2 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
3 THE MATERIAL BALANCE EQUATION The equation can be written on volumetric basis as:Initial volume = volume remaining + volume removedBefore 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.
4 THE MATERIAL BALANCE EQUATION TermsSymbolsInitial reservoir pressure, psiPiChange in reservoir pressure = pi − p, psiΔpBubble point pressure, psipbInitial (original) oil in place, STBNCumulative oil produced, STBNpCumulative water produced, bblWpCumulative gas produced, scfGpCumulative gas-oil ratio, scf/STBRpInstantaneous gas-oil ratio, scf/STBGORInitial gas solubility, scf/STBRsiGas solubility, scf/STBRs
5 THE MATERIAL BALANCE EQUATION TermsSymbolsInitial oil formation volume factor, bbl/STBBoiOil formation volume factor, bbl/STBBoInitial gas formation volume factor, bbl/scfBgiGas formation volume factor, bbl/scfBgCumulative water injected, STBWinjCumulative gas injected, scfGinjCumulative water influx, bblWeRatio of initial gas-cap-gas reservoir volume to initial reservoir oil volume , bbl/bblmInitial gas-cap gas, scfGPore volume, bblP.V
6 THE MATERIAL BALANCE EQUATION TermsSymbolsWater compressibility, psi−1cwFormation (rock) compressibility, psi−1cfGas formation volume factor of the gas cap gas ,bbl/scfBg cGas formation volume factor of the solution gas ,bbl/scfBg scumulative gas production from gas capGpccumulative gas production from solution gas.Gps
7 Derivation of the oil MBE Let us define the following quantities:N= the initial oil in place(STB) = VBO (*1-Swi) /BoiGas cap expansion = New gas cap volume -original gas cap volume New gas cap volume = (G-Gpc) B gcOriginal gas cap volume = G x Bgci(1)Gas cap expansion = (G-Gpc) B gc - G x Bgci
8 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.
9 THE MATERIAL BALANCE EQUATION Condition I Condition IIPressure = Pi P<Pi NP = Zero NP = +iveGP = Zero GP = +iveWP = Zero WP = +ivepCondition I Condition IIPressure = Pi P<Pi NP = Zero NP = +iveGP = Zero GP = +iveWP = Zero WP = +ive
10 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)
11 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)
12 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)BgTherefore ,Np [ Bo - RsBg]+Gp Bg =N[(Bo-Boi)+(Rsi-Rs)Bg] G(Bg-Bgi)+(We-WpBw)
13 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)
14 THE MATERIAL BALANCE EQUATION And finally : Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti ) + mNBoi(Bg - Bgi) + (We-WpBw) …..(1) Bgi
15 THE MATERIAL BALANCE EQUATION Np [Bt +(Rp - Rsi) Bg]=N (Bt – Bti )+mNBoi(Bg - Bgi) + (We-WpBw)BgiCumulative oil Depletion Drive Gas cap drive Water drive withdrawal mechanism mechanism mechanismAnd this is the generalized material balance equation for combination drive reservoir neglecting the rock and connate water expansion.
16 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
17 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.
18 THE MATERIAL BALANCE EQUATION Driving indices Driving indexes in a combination-drive reservoir
20 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 bederived by conveniently introducing the parameter m into the relationship as follows:
21 THE MATERIAL BALANCE EQUATION Change in Pore Volume Due to Initial Water :and Rock Expansion
22 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:
23 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:
24 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
25 THE MATERIAL BALANCE EQUATION Similarly, the reduction in the pore volume due to the expansion of the reservoir rock is given by:
26 THE MATERIAL BALANCE EQUATION The total pore volume occupied by the two injected fluids is given by:Total volume = Ginj Bginj +Winj Bw
27 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 )
28 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)
29 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)
30 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)
31 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)
32 THE MATERIAL BALANCE EQUATION Where : Ce = Cf + Cw Sw +So So 1-Swi Ce = effective oil compressibility.
33 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
34 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 )
35 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
36 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).
37 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
38 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).