Material Balance Equations - NTNU

1 TPG4150 Reservoir Recovery Techniques 2017 Material Balance Equations Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 21, 2017 1 Material Balance Equations To illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start with derivation of so-called Material Balance Equations . This type of model excludes fluid flow inside the reservoir, and considers fluid and rock expansion/compression effects only, in addition, of course, to fluid injection and production.

2 First, let us define the symbols used in the Material Balance Equations : Symbols used in Material Balance Equations Bg Formation volume factor for gas ( ) Bo Formation volume factor for oil ( ) Bw Formation volume factor for water ( ) Cr Pore compressibility (pressure-1) Cw Water compressibility (pressure-1) P PP21 Gi Cumulative gas injected ( ) Gp Cumulative gas produced ( ) m Initial gas cap size ( of gas cap)/( of oil zone) N Original oil in place ( ) Np Cumulative oil produced ( ) P Pressure Rp Cumulative producing gas-oil ratio ( )

3 = GNpp/ Rso Solution gas-oil ratio ( oil) Sg Gas saturation So Oil saturation Sw Water saturation T Temperature Vb Bulk volume ( ) Vp Pore volume ( ) We Cumulative aquifer influx ( ) Wi Cumulative water injected ( ) Wp Cumulative water produced ( ) Density (mass/vol.) Porosity Then, the Black Oil fluid phase behavior is illustrated by the following figures: Fluid phase behavior parameters (Black Oil) BoRsoPPPPBwB g Oil density: ooSgSsooRB=+ TPG4150 Reservoir Recovery Techniques 2017 Material Balance Equations Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 21, 2017 2 Water compressibility: CVVPwwwT= ()()1 Water volume change.

4 BBeB1cPw2w1cPw1ww= () Finally, we need to quantify the behavior of the pores during pressure change in the reservoir. The rock compressibility used in the following is the pore compressibility, and assumes that the bulk volume of the rock itself does not change. Pore volume behavior Rock compressibility: CPrT=()()1 Porosity change: w2w1cPw1re1cPr= + () The Material Balance Equations are based on simple mass balances of the fluids in the reservoir, and may in words be formulated as follows: Principle of Material conservation Amount of fluids presentin the reservoir initially (st.)

5 Vol.)Amount of fluids produced(st. vol.)Amount of fluids remainingin the reservoir finally(st. vol.) = We will define our reservoir system in terms of a simple block diagram, with an initial reservoir stage before production/injection starts, and a final stage at which time we would like to determine pressure and/or production. Block diagram of reservoir Gas Oil WaterInitial stage (1)Gas Oil WaterFinal stage (2)oil production: Npgas production: RpNpwater production: Wpaquifer influx: Wegas injection: Giwater injection.

6 Wi TPG4150 Reservoir Recovery Techniques 2017 Material Balance Equations Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 21, 2017 3 The two stages on the block diagram are reflected in the fluid phase behavior plots as follows: Initial and final fluid conditions BBPPoRsoPwgP||||||||11112222 BNote: If a gas cap is present ini-tially, then the initialpressure isequal to the bubble point pressure Now, we will apply the above Material Balance equation to the three fluids involved, oil, gas and water: Equation 1: Oil Material Balance Oil presentin the reservoirinitially(st.)

7 Vol.) Oilproduced(st. vol.)Oil remainingin the reservoirfinally(st. vol.) = or N - Np = Vp2So2/Bo2 yielding SNNBVopo2p22= () Equation 2: Water Material Balance Water presentin the reservoirinitially(st. vol.) Waterproduced(st. vol.) Waterinjected(st. vol.) Aquiferinflux(st. vol.)Water remainingin the reservoirfinally(st. vol.) + + = or VS/B - W + W + W= VS/Bpwwpiepww111222 yielding ()()S1mNBSS1 BWWWBVw2o1w1w1w1iepw2p2=+ ++ 1 TPG4150 Reservoir Recovery Techniques 2017 Material Balance Equations Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 21, 2017 4 Equation 3: Gas Material Balance Solution gaspresent in thereservoir initially(st.

8 Vol.)Free gas present in the reservoir initially(st. vol.) Gasproduced(st. vol.) Gasinjected(st. vol.)Solution gaspresent in the reservoir finally(st. vol.)Free gaspresent in thereservoir finally(st. vol.) + + = + or NRso1+mNBo1/Bg1 RpNp+Gi=(N Np)Rso2+Vp2Sg2/Bg2 yielding Sg2=N(Rso1 Rso2)+m(Bo1Bg1) Np(Rp Rso2)+Gi (Bg2Vp2) In addition to these three fluid balances, we have the following relationships for fluid saturations and pore volume change: Equation 4: Sum of saturations ++= Equation 5.

9 Pore volume change V=V(+cP)ppr211 TPG4150 Reservoir Recovery Techniques 2017 Material Balance Equations Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 21, 2017 5 By combining the 5 Equations above, and grouping terms, we obtain the Material Balance relationships, as shown below: THE COMPLETE BLACK OIL Material Balance EQUATION: ()()FNEmEEWWBGB ogfwiew2ig2=+++++, where production terms are ()[]FNBRRBWBpo2pso2g2pw2=+ + oil and solution gas expansion terms are ()()EBBRRBoo2o1so1so2g2= + gas cap expansion terms are EBBB1go1g2g1= and rock and water compression/expansion terms are ()

10 E1mBCCS1 SPfwo1rww1w1,= ++ TPG4150 Reservoir Recovery Techniques 2017 Material Balance Equations Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 21, 2017 6 Material Balance EQUATION FOR A CLOSED GAS RESERVOIR The Material Balance equation for a closed gas reservoir is very simple. Applying the mass Balance principle to a closed reservoir with 100% gas, we may derive the general eguation GBg1=(G Gp)Bg2 where G is gas initially in place, Gp is cumulative gas production, and Bg is the formation-volume-factor for gas.