advanced gas material balance in simplified format · 90 journal of canadian petroleum technology...
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Journal of Canadian Petroleum Technology90
AbstractMaterial balance has long been used in reservoir engineering prac-tice as a simple yet powerful tool to determine the original gas in place (G). The conventional format of the gas material balance equation is the simple straight line plot of p/Z vs. cumulative gas production (Gp), which can be extrapolated to zero p/Z to obtain G. The graphical simplicity of this method makes it popular. The method was developed for a “volumetric” gas reservoir. It assumes a constant pore volume (PV) of gas and accounts for the energy of gas expansion, but it ignores other sources of energy, such as the effects of formation compressibility, residual fluids expansion and aquifer support. It also does not include other sources of gas storage, such as connected reservoirs or adsorption in coal/shale. In the past, researchers have introduced modified gas material bal-ance equations to account for these other sources of energy. How-ever, the simplicity of the p/Z straight line is lost in the resulting complexity of these equations.
In this paper, a new format of the gas material balance equation is presented, which recaptures the simplicity of the straight line while accounting for all the drive mechanisms. This new method uses a p/Z** instead of p/Z. The effect of each of the previously mentioned drive mechanisms appears as an effective compressibility term in the new gas material balance equation. Also, the physical meaning of the effective compressibilities are explained and compared with the concept of drive indices. Furthermore, the gas material balance is used to derive a generalized rigorous total compressibility in the presence of all the previously mentioned drive mechanisms, which is important in calculating the pseudotime used in rate transient analysis of production data.
IntroductionIt has been of great interest to find G by using material balance. The conventional gas material balance equation was developed for a “volumetric” gas reservoir. Therefore, the p/Z vs. cumula-tive gas production plot may give misleading results in some situa-tions [e.g., when the formation compressibility is of the same order of magnitude as gas compressibility (overpressured reservoirs) or where desorption plays a role (coalbed methane/shale)]. Fig. 1 shows p/Z vs. Gp for several scenarios with the same G. It can be seen from this figure that except for the volumetric reservoir, the plot is not a straight line because gas expansion is not the only drive mechanism. In fact, water encroachment in water-drive reservoirs, formation and residual fluid expansion in overpressured reservoirs and gas desorption in coalbed methane (CBM) or shale reservoirs can have a significant role as a driving force in these cases. In these situations, where the gas expansion is not the dominant driving force, modified material balance equations have been developed by several researchers. Among them, Ramagost and Farshad(1) modi-fied the conventional material balance equation to account for PV shrinkage caused by formation and residual fluid expansion and in-
troduced a new plotting function that keeps the material balance as a straight line. Therefore, the modified material balance equation can be used for overpressured reservoirs. Later, Rahman et al.(2) introduced a rigorous form of material balance equation that con-siders the effect of the formation and residual fluid expansion.
The attempt to find a material balance equation for unconven-tional gas reservoirs started when these resources became more popular. Jensen and Smith(3) proposed a simplified material bal-ance equation for unconventional gas reservoirs by assuming that the stored free gas is negligible and consequently omitted the ef-fect of water saturation completely. However, King(4) derived a comprehensive material balance equation for unconventional gas reservoirs that accounts for the free and adsorbed gas, water en-croachment/production and water and formation compressibility. Seidle(5) suggested that the water saturation change does not have a significant effect on material balance and substituted constant water saturation in King’s material balance.
This study presents an advanced, rigorous gas material balance equation and its plotting function that unifies all the previously mentioned modifications in one equation. The new gas material balance equation has the same format as traditional material bal-ance and can be plotted as a straight line with pi/Zi as y-intercept and G as x-intercept. A significant advantage of this material bal-ance equation is that it can be used to define the total compress-ibility of the system; therefore, the pseudotime calculated with this total compressibility honours material balance in all situations.
Volumetric ReservoirThe conventional gas material balance was derived based on the fact that the remaining gas in the reservoir at any pressure expands to fill the reservoir volume, which was initially occupied by G at the initial pressure [Fig. 2(a)]. In other words, the reservoir volume occupied by gas stays constant. In this situation, gas compress-ibility is the only production mechanism.
GB G G Bgi p g= −( ) , ....................................................................... (1)
where Bgi is initial gas formation volume factor, Bg is gas forma-tion volume factor at pressure p, G is original-gas-in-place and Gp is cumulative gas produced.
Substituting for Bg from the real gas law, at constant tempera-ture, results in:
pZ
pZ
GG
=
i
i
p1- ......................................................................... (2)
The previously described equation is the well-known conven-tional gas material balance equation.
Generalized EquationIn this paper, we derive the advanced gas material equation to ac-count for water encroachment in waterdrive reservoirs, expansion of formation and of residual liquids in overpressured reservoirs and gas desorption in CBM and shale gas reservoirs in the same
Advanced Gas Material Balance in Simplified Format
S. Moghadam, O. Jeje, and L. Mattar, Fekete Associates Inc.
This paper (2009-149) was accepted for presentation at the 10th Canadian International Petroleum Conference (the 60th Annual Technical Meeting of the Petroleum Society), Calgary, 16-18 June, 2009, and revised for publication. Original manuscript received for review 23 March 2009. Revised paper received for review 26 April 2010. Paper peer ap-proved 28 April 2010 as SPE Paper 139428.
January 2011 91
simple format of Equation (2). However, the modification needs to be started from Equation (1).
Each of the previously mentioned effects can be added to the right side of Equation (1) as a volume change term.
GB G G B V V Vgi p g ep d= − + + +( ) ∆ ∆ ∆wip ......................................... (3)
The explanation of each of the volume change terms is provided in the following sections.
Water Influx and Production. In a waterdrive reservoir, the aqui-fer provides pressure support for the reservoir by encroachment of water into the gas reservoir. The encroached water (We) decreases the PV available for the remaining gas [Fig. 2(b)]. The reservoir volume changes because the net encroached water (DVwip) can be calculated from(6):
∆V W W Be p wwip = −( )5 615. , ............................................................ (4)
where We is the water encroachment into the gas reservoir, Wp is water produced at surface and Bw is the water formation volume factor (5.615 is a constant used only in oilfield units).
The encroached water may be determined by using the aquifer models in the literature, such as Schilthuis (steady state)(6), Fet-kovich (pseudosteady state)(7), Carter and Tracy(8) and Van Everdingen and Hurst (unsteady-state)(9). Each of these has its own assumptions and applications.
Overpressured Reservoir. Formation and residual fluid compress-ibility are usually small in comparison with gas compressibility. Therefore, in general, ignoring the formation and the residual fluid expansion does not affect the gas material balance significantly. However, at high pressures the gas compressibility is of the same order of magnitude as that of the formation and residual liquids. Overpressured reservoirs are the most common example of this situation, in which ignoring the effect of formation and residual fluid expansion may result in serious overprediction of G. In over-pressured reservoirs, the p/Z vs. Gp plot yields two distinct slopes. The first slope (shallow) is in the pressure range where formation
00 Gp
p/Z
Overpressured ReservoirWaterdrive ReservoirVolumetric ReservoirCBM Reservoir
G
pi /Zi
Fig. 1—Conventional plot p/Z vs. cumulative gas production.
Reservoir @ pi Reservoir @ p Producedvolume @ p
=
=
=
(a)
(b)
(c)
=
(d)
(e)
∆Vd
∆Vwip
∆Vep
(G−Gp)Bg
(G−Gp)Bg(We−WpBw)+
(G−Gp)Bg+
GBgi
GBgi
GBgi
GfBgi
= GfBgi
(GBgi /Sgi)(cf+cwSwi+coSoi)(pi−p)
ρBVBVL(pi /(pL+pi)−p /(pL+p ))Bg
+ρBVBVL(pi /(pL+pi)−p /(pL+p ))Bg
+(Gf −Gp)Bg
+(Gf −Gp)Bg
+(We−WpBw)(GfBgi /Sgi)(cf+cwSwi+coSoi)(pi−p)
Fig. 2—Schematic of reservoir volume at initial pressure and a lower pressure.
Journal of Canadian Petroleum Technology92
and residual fluid expansion play a significant role, while the sec-ond slope (steep) reflects the region where gas expansion is the dominant production mechanism(1). Ramagost and Farshad(1) con-sidered the effect of formation and residual fluid expansion by a volume change equal to
GB S c cS
p pgi w w f
wi
+( )−
−( )1
.
Later, Rahman et al.(2) introduced a rigorous form of this volume change by integrating the compressibility equation for any sub-stance in the reservoir. The total effect of formation and the residual fluids compressibility can be added together as(2):
∆VB GS
e S e S eepgi
gi
c dp
wi
c dp
oi
c dpfp
pi
wp
pi
op
pi
= −∫ ∫
−∫
−−
( )+ ( )+ (1 1 11)
.................. (5)
When matrix shrinkage occurs during CBM production, the (fracture) porosity containing the free gas increases. In that situa-tion, cf has a negative value and is a complex function of pressure.
If cf, cw and co are constant values, a simplified form of Equa-tion (5) can be written as:
∆VB GS
e S eepgi
gi
c p pwi
c p pf i w i= − + − +− − −( ) ( )( ) ( )1 1
S eoic p po i( )( )− − 1
......................... (6)
The approximate form of Equation (6) can be found considering ex≈1+x as:
∆VB GS
c S c S c p pepgi
gif wi w oi o i= + + −( )( ) ................................................ (7)
Equation (7) is the format used by Ramagost and Farshad(1). Because of its simplicity, it is also the format that is used in this paper, but for a more rigorous calculation, DVep from Equation (5) should be used. The effect of formation and residual liquids expan-sions, DVep, is depicted in Fig. 2(c).
CBM/Shale Gas Desorption. The gas storage mechanism in a CBM (or shale gas) reservoir is unlike that of a conventional gas reservoir. In a typical gas reservoir, gas is stored in the pores by compression. In a CBM/shale reservoir, in addition to the free gas (Gf) stored in the fracture network, gas is stored within the coal/shale matrix by adsorption. As the reservoir pressure is reduced, gas is desorbed from the surface of the matrix. The amount of gas stored by adsorption can exceed the gas stored by compres-sion. Desorption of gas is commonly described by the Langmuir Isotherm as specific gas content
= V pp p
L
L +,
VL is the Langmuir volume parameter and pL is the Langmuir pres-sure parameter.
Specific gas content is the volume of gas per unit mass of coal. Therefore, the total amount of gas adsorbed can be calculated from:
Adsorbed gas= G V V pp pa B B
L
L
=+
ρ ,
where rB and VB are the density and volume of the coal, respec-tively, and VL is on a “dry, ash-free” basis.
The material balance equation is based on the reservoir volume that the free gas occupies at the initial pressure. For CBM, this is equal to Gf Bgi. In a conventional gas reservoir, G=Gf , but for a CBM reservoir, the total G includes the Gf and the adsorbed gas
(Ga). The red-dashed box in Fig. 2(d) shows the volume of “de-sorbed” gas at reservoir pressure p, which is added to the “free” gas. The desorbed gas volume, which needs to be added to the right side of Equation (1), can be calculated from (for Sgi>0):
∆V BG BS
V pp p
V pp pd B g
f gi
gi
L i
L i
L
L
=+
−+
ρφ
( ) ........................................ (8)
Advanced Material Balance EquationThe advanced material balance equation, with consideration of water encroachment/production, formation and residual fluids ex-pansions and gas desorption, can be derived by substituting DVwip, DVep, and DVd from Equations (4), (7) and (8) into Equation (3):
G B G G B W W BG BS
c c S c
f gi f p g e p w
f gi
gif w wi o
= − + −
+ + +
( ) ( )
( SS p p
BG BS
V pp p
V pp p
oi i
B gf gi
gi
L i
L i
L
L
)( )
( )
−
++
−+
ρφ
............................... (9)
If both sides of the previously described equation are divided by G BSf gi
gi
(reservoir PV), it can be reduced to:
pZS c c c p
ZGG
Sgi ep di
i
p
fgi( ) ( )− − − = −wip 1 , ...................................... (10)
where cwip, cep and cd are defined as cwip, the change in PV caused by the water encroachment/production relative to the reservoir PV:
cV
G B SW W B
G B Sf gi gi
e p w
f gi giwip
wip= =−∆ 5 615. ( ) .................................. (10A)
cep is the relative change of the PV caused by the formation and residual fluid expansion (approximate form):
cV
G B Sc c S c S p pep
ep
f gi gif w wi o oi i= = + + −
∆( )( ) ........................ (10B)
cd is the relative change of the PV caused by gas desorption:
c VG B S
B V pp p
V pp pd
d
f gi gi
B g L i
L i
L= =+
−+
∆ ρφ
( )L
........................... (10C)
Note that the variables cwip, cep and cd are not compressibilities (as implied by their symbol), but they represent the relative change in the PV caused by the specific mechanism.
Plotting Function of Advanced Gas Material BalanceEquation (10) is an easy formulation for a general material balance equation and can be plotted as (p/Z)(Sgi-cwip-cep-cd) vs. Gp to give a straight line. However, it is derived based on the PV of the free gas. Therefore, the straight line crosses the abscissa at Gf (free gas volume), not G. This is an inconvenience, and is a disadvantage of this plotting format (see Fig. 3) when compared to the conven-tional material balance (Fig. 1), where the abscissa is G. It is worth mentioning that G can be found easily if Gf is known,
G GG BS
V pp pf
f gi
gi
B L i
L i
= ++
ρφ
.
January 2011 93
Note also that Equation (10) must be solved iteratively in the case of water encroachment/production because Gf appears in the cwip term.
In his work explaining CBM material balance, King(4) intro-duced Z* as:
Z Z
S c c S p pW W BG G S
BV p p pgi f w wi i
e p w
f gi gi
B g
L L
* =− +( ) −( ) −
−+
+( )ρ φ
and reformatted Equation (1) as:
pZ
GG
pZ
p i
i* *= −
1 ............................................................... (11)
This equation has the same format as the conventional gas ma-terial balance equation, and can be plotted as a straight line of p/Z* vs. Gp, which extrapolates to G, as can be seen in Fig. 4. This format has a clear advantage over that of Fig. 3 in that it extrapo-lates to the greater practical value of G rather than Gf. Whereas this format is theoretically applicable to gas reservoirs other than CBM, the fact that the p/Z* values bear little resemblance to the conven-tional p/Z values detracts from its utility.
In an effort to generalize the gas material balance equation for all reservoirs (conventional, overpressured and CBM/shale), we have developed a Z** variable to replace King’s Z* and have re-written the gas material balance equation, Equation (10), as:
pZ
GG
pZ
p i
i** **= −
1 ............................................................. (12)
The advantage of the Z** format is that the p/Z** values are similar in magnitude to the conventional p/Z values. As shown in Fig. 5, p/Z** vs. Gp is a straight line that starts from the conven-tional pi/Zi and extrapolates to G. This formulation and presenta-tion has simplified the applicability of the general material balance equation. The definition of Z** was derived from Equations (10) and (12) as:
Z p
SpZS c c c p
ZGG
GGgi
gi ep di
i f
f
**
( )
=
− − − + −
1 1wip
........... (13)
Also, Z** is related to King’s Z* by the following relationship:
Z Z ZZ
i
i
** **
=
.
Equation (12) is the general material balance equation for all gas reservoirs (conventional, overpressured and CBM/shale). When plotted as p/Z** vs. Gp it yields a straight line, which, similar to the conventional p/Z plot, starts from the conventional pi/Zi and ex-trapolates to G.
Analysis ProcedureTo interpret field data, when multiple drive mechanisms exist it is good engineering practice to select for modification the most im-portant of these mechanisms and to specify constant values of the other mechanisms.
In the following example, the rock compressibility was speci-fied as constant and except for gas expansion, waterdrive is consid-ered to be the important mechanism to be accounted for. A sample procedure is presented:
1. Specify cf and pi. Select the aquifer model (in this example we chose Fetkovich).
2. For each data point, calculate cep [Equation (10B)]. 3. Assume cwip = 0.4. Calculate Z** using Equation (13) (note: when there is no ad-
sorption, as in this example, G/Gf is equal to 1. If adsorption is present
GG
BS
V pp pf
gi
gi
B L i
L i
= ++
1 ρφ ).
5. Plot p/Z** vs. Gp (Fig. 6).
(p /Z )(Sgi−cwip−cep−cd)Gf not G
p /Z
pi /Zip
/Z
00 Gp
Fig. 3 —(p/Z)(1-cwip-cep-cd) and p/Z plots.
p /Z *
p /ZG
pi /Zi
pi /Zi
p/Z
00 Gp
Fig. 4 —King’s p/Z* and p/Z plots.
pi /Zi
p/Z
00 Gp
G
p /Z p /Z**
Fig. 5—p/Z** and p/Z plots.
Journal of Canadian Petroleum Technology94
6. Choose/adjust aquifer parameters; calculate We.7. Draw best straight line though p/Z** vs. Gp data (this data
may be curved) and extrapolate to G (on the x-axis).8. Calculate cwip [Equation (10A)]. 9. Return to Step 4.10. Repeat until acceptable straight line is obtained (Fig. 7).
Drive Indices Drive indices were defined to indicate the relative contribution of different drive mechanisms in oil recovery(9). The same concept can also be applied for gas reservoirs(10). The drive indices are defined relative to the produced gas volume:
Gas (Compressibility) Drive Index
GDIG B BG B
VG B
g gi
p g
G
p g
=−( )
= ∆
Formation and Residual Liquids Saturation Compressibility Drive Index
CDIV
G Bep
p g
=∆
Waterdrive Index
WDIVG Bp g
=∆ wip
Desorption drive index can be added to the previously men-tioned indices as:
DDI VG B
d
p g
= ∆
Theoretically, the sum of the drive indices should equal 1. This is identical to GpBg=DVG+DVwip+DVep+DVd, as is shown sche-matically in the right side of Fig. 2(e).
The variables cwip, cep and cd defined in Equation (10) are re-lated to the drive indices. The denominator in those variables is the reservoir PV, whereas the denominator in the drive indices is the produced gas volume at pressure p, GpBg.
It is worth noting that in a conventional gas reservoir, the gas compressibility drive, GDI, is the dominant drive mechanism, whereas in an unconventional gas reservoir such as a CBM or shale
gas reservoir, GDI can be fairly small (or even negligible) in com-parison with the desorption drive index, DDI.
Total Compressibility and PseudotimeIn fluid-flow and pressure transient analysis of gas reservoirs, pseu-dovariables (pseudotime and pseudopressure) are used to linearize the diffusivity equation. Pseudotime is defined as:
t dtc
t
at
= ∫ µ0
,
where m is the viscosity of the gas and ct is the total compressibility of the system. The conventional definition of ct is(11):
c c S c S c S ct f oi o w gi g= + + +wi.
The problem with the traditional definition of the total com-pressibility is that it does not always honour the material bal-ance equation. Therefore, the computed pseudotime may contain a considerable error. Rahman et al.(12) introduced a rigorous pseudotime definition, which is defined by manipulating the material balance equation. Their major assumption is that gas was the only mobile phase in a conventional gas reservoir. In this paper, a more generalized form of the gas material bal-ance equation is used, which considers water production and is not limited to conventional gas reservoirs. Therefore, it can be used for waterdrive and also unconventional gas reservoirs (e.g., CBM/shale gas reservoirs). The detailed derivation of this pseu-dotime is given in the Appendix. The total compressibility is de-fined as:
c c S c c ccp
cp
cpt g gi ep d
ep d= − − − −∂∂
−∂∂
− ∂∂
wip
wip ,
where cwip, cep, cd and their derivatives are defined in the Table 1.
ConclusionsAn advanced gas material balance equation has been presented and the corresponding plotting function introduced; therefore, the ma-terial balance equation can be plotted as a straight line with pi/Zi as y-intercept and G as x-intercept.
The similarity of the recommended plotting procedure, pi/Z** vs. Gp, to the more commonly used p/Z format is a great practical advantage. It allows the use of a rigorous material balance formula-tion for complex and unconventional gas reservoirs, while retaining the simplicity and familiarity of the commonly used p/Z format.
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Curved data First estimate of OGIP=30.5 Bscf
Fig. 6—p/Z** vs. Gp (cwip is assumed to be 0 in Z** calculation).
January 2011 95
The advanced gas material balance equation is used to derive a rigorous definition for total compressibility that can be used for an-alyzing fluid-flow in unconventional gas reservoirs, or when gas is not the only mobile phase.
AcknowledgementsThe authors acknowledge the contribution of N.M. Anisur Rahman to this development. JCPT
Nomenclature Bg = gas formation volume factor at time, t, ft3/scf, m3/m3
Bgi = initial gas formation volume factor, ft3/scf, m3/m3
Bw = water formation volume factor, bbl/stb, m3/m3
c = compressibility, 1/psia, 1/Pa c = in the Appendix, the summation of cwip, cep and cd cd = relative volume change caused by CBM gas
desorption,
∆VGB
Sd
gigi
cep = relative volume change caused by residual fluid and formation,
∆VGB
Sep
gigi
cf = formation compressibility, 1/psia, 1/Pa cg = gas compressibility, 1/psia, 1/Pa co = oil compressibility, 1/psia, 1/Pa cs = CBM sorption compressibility (Appendix) ct = total compressibility, 1/psia, 1/Pa cw = water compressibility, 1/psia, 1/Pa cwip = relative volume change caused by water influx and
production,
∆VGB
Sgi
giwip
G = original-gas-in-place, Bcf, m3
Ga = adsorbed-gas-in-place, Bcf, m3
Gf = free-gas-in-place, Bcf, m3
Gp = cumulative gas produced to time t, Bcf, m3
k = permeability, md, m2
kr = permeability, md, m2
p = pressure, psia, Pa psc = standard conditions reservoir pressure, psia, Pa PL = Langmuir pressure, psia, Pa q = flow rate, MMscfd, m3/s S = gas saturation, % Sgi = initial gas saturation, % So = oil saturation, % Soi = initial oil saturation, % Sw = water saturation, %
Swi = initial water saturation, % t = time, hours, s ta = pseudotime, hours, s T = reservoir temperature, °F, K Tsc = standard conditions temperature, °F, K VB = bulk volume, ft3, m3
Vi = initial volume, ft3, m3
VL = Langmuir volume, scf/ton, m3/kg We = Water encroachment into formation, bbl, m3
Wp = cumulative water produced, Bbl, m3
Z = Gas compressibility factor, no units Zi = initial gas compressibility factor, no units 5.615 = Conversion constant in oilfield units, ft3/bbl 3 DV = change in volume, scf, m3
DVd = change in volume caused by CBM gas desorption, ft3, m3
DVep = change in volume caused by formation and the residual fluids expansion, ft3, m3
DVwip = change in volume caused by water encroachment/production, bbl, m3
m = viscosity, cp, Pa.s rB = bulk density, lb/ft3, kg/m3
f = porosity, % y = pseudopressure, psia2/cp, Pa/s
Subscripts a = adsorbed B = bulk d = desorption e = encroachment ep = expansion f = free g = gas i = initial L = Langmuir o = oil p = production r = relative s = sorption sc = standard conditions t = total w = water wip = water influx and production
SI Metric Conversion Factors bbl × 1.589 873 E-01 = m3
cp × 1.0* E-03 = Pa·s ft × 3.048* E-01 = m
p/Z
**, p
sia
Net Cumulative Gas Production, MMscf
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Straightened data OGIP=19.6 Bscf
Fig. 7—p/Z** vs. Gp.
Journal of Canadian Petroleum Technology96
hp × 7.460 43 E-01 = kW °F (°F-32)/1.8 = °C lb × 4.535 924 E-01 = kilogram (kg) ton × 9.071 847 E-01 = Mg
*Conversion factor is exact.
References 1. Ramagost, B.P. and Farshad, F.F. 1981. P/Z Abnormally Pressured Gas
Reservoirs. Paper SPE 10125 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 4–7 October. doi: 10.2118/10125-MS.
2. Rahman, N.M.A., Anderson, D.M., and Mattar, L. 2006. New, Rig-orous Material Balance Equation for Gas Flow in a Compressible Formation. Paper SPE 100563 presented at the SPE Symposium on Gas Technology, Calgary, 15–17 May. doi: 10.2118/100563-MS.
3. Jensen, D. and Smith, L.K. 1997. A Practical Approach to Coalbed Methane Reserve Prediction Using A Modified Material Bal-ance Technique. Paper 9765 presented at the International Coalbed Methane Symposium, Tuscaloosa, Alabama, USA, 12–17 May.
4. King, G.R. 1993. Material-Balance Techniques for Coal-Seam and Devonian Shale Gas Reservoirs with Limited Water Influx. SPE Res Eng 8 (1): 67–72; Trans., AIME, 295. SPE-20730-PA. doi: 10.2118/20730-PA.
5. Seidle, J.P. 1993. Long-Term Gas Deliverability of a Dewatered Coalbed. J Pet Technol 45 (6): 564–569; Trans., AIME, 295. SPE-21488-PA. doi: 10.2118/21488-PA.
6. Schilthuis, R.J. 1936. Active Oil and Reservoir Energy. SPE-936033-G. Trans., AIME, 118: 33–52.
7. Fetkovich, M.J. 1980. Decline Curve Analysis Using Type Curve. J Pet Technol 32 (6): 1065–1077. SPE-4629-PA. doi: 10.2118/4629-PA.
8. Carter, R.D. and Tracy, G.W. 1960. An Improved Method for Calcu-lating Water Influx. SPE-1626-G. Trans., AIME, 219: 415–417.
9. Van Everdingen, A.F. and Hurst, W. 1949. The Application of the La-place Transformation to Flow Problems in Reservoirs. SPE-949305-G. Trans., AIME, 186: 305–324.
10. Pletcher, J.L. 2000. Improvements to Reservoir Material Balance Methods. Paper SPE 62882 presented at the SPE Annual Technical Con-ference and Exhibition, Dallas, 1–4 October. doi: 10.2118/62882-MS.
11. Ramey, H.J. Jr. 1964. Rapid Methods for Estimating Reservoir Com-pressibilities. J Pet Technol 16 (4): 447–454; Trans., AIME, 231. SPE-772-PA. doi: 10.2118/772-PA.
TABLE 1—SUMMARY OF TOTAL COMPRESSIBILITY EQUATIONS FOR ALL RESERVOIR TYPES
Definition Comment
Waterdrive reservoir
co, cw and c f are constant.
cep is approximate form.
co, cw and c f are constant.
cep is simple form.
co, cw and c f are pressure-sensitive,
without matrix shrinkage.
cep is rigorous form.
co, cw and cf are pressure-sensitive, with
matrix shrinkage.
cep is rigorous form.
CBM/shale gas reservoir
cW W B
G B Se p w
f gi giwip =
−5 615. ( )
∂∂
= ∂∂
cp G B S
Wpf gi gi
ewip 5 615.
c c c S c S p p c p pep f w wi o oi i e i= + + − = −( )( ) ( )
∂∂
= − − −( )( ) + + −( )( ) + + −( )cp
c c p p S c c p p S c c p pi wi w w i oi o o iep
f f1 1 1(( )c e S e S eep
c p pwi
c p poi
c p pf i w i o i= − − −− − − −( )+ ( )+ ( )( ) ( ) ( )1 1 1
∂∂
= − + +( )− − − −cp
S c e S c e c eepoi o
c p pwi w
c p pf
c p po i w i f i( ) ( ) ( ) )
c e S e S eep
c dp
wi
c dp
oi
c dpfp
pi
wp
pi
op
pi
= −∫ ∫
−∫
−−
( )+ ( )+ ( )1 1 1
∂∂
= −∫ ∫ ∫
−c
pc e S c e S c eepf
c dp
wi w
c dp
oi o
c dpfp
pi
wp
pi
op
pi
+ +
c F p S e S eep wi
c dp
oi
c dpwp
pi
op
pi
= −∫
−∫
−( ( ))+ ( )+ ( )1 1 1φ
∂∂
= − −∫ ∫
cp
dF pdp
S c e S c eepwi w
c dp
oi o
c dpwp
pi
op
pi
φ ( )+ +
cB
a m V pp p
V pp pd
B g L i
L i
L
L
= − −( )+
−+
ρφ
1 ( )
∂∂
= − −( ) −+
−+
+ −+
++( )
cp
B Va m c p
p pp
p p p pp
p pd B g L
gi
L i L L L
ρφ
1 12
January 2011 97
12. Rahman, N.M.A., Mattar, L., and Zaoral, K. 2006. A New Method for Computing Pseudo-Time for Real Gas Flow Using the Material Bal-ance Equation. J Can Pet Technol 45 (10): 36–44. JCPT Paper No. 06-10-03. doi: 10.2118/06-10-03.
13. Bumb, A.C. and McKee, C.R. 1988. Gas Well Testing in the Presence of Desorption for Coalbed Methane and Devonian Shale. SPE Form Eval 3 (1): 179–185. SPE-15227-PA. doi: 10.2118/15227-PA.
Appendix: Total Compressibility and Pseudotime The purpose of this Appendix is to define ct for any reservoir type in a general form that honours material balance. The gas material balance in Equation (10) can be written as:
pZS c p
ZG GG
Sgii
i
f p
fgi−( ) =
−( ) , ................................................ (1A)
where p is reservoir pressure at time, t, Z is the compressibility factor, Sgi is the initial gas saturation, c is the summation of cwip, cep and cd, pi is the initial reservoir pressure, Zi is the initial com-pressibility factor, Gf is the original-free-gas-in-place and Gp is the gas produced to time, t.
Equation (1A) is rearranged to:
pZ
pZ G
G GS c
Si
i f
f p
gigi=
−−
( ) ................................................................ (2A)
Next, Equation (2A) is differentiated with respect to time con- sidering
∂∂ ( ) =tG qp and ∂
∂= ∂
∂∂∂
ct
cppt, where q represents the rate at
time, t. This results in the following equation:
∂∂
= −−
−− − ∂
∂∂∂
−tpZ
p SZ G
qS c
G G cppt
Si gi
i gi
f p
gi
( )
cc
2 ................. (3A)
The chain rule can be applied to the right side of Equation (3A) as:
∂∂
= ∂∂
∂∂
= − ∂∂
∂∂t
pZ p
pZ
pt Z
pZ
Zp
pt
12
.
Also, remembering that cp Z
Zpg = − ∂
∂1 1
, and then multiplying by pZ
, results in:
∂∂
= ∂∂t
pZ
pZc p
tg ................................................................... (4A)
Substitute Equation (4A) into (3A):
pZcZ Gp S
pt
qS c
G G cpptc
S cg
i f
i gi gi
f p
gi
∂∂
= −−
−− − ∂
∂∂∂
−
( )
2
................ (5A)
Equation (1A) can be rearranged as ( )G GS c
pZZ Gp S
f p
gi
i
i gi
−−
= .
Therefore, Equation (5A) becomes
pZZ Gp S
c
cp
S cpt
qS
i f
i gig
gi
−
∂∂
−
∂∂
= − ggi c−
.
The previously described equation can be solved for ∂∂pt:
∂∂
=−
− − ∂∂
pt
qSG
pZZp
c S c cp
gi
f
i
i
g gi
...................................................... (6A)
The next step is involving pseudopressure, ψµ
=
∫2 k pzdpr
p
p
o
. Therefore,
∂∂
=ψµpk pzr2 ............................................................................. (7A)
Combining Equations (6A) and (7A) together results in
∂∂
= ∂∂
∂∂
=
−
− − ∂∂
ψ ψµt p
Pt
k pZ
qSG
pZZp
c S c cp
r
gi
f
i
i
g gi
2 ,
which expands to:
∂∂
= ∂∂
∂∂
=
−
− − ∂∂
=
ψ ψ µt p
Pt
k qSG
pZ
c S c cp
k Sr gi
f
i
i
g gi
r gi2 2µµG
pZ
q
c S c cp
f
i
i
g gi
−
− − ∂∂
( )
............................................... (8A)
There is a relationship between pseudopressure and pseudo-time(12) as:
∂∂
= −ψt
p qk SGZa
i r gi
i
2 .................................................................... (9A)
Therefore, Equations (8A) and (9A) together results in:
∂∂
= ∂∂
∂∂
=−( )
− − ∂∂
−tt
tt
k SG
pZ
q
c S c cp
G Zp q
a a
r gi
f
i
i
g gi
f i
iψψ µ
2
2 kk Sr gi
∂∂
=− − ∂
∂
tt
c S c cp
a
g gi
1
µ .................................................... (10A)
Also, pseudotime is defined as t dtcat
t= ∫ µ0
, which gives, ∂∂
=tt ca
t
1µ
. Therefore,
c c S c cpt g gi= − − ∂
∂
........................................................ (11A)
Depending on what kinds of assumptions are used for reservoir type, the previously described equation changes to match the reser-voir. The most general form of the total compressibility equation is:
Journal of Canadian Petroleum Technology98
c c S c c ccp
cp
cpt g gi ep d
ep d= − − − −∂∂
−∂∂
− ∂∂
wip
wip
For CBM reservoirs, c c cpg dd−( ) − ∂
∂ can be further simplified to:
c c cp
B V pp pg d
d B g L L
L
−( ) − ∂∂
=+( )
ρφ 2
.
The previously described formulation removes the dependence of the CBM terms on initial pressure. It is defined as the sorption compressibility, cs
(13).This changes the form of the total compressibility equation to
the following:
c c S c ccp
cp
ct g gi epep
s= − − −∂∂
−∂∂
+
wip
wip .
Summary of Equations
pZS c c c p
ZGG
Sgi ep di
i
p
fgi− − −( ) = −
wip 1
pZ
pZ
GG
i
i
p
** **= −
1
Z p
SpZS c c c p
ZGG
GGgi
gi ep di
i f
f
**
( )
=
− − − + −
1 1wip
c c S c c ccp
cp
cpt g gi ep d
ep d= − − − −∂∂
−∂∂
− ∂∂
wip
wip
Authors
Samane Moghadam is a reservoir engineer with Fekete Associates Inc., where she is working on software development-related R&D projects. She holds B.Sc. degrees in mechanical engineering and petroleum engi-neering from Sharif University of Technology in Iran and an M.A.Sc. degree in petroleum systems engineering from the University of Regina.
Oluyemisi Jeje has been with Fekete and As-sociates Inc. for over 8 years as a member of the R&D department. He holds a B.Sc. de-gree in chemical engineering from the Uni-versity of Calgary.
Louis Mattar is the president of Fekete As-sociates Inc. He specializes in the analysis of production data and well tests. He has au-thored over 60 technical publications. He has received the Society of Petroleum Engineers’ Distinguished Author Award and the Out-standing Service Award. In 2003, Mattar was the SPE International Distinguished Lecturer in well testing.