Simultaneous Flow of Immiscible Fluids • Purpose: predict the displacement of oil by water • Organization: – development of equations of multiphase, immiscible flow • concluding with the frontal advance and Buckley-Leverett equations. – factors that control displacement efficiency – limitations of immiscible displacement solutionsSimultaneous Flow of Immiscible Fluids (1). Change in saturation (2). Variation of density with temperature and pressure (3). Change in porosity due to a change in confining stress. increment in time saccumulate thatphase of massincrement in timeleaving phase of massincrement in timeentering phase of mass uox│x uox│x+x x y z Development of Equations tAutAutAuzzozoyyoyoxxoxotAutAutAuzzzozoyyyoyoxxxoxotoottooVSVSSimultaneous Flow of Immiscible Fluids • Phase dependent continuity equations • Apply Darcy’s Law Development of Equations      ooozooyooxoStuzuyux      wwwzwwywwxwStuzuyux xkuiiiixix oooooozooooyooooxoStgzpkzypkyxpkx  wwwwwwzwwwwywwwwxwStgzpkzypkyxpkx Simultaneous Flow of Immiscible Fluids • To combine requires: So + Sw = 1.0 And Development of Equations wPoPorwPnwPcP Simultaneous Flow of Immiscible Fluids • Oil and water are injected simultaneously • rates and pressures are measured • core saturation is determined gravimetrically. • Permeability is unknown. • Steady state, incompressible diffusivity Eqs. • Assume water saturation is uniform throughout the core Steady state, linear solution qo qw L poi Pwi poL PwL D 00dxdpkdxddxdpkdxdwwoo)(oLoioooppALqkSimultaneous Flow of Immiscible Fluids Methods to avoid 1. inject at a sufficiently high rate 2. The second method is to attach a thin, (high porosity and high permeability) Berea sandstone plug in series Capillary End Effects gap Pc=0 Sw 0 0 L L Po Pw Pc=0+ Swc Sor PSimultaneous Flow of Immiscible Fluids Frontal advance – USS, 1D SwiSorSwASwiSorSwC01x/LSwiSorSwBSwiSorSwD01x/LSwiSorSwASwiSorSwASwiSorSwC01x/LSwiSorSwBSwiSorSwBSwiSorSwD01x/LProgression of water displacing oil for immiscible, 1DSimultaneous Flow of Immiscible Fluids • The derivation begins from the 1D, multiphase continuity equations. • In terms of volumetric flow rate, • Assume the fluids are incompressible and the porosity is constant. • Combining, Buckley - Leverett    oooxoStux    wwwxwStux tSAxqootSAxqww   ooooStAqx    wwwwStAqx    0tSSAxqqowowqT = qo + qw = constantSimultaneous Flow of Immiscible Fluids • From the definition of fractional flow, • Substitute into Darcy’s equation for each phase, • complete fractional flow equation. Buckley - Leverett TwoTwwqfqqfq)1( sin)1( gxpAkqfqooooTwosingxpAkqfqwwwwTwwowwocTooowwowkkgxpqAkkkf1sin11Simultaneous Flow of Immiscible Fluids • fractional flow equation reduces to, • If we define mobility ratio as, • then fw = 1/(1+1/M) Buckley - Leverett owwocTooowwowkkgxpqAkkkf1sin11-xSSpxpwwcc Sw Pc 0wcSpowwowkkf11woowkkMSimultaneous Flow of Immiscible Fluids • From • substitute for qw, • reduce to one dependent variable. Observe, Sw = Sw(x,t) or, • Let dSw(x,t)/dt = 0, then the velocity of the saturation front is given by Frontal Advance tSAxqwwtSqAxfwTwdttSdxxSdStwtwwtwxwSxStSdtdxwSimultaneous Flow of Immiscible Fluids • Observe fw = fw(Sw) only, then, • Substitution results in the frontal advance equation Frontal Advance twtwwtwxSSfxftwwTSSfAqdtdxwRepresents the velocity of the saturation front. Basic assumptions in the derivation are : 1. incompressible fluid, fw(Sw) only 2. immiscible fluids. 3. only oil is displaced; i.e., the initial water saturation is immobile, and 4. no initial free gas saturation exists; i.e., not a depleted reservoir.Simultaneous Flow of Immiscible Fluids • injection rate is constant and if the dfw/dSw = f(Sw) only, then the location of the front is given by: Frontal Advance Swf Sw Swc fw fwf Swbt wwSwwTSSfAtqxfractional flow of water at the front water saturation at the front average water saturation behind the front at breakthroughSimultaneous Flow of Immiscible Fluids Prior to breakthrough Volume of oil produced (Np) = Volume of water injected (Wi) Displacement performance Constant injection rate Np Qi breakthrough after breakthrough • Water saturation gradients exists • Thus the rate of oil recovery decreases • Apply Welge’s solution to predict waterflood performanceSimultaneous Flow of Immiscible Fluids • Average water saturation where fw1 is assumed to be one at the inlet. • pore volumes injected, Qi, • Thus in terms of Qi, • The cumulative oil displaced, Np, can be expressed in terms of the difference in the average water saturation and the exit end saturation, i.e., Displacement performance Constant