Orr, F. M. - Theory of Gas Injection Processes Episode 2

2.2. ONE-DIMENSIONAL FLOW 11 Table 2.1: Inventory of Equations and Unknowns Unknowns Equations Variable Compositions, xij Saturations, Sj Pressures, Pj Velocities, vj Total Number npnc np np np np(nc + 3) Equation Material balances Phase equilibrium Capillary pressure Darcy’s law Saturation sum Mole fraction sum Total Eq. Number 2.1.18 nc 2.1.21 nc(np −1) 2.1.20 np −1 2.1.19 np 2.1.22 1 2.1.23 np np(nc + 3) If capillary pressure differences are neglected, Eq. 2.2.1 can be simplified by eliminating the pressure gradient from the expressions for the flow velocities. Phase flow velocities can then be written easily in terms of fractional flow functions , fj defined by np vj = fjv = fj vk, j = 1,np. (2.2.2) k=1 In Eq. 2.2.2 v is the total flow velocity , defined as the sum of the phase flow velocities, vj. In one-dimensional flow in the absence of capillary pressure Darcy’s law (Eq. 2.1.19) becomes vj = −kµrj ∂x + ρmjgsinθ, j = 1,np. (2.2.3) In Eq. 2.2.3, θ is the dip angle measured as the angle between the flow direction and a horizontal line. An expression for the fractional flow function, fj, is obtained by eliminating the pressure gradient from Eqs. 2.2.2 and Eq. 2.2.3. The expression for the pressure gradient can be obtained from any one of the np expressions for the phase flow velocities (Eq. 2.2.3). For the jth phase, for example, ∂x = −kkvj − ρmjgsinθ. Substitution of Eq. 2.2.4 into Eq. 2.2.3 written for the nth phase gives vn = krn µj vj + kkrngsinθ(ρmj −ρmn). rj n n (2.2.4) (2.2.5) Substitution of Eq. 2.2.5 into Eq. 2.2.2 gives the expression for the fractional flow of phase j [62] , krj/µj j n=1(krn/µn) 1 − kgsinθ np krn(ρmj −ρmn)!. (2.2.6) n=1 n Substitution of Eq. 2.2.2 into Eq. 2.2.1 gives the one-dimensional version of the convection-dispersion equation for multicomponent, multiphase flow, 12 CHAPTER 2. CONSERVATION EQUATIONS ∂ φ np xijρjSj + ∂ np xijρjfjv − φKij ∂ρjxij = 0, i = 1,nc. (2.2.7) j=1 j=1 2.3 Pure Convection If the effects of dispersion can be neglected, then Eq. 2.2.7 reduces to a set of equations that describes the interaction of pure convection with equilibrium phase behavior, np np φ xijρjSj + v xijρjfj = 0, i = 1,nc. (2.3.1) j=1 j=1 It is also convenient to write Eq. 2.3.1 in dimensionless form based on the following scaled variables: τ = ξ = vD = ρjD = vinjt φL x, v , inj ρj . inj (2.3.2) (2.3.3) (2.3.4) (2.3.5) where vinj and ρinj are the flow velocity and density of the injected fluid, and L is the length of the one-dimensional flow system. In Eq. 2.3.2, the time scale is the length of time required to displace one pore volume of fluid at the flow velocity and density of the injected fluid (the volume per unit of area available for flow over length L is φL, and the volumetric flow rate per unit area is vinj, so the time required to flow length L is φL/vinj). Thus, τ is a dimensionless time measured in pore volumes. For simplicity, we also assume that the porosity, φ, is constant, though that assumption can be relaxed easily. The result is ∂ XxijρjDSj + ∂ vD XxijρjDfj = 0, i = 1,nc. (2.3.6) j=1 j=1 The notation of Eq. 2.3.6 can be simplified by defining two additional functions, Gi and Hi, as np Gi = xijρjDSj, j=1 and np Hi = vD xijρjDfj. j=1 (2.3.7) (2.3.8) Gi is an overall concentration (in moles per unit volume) of component i. Hi is an overall molar flow of component i . The final version of the equations for multicomponent, multiphase convection is, therefore, 2.4. NO VOLUME CHANGE ON MIXING 13 ∂Gi + ∂Hi = 0, i = 1,nc. (2.3.9) The local flow velocity , vD, in the definition of Hi (Eq. 2.3.8) can vary with spatial location because volume is not conserved if components change volume as they transfer between phases. If, for example, CO2 displaces oil at modest pressure, it often occupies much less volume when dissolved in a liquid hydrocarbon phase than it does in a vapor phase. In those systems, the local flow velocity can vary substantially over the displacement length [22, 82, 19]. Thus, for some gas displacements, it will be important to include the effects of volume change on mixing . 2.4 No Volume Change on Mixing If the displacement pressure is high enough, then the volume occupied by a component in the gas phase may not change greatly when that component transfers to the liquid phase. Components in the liquid/liquid systems that describe surfactant flooding processes also exhibit minimal volume change on mixing. In such systems, it is reasonable to assume that the partial molar volume of each component is a constant (independent of composition or phase) and hence that ideal mixing applies. In other words, the volume occupied by a given amount of a component is constant no matter what phase the component appears in . Under the assumption that each component has a constant molar density, ρci, in any phase Eq. 2.3.9 can be simplified further. The local flow velocity is constant everywhere and equal to the injection velocity, so vD = 1. Furthermore, the volume occupied by component i in one mole of phase j is xij/ρci, and the volume fraction of component i in phase j is xij/ρci ij k=1 xkj/ρck The molar density of a phase is (2.4.1) nc !−1 ρj = xij/ρci . (2.4.2) i=1 Comparison of Eqs. 2.4.1 and 2.4.2 indicates that ρcicij = ρjxij. (2.4.3) Division of Eq. 2.4.3 by ρinj followed by substitution of Eq. 2.4.3 into Eq. 2.3.6, with vD = 1, and division by ρci/ρinj yields the set of conservation equations for pure convection with no volume change on mixing, ∂τi + ∂ξi = 0, i = 1,nc − 1. (2.4.4) where Ci is an overall volume fraction of component i given by np Ci = cijSj, (2.4.5) j=1 14 CHAPTER 2. CONSERVATION EQUATIONS and Fi is an overall fractional volumetric flow of component i given by np Fi = cijfj. (2.4.6) j=1 2.5 Classification of Equations The convection-dominated conservation equations, Eqs. 2.3.6 and 2.4.4 are systems of first order partial differential equations. Those equations have the general form, P(x,t,z)∂z + Q(x,t,z)∂z = R(x,t,z). (2.5.1) Equations like 2.5.1 are called quasilinear. If P, Q, and R are independent of z, the equation is strictly linear. It is called linear if R depends linearly on z and semilinear if R is a nonlinear function of z. In the problems considered here, R(x,t,z) = 0. Such equations are called homogeneous. In Eqs. 2.3.9 and 2.4.4 the dependent variables that correspond to z in Eq. 2.5.1 are the overall concentrations Gi or Ci. Because the phase saturations, Sj, densities, ρj, and fractional flows, fj, all depend nonlinearly on those concentrations, Eqs. 2.3.9 and 2.4.4 are homogeneous, quasilinear systems of first order equations. 2.6 Initial and Boundary Conditions Before Eqs. 2.3.9 and 2.4.4 can be solved, initial and boundary conditions must be imposed. In the chapters that follow, solutions will be derived for initial compositions that are constant throughout a semi-infinite domain, Gi(ξ,0) = Ginit, 0 < ξ < ∞, i = 1,nc, (2.6.1) or Ci(ξ,0) = Cinit, 0 < ξ < ∞, i = 1,nc. (2.6.2) The only boundary condition required is the composition of the injected fluid, Gi(0,τ) = Ginj, τ > 0, i = 1,nc, (2.6.3) or Ci(0,τ) = Cinj, τ > 0, i = 1,nc. (2.6.4) Thus, at time τ = 0, the composition of the fluid at the inlet changes discontinuously from the initial value to the injected value. Problems in which the initial state (sometimes referred to as the right state) is constant and the upstream boundary condition (sometimes called the left state) is also constant are known as Riemann problems. Such problems can be viewed as a description of the propagation of a discontinuity, initially placed at ξ = 0, between constant initial states for −∞ < ξ < 0, the 2.7. CONVECTION-DISPERSION EQUATION 15 injection composition, and for 0 < ξ < ∞, the initial composition. Given the fact that the flow problem begins with the propagation of a discontinuity, it is no surprise that the solutions may also display discontinuities known as shocks. At a shock, the differential material balances derived in this chapter must be replaced by integral balances across the shock. The properties and behavior of shocks are considered in some detail in Chapter 4 for two-component flow problems, and again in Chapter 5 for multicomponent problems. 2.7 Convection-Dispersion Equation If only two components and one phase are present, and the assumptions of constant porosity and no volume change on mixing apply, then Eq. 2.2.1 simplifies considerably to 2 ∂t + φ ∂x − K` ∂x2 = 0, (2.7.1) where C is the volume fraction of one component, and K` is the longitudinal dispersion coefficient, assumed here to be independent of composition. If Eq. 2.7.1 is made dimensionless with the scaled length and time given in Eqs. 2.3.2 and 2.3.3, the result is ∂C ∂C 1 ∂2C ∂τ ∂ξ Pe ∂ξ2 (2.7.2) where Pe = vL/φK` is the Peclet number . The Peclet number is a ratio of a characteristic time for dispersion, L2/K`, to a characteristic time for convection, φL/v. When the Peclet number is large, the effects of dispersion are small, and convection dominates. Thus, Eqs. 2.3.9 and 2.4.4 can be viewed as applicable in the limit of large Peclet number. If Kl is a constant (independent of composition) then the Peclet number is a constant as well, and Eq. 2.7.2 can be solved easily by Laplace transforms. If the domain is chosen to be 0 ≤ ξ ≤ ∞, the initial concentration is C(ξ,0) = 0 for 0 ≤ ξ ≤ ∞, and fluid with concentration C(0,τ) = 1 is injected for τ > 0, the solution is [12] C(ξ,τ) = 1erfc √Pe(ξ − τ)! + 2 exp(Peξ)erfc ! Pe(ξ + τ) 2 τ (2.7.3) The first term on the right side of Eq. 2.7.3 is usually significantly larger than the second term. The second term is significant only at early times near the inlet when the Peclet number is small. For large Peclet number (say Pe > 1000), however, the second term can be neglected. Hence, many investigators have used the approximate solution, C(ξ,τ) = 1erfc √Pe(ξ −τ)! 2 τ (2.7.4) Fig. 2.1 is a plot of Eq. 2.7.3 for three Peclet numbers (Pe = 10, 100, and 1000) at times, τ = 0.25 and 0.75. Fig. 2.1 shows that at each Peclet number, a transition zone from the injected composition (c = 1) to the initial composition (c = 0) moves downstream and increases in length as the flow proceeds. The width of the transition zone increases as the Peclet number is reduced. At τ = 0.75, for example, detectable amounts of the injected fluid have reached the outlet for Pe = 10 and 100 but have not yet done so for Pe = 1000. ... - --nqh--