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**MATERIAL BALANCE EQUATION**

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**THE MATERIAL BALANCE EQUATION**

The material balance equation (MBE) is one of the basic tools of reservoir engineers for interpreting and predicting reservoir performance. The MBE can be used to: • Estimate initial hydrocarbon volumes in place • Predict future reservoir performance • Predict ultimate hydrocarbon recovery under various types of primary driving mechanisms

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**THE MATERIAL BALANCE EQUATION**

The equation can be written on volumetric basis as: Initial volume = volume remaining + volume removed Before deriving the material balance, it is convenient to denote certain terms by symbols for brevity. The symbols used conform where possible to the standard nomenclature adopted by the Society of Petroleum Engineers.

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**THE MATERIAL BALANCE EQUATION**

Terms Symbols Initial reservoir pressure, psi Pi Change in reservoir pressure = pi − p, psi Δp Bubble point pressure, psi pb Initial (original) oil in place, STB N Cumulative oil produced, STB Np Cumulative water produced, bbl Wp Cumulative gas produced, scf Gp Cumulative gas-oil ratio, scf/STB Rp Instantaneous gas-oil ratio, scf/STB GOR Initial gas solubility, scf/STB Rsi Gas solubility, scf/STB Rs

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**THE MATERIAL BALANCE EQUATION**

Terms Symbols Initial oil formation volume factor, bbl/STB Boi Oil formation volume factor, bbl/STB Bo Initial gas formation volume factor, bbl/scf Bgi Gas formation volume factor, bbl/scf Bg Cumulative water injected, STB Winj Cumulative gas injected, scf Ginj Cumulative water influx, bbl We Ratio of initial gas-cap-gas reservoir volume to initial reservoir oil volume , bbl/bbl m Initial gas-cap gas, scf G Pore volume, bbl P.V

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**THE MATERIAL BALANCE EQUATION**

Terms Symbols Water compressibility, psi−1 cw Formation (rock) compressibility, psi−1 cf Gas formation volume factor of the gas cap gas ,bbl/scf Bg c Gas formation volume factor of the solution gas ,bbl/scf Bg s cumulative gas production from gas cap Gpc cumulative gas production from solution gas. Gps

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**Derivation of the oil MBE**

Let us define the following quantities: N= the initial oil in place(STB) = VBO (*1-Swi) /Boi Gas cap expansion = New gas cap volume - original gas cap volume New gas cap volume = (G-Gpc) B gc Original gas cap volume = G x Bgci (1)Gas cap expansion = (G-Gpc) B gc - G x Bgci

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**THE MATERIAL BALANCE EQUATION**

(2) Remaining release gas = original soluble gas – remaining soluble gas – cumulative produced gas = [N Rsi -(N-Np)Rs -Gps] Bgs (3) Remaining oil volume = (N-Np) Bo (4) Net water influx = (We -Wp Bw) (5) Rock and water expansion is neglected in the presence of gas.

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**THE MATERIAL BALANCE EQUATION**

Condition I Condition II Pressure = Pi P<Pi NP = Zero NP = +ive GP = Zero GP = +ive WP = Zero WP = +ive p Condition I Condition II Pressure = Pi P<Pi NP = Zero NP = +ive GP = Zero GP = +ive WP = Zero WP = +ive

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**THE MATERIAL BALANCE EQUATION**

Now, by equating the initial conditions , to the final conditions resulting from a finite pressure drop: N Boi = (N-NP) Bo + [(G-GPC) Bgc -G Bgci] +[NRsi – -(N-NP)Rs–GPS ]Bgs+(We -WPBw) N Boi = NBo – NPBo + G Bgc- GPC Bgc - G Bgci + N RSi Bgs - N RS Bgs+ NP Rs Bgs - GPS Bgs + + (We-WP Bw)

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**THE MATERIAL BALANCE EQUATION**

NBoi-NBo-NRsi Bgs +NRsBgs=-NPBo+NPRsBgs - GPSBgs+ +GBgc-GPcBgc-GBgci+(We-WPBw) N[Boi-Bo-RsiBgs+RsBgs] =Np[-Bo+RsBgs]-Gps Bgs+GBgc- Gpc Bgc-GBgci+(We-WpBw) But : Bgc = Bgs = Bg and Bgci = Bgsi Bgi N[Boi-Bo+(-Rsi+Rs)Bg] = Np[-Bo+RsBg]-Gps Bg+GBg –GpcBg-GBgi+(We-WpBw)

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**THE MATERIAL BALANCE EQUATION**

Np[Bo-RsBg]+(Gps+Gpc)Bg =N[Bo-Boi+(Rsi-Rs)Bg]+ +G(Bg-Bgi)+(We-WpBw) But : (Gpc+Gps)Bg = GpBg and Rp = Gp/Np and Bt = Bo+(Rsi-Rs)Bg Therefore , Np [ Bo - RsBg]+Gp Bg =N[(Bo-Boi)+(Rsi-Rs)Bg] G(Bg-Bgi)+(We-WpBw)

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**THE MATERIAL BALANCE EQUATION**

By adding and subtracting RsiBg –RsiBg: Np [Bo -(Rsi-Rs)Bg]- RsiBg]+Gp Bg =N[(Bo Boi)+ + (Rsi-Rs)Bg]+ G(Bg-Bgi)+(We-WpBw) Np[Bt-RsiBg]+Np Rp Bg =N[(Bo-Boi)+(Rsi-Rs)Bg] + G(Bg-Bgi)+(We-WpBw) Np[Bt – Rsi Bg + Rp Bg]= N[(Bo - Boi)+(Rsi-Rs) Bg] + G(Bg-Bgi) +(We-WpBw)

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**THE MATERIAL BALANCE EQUATION**

And finally : Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti ) + mNBoi(Bg - Bgi) + (We-WpBw) …..(1) Bgi

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**THE MATERIAL BALANCE EQUATION**

Np [Bt +(Rp - Rsi) Bg]=N (Bt – Bti )+mNBoi(Bg - Bgi) + (We-WpBw) Bgi Cumulative oil Depletion Drive Gas cap drive Water drive withdrawal mechanism mechanism mechanism And this is the generalized material balance equation for combination drive reservoir neglecting the rock and connate water expansion.

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**THE MATERIAL BALANCE EQUATION**

N ( B t – B t I ) = D.D.I Np[Bt+(Rp-Rsi)Bg] (m N Boi/ Bgi) (Bg - Bgi) = GCDI Np [Bt + (Rp - Rsi) Bg] ( We- Wp Bw) = WDI

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**THE MATERIAL BALANCE EQUATION**

Driving indices : The driving index of any mechanism represents the fractional contribution of the total oil withdrawal produced by that mechanism.

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**THE MATERIAL BALANCE EQUATION Driving indices**

Driving indexes in a combination-drive reservoir

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Driving indices

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**THE MATERIAL BALANCE EQUATION**

Several of the material balance calculations require the total pore volume (P.V) as expressed in terms of the initial oil volume N and the volume of the gas cap. The expression for the total pore volume can be derived by conveniently introducing the parameter m into the relationship as follows:

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**THE MATERIAL BALANCE EQUATION**

Change in Pore Volume Due to Initial Water :and Rock Expansion

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**THE MATERIAL BALANCE EQUATION**

where ΔV represents the net changes or expansion of the material as a result of changes in the pressure. Therefore, the reduction in the pore volume due to the expansion of the connate water in the oil zone and the gas cap is given by:

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**THE MATERIAL BALANCE EQUATION**

Connate water expansion = [(pore volume) Swi] x cw Δp Substituting for the pore volume (P.V) with Equation 11-1 gives:

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**THE MATERIAL BALANCE EQUATION**

The total volume of the hydrocarbon system is then given by: Initial oil volume + initial gas cap volume = (P.V)(1− Swi) N Boi + m N Boi = (P.V) (1 − Swi) or

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**THE MATERIAL BALANCE EQUATION**

Similarly, the reduction in the pore volume due to the expansion of the reservoir rock is given by:

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**THE MATERIAL BALANCE EQUATION**

The total pore volume occupied by the two injected fluids is given by: Total volume = Ginj Bginj +Winj Bw

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**THE MATERIAL BALANCE EQUATION**

The most general form of Material Balance Equation is Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti ) + mNBoi(Bg - Bgi) + (We-WpBw) + Bgi + (Cf +CwSw) (NBoi ) (Pi-P) ……(2) (1-Swi )

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**THE MATERIAL BALANCE EQUATION**

Case (1): Water drive reservoir: A-Below the bubble point pressure: The driving mechanisms involved are : 1.Water drive mechanism 2.Depletion drive mechanism The material balance equation is : Np [Bt + (Rp - Rsi) Bg]=N(Bt – Bti ) +(We-WpBw) ………………..(3)

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**THE MATERIAL BALANCE EQUATION**

B) Above the bubble point pressure: The driving mechanisms involved are : 1.Water drive mechanism 2.Depletion drive mechanism and 3.Rock and water expansion mechanism. The material balance equation is : Np Bo = N (Bo – Boi ) + ( We –Wp Bw ) + (Cf +CwSw) (NBoi ) (Pi-P ) (4) (1-Swi)

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**THE MATERIAL BALANCE EQUATION**

(Since Rp = Rsi = Rs = Constant) . Effective oil compressibility : Co = - Vo 1 = Bo-Boi 1 Vo P Boi (Pi-P) Bo - Boi = Co Boi (Pi-P) Substitute this value in equation (4): Np Bo = N Co Boi (Pi-P) + ( We –Wp Bw ) + (Cf +CwSw) (NBoi ) (Pi-P ) ……………………….(5) (1-Swi)

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**THE MATERIAL BALANCE EQUATION**

Np Bo + Wp Bw = N Boi (Pi-P) Cf +CwSw + Co + We ( 1-Swi ) ……………………….………………………..………………………..(6) Np Bo + Wp Bw = N Boi (Pi-P) Cf +CwSw +Co So (1-Swi ) + We …....……………………………………………………...(7) Np Bo + Wp Bw = N Boi Ce (Pi-P) + We ……….(8)

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**THE MATERIAL BALANCE EQUATION**

Where : Ce = Cf + Cw Sw +So So 1-Swi Ce = effective oil compressibility.

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**THE MATERIAL BALANCE EQUATION**

Case (2): Gas cap drive reservoir: The driving mechanisms involved are : 1.Gas cap drive mechanism and 2.Depletion (solution gas) drive mechanism Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti ) + + mNBoi (Bg - Bgi) Bgi

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**THE MATERIAL BALANCE EQUATION**

Case (3): Depletion drive reservoir: A) Below the bubble point pressure: The driving mechanism involved is : Depletion (solution gas) drive mechanism only. The material balance equation is: Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti )

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**THE MATERIAL BALANCE EQUATION**

B-Above the bubble point pressure The driving mechanism involved is : Rock and fluid expansion only. The material balance equation is: Np Bo = N(Bo-Boi) + Cf +CwSw +Co So (1-Swi ) +N Boi ( Pi - P ) + We

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**THE MATERIAL BALANCE EQUATION**

Bo-Boi = Co Boi (Pi –P ) , finally: Np Bo = N Boi Ce P If there is water production :, the equation form becomes : Np Bo + Wp Bw = N Boi Ce P This is the material balance equation for depletion drive reservoir (DDR) producing above the bubble point pressure (under- saturated reservoir).

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**THE MATERIAL BALANCE EQUATION**

When the rock and water expansion mechanism cancelled. Therefore , the equation will be as follows: Np Bo = N (Bo - Boi ) Np Bo = N Bo - N Boi and therefore : (N – Np ) Bo = N Boi

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**THE MATERIAL BALANCE EQUATION**

this is the simplest form of the material balance equation which represents a depletion drive reservoir (DDR) producing above the bubble point pressure (under- saturated reservoir) neglecting the rock and water expansion mechanism. The last equation can be driven simply by considering the initial and remaining oil in-place only ( of course , in addition to the connate water).

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**THE MATERIAL BALANCE EQUATION**

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