Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 05:
Gas Material Balance
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
+1.979.845.2292 — t-blasingame@tamu.edu
PETE 613
(2005A)
Gas Material
Balance
Slide — 1
Material Balance
 "Accounting" Concept of Material Balance:
 Require all inflows/outflows/generations.
 (Average) reservoir pressure profile is REQUIRED.
 Require rock, fluid, and rock-fluid properties (at some scale).
 Oil Material Balance:
 Less common than gas material balance (pressure required).
 Gas Material Balance:
 Volumetric dry gas reservoir (p/z versus Gp (straight-line)).
 Abnormally-pressured gas reservoirs (various techniques).
 Waterdrive/water influx cases (always problematic) (i.e., we
don't know the influx, so we use a model).
 Material Balance yields RESERVOIR VOLUME!
PETE 613
(2005A)
Gas Material
Balance
Slide — 2
Material Balance of a Petroleum Reservoir
 General Concept of Material Balance...
a. Initial reservoir conditions.
 From: Petroleum Reservoir Engineering
b. Conditions after producing Np STB of oil,
and Gp SCF of gas, and Wp STB of water.
— Amyx, Bass, and Whiting (1960).
 Material Balance: Key Issues
 Must have accurate production measurements (oil, water, gas).
 Estimates of average reservoir pressure (from pressure tests).
 Suites of PVT data (oil, gas, water).
 Reservoir properties: saturations, formation compressibility, etc.
PETE 613
(2005A)
Gas Material
Balance
Slide — 3
Average Reservoir Pressure for Material Balance
 Average Reservoir Pressure

From: Engineering Features of the Schuler Field and
Unit Operation — Kaveler (SPE-AIME, 1944).
 Average Reservoir Pressure: Key Issues
 Must "average" pressures over volume or area (approximation).
 Pressure tests must be representative (pavg extrapolation valid).
 Can average using cumulative production (surrogate for volume).
PETE 613
(2005A)
Gas Material
Balance
Slide — 4
Gas Material Balance Case
(1/3)
General Gas Material Balance:
p
1  ce ( p )( pi  p ) 
z
pi pi 1 
1

(Wp  Winj ) Bw  We
Gp  Ginj  Wp Rsw  5.615
zi zi G 
Bg





"Dry Gas" Material Balance: (no reservoir liquids )
p pi

z zi
PETE 613
(2005A)
 1

1

G
p
 G

Gas Material
Balance
Slide — 5
Gas Material Balance Case
(2/3)
General Gas Material Balance:
p
1  ce ( p )( pi  p ) 
z
pi pi 1 
1

(Wp  Winj ) Bw  We
Gp  Ginj  Wp Rsw  5.615
zi zi G 
Bg





"Abnormal Pressure" Material Balance: (cf=f(p))
 Gp 
p pi
1

1
z zi 1  ce ( p )( pi  p ) 
G 

 VpNNP  VpAQ  
1
ce ( p ) 
 S wi cw  c f   

  (cw  c f
(1  S wi ) 
  VpR   VpR  

)

"Quadratic Cumulative" Approximation:
p pi 
1
 2

1

(


)
G

Gp 
p

z
zi 
G
G

PETE 613
(2005A)
Gas Material
Balance
Slide — 6
Gas Material Balance Case
(3/3)
General Gas Material Balance:
p
1  ce ( p )( pi  p ) 
z
pi pi 1 
1

(Wp  Winj ) Bw  We
Gp  Ginj  Wp Rsw  5.615
zi zi G 
Bg





"Water Influx" Material Balance:
 Gp
p/z  pi /zi
1 
G
 W B  
e
w
1 

 GB gi 
1



"Cubic Cumulative" Approximation: (Current Research)
p pi

z zi
PETE 613
(2005A)

1  (1   )  Gp


 G


 Gp
  (   ) 

 G
2

 

Gas Material
Balance
 Gp

 G
3
 
 
 

Slide — 7
Volumetric Gas Material Balance
 "Dry Gas" Material Balance: Normally Pressured Reservoir Example
 Volumetric reservoir — no external energy (gas expansion only).
 p/z versus Gp yields unique straight-line trend.
 Linear extrapolation yield gas-in-place (G).
PETE 613
(2005A)
Gas Material
Balance
Slide — 8
Gas MBE Abnormally-Pressured Reservoir
 "Dry Gas" Material Balance: Abnormally Pressured Reservoir Example
 Volumetric reservoir — no water influx or leakage.
 p/z versus Gp yields unique quadratic trend (from approximated MBE).
 Quadratic extrapolation yield gas-in-place (G).
PETE 613
(2005A)
Gas Material
Balance
Slide — 9
Gas MBE "Water Influx" Case
a. Gas Material Balance Plot: p/z vs. Gp — simulated
performance. Note effect of aquifer permeability
on field performance.
b. Gas Material Balance Plot: p/z vs. Gp — simulated
performance. Note effect of displacement
efficiency (Ep).
 Gas Material Balance: Water Drive Gas Reservoir
 Pressure (hence p/z) is maintained during production via communication
with an unsteady-state aquifer (this study).
 From: Unsteady-State Performance of Water Drive Gas Reservoirs, Agarwal
(Texas A&M Ph.D., 1967).
PETE 613
(2005A)
Gas Material
Balance
Slide — 10
Concept: p/z vs. Gp — Water Influx Case
 Simulated Performance: Agarwal Dissertation (1967)
 Recovery is a function of production rate, Ep, and kaquifer.
 p/z vs. Gp performance appears to be cubic (i.e., f(Gp3)).
PETE 613
(2005A)
Gas Material
Balance
Slide — 11
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 05:
Gas Material Balance
(End of Lecture)
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
+1.979.845.2292 — t-blasingame@tamu.edu
PETE 613
(2005A)
Gas Material
Balance
Slide — 12
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
A Quadratic Cumulative Production Model
for the Material Balance of
Abnormally-Pressured Gas Reservoirs
F.E. Gonzalez
M.S. Thesis (2003)
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
PETE 613
(2005A)
Gas Material
Balance
Slide — 13
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 Gp=ce(p)(pi-p) and
assuming that Gp<1, then rearranging gives the following result:
p pi

z
zi
PETE 613
(2005A)
1
 2

1

(


)
G

Gp 
p

G
G


Gas Material
Balance
Slide — 14
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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 15
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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 16
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
PETE 613
(2005A)
Gas Material
Balance
Slide — 17
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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 18
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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 19
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  Gp
PETE 613
(2005A)
 Gp
1 
) 
G

 [where Gp  ce ( p)( pi  p)]

Gas Material
Balance
Slide — 20
p/z-Cumulative Model:
(2/3)
Considering the condition where:
 D  Gp  1
Then we can use a geometric series to represent the D
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  Gp
(1  Gp )
(1  Gp  1)
Substituting this result into the material balance relation,
we obtain:
p pi 
1
 2

1

(


)
G

Gp 
p

z
zi 
G
G

PETE 613
(2005A)
Gas Material
Balance
Slide — 21
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
)
G
pi
zi

 pi
G zi
We note that these parameters presume that  is constant. Presuming that  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
where  a  bGp
Obviously, one of our objectives will be the study of the
behavior of  vs. Gp (based on a prescribed value of G).
PETE 613
(2005A)
Gas Material
Balance
Slide — 22
-Gp Performance (Case 1)
a. Case 1: Simulated Performance Case — Plot of
 versus Gp (requires an estimate of gas-inplace). Note the apparent linear trend of the
data. Recall that Gp=ce(pp-p).
(1/2)
b. Case 1: Simulated Performance Case — Plot of
p/z versus Gp. The constant and linear  trends
match well with the data — essentially a confirmation of both models.
Simulated Dry Gas Reservoir Case — Abnormal Pressure:
 A linear trend of  vs. Gp is 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 of
1/volume, which suggests  is a scaling variable for G.
PETE 613
(2005A)
Gas Material
Balance
Slide — 23
-Gp Performance (Case 3)
a. Case 3: Anderson L Reservoir Case (South
Texas, USA) — Plot of  versus Gp (requires an
estimate of gas-in-place). Some data scatter
exists, but a linear trend is evident (recall that
Gp=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  approximation is based on the "best fit" of several data functions.
 The linear approximation for  is reasonable (should favor later data).
PETE 613
(2005A)
Gas Material
Balance
Slide — 24
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.
 -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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 25
p/z-Gp2 Plotting Functions: Case 1
a.
d.
p
p
Δ( p/z )   i   vs. Gp
 zi z 
1
G 2p
Gp
0
PETE 613
(2005A)
Δ( p/z ) dGp vs. Gp
b.
1
Δ( p/z ) vs. Gp
Gp
1
e. Δ( p/z ) 
Gp
Gp
0
Δ( p/z ) dGp vs. Gp
Gas Material
Balance
(1/5)
1
c.
Gp
1
f.
Gp
Gp
0
Δ( p/z ) dGp vs. Gp

1
Δ( p/z ) 
Gp

Gp
0

Δ( p/z ) dGp  vs. Gp

Slide — 26
-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 
versus Gp (requires estimate of G).
PETE 613
(2005A)
(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 
versus Gp/G (requires estimate of G).
Gas Material
Balance
Slide — 27
-Gp Plotting Functions: Case 1
(3/5)
Simulated Dry Gas Reservoir Case — Abnormal Pressure:
 Summary p/z—Gp plot for  =constant and  =linear cases.
 Good comparison of trends,  =linear trend appears slightly conservative as it
emerges from data trend — but both solutions appear to yield same G estimate.
PETE 613
(2005A)
Gas Material
Balance
Slide — 28
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).
PETE 613
(2005A)
 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.
Gas Material
Balance
Slide — 29
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.
PETE 613
(2005A)
(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.
Gas Material
Balance
Slide — 30
p/z-Gp2 Plotting Fcns: Case 3 (Anderson L)
a.
d.
p
p
Δ( p/z )   i   vs. Gp
 zi z 
1
G 2p
Gp
0
PETE 613
(2005A)
Δ( p/z ) dGp vs. Gp
b.
1
Δ( p/z ) vs. Gp
Gp
1
e. Δ( p/z ) 
Gp
Gp
0
Δ( p/z ) dGp vs. Gp
Gas Material
Balance
1
c.
Gp
1
f.
Gp
Gp
0
(1/5)
Δ( p/z ) dGp vs. Gp

1
Δ( p/z ) 
Gp

Gp
0

Δ( p/z ) dGp  vs. Gp

Slide — 31
-Gp Plotting Functions: Case 3
a. Case 3: Anderson L (South Texas) — Plot of
ce(p)(pi-p) versus Gp (requires estimate of G).
b. Case 3: Anderson L (South Texas) — Plot of
1/ce(p)(pi-p) versus Gp (requires estimate of G).
c. Case 3: Anderson L (South Texas) — Plot of 
versus Gp (requires estimate of G).
PETE 613
(2005A)
(2/5)
d. Case 3: Anderson L (South Texas) — Plot of 
versus Gp/G (requires estimate of G).
Gas Material
Balance
Slide — 32
-Gp Plotting Functions: Case 3
(3/5)
Case 3 — Anderson L Reservoir (South Texas (USA))
 Summary p/z—Gp plot for  =constant and  =linear cases.
 Good comparison of trends,  =constant and  =linear cases in good agreement.
 Data trend is very consistent.
PETE 613
(2005A)
Gas Material
Balance
Slide — 33
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).
PETE 613
(2005A)
 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.
Gas Material
Balance
Slide — 34
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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 35
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.
 -Gp performance plots.
 Gan analysis (2-straight line trends on a p/z-Gp plot).
 pD-GpD type curve approach.
PETE 613
(2005A)
Gas Material
Balance
Slide — 36
Summary:
(1/3)
 Developed a new p/z-Gp2 material balance model for
the analysis of abnormally pressured gas reservoirs:
where:
p pi 
1
 2

1

(


)
G

Gp 
p

z
zi 
G
G

1

ce ( p)( pi  p)
Gp
The -function is presumed (based on graphical
comparisons) to be either constant, or approximately
linear with Gp. For the =constant case, we have:
p pi

  Gp   G 2p
z zi
 (
PETE 613
(2005A)
1
)
G
pi
zi

Gas Material
Balance
 pi
G zi
Slide — 37
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
)
G
pi
zi

G zi
d. Plotting Function 4 :
(quadratic)
(linear)
p
p
Δ( p/z )   i   vs. Gp
 zi z 
b. Plotting Function 2:
1
G 2p
c. Plotting Function 3:
PETE 613
(2005A)
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
 pi
(linear)
1
Gp

1
Δ( p/z ) 
Gp

Gas Material
Balance
Gp
0

Δ( p/z ) dGp  vs. Gp

Slide — 38
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, but we
recommend a combined/simultaneous analysis.
 The -Gp plots are useful for checking data consistency and for guiding the selection of the -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  or (D ) value.
PETE 613
(2005A)
Gas Material
Balance
Slide — 39
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.
PETE 613
(2005A)
Gas Material
Balance
Slide — 40
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
A Quadratic Cumulative Production Model
for the Material Balance of
Abnormally-Pressured Gas Reservoirs
(End of Presentation)
F.E. Gonzalez
M.S. Thesis (2003)
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
PETE 613
(2005A)
Gas Material
Balance
Slide — 41