Transcript No Slide Title

A Quadratic Cumulative Production
Model for the Material Balance of
Abnormally-Pressured Gas Reservoirs
F.E. Gonzalez, M.S. Thesis Defense
17 October 2003
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 1
Executive Summary — "p/z-Gp2" Relation
(1/4)
The rigorous relation for the material balance of a dry
gas reservoir system is given by Fetkovich, et al. as:
p
1  ce ( p)( pi  p) 
z
pi
zi

pi 1 
5.615

(W p Bw  Winj Bw  We )
Gp  Ginj  W p Rsw 
zi G 
Bg

Eliminating the water influx, water production/injection,
and gas injection terms; defining wGp=ce(p)(pi-p) and
assuming that wGp<1, then rearranging gives the following result:
p pi

z
zi
1
w 2

1

(

w
)
G

Gp 
p

G
G


A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 2
Executive Summary — "p/z-Gp2" Relation
(2/4)
 Simulated Dry Gas Reservoir Case — Abnormal Pressure:
 Volumetric, dry gas reservoir — with cf(p) (from Fetkovich).
 Note extrapolation to the "apparent" gas-in-place (previous approaches).
 Note comparison of data and the new "Quadratic Cumulative Production" model.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 3
Executive Summary — "p/z-Gp2" Relation
(3/4)
 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.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 4
Executive Summary — "p/z-Gp2" Relation
(4/4)
 Simulated Dry Gas Reservoir Case — No Abnormal Pressure:
 Volumetric, dry gas reservoir — note that all analyses lie on the same trend.
 Purpose is to orient the analyst that the new methodology degenerates into the
traditional p/z—Gp straight-line for the case of no abnormal pressure effects.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 5
Presentation Outline
 Executive Summary
 Objectives and Rationale
 Rigorous technique for abnormal pressure analysis.
 Development of the p/z-Gp2 model
 Derivation from the rigorous material balance.
 Validation — Field Examples
 Case 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
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 6
Objectives and Rationale
Objectives:
 Develop a rigorous functional form (i.e., a model) for
the p/z vs. Gp behavior demonstrated by a typical
abnormally pressured gas reservoir.
 Develop a sequence of plotting functions for the
analysis of p/z—Gp data (multiple plots).
 Provide an exhaustive validation of this new model
using field data.
Rationale: The analysis of p/z—Gp data for abnormally pressured gas reservoirs has evolved from empirical models and idealized assumptions (e.g., cf(p)=
constant). We would like to establish a rigorous approach — one where any approximation is based on
the observation of some characteristic behavior, not
simply a mathematical/graphical convenience.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 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 where
cf(p) is NOT presumed constant.
 Use some observed limiting behavior to construct a
semi-analytical relation for p/z—Gp behavior.
Implementation:
 Develop and apply a series of data plotting functions
which clearly exhibit unique behavior relative to the
p/z—Gp data.
 Use a "multiplot" approach which is based on the
dynamic updating of the model solution on each
data plot.
 Develop a "dimensionless" type curve approach that
can be used to validate the model and estimate G.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 8
p/z-Cumulative Model:
(1/3)
The rigorous relation for the material balance of a dry
gas reservoir system is given by Fetkovich, et al. as:
p
1  ce ( p)( pi  p) 
z
pi
zi

pi 1 
5.615

(W p Bw  Winj Bw  We )
Gp  Ginj  W p Rsw 
zi G 
Bg

Eliminating the water influx, water production/injection,
and gas injection terms, then rearranging gives the
following definition:
pi /z i
p/z 
(1  wGp
 Gp
1 
) 
G

 [wherewGp  ce ( p)( pi  p)]

A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 9
p/z-Cumulative Model:
(2/3)
Considering the condition where:
wGp  1
Then we can use a geometric series to represent the wGp
term in the governing material balance. The appropriate
geometric series is given by:
1 / 1  x  1  x  x 2  x3  ...
(1  x  1)
or, for our problem, we have:
1
 1  wGp
(1  wGp )
(1  wGp  1)
Substituting this result into the material balance relation,
we obtain:
p pi 
1
w 2

1

(

w
)
G

Gp 
p

z
zi 
G
G

A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 10
p/z-Cumulative Model:
(3/3)
A more convenient form of the p/z-cumulative model is:
p pi

  Gp   G 2p
z
zi
 (
1
w)
G
pi
zi

w pi
G zi
We note that these parameters presume that w is constant. Presuming that w is linear with Gp, we can derive
the following form:
p pi
1

 (  a)
z zi
G
pi
a
Gp  (  b )
zi
G
pi 2 b pi 3
Gp 
Gp
zi
G zi
wherew  a  bGp
Obviously, one of our objectives will be the study of the
behavior of w vs. Gp (based on a prescribed value of G).
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 11
w-Gp Performance (Case 1)
a. Case 1: Simulated Performance Case — Plot of
w versus Gp/G (requires an estimate of gas-inplace). Note the apparent linear trend of the
data. Recall that wGp=ce(pp-p).
(1/2)
b. Case 1: Simulated Performance Case — Plot of
p/z versus Gp. The constant and linear w trends
match well with the data — essentially a confirmation of both models.
 Simulated Dry Gas Reservoir Case — Abnormal Pressure:
 A linear trend of w vs. Gp is reasonable and should yield an accurate model.
 w is approximated by a constant value within the trend.
 A physical definition of w is elusive — wGp=ce(p)(pi-p) implies that w has units of
1/volume, which suggests w is a scaling variable for G.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 12
w-Gp Performance (Case 3)
a. Case 3: Anderson L Reservoir Case (South
Texas, USA) — Plot of w versus Gp/G (requires
an estimate of gas-in-place). Some data scatter
exists, but a linear trend is evident (recall that
wGp=ce(p )(pi-p)).
(2/2)
b. Case 3: Anderson L Reservoir Case (South
Texas, USA) — Plot of p/z versus Gp. Both
models are in strong agreement.
 Anderson L Reservoir Case — Abnormal Pressure:
 Field data will exhibit some scatter, method is relatively tolerant of data scatter.
 Constant w approximation is based on the "best fit" of several data functions.
 The linear approximation for w is reasonable (should favor later data).
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 13
Validation of the p/z-Gp2 model: Orientation
Methodology:
 All analyses are "dynamically" linked in a spread-
sheet program (MS Excel). Therefore, all analyses
are consistent — should note that some functions/
plots perform better than others — but the model
results are the same for every analysis plot.
Validation: Illustrative Analyses
 p/z-Gp2 plotting functions — based on the proposed
material balance model.
 w-Gp performance plots — used to calibrate analysis.
 Gan analysis — presumes 2-straight line trends on a
p/z-Gp plot for an abnormally pressured reservoir.
 pD-GpD type curve approach — use p/z-Gp2 material
balance model to develop type curve solution — this
approach is most useful for data validation.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 14
p/z-Gp2 Plotting Functions: Case 1
a.
d.
 pi p 
Δ( p/z )     vs. Gp
 zi z 
1
G 2p
Gp
0
Δ( p/z ) dGp vs. Gp
1
Δ( p/z ) vs. Gp
b.
Gp
1
e. Δ( p/z ) 
Gp
Gp
0
c.
1
Δ( p/z ) dGp vs. Gp f.
Gp
(1/5)
1
Gp
Gp
0
Δ( p/z ) dGp vs. Gp

1
Δ( p/z) 
Gp

A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs


Δ( p/z ) dGp  vs. Gp
0

Gp
Slide — 15
w-Gp Plotting Functions: Case 1
a. Case 1: Simulated Performance Case — Plot of
ce(p)(pi-p) versus Gp (requires estimate of G).
c. Case 1: Simulated Performance Case — Plot of w
versus Gp (requires estimate of G).
(2/5)
b. Case 1: Simulated Performance Case — Plot of
1/ce(p)(pi-p) versus Gp (requires estimate of G).
d. Case 1: Simulated Performance Case — Plot of w
versus Gp/G (requires estimate of G).
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 16
w-Gp Plotting Functions: Case 1
(3/5)
 Simulated Dry Gas Reservoir Case — Abnormal Pressure:
 Summary p/z—Gp plot for w =constant and w =linear cases.
 Good comparison of trends, w =linear trend appears slightly conservative as it
emerges from data trend — but both solutions appear to yield same G estimate.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 17
Gan-Blasingame Analysis (2001): Case 1
a. Case 1: Simulated Performance Case — Gan Plot 1
ce(p)(pi-p) versus (p/z)/(pi/zi) (requires est. of G).
(4/5)
b. Case 1: Simulated Performance Case — Gan Plot 2
(p/z)/(pi /zi ) versus (Gp/G) (requires est. of G).
 Gan-Blasingame Analysis:
c. Case 1: Simulated Performance Case — Gan Plot 3
(p/z) versus Gp (results plot).
 Approach considers the "match"
of the ce(p)(pi-p) — (p/z)/(pi/zi)
data and "type curves."
 Assumes that both abnormal
and normal pressure p/z trends
exist.
 Straight-line extrapolation of the
"normal" p/z trend used for G.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 18
pD-GpD Type Curve Approach: Case 1
a. pD-GpD Type curve solution based on the p/z-Gp2
model. pD= [(pi/zi)-(p/z)]/(pi/zi) and GpD=Gp/G —
both pD and pDi functions are plotted.
(5/5)
b. Case 1: Simulated Performance Case — Type
curve analysis of (p/z)-Gp data, this case is
"force matched" to the same results as all of the
other plotting functions.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 19
p/z-Gp2 Plotting Fcns: Case 3 (Anderson L)
a.
d.
 pi p 
Δ( p/z )     vs. Gp
 zi z 
1
G 2p
Gp
0
Δ( p/z ) dGp vs. Gp
1
Δ( p/z ) vs. Gp
b.
Gp
1
e. Δ( p/z ) 
Gp
Gp
0
c.
1
Δ( p/z ) dGp vs. Gp f.
Gp
1
Gp
Gp
0
(1/5)
Δ( p/z ) dGp vs. Gp

1
Δ( p/z) 
Gp

A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs


Δ( p/z ) dGp  vs. Gp
0

Gp
Slide — 20
w-Gp Plotting Functions: Case 3
a. Case 3: Anderson L (South Texas) — Plot of
ce(p)(pi-p) versus Gp (requires estimate of G).
c. Case 3: Anderson L (South Texas) — Plot of w
versus Gp (requires estimate of G).
(2/5)
b. Case 3: Anderson L (South Texas) — Plot of
1/ce(p)(pi-p) versus Gp (requires estimate of G).
d. Case 3: Anderson L (South Texas) — Plot of w
versus Gp/G (requires estimate of G).
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 21
w-Gp Plotting Functions: Case 3
(3/5)
 Case 3 — Anderson L Reservoir (South Texas (USA))
 Summary p/z—Gp plot for w =constant and w =linear cases.
 Good comparison of trends, w =constant and w =linear cases in good agreement.
 Data trend is very consistent.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 22
Gan-Blasingame Analysis (2001): Case 3
a. Case 3: Anderson L Reservoir — Gan Plot 1 ce(p)(pi-p)
versus (p/z)/(pi/zi) (requires est. of G).
(4/5)
b. Case 3: Anderson L Reservoir — Gan Plot 2 (p/z)/(pi /zi )
versus (Gp/G) (requires est. of G).
 Gan-Blasingame Analysis:
c. Case 3: Anderson L Reservoir — Gan Plot 3 (p/z)
versus Gp (results plot).
 We note an excellent "match" of
the ce(p)(pi-p) — (p/z)/(pi/zi) data
and the "type curves."
 Both the abnormal and normal
pressure p/z trends appear accurate and consistent.
 Straight-line extrapolation of the
"normal" p/z trend used for G.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 23
pD-GpD Type Curve Approach: Case 3
(5/5)
 Case 3 — Anderson L Reservoir (South Texas (USA))
 pD-GpD type curve solution matched using field data.
 Note the "tail" in the pD trend for small values of GpD (common field data event).
 "Force matched" to the same results as each of the other plotting functions.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 24
Example Analysis Using MS Excel: Case 3
 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-Gp2 plotting functions.
 w-Gp performance plots.
 Gan analysis (2-straight line trends on a p/z-Gp plot).
 pD-GpD type curve approach.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 25
Summary:
(1/3)
 Developed a new p/z-Gp2 material balance model for
the analysis of abnormally pressured gas reservoirs:
where:
p pi 
1
w 2

1

(

w
)
G

Gp 
p

z
zi 
G
G

w
1
ce ( p)( pi  p)
Gp
The w-function is presumed (based on graphical
comparisons) to be either constant, or approximately
linear with Gp. For the w=constant case, we have:
p pi

  Gp   G 2p
z
zi
 (
1
w)
G
pi
zi

w pi
G zi
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 26
Summary:
(2/3)
 Base relation: p/z-Gp2 form of the gas material balance
p pi

  Gp   G 2p
z
zi
a. Plotting Function 1:
 (
1
w)
G
pi
zi

(linear)
p
p
Δ( p/z )   i   vs. Gp
 zi z 
1
G 2p
c. Plotting Function 3:
Gp
0
Δ( p/z ) dGp vs. Gp
Δ( p/z ) dGp vs. Gp
e. Plotting Function 5 :
(quadratic)
Δ( p/z ) 
1
Gp
Gp
0
Δ( p/z ) dGp vs. Gp
f. Plotting Function 6:
(quadratic)
1
Gp
Gp
0
(linear)
1
Δ( p/z ) vs. Gp
Gp
G zi
d. Plotting Function 4 :
(quadratic)
b. Plotting Function 2:
w pi
(linear)
1
Gp

1
Δ( p/z) 
Gp



Δ( p/z ) dGp  vs. Gp
0

Gp
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 27
Summary:
(3/3)
 The plotting functions developed in this work have
been validated as tools for the analysis reservoir
performance data from abnormally pressured gas
reservoirs. Although the straight-line functions (PF2,
PF4, and PF6) could be used independently, we
recommend a combined/simultaneous analysis.
 The w-Gp plots are useful for checking data consistency and for guiding the selection of the w-value.
These plots represent a vivid and dynamic visual
balance of all of the other analyses.
 The Gan analysis sequence is also useful for orienting the overall analysis — particularly the ce(p)(pi-p)
versus (p/z)/(pi/zi) plot.
 The pD-GpD type curve is useful for orientation —
particularly for estimating the w or (wD ) value.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 28
Recommendations for Future Work:
 Consider the extension of this methodology for
cases of external drive energy (e.g., water influx, gas
injection, etc.).
 Continue the validation of this approach by applying
the methodology to additional field cases.
 Implementation into a stand alone software.
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 29
A Quadratic Cumulative Production
Model for the Material Balance of
Abnormally-Pressured Gas Reservoirs
End of Presentation
F.E. Gonzalez, M.S. Thesis Defense
17 October 2003
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
A New p/z-Gp2 Material Balance for Abnormally-Pressured Reservoirs
Slide — 30