi-rical models and idealized assumptions (e.g., cf(p)=

constant). We would like to establish a rigorous ap-proach one where any approximation is based onthe observation of some characteristic behavior, notsimply a mathematical/graphical convenience.


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7

Development of the p/z-Gp2 model

Concept: Use the rigorous material balance relation given by

Fetkovich,et al. for the case of a reservoir wherecf(p) is NOT presumed constant.

Use some observed limiting behavior to construct asemi-analytical relation forp/zGpbehavior.

Implementation: Develop and apply a series of data plotting functions

which clearly exhibit unique behavior relative to thep/zGpdata.

Use a "multiplot" approach which is based on the

dynamic updating of the model solution on eachdata plot.

Develop a "dimensionless" type curve approach thatcan be used to validate the model and estimate G.


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A New p/z-Gp2 M a t er i a l B a l a n c e f o r A b n o r m a l l y -P r es s u r e d R e s er v o i r s Slide 8

p/z-Cumulative Model: (1/3)

The rigorous relation for the material balance of a drygas reservoir system is given by Fetkovich, et al. as:

)(61551

))((1

ewinjwpg

swpinj WBWBWB

.RWGpG

Giz

ip

iz

ippippecz

p

Eliminating the water influx, water production/injection,and gas injection terms, then rearranging gives thefollowing definition:

)])(([where1)1(

// pippecG

G

G

G

zpp

p

p

iizp


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Considering the condition where:

Then we can use a geometric series to represent the Dterm in the governing material balance. The appropriategeometric series is given by:

1 pD G

1)1(3

111

2

x...xxxx/

1)1(1)1(

1

pp

p

GGG

or, for our problem, we have:

2)

1(1 pp

i

i GG

GGz

p

z

p

Substituting this result into the material balance relation,we obtain:



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A more convenient form of the p/z-cumulative model is:

We note that these parameters presume that

is con-stant. Presuming that is linear with Gp, we can derivethe following form:


i

i

z

p

G)

1(

i

i

z

p

G

32)()

1( p

i

ip

i

ip

i

i

i

i Gz

p

G

bG

z

pb

G

aG

z

pa

Gz

p

z

p

pbGa where

Obviously, one of our objectives will be the study of thebehavior of vs.Gp(based on a prescribed value ofG).

2ppii GG

z

p

z

p


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Simulated Dry Gas Reservoir Case A b n o r m al P r es s u r e :A linear trend of vs.Gpis reasonable and should yield an accurate model.

is approximated by a constant value within the trend.

A physical definition of is elusive Gp=ce(p )(pi-p) implies that has units of1/volume, which suggests is a scaling variable forG.

-GpPerformance (Case 1) (1/2)

a. Case 1: Simulated Performance Case Plot of

versusGp(requires an estimate of gas-in-

place). Note the apparent linear trend of the

data. Recall that Gp=ce(pp-p).

b. Case 1: Simulated Performance Case Plot of

p/zversus Gp. The constant and linear trends

match well with the data essentially a con-

firmation ofb o t h models.


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Anderson L Reservoir Case Abnormal Pressure: Field data will exhibit some scatter, method is relatively tolerant of data scatter.

Constant approximation is based on the "best fit" of several data functions.

The linear approximation for is reasonable (should favor later data).

-GpPerformance (Case 3) (2/2)

a. Case 3: Anderson L Reservoir Case (South

Texas, USA) Plot of versusGp(requires an

estimate of gas-in-place). Some data scatter

exists, but a linear trend is evident (recall that

Gp=ce(p)(pi-p)).

b. Case 3: Anderson L Reservoir Case (South

Texas, USA) Plot ofp/zversus Gp. Both

models are in strong agreement.


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Validation of the p/z-Gp2 model: Orientation

Methodology: All analyses are "dynamically" linked in a spread-

sheet program (MS Excel). Therefore, all analysesare consistent should note that some functions/plots perform better than others b u t t h e m o d el r e s u l t s a r e t h e s am e f o r e v e r y a n al y s i s p l o t .

Validation: Illustrative Analyses p/z-Gp

2 plotting functions based on the proposedmaterial balance model.

-Gpperformance plots used to calibrate analysis.

Gan analysis presumes 2-straight line trends on a

p/z-Gpplot for an abnormally pressured reservoir. pD-Gp Dtype curve approach usep/z-Gp

2 materialbalance model to develop type curve solution thisapproach is most useful for data validation.


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p/z-Gp2 Plotting Functions: Case 1 (1/5)

a. b. c. ppp

p

GdGzpG

Gvs.)/(

0

1

pp

GzpG

vs.)/(1

d. e.ppp

p

GdGzpG

Gvs.

2 )/(

0

1

ppp

p

GdGzpG

Gzp vs.)/(

0

1)/( f. pp

p

pp

GdGzpG

Gzp

Gvs.)/(

0

1)/(

1

pG

z

p

iz

ipzp vs.)/(


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-GpPlotting Functions: Case 1 (2/5)

a. Case 1: Simulated Performance Case Plot of

ce(p)(pi-p) versus Gp(requires estimate ofG).

b. Case 1: Simulated Performance Case Plot of

1/ce(p)(pi-p) versus Gp(requires estimate ofG).

c. Case 1: Simulated Performance Case Plot of

versusGp(requires estimate ofG).

d. Case 1: Simulated Performance Case Plot of

versusGp/G(requires estimate ofG).


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Simulated Dry Gas Reservoir Case A b n o r m al P r es s u r e : Summary p/zGpplot for =constant and =linear cases.

Good comparison of trends, =linear trend appears slightly conservative as itemerges from data trend but both solutions appear to yield same Gestimate.



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Gan-Blasingame Analysis (2001): Case 1 (4/5)

a. Case 1: Simulated Performance Case Gan Plot 1

ce(p)(pi-p) versus (p/z)/(pi/zi) (requires est. ofG).

b. Case 1: Simulated Performance Case Gan Plot 2

(p/z)/(pi/zi) versus (Gp/G) (requires est. ofG).

c. Case 1: Simulated Performance Case Gan Plot 3

(p/z) versus Gp(results plot).

Gan-Blasingame Analysis:Approach considers the "match"

of the ce(p)(pi-p) (p/z)/(pi/zi)data and "type curves."

Assumes that b o t h abnormaland normal pressurep/ztrendsexist.

Straight-line extrapolation of the"normal"p/ztrend used forG.


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pD-Gp DType Curve Approach: Case 1 (5/5)

a. pD-GpDType curve solution based on the p/z-Gp2

model. pD= [(pi/zi)-(p/z)]/(pi/zi) andGpD=Gp/G

both pDandpDifunctions are plotted.

b. Case 1: Simulated Performance Case Type

curve analysis of (p/z)-Gpdata, this case is

"force matched" to the same results as all of the

other plotting functions.


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19

p/z-Gp2 Plotting Fcns: Case 3 (Anderson L) (1/5)

a. b. c. ppp

p

GdGzpG

Gvs.)/(

0

1

pp

GzpG

vs.)/(1

d. e.ppp

p

GdGzpG

Gvs.

2 )/(

0

1

ppp

p

GdGzpG

Gzp vs.)/(

0

1)/( f. pp

p

pp

GdGzpG

Gzp

Gvs.)/(

0

1)/(

1

pGz

p

iz

ipzp vs.)/(


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a. Case 3: Anderson L (South Texas) Plot of

ce(p)(pi-p) versus Gp(requires estimate ofG).

b. Case 3: Anderson L (South Texas) Plot of

1/ce(p)(pi-p) versus Gp(requires estimate ofG).

c. Case 3: Anderson L (South Texas) Plot of

versusGp(requires estimate ofG).

d. Case 3: Anderson L (South Texas) Plot of

versusGp/G(requires estimate ofG).


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Case 3 Anderson L Reservoir (South Texas (USA)) Summary p/zGpplot for =constant and =linear cases.

Good comparison of trends, =constant and =linear cases in good agreement.

Data trend is very consistent.



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Gan-Blasingame Analysis (2001): Case 3 (4/5)

a. Case 3: Anderson L Reservoir Gan Plot 1 ce(p)(pi-p)

versus (p/z)/(pi/zi) (requires est. ofG).

b. Case 3: Anderson L Reservoir Gan Plot 2 (p/z)/(pi/zi)

versus (Gp/G) (requires est. ofG).

c. Case 3: Anderson L Reservoir Gan Plot 3 (p/z)

versusGp(results plot).

Gan-Blasingame Analysis:We note an e x c e l l en t "match" of

thece(p)(pi-p) (p/z)/(pi/zi) dataand the "type curves."

Both the abnormal and normalpressurep/ztrends appear ac-curate and consistent.

Straight-line extrapolation of the"normal"p/ztrend used forG.


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pD-Gp DType Curve Approach: Case 3 (5/5)

Case 3 Anderson L Reservoir (South Texas (USA)) pD-Gp Dtype curve solution matched using field data.

Note the "tail" in thepDtrend for small values ofGp D(common field data event).

"Force matched" to the same results as each of the other plotting functions.


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24

Case 3 Anderson L (South Texas (USA))Literature standard case.

A 3-well reservoir, delimited by faults.

Good quality data.

Evidence of overpressure from static pressure tests.

Analysis: (Implemented using MS Excel)p/z-Gp

2 plotting functions.

-Gpperformance plots.

Gan analysis (2-straight line trends on a p/z-Gpplot).

pD-Gp Dtype curve approach.

Example Analysis Using MS Excel: Case 3


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Summary: (1/3)

Developed a new p/z-Gp2 material balance model for

the analysis of abnormally pressured gas reservoirs:

The -function is presumed (based on graphical

comparisons) to be either constant, or approximatelylinear with Gp. For the =constant case, we have:

where:

2)

1(1 pp

i

i GG

GGz

p

z

p ))((

1pippec

Gp

2pp

i

i GGz

p

z

p

i

i

z

p

G)

1(

i

i

z

p

G


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Base relation: p/z-Gp2 form of the gas material balance

a. P l o t t i n g F u n c t i o n 1:(quadratic)

b. P l o t t i n g F u n c t i o n 2:(linear)

c. P l o t t i n g F u n c t i o n 3:(quadratic)

ppp

pGdGzp

G

Gvs.)/(

0

1

pp

GzpG

vs.)/(1

pG

z

p

iz

ipzp vs.)/(

pp

p

p

GdGzpG

G

vs.2

)/(

0

1

ppp

p

GdGzpG

Gzp vs.)/(

0

1)/(

ppp

pp

GdGzpG

Gzp

Gvs.)/(

0

1)/(

1

2ppii GG

zp

zp

ii

zp

G)1(

ii

zp

G

d. P l o t t i n g F u n c t i o n 4 :(linear)

e. P l o t t i n g F u n c t i o n 5 :(quadratic)

f.P l o t t i n g F u n c t i o n 6:(linear)

Summary: (2/3)


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The plotting functions developed in this work havebeen validated as tools for the analysis reservoirperformance data from abnormally pressured gasreservoirs. Although the straight-line functions (PF2,PF4, and PF6) could be used independently, but werecommend a combined/simultaneous analysis.

The

-Gpplots are useful for checking data con-sistency and for guiding the selection of the -value.These plots represent a vivid and dynamic visualbalance of all of the other analyses.

The Gan analysis sequence is also useful for orient-

ing the overall analysis

particularly the ce(p)(pi-p)versus (p/z)/(pi/zi) plot.

ThepD-Gp Dtype curve is useful for orientationparticularly for estimating the or (D) value.

Summary: (3/3)


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Recommendations for Future Work:

Consider the extension of this methodology forcases of external drive energy (e.g., water influx, gasinjection, etc.).

Continue the validation of this approach by applyingthe methodology to additional field cases.

Implementation into a stand alone software.


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A New p/z-Gp 2 M a t er i a l B a l a n c e f o r A b n o r m a l l y -P r es s u r e d R e s er v o i r s Slide 29

A Q u ad r a t ic C u m u l at i v e Pr o d u c t i o n

M o d el f o r t h e M a ter i al B al an c e o f A b n o r m a l l y - P r e s s u r e d G a s R e s e r v o i r s

F.E. Gonzalez, M.S. Thesis Defense

17 October 2003Department of Petroleum Engineering

Texas A&M University

College Station, TX 77843-3116

pdf p324 07a (for class) lec mod2 01b poz for abnprs (gonzalez)

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