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The material-balance equation is the simplest expression of the conservation of mass in a reservoir. The equation mathematically defines the different producing mechanisms and effectively relates the reservoir fluid and rock expansion to the subsequent fluid withdrawal.  
The material-balance equation is the simplest expression of the conservation of mass in a reservoir. The equation mathematically defines the different producing mechanisms and effectively relates the reservoir fluid and rock expansion to the subsequent fluid withdrawal.


==Material balance equation==
== Material balance equation ==
The applicable equation for initially saturated volatile- and black-oil reservoirs is<ref name="r1" /><ref name="r2" /><ref name="r3" /><ref name="r4" />


[[File:Vol5 page 0906 eq 001.png]]....................(1)
The applicable equation for initially saturated volatile- and black-oil reservoirs is<ref name="r1">Walsh, M.P. 1995. A Generalized Approach to Reservoir Material Balance Calculations. J Can Pet Technol 34 (1). PETSOC-95-01-07. http://dx.doi.org/10.2118/95-01-07</ref><ref name="r2">Walsh, M.P. 1994. New, Improved Equation Solves for Volatile Oil and Condensate Reserves. Oil & Gas J. (22 August): 72.</ref><ref name="r3">Walsh, M.P., Ansah, J., and Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 2 - Applications to Saturated and Non-Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27728-MS. http://dx.doi.org/10.2118/27728-MS</ref><ref name="r4">Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.</ref>
 
[[File:Vol5 page 0906 eq 001.png|RTENOTITLE]]....................(1)


where:
where:
* ''G''<sub>''fgi''</sub>, ''N''<sub>''foi''</sub>, and ''W'' are the initial free gas, oil, and water in place, respectively
* ''G''<sub>''p''</sub>, ''N''<sub>''p''</sub>, and ''W''<sub>''p''</sub> are the cumulative produced gas, oil, and water, respectively
* ''G''<sub>''I''</sub> and ''W''<sub>''I''</sub> are the cumulative injected gas and water respectively 
* ''E''<sub>''g''</sub>, ''E''<sub>''o''</sub>, ''E''<sub>''w''</sub>, and ''E''<sub>''f''</sub> are the gas, oil, water, and rock (formation) expansivities


Most of the equations regarding [[Primary drive mechanisms|primary drive mechanisms]] for oil reservoirs apply to any consistent set of units. A few equations, however, are written assuming English or customary units. Those equations are expressed in SI units:
*''G''<sub>''fgi''</sub>, ''N''<sub>''foi''</sub>, and ''W'' are the initial free gas, oil, and water in place, respectively
*''G''<sub>''p''</sub>, ''N''<sub>''p''</sub>, and ''W''<sub>''p''</sub> are the cumulative produced gas, oil, and water, respectively
*''G''<sub>''I''</sub> and ''W''<sub>''I''</sub> are the cumulative injected gas and water respectively
*''E''<sub>''g''</sub>, ''E''<sub>''o''</sub>, ''E''<sub>''w''</sub>, and ''E''<sub>''f''</sub> are the gas, oil, water, and rock (formation) expansivities
 
Most of the equations regarding [[Primary_drive_mechanisms|primary drive mechanisms]] for oil reservoirs apply to any consistent set of units. A few equations, however, are written assuming English or customary units. Those equations are expressed in SI units:


[[File:Vol5 page 0980 eq 001.png]]....................(2)
[[File:Vol5 page 0980 eq 001.png|RTENOTITLE]]....................(2)


[[File:Vol5 page 0980 eq 002.png]]....................(3)
[[File:Vol5 page 0980 eq 002.png|RTENOTITLE]]....................(3)


[[File:Vol5 page 0980 eq 003.png]]....................(4)
[[File:Vol5 page 0980 eq 003.png|RTENOTITLE]]....................(4)


[[File:Vol5 page 0980 eq 004.png]]....................(5)
[[File:Vol5 page 0980 eq 004.png|RTENOTITLE]]....................(5)


[[File:Vol5 page 0980 eq 005.png]]....................(6)
[[File:Vol5 page 0980 eq 005.png|RTENOTITLE]]....................(6)


[[File:Vol5 page 0980 eq 006.png]]....................(7)
[[File:Vol5 page 0980 eq 006.png|RTENOTITLE]]....................(7)


and [[File:Vol5 page 0980 eq 007.png]]....................(8)
and [[File:Vol5 page 0980 eq 007.png|RTENOTITLE]]....................(8)


''N''<sub>''foi''</sub> and ''G''<sub>''fgi''</sub> are related to the total original oil in place (OOIP) and original gas in place (OGIP), ''N'' and ''G'', according to ''N'' = ''N''<sub>''foi''</sub> + ''G''<sub>''fgi''</sub> ''R''<sub>''vi''</sub> and ''G'' = ''G''<sub>''fgi''</sub> + ''N''<sub>''foi''</sub> ''R''<sub>''si''</sub>.  
''N''<sub>''foi''</sub> and ''G''<sub>''fgi''</sub> are related to the total original oil in place (OOIP) and original gas in place (OGIP), ''N'' and ''G'', according to ''N'' = ''N''<sub>''foi''</sub> + ''G''<sub>''fgi''</sub> ''R''<sub>''vi''</sub> and ''G'' = ''G''<sub>''fgi''</sub> + ''N''<sub>''foi''</sub> ''R''<sub>''si''</sub>.


The expansivities are defined as
The expansivities are defined as


[[File:Vol5 page 0906 eq 002.png]]....................(9)
[[File:Vol5 page 0906 eq 002.png|RTENOTITLE]]....................(9)
 
[[File:Vol5 page 0906 eq 003.png|RTENOTITLE]]....................(10)


[[File:Vol5 page 0906 eq 003.png]]....................(10)
[[File:Vol5 page 0906 eq 004.png|RTENOTITLE]]....................(11)


[[File:Vol5 page 0906 eq 004.png]]....................(11)
and [[File:Vol5 page 0906 inline 001.png|RTENOTITLE]], where B to and B tg are the two-phase formation volume factors (FVFs),


and [[File:Vol5 page 0906 inline 001.png]], where B to and B tg are the two-phase formation volume factors (FVFs),
[[File:Vol5 page 0907 eq 001.png|RTENOTITLE]]....................(12)


[[File:Vol5 page 0907 eq 001.png]]....................(12)
and [[File:Vol5 page 0907 eq 002.png|RTENOTITLE]]....................(13)


and [[File:Vol5 page 0907 eq 002.png]]....................(13)
The rock expansivity is obtained from direct measurement. See [[Compaction_drive_reservoirs|compaction driving oil reservoir]] for a greater discussion.


The rock expansivity is obtained from direct measurement. See [[Compaction drive reservoirs|compaction driving oil reservoir]] for a greater discussion.  
Physically, the two-phase FVF is the total hydrocarbon volume per unit volume of oil or gas at standard conditions. The two-phase FVF mimics the observations noted during a constant-composition expansion test. For instance, the two-phase oil FVF is the total hydrocarbon (oil + gas) volume of a saturated oil sample per unit volume of oil at standard conditions. In contrast, the two-phase gas FVF is the total hydrocarbon volume of a saturated gas sample per unit volume of gas at standard conditions. ''B''<sub>''to''</sub> and ''B''<sub>''tg''</sub> typically are expressed in units of RB/stock tank barrel (STB) and RB/Mscf, respectively.


Physically, the two-phase FVF is the total hydrocarbon volume per unit volume of oil or gas at standard conditions. The two-phase FVF mimics the observations noted during a constant-composition expansion test. For instance, the two-phase oil FVF is the total hydrocarbon (oil + gas) volume of a saturated oil sample per unit volume of oil at standard conditions. In contrast, the two-phase gas FVF is the total hydrocarbon volume of a saturated gas sample per unit volume of gas at standard conditions. ''B''<sub>''to''</sub> and ''B''<sub>''tg''</sub> typically are expressed in units of RB/stock tank barrel (STB) and RB/Mscf, respectively.
*For undersaturated oils, the two-phase oil FVF is equal to the oil FVF
* For undersaturated oils, the two-phase oil FVF is equal to the oil FVF
*For undersaturated gases, the two-phase gas FVF is equal to the gas FVF.
* For undersaturated gases, the two-phase gas FVF is equal to the gas FVF.  


'''Eqs. 12''' and '''13''' account for volatilized oil in the equilibrium gas phase. If volatilized oil is negligible, these equations are simplified. For instance, ''B''<sub>''to''</sub> = ''B''<sub>''o''</sub> + ''B''<sub>''g''</sub> (''R''<sub>''si''</sub> – ''R''<sub>''s''</sub>) and ''B''<sub>''tg''</sub> = ''B''<sub>''g''</sub>. These equations apply for black oils. '''Eq.11''' ignores dissolved gas in the aqueous phase.  
'''Eqs. 12''' and '''13''' account for volatilized oil in the equilibrium gas phase. If volatilized oil is negligible, these equations are simplified. For instance, ''B''<sub>''to''</sub> = ''B''<sub>''o''</sub> + ''B''<sub>''g''</sub> (''R''<sub>''si''</sub> – ''R''<sub>''s''</sub>) and ''B''<sub>''tg''</sub> = ''B''<sub>''g''</sub>. These equations apply for black oils. '''Eq.11''' ignores dissolved gas in the aqueous phase.


'''Eq.1''' broadly states that net expansion equals net withdrawal. More specifically, it shows the different forms of expansion and withdrawal. The different forms of expansion such as gas expansion are responsible for the different producing mechanisms.  
'''Eq.1''' broadly states that net expansion equals net withdrawal. More specifically, it shows the different forms of expansion and withdrawal. The different forms of expansion such as gas expansion are responsible for the different producing mechanisms.


For the sake of simplicity, '''Eq.1''' is often written in the abbreviated form of
For the sake of simplicity, '''Eq.1''' is often written in the abbreviated form of


[[File:Vol5 page 0907 eq 003.png]]....................(14)
[[File:Vol5 page 0907 eq 003.png|RTENOTITLE]]....................(14)


where:
where:
* ''F'' = total net fluid withdrawal or production
 
* ''E''<sub>''gwf''</sub> = composite gas expansivity  
*''F'' = total net fluid withdrawal or production
* ''E''<sub>''owf''</sub> = composite oil expansivities  
*''E''<sub>''gwf''</sub> = composite gas expansivity
*''E''<sub>''owf''</sub> = composite oil expansivities


''F'', ''E''<sub>''gwf''</sub>, and ''E''<sub>''owf''</sub> are defined in
''F'', ''E''<sub>''gwf''</sub>, and ''E''<sub>''owf''</sub> are defined in


[[File:Vol5 page 0908 eq 001.png]]....................(15)
[[File:Vol5 page 0908 eq 001.png|RTENOTITLE]]....................(15)


[[File:Vol5 page 0908 eq 002.png]]....................(16)
[[File:Vol5 page 0908 eq 002.png|RTENOTITLE]]....................(16)


and [[File:Vol5 page 0908 eq 003.png]]....................(17)
and [[File:Vol5 page 0908 eq 003.png|RTENOTITLE]]....................(17)


The composite expansivities include the connate-water and rock expansivities. '''Eq.15''' includes ''G''<sub>''ps''</sub>, which is the cumulative produced sales gas and is defined as (''G''<sub>''p''</sub> – ''G''<sub>''I''</sub>).  
The composite expansivities include the connate-water and rock expansivities. '''Eq.15''' includes ''G''<sub>''ps''</sub>, which is the cumulative produced sales gas and is defined as (''G''<sub>''p''</sub> – ''G''<sub>''I''</sub>).
* ''F'' is expressed in reservoir volume units (e.g., RB or res m<sup>3</sup>)
* ''E''<sub>''gwf''</sub> is expressed in reservoir volume units per standard unit volume of gas (e.g., RB/scf)
* ''E''<sub>''owf''</sub> is expressed in reservoir volume units per standard unit volume of oil (e.g., RB/STB)


For strictly undersaturated oil reservoirs, no free gas exists (i.e., ''G''<sub>''fgi''</sub> = 0) and the initial free oil in place is equal to the OOIP (i.e., ''N''<sub>''foi''</sub> = ''N'') and '''Eqs.1 , 14''', and '''15''' simplify, respectively, to<ref name="r1" /><ref name="r4" /><ref name="r5" />
*''F'' is expressed in reservoir volume units (e.g., RB or res m<sup>3</sup>)
*''E''<sub>''gwf''</sub> is expressed in reservoir volume units per standard unit volume of gas (e.g., RB/scf)
*''E''<sub>''owf''</sub> is expressed in reservoir volume units per standard unit volume of oil (e.g., RB/STB)


[[File:Vol5 page 0909 eq 001.png]]
For strictly undersaturated oil reservoirs, no free gas exists (i.e., ''G''<sub>''fgi''</sub> = 0) and the initial free oil in place is equal to the OOIP (i.e., ''N''<sub>''foi''</sub> = ''N'') and '''Eqs.1 , 14''', and '''15''' simplify, respectively, to<ref name="r1">Walsh, M.P. 1995. A Generalized Approach to Reservoir Material Balance Calculations. J Can Pet Technol 34 (1). PETSOC-95-01-07. http://dx.doi.org/10.2118/95-01-07</ref><ref name="r4">Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.</ref><ref name="r5">Walsh, M.P., Ansah, J., and Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 1 - Applications to Undersaturated, Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27684-MS. http://dx.doi.org/10.2118/27684-MS</ref>


[[File:Vol5 page 0910 eq 001.png]]....................(18)
[[File:Vol5 page 0909 eq 001.png|RTENOTITLE]]


[[File:Vol5 page 0910 eq 002.png]]....................(19)
[[File:Vol5 page 0910 eq 001.png|RTENOTITLE]]....................(18)


[[File:Vol5 page 0910 eq 003.png]]....................(20)
[[File:Vol5 page 0910 eq 002.png|RTENOTITLE]]....................(19)


'''Eqs.18''' through '''20''' ignore gas reinjection.  
[[File:Vol5 page 0910 eq 003.png|RTENOTITLE]]....................(20)
 
'''Eqs.18''' through '''20''' ignore gas reinjection.


The material balance equation also helps explain most oil-recovery strategies. If the material-balance equation is solved for the produced fraction of the original free oil in place, then
The material balance equation also helps explain most oil-recovery strategies. If the material-balance equation is solved for the produced fraction of the original free oil in place, then


[[File:Vol5 page 0910 eq 004.png]]....................(21)
[[File:Vol5 page 0910 eq 004.png|RTENOTITLE]]....................(21)


'''Eq.21''' succinctly shows that oil recovery increases with:
'''Eq.21''' succinctly shows that oil recovery increases with:
* [[Water influx models|Water influx (''W''<sub>''e''</sub>)]]
* [[Gas cap drive reservoirs|Initial free-gas-cap volume (which is proportional to ''G''<sub>''fgi''</sub>)]]
* [[Surface water treatment for injection|Surface water injection (''W''<sub>''I''</sub>)]]
* Surface gas injection (by minimizing gas sales through ''G''<sub>''ps''</sub>)


It also shows that oil recovery increases by minimizing water production (''W''<sub>''p''</sub>).
*[[Water_influx_models|Water influx (''W''<sub>''e''</sub>)]]
*[[Gas_cap_drive_reservoirs|Initial free-gas-cap volume (which is proportional to ''G''<sub>''fgi''</sub>)]]
*[[Surface_water_treatment_for_injection|Surface water injection (''W''<sub>''I''</sub>)]]
*Surface gas injection (by minimizing gas sales through ''G''<sub>''ps''</sub>)


The material balance equation and its many different forms have many uses including:  
It also shows that oil recovery increases by minimizing water production (''W''<sub>''p''</sub>).
* Confirming the producing mechanism
 
* Estimating the OOIP and OGIP
The material balance equation and its many different forms have many uses including:
* Estimating gas cap sizes
 
* Estimating water influx volumes
*Confirming the producing mechanism
* Identifying [[Water influx models|water influx model]] parameters  
*Estimating the OOIP and OGIP
* Estimating producing indices
*Estimating gas cap sizes
*Estimating water influx volumes
*Identifying [[Water_influx_models|water influx model]] parameters
*Estimating producing indices
 
== Nomenclature ==


==Nomenclature==
{|
{|
|''B''<sub>''g''</sub>
|=
|gas FVF, RB/scf
|-
|-
|''B''<sub>''o''</sub>  
| ''B''<sub>''g''</sub>
|=  
| =
|oil FVF, RB/STB
| gas FVF, RB/scf
|-
|-
|''B''<sub>''tg''</sub>  
| ''B''<sub>''o''</sub>
|=  
| =
|two-phase gas FVF, RB/scf
| oil FVF, RB/STB
|-
|-
|''B''<sub>''to''</sub>  
| ''B''<sub>''tg''</sub>
|=  
| =
|two-phase oil FVF, RB/STB
| two-phase gas FVF, RB/scf
|-
|-
|''B''<sub>''tw''</sub>  
| ''B''<sub>''to''</sub>
|=  
| =
|two-phase water/gas FVF, RB/STB  
| two-phase oil FVF, RB/STB
|-
|-
|''B''<sub>''w''</sub>  
| ''B''<sub>''tw''</sub>
|=  
| =
|water FVF, RB/STB  
| two-phase water/gas FVF, RB/STB
|-
|-
|''c''<sub>''f''</sub>  
| ''B''<sub>''w''</sub>
|=  
| =
|rock compressibility, Lt<sup>2</sup>/m, 1/psi
| water FVF, RB/STB
|-
|-
|''c''<sub>''t''</sub>  
| ''c''<sub>''f''</sub>
|=  
| =
|total aquifer compressibility, Lt<sup>2</sup>/m, 1/psi  
| rock compressibility, Lt<sup>2</sup>/m, 1/psi
|-
|-
|''E''<sub>''f''</sub>  
| ''c''<sub>''t''</sub>
|=  
| =
|rock (formation) expansivity
| total aquifer compressibility, Lt<sup>2</sup>/m, 1/psi
|-
|-
|''E''<sub>''g''</sub>  
| ''E''<sub>''f''</sub>
|=  
| =
|gas expansivity, RB/scf
| rock (formation) expansivity
|-
|-
|''E''<sub>''gw''</sub>  
| ''E''<sub>''g''</sub>
|=  
| =
|expansivity for McEwen method, RB/scf  
| gas expansivity, RB/scf
|-
|-
|''E''<sub>''gwf''</sub>  
| ''E''<sub>''gw''</sub>
|=  
| =
|composite gas/water/rock FVF, RB/scf  
| expansivity for McEwen method, RB/scf
|-
|-
|''E''<sub>''o''</sub>  
| ''E''<sub>''gwf''</sub>
|=  
| =
|oil expansivity, RB/STB
| composite gas/water/rock FVF, RB/scf
|-
|-
|''E''<sub>''ow''</sub>  
| ''E''<sub>''o''</sub>
|=  
| =
|expansivity for McEwen method, RB/STB  
| oil expansivity, RB/STB
|-
|-
|''E''<sub>''owf''</sub>  
| ''E''<sub>''ow''</sub>
|=  
| =
|composite oil/water/rock FVF, RB/STB  
| expansivity for McEwen method, RB/STB
|-
|-
|''E''<sub>''w''</sub>  
| ''E''<sub>''owf''</sub>
|=  
| =
|water expansivity, RB/STB  
| composite oil/water/rock FVF, RB/STB
|-
|-
|''F''  
| ''E''<sub>''w''</sub>
|=  
| =
|total fluid withdrawal, L<sup>3</sup>, RB
| water expansivity, RB/STB
|-
|-
|''G''  
| ''F''
|=  
| =
|total original gas in place, L<sup>3</sup>, scf
| total fluid withdrawal, L<sup>3</sup>, RB
|-
|-
|''G''<sub>''fgi''</sub>
| ''G''
|=  
| =
|initial free gas in place, L<sup>3</sup>, scf
| total original gas in place, L<sup>3</sup>, scf
|-
|-
|''G''<sub>''i''</sub>  
| ''G''<sub>''fgi''</sub>
|=  
| =
|cumulative gas injected, L<sup>3</sup>, scf  
| initial free gas in place, L<sup>3</sup>, scf
|-
|-
|''G''<sub>''p''</sub>  
| ''G''<sub>''i''</sub>
|=  
| =
|cumulative produced gas, L<sup>3</sup>, scf  
| cumulative gas injected, L<sup>3</sup>, scf
|-
|-
|''h''  
| ''G''<sub>''p''</sub>
|=  
| =
|pay thickness, L, ft
| cumulative produced gas, L<sup>3</sup>, scf
|-
|-
|''k''  
| ''h''
|=  
| =
|permeability, L<sup>2</sup>, md
| pay thickness, L, ft
|-
|-
|''k''<sub>''a''</sub>
| ''k''
|=  
| =
|aquifer permeability, L<sup>2</sup>, md  
| permeability, L<sup>2</sup>, md
|-
|-
|''k''<sub>''H''</sub>  
| ''k''<sub>''a''</sub>
|=  
| =
|horizontal permeability, L<sup>2</sup>, md  
| aquifer permeability, L<sup>2</sup>, md
|-
|-
|''k''<sub>''t''</sub>  
| ''k''<sub>''H''</sub>
|=  
| =
|time constant, 1/t, 1/years
| horizontal permeability, L<sup>2</sup>, md
|-
|-
|''k''<sub>''v''</sub>  
| ''k''<sub>''t''</sub>
|=  
| =
|vertical permeability, L<sup>2</sup>, md
| time constant, 1/t, 1/years
|-
|-
|''L''<sub>''a''</sub>  
| ''k''<sub>''v''</sub>
|=  
| =
|aquifer length, L, ft
| vertical permeability, L<sup>2</sup>, md
|-
|-
|''N''  
| ''L''<sub>''a''</sub>
|=  
| =
|total original oil in place, L<sup>3</sup>, STB
| aquifer length, L, ft
|-
|-
|''N''<sub>''foi''</sub>
| ''N''
|=  
| =
|initial free oil in place, L<sup>3</sup>, STB  
| total original oil in place, L<sup>3</sup>, STB
|-
|-
|''N''<sub>''g''</sub>  
| ''N''<sub>''foi''</sub>
|=  
| =
|dimensionless gravity number
| initial free oil in place, L<sup>3</sup>, STB
|-
|-
|''N''<sub>''p''</sub>  
| ''N''<sub>''g''</sub>
|=  
| =
|cumulative produced oil, L<sup>3</sup>, STB
| dimensionless gravity number
|-
|-
|''p''  
| ''N''<sub>''p''</sub>
|=  
| =
|pressure, m/Lt<sup>2</sup>, psi
| cumulative produced oil, L<sup>3</sup>, STB
|-
|-
|''p''<sub>''e''</sub>
| ''p''
|=  
| =
|pressure at drainage radius, m/Lt<sup>2</sup>, psi  
| pressure, m/Lt<sup>2</sup>, psi
|-
|-
|''p''<sub>''w''</sub>  
| ''p''<sub>''e''</sub>
|=  
| =
|wellbore pressure, m/Lt<sup>2</sup>, psi  
| pressure at drainage radius, m/Lt<sup>2</sup>, psi
|-
|-
|''q''  
| ''p''<sub>''w''</sub>
|=  
| =
|producing rate at reservoir conditions (RB/D) or surface conditions (STB/D),v L<sup>3</sup>/t
| wellbore pressure, m/Lt<sup>2</sup>, psi
|-
|-
|''q''<sub>''c''</sub>
| ''q''
|=  
| =
|critical coning rate, STB/D, L<sup>3</sup>/t  
| producing rate at reservoir conditions (RB/D) or surface conditions (STB/D),v L<sup>3</sup>/t
|-
|-
|''q''<sub>''Dc''</sub>  
| ''q''<sub>''c''</sub>
|=  
| =
|dimensionless critical coning rate  
| critical coning rate, STB/D, L<sup>3</sup>/t
|-
|-
|''r''<sub>''e''</sub>  
| ''q''<sub>''Dc''</sub>
|=  
| =
|reservoir drainage radius
| dimensionless critical coning rate
|-
|-
|''r''<sub>''w''</sub>  
| ''r''<sub>''e''</sub>
|=  
| =
|wellbore radius, L, ft
| reservoir drainage radius
|-
|-
|''R''  
| ''r''<sub>''w''</sub>
|=  
| =
|instantaneous producing GOR, scf/STB
| wellbore radius, L, ft
|-
|-
|''R''<sub>''s''</sub>
| ''R''
|=  
| =
|dissolved GOR, scf/STB  
| instantaneous producing GOR, scf/STB
|-
|-
|''R''<sub>''sw''</sub>  
| ''R''<sub>''s''</sub>
|=  
| =
|dissolved-gas/water ratio, scf/STB  
| dissolved GOR, scf/STB
|-
|-
|''R''<sub>''v''</sub>  
| ''R''<sub>''sw''</sub>
|=  
| =
|volatilized-oil/gas ratio, STB/MMscf
| dissolved-gas/water ratio, scf/STB
|-
|-
|''S''<sub>''wi''</sub>  
| ''R''<sub>''v''</sub>
|=  
| =
|initial water saturation, fraction
| volatilized-oil/gas ratio, STB/MMscf
|-
|-
|''t''  
| ''S''<sub>''wi''</sub>
|=  
| =
|time, t, years
| initial water saturation, fraction
|-
|-
|''t''<sub>max</sub>
| ''t''
|=  
| =
|maximum time, t, years  
| time, t, years
|-
|-
|''t''<sub>''D''</sub>  
| ''t''<sub>max</sub>
|=  
| =
|dimensionless time  
| maximum time, t, years
|-
|-
|''t''<sub>''D''max</sub>  
| ''t''<sub>''D''</sub>
|=  
| =
|maximum dimensionless time  
| dimensionless time
|-
|-
|''U''  
| ''t''<sub>''D''max</sub>
|=  
| =
|aquifer constant, L<sup>4</sup>t<sup>2</sup>/m, RB/psi
| maximum dimensionless time
|-
|-
|''V''<sub>''pi''</sub>
| ''U''
|=  
| =
|initial reservoir PV, L<sup>3</sup>, RB  
| aquifer constant, L<sup>4</sup>t<sup>2</sup>/m, RB/psi
|-
|-
|''w''  
| ''V''<sub>''pi''</sub>
|=  
| =
|reservoir width, L, ft
| initial reservoir PV, L<sup>3</sup>, RB
|-
|-
|''W''  
| ''w''
|=  
| =
|initial water in place, L<sup>3</sup>, STB
| reservoir width, L, ft
|-
|-
|''W''<sub>''D''</sub>
| ''W''
|=  
| =
|dimensionless cumulative water influx
| initial water in place, L<sup>3</sup>, STB
|-
|-
|''W''<sub>''e''</sub>  
| ''W''<sub>''D''</sub>
|=  
| =
|cumulative water influx, L<sup>3</sup>, RB
| dimensionless cumulative water influx
|-
|-
|''W''<sub>''I''</sub>  
| ''W''<sub>''e''</sub>
|=  
| =
|cumulative injected water, L<sup>3</sup>, STB
| cumulative water influx, L<sup>3</sup>, RB
|-
|-
|''W''<sub>''p''</sub>  
| ''W''<sub>''I''</sub>
|=  
| =
|cumulative produced water, L<sup>3</sup>, STB  
| cumulative injected water, L<sup>3</sup>, STB
|-
|-
|Δ''p''  
| ''W''<sub>''p''</sub>
|=  
| =
|difference of time-averaged pressure, m/Lt<sup>2</sup>, psi
| cumulative produced water, L<sup>3</sup>, STB
|-
|-
|Δ''ρ''  
| Δ''p''
|=  
| =
|density difference, m/L<sup>3</sup>, lbm/ft<sup>3</sup> and g/cm<sup>3</sup>
| difference of time-averaged pressure, m/Lt<sup>2</sup>, psi
|-
|-
|''μ''<sub>''g''</sub>
| Δ''ρ''
|=  
| =
|gas viscosity, m/Lt, cp
| density difference, m/L<sup>3</sup>, lbm/ft<sup>3</sup> and g/cm<sup>3</sup>
|-
|-
|''μ''<sub>''o''</sub>  
| ''μ''<sub>''g''</sub>
|=  
| =
|oil viscosity, m/Lt, cp  
| gas viscosity, m/Lt, cp
|-
|-
|''μ''<sub>''w''</sub>  
| ''μ''<sub>''o''</sub>
|=  
| =
|water viscosity, m/Lt, cp  
| oil viscosity, m/Lt, cp
|-
|-
| ''μ''<sub>''w''</sub>
| =
| water viscosity, m/Lt, cp
|}
|}


==References==
== References ==
<references>
<ref name="r1">Walsh, M.P. 1995. A Generalized Approach to Reservoir Material Balance Calculations. ''J Can Pet Technol'' '''34''' (1). PETSOC-95-01-07. http://dx.doi.org/10.2118/95-01-07</ref>


<ref name="r2">Walsh, M.P. 1994. New, Improved Equation Solves for Volatile Oil and Condensate Reserves. ''Oil & Gas J.'' (22 August): 72. </ref>
<references />


<ref name="r3">Walsh, M.P., Ansah, J., and  Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 2 - Applications to Saturated and Non-Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27728-MS. http://dx.doi.org/10.2118/27728-MS  </ref>
== Noteworthy papers in OnePetro ==
 
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read


<ref name="r4"> Walsh, M.P. and Lake, L.W. 2003. ''A Generalized Approach to Primary Hydrocarbon Recovery''. Amsterdam: Elsevier. </ref>
== External links ==


<ref name="r5">Walsh, M.P., Ansah, J., and  Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 1 - Applications to Undersaturated, Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27684-MS. http://dx.doi.org/10.2118/27684-MS  </ref>
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
</references>


==Noteworthy papers in OnePetro==
== See also ==
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read


==External links==
[[Material_balance_in_water_drive_reservoirs|Material balance in water drive reservoirs]]
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro


==See also==
[[Primary_drive_mechanisms|Primary drive mechanisms]]
[[Material balance in water drive reservoirs]]


[[Primary drive mechanisms]]
[[Oil_fluid_characteristics|Oil fluid characteristics]]


[[Oil fluid characteristics]]
[[Oil_fluid_properties|Oil fluid properties]]


[[Oil fluid properties]]
[[PEH:Oil_Reservoir_Primary_Drive_Mechanisms]]


[[PEH:Oil Reservoir Primary Drive Mechanisms]]
[[Category:5.5 Reservoir simulation]]

Revision as of 11:33, 12 June 2015

The material-balance equation is the simplest expression of the conservation of mass in a reservoir. The equation mathematically defines the different producing mechanisms and effectively relates the reservoir fluid and rock expansion to the subsequent fluid withdrawal.

Material balance equation

The applicable equation for initially saturated volatile- and black-oil reservoirs is[1][2][3][4]

RTENOTITLE....................(1)

where:

  • Gfgi, Nfoi, and W are the initial free gas, oil, and water in place, respectively
  • Gp, Np, and Wp are the cumulative produced gas, oil, and water, respectively
  • GI and WI are the cumulative injected gas and water respectively
  • Eg, Eo, Ew, and Ef are the gas, oil, water, and rock (formation) expansivities

Most of the equations regarding primary drive mechanisms for oil reservoirs apply to any consistent set of units. A few equations, however, are written assuming English or customary units. Those equations are expressed in SI units:

RTENOTITLE....................(2)

RTENOTITLE....................(3)

RTENOTITLE....................(4)

RTENOTITLE....................(5)

RTENOTITLE....................(6)

RTENOTITLE....................(7)

and RTENOTITLE....................(8)

Nfoi and Gfgi are related to the total original oil in place (OOIP) and original gas in place (OGIP), N and G, according to N = Nfoi + Gfgi Rvi and G = Gfgi + Nfoi Rsi.

The expansivities are defined as

RTENOTITLE....................(9)

RTENOTITLE....................(10)

RTENOTITLE....................(11)

and RTENOTITLE, where B to and B tg are the two-phase formation volume factors (FVFs),

RTENOTITLE....................(12)

and RTENOTITLE....................(13)

The rock expansivity is obtained from direct measurement. See compaction driving oil reservoir for a greater discussion.

Physically, the two-phase FVF is the total hydrocarbon volume per unit volume of oil or gas at standard conditions. The two-phase FVF mimics the observations noted during a constant-composition expansion test. For instance, the two-phase oil FVF is the total hydrocarbon (oil + gas) volume of a saturated oil sample per unit volume of oil at standard conditions. In contrast, the two-phase gas FVF is the total hydrocarbon volume of a saturated gas sample per unit volume of gas at standard conditions. Bto and Btg typically are expressed in units of RB/stock tank barrel (STB) and RB/Mscf, respectively.

  • For undersaturated oils, the two-phase oil FVF is equal to the oil FVF
  • For undersaturated gases, the two-phase gas FVF is equal to the gas FVF.

Eqs. 12 and 13 account for volatilized oil in the equilibrium gas phase. If volatilized oil is negligible, these equations are simplified. For instance, Bto = Bo + Bg (Rsi – Rs) and Btg = Bg. These equations apply for black oils. Eq.11 ignores dissolved gas in the aqueous phase.

Eq.1 broadly states that net expansion equals net withdrawal. More specifically, it shows the different forms of expansion and withdrawal. The different forms of expansion such as gas expansion are responsible for the different producing mechanisms.

For the sake of simplicity, Eq.1 is often written in the abbreviated form of

RTENOTITLE....................(14)

where:

  • F = total net fluid withdrawal or production
  • Egwf = composite gas expansivity
  • Eowf = composite oil expansivities

F, Egwf, and Eowf are defined in

RTENOTITLE....................(15)

RTENOTITLE....................(16)

and RTENOTITLE....................(17)

The composite expansivities include the connate-water and rock expansivities. Eq.15 includes Gps, which is the cumulative produced sales gas and is defined as (Gp – GI).

  • F is expressed in reservoir volume units (e.g., RB or res m3)
  • Egwf is expressed in reservoir volume units per standard unit volume of gas (e.g., RB/scf)
  • Eowf is expressed in reservoir volume units per standard unit volume of oil (e.g., RB/STB)

For strictly undersaturated oil reservoirs, no free gas exists (i.e., Gfgi = 0) and the initial free oil in place is equal to the OOIP (i.e., Nfoi = N) and Eqs.1 , 14, and 15 simplify, respectively, to[1][4][5]

RTENOTITLE

RTENOTITLE....................(18)

RTENOTITLE....................(19)

RTENOTITLE....................(20)

Eqs.18 through 20 ignore gas reinjection.

The material balance equation also helps explain most oil-recovery strategies. If the material-balance equation is solved for the produced fraction of the original free oil in place, then

RTENOTITLE....................(21)

Eq.21 succinctly shows that oil recovery increases with:

It also shows that oil recovery increases by minimizing water production (Wp).

The material balance equation and its many different forms have many uses including:

  • Confirming the producing mechanism
  • Estimating the OOIP and OGIP
  • Estimating gas cap sizes
  • Estimating water influx volumes
  • Identifying water influx model parameters
  • Estimating producing indices

Nomenclature

Bg = gas FVF, RB/scf
Bo = oil FVF, RB/STB
Btg = two-phase gas FVF, RB/scf
Bto = two-phase oil FVF, RB/STB
Btw = two-phase water/gas FVF, RB/STB
Bw = water FVF, RB/STB
cf = rock compressibility, Lt2/m, 1/psi
ct = total aquifer compressibility, Lt2/m, 1/psi
Ef = rock (formation) expansivity
Eg = gas expansivity, RB/scf
Egw = expansivity for McEwen method, RB/scf
Egwf = composite gas/water/rock FVF, RB/scf
Eo = oil expansivity, RB/STB
Eow = expansivity for McEwen method, RB/STB
Eowf = composite oil/water/rock FVF, RB/STB
Ew = water expansivity, RB/STB
F = total fluid withdrawal, L3, RB
G = total original gas in place, L3, scf
Gfgi = initial free gas in place, L3, scf
Gi = cumulative gas injected, L3, scf
Gp = cumulative produced gas, L3, scf
h = pay thickness, L, ft
k = permeability, L2, md
ka = aquifer permeability, L2, md
kH = horizontal permeability, L2, md
kt = time constant, 1/t, 1/years
kv = vertical permeability, L2, md
La = aquifer length, L, ft
N = total original oil in place, L3, STB
Nfoi = initial free oil in place, L3, STB
Ng = dimensionless gravity number
Np = cumulative produced oil, L3, STB
p = pressure, m/Lt2, psi
pe = pressure at drainage radius, m/Lt2, psi
pw = wellbore pressure, m/Lt2, psi
q = producing rate at reservoir conditions (RB/D) or surface conditions (STB/D),v L3/t
qc = critical coning rate, STB/D, L3/t
qDc = dimensionless critical coning rate
re = reservoir drainage radius
rw = wellbore radius, L, ft
R = instantaneous producing GOR, scf/STB
Rs = dissolved GOR, scf/STB
Rsw = dissolved-gas/water ratio, scf/STB
Rv = volatilized-oil/gas ratio, STB/MMscf
Swi = initial water saturation, fraction
t = time, t, years
tmax = maximum time, t, years
tD = dimensionless time
tDmax = maximum dimensionless time
U = aquifer constant, L4t2/m, RB/psi
Vpi = initial reservoir PV, L3, RB
w = reservoir width, L, ft
W = initial water in place, L3, STB
WD = dimensionless cumulative water influx
We = cumulative water influx, L3, RB
WI = cumulative injected water, L3, STB
Wp = cumulative produced water, L3, STB
Δp = difference of time-averaged pressure, m/Lt2, psi
Δρ = density difference, m/L3, lbm/ft3 and g/cm3
μg = gas viscosity, m/Lt, cp
μo = oil viscosity, m/Lt, cp
μw = water viscosity, m/Lt, cp

References

  1. ↑ 1.0 1.1 Walsh, M.P. 1995. A Generalized Approach to Reservoir Material Balance Calculations. J Can Pet Technol 34 (1). PETSOC-95-01-07. http://dx.doi.org/10.2118/95-01-07
  2. ↑ Walsh, M.P. 1994. New, Improved Equation Solves for Volatile Oil and Condensate Reserves. Oil & Gas J. (22 August): 72.
  3. ↑ Walsh, M.P., Ansah, J., and Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 2 - Applications to Saturated and Non-Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27728-MS. http://dx.doi.org/10.2118/27728-MS
  4. ↑ 4.0 4.1 Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.
  5. ↑ Walsh, M.P., Ansah, J., and Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 1 - Applications to Undersaturated, Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27684-MS. http://dx.doi.org/10.2118/27684-MS

Noteworthy papers in OnePetro

Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read

External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

See also

Material balance in water drive reservoirs

Primary drive mechanisms

Oil fluid characteristics

Oil fluid properties

PEH:Oil_Reservoir_Primary_Drive_Mechanisms

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