Chapter 9 – Simultaneous Flow of Immiscible Fluids 9.1 An important problem in petroleum engineering is the prediction of oil recovery during displacement by water. Two common examples are a natural water drive and secondary waterflood. The latter is displacement of oil by bottom or edge water, the former is the injection of water to enhance production. In this chapter we will begin with the development of equations of multiphase, immiscible flow, concluding with the frontal advance and Buckley-Leverett equations. Next, we will discuss factors that control displacement efficiency followed by limitations of immiscible displacement solutions. 9.1 Development of equations The development of equations for describing multiphase flow in porous media follows a similar derivation as given previously for single phase, i.e., combination of continuity equation, momentum equation and equation of state. The mass balance of each phase can be written as: increment in time saccumulate thatphase of massincrement in timeleaving phase of massincrement in timeentering phase of mass Shown in Figure 9.1 is the differential element of porous media for oil. uox│x uox│x+x x y z Figure 9.1 Differential element in Cartesian coordinates. Only x-direction velocity is shown. As an example, the mass of oil entering and leaving the element is given by: Entering: tAutAutAuzzozoyyoyoxxoxo (9.1) Leaving: tAutAutAuzzzozoyyyoyoxxxoxo (9.2) Oil can accumulate by: (1). Change in saturation, (2). Variation of density with temperature and pressure, and (3). Change in porosity due to a change in confining stress. Thus we can write, toottooVSVS (9.3)Chapter 9 – Simultaneous Flow of Immiscible Fluids 9.2 Substitute Eqs. (9.1-9.3) into the conservation of mass expression, rearrange terms, and take the derivative as t, x, y, z  0, then the phase dependent continuity equations can be written as;      ooozooyooxoStuzuyux  (9.4)      wwwzwwywwxwStuzuyux  (9.5) The oil and water continuity equations assume no dissolution of oil in the water phase. That is, no mass transfer occurs between phases and thus flow is immiscible. The next step is to apply Darcy’s Law to each phase, i. For example in the x-direction, xkuiiiixix (9.6) where uix is the superficial velocity of phase i in the x-direction, kix, is the effective permeability to phase i in the x-direction, and is the phase potential. Substitute Eq. (9.6) into (9.4), apply Leibnitz rule of differentiation, and combine terms, results in,  oooooozooooyooooxoStgzpkzypkyxpkx  (9.7)  wwwwwwzwwwwywwwwxwStgzpkzypkyxpkx  (9.8) Even though Eqs. (9.7) and (9.8) are written in Cartesian coordinates, they both can be solved for a particular geometry. The solution will provide not only pressure and saturation distributions, but also phase velocities at any point in the porous media. To combine Eqs. (9.7) and (9.8) requires a relationship between phase pressures and between phase saturations. The latter is easily understood from the definition of saturations in Chapter 4, So + Sw = 1.0. The relationship between pressures was developed in Chapter 5, and is known as capillary pressure. wPoPorwPnwPcP  (9.9)Chapter 9 – Simultaneous Flow of Immiscible Fluids 9.3 9.2 Steady state, 1D solution As a simple example, let’s consider the steady state solution to fluid flow in a linear system as shown in Figure 9.2. This example is of primary interest in lab experiments to determine relative permeabilities. qo qw L poi Pwi poL PwL D Figure 9.2 Steady state core flood of oil and water. Oil and water are injected simultaneously, rates and pressures are measured, and core saturation is determined gravimetrically. Permeability is unknown. The steady state, incompressible fluid diffusivity equations are given by: 00dxdpkdxddxdpkdxdwwoo (9.10) Integrating and combining with Darcy’s equations, AqcdxdpkAqcdxdpkwwwwwooooo (9.11) If water saturation is uniform throughout the core, then effective permeability is independent of x. Therefore, for oil, LoooppodxkcdpoLoi (9.12) which upon integrating, becomes, )(oLoioooppALqk (9.13) If kbase = ko at Swi is known, then it is possible to calculate relative permeability.Chapter 9 – Simultaneous Flow of Immiscible Fluids 9.4 9.3 Capillary End Effect During laboratory experiments, capillary equilibrium must be maintained; that is, Pc = Po – Pw. Unfortunately, under certain conditions capillary end effects occur due to a thin gap existing between the end of the core and the core holder. As shown in Figure 9.3, capillary pressure in this gap is zero. gap Pc=0 Figure 9.3 Schematic of gap between core and holder The result is a rapid change in capillary pressure from a finite value immediately adjacent to the outlet to zero in the gap. As a consequence, the saturation of the wetting phase must increase to a value of Pc = 0. A generic core profile is shown in Figure 9.4 for both pressures and saturation. Sw 0 0 L L Po Pw Pc=0+ Swc Sor P Figure 9.4 Pressure and saturation profile through a core of length, L, with capillary end effect. Mathematically, we can describe this effect by investigating Darcy’s Law for the non-wetting phase. xSSpxpAkqwwcwnwnwnw (9.14)Chapter 9 – Simultaneous Flow of Immiscible Fluids 9.5 At the outlet, knw  0, but qnw ≠ 0; therefore, xSLxwlim (9.15) Two plausible methods have been