2.1. Problem Formulation and Assumptions
 We consider immiscible, incompressible, isothermal two-phase flow through a homogeneous, horizontal, one-dimensional porous medium where the fluid phases additionally transport components. Material balance for the two phases leads to the equations [Bear, 1972]
where Sw is the wetting phase saturation, Sn is the nonwetting phase saturation, and is the porosity, which is assumed to be constant throughout the whole domain. Furthermore, we assume that the volume flux of the wetting and the nonwetting phases, qw and qnw, can be described by the multiphase extension of Darcy's equation [Muskat, 1949], which describes the volume flux because of a gradient in the phase pressures pw and pnw:
 Here K is the absolute permeability, is the viscosity of the wetting phase, is the viscosity of the nonwetting phase, and knw = knw(Sw) and kw = kw(Sw) are the relative permeability of the nonwetting and wetting fluids, respectively, which describe the impairment of the one fluid phase by the other. The two-phase pressures pw and pnw are related through the capillary pressure pc = pnw − pw. Combining the definition of capillary pressure with equations (1) and (2), one can rewrite qw as an expression of the total volume flux qt = qnw + qw, which yields
 Here D can be thought of as a capillary dispersion coefficient for the fluid phases, and together with f, it describes the capillary-hydraulic properties of the fluid-porous medium system and is defined through
 We consider components Cj, j = 1,…, n and k = 1,…, m that are transported in the wetting phase and the nonwetting phase, respectively. In the following, we assume that (1) they do not alter the porous medium (e.g., through chemical reactions), (2) they do not change the flow parameters, (3) they do not partition into the other phase, (4) the solute mass flux due to hydrodynamic dispersion within a phase is described by a Fickian model, and (5) density effects can be ignored. The continuity equation can then be written as [Acs et al., 1985; Gerritsen and Durlofsky, 2005]
 As stated, the components are assumed to not change the flow field. If chemical flooding with surfactants, polymers, foams, etc., is considered, the constitutive relationships depend on both saturation and component concentration. For this case, analytical solutions can be derived if both capillarity and hydrodynamic dispersion are ignored. This leads to a system of hyperbolic conservation laws, and the method of characteristics can be used to derive analytical solutions [e.g., Pope, 1980; Johansen and Winther, 1988; Juanes and Blunt, 2006; LaForce and Johns, 2005; Seto and Orr, 2009]. As explained in section 1, our primary interest is the mixing of the inert components (Figure 1). We hence assume that the capillary-hydraulic properties are functions of saturation only.
Figure 1. Schematic representation of one-dimensional, unidirectional displacement of a nonwetting phase by a wetting phase with an initial wetting saturation Si. Behind the wetting front, a mixing zone between the old composition of the wetting phase and the new one of length develops. Note that the solute front always trails the saturation front if Swr > 0 (see Figure 3).
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 For the solutes in the water phase, equation (6) assumes that the volume fraction of the components is small compared to that of the wetting phase, which, for most practical applications, such as different ion-compositions, is an excellent approximation [Pope, 1980]. The solutes in the nonwetting phase can consume any arbitrary fraction of the nonwetting phase volume [Lie and Juanes, 2005]. The DH is the hydrodynamic dispersion coefficient and for the one-dimensional case becomes [Bear, 1972]
where is the coefficient of hydrodynamic dispersion and accounts for effects of the flow field and Dmol is the effective molecular diffusion coefficient. In the analysis that follows, we will assume that j = 1, but the entire analysis immediately carries over to the case where more than one component is present. Similarly, we mainly will focus on the case where the nonwetting phase has a homogeneous composition, i.e. k = 1, and is completely described by the restriction (2). Again, the analysis that follows can easily be extended to multiple solutes k = 1, …, m. To simplify notation, we will write DH instead of DH,w.
 Both the conservation equation for the fluid phase and the solutes are of parabolic type and, consequently, the resulting solutions are smooth. Therefore, we can expand equation (6) to arrive at
 We derive an analytical solution for the transport equation (6) that fully considers both capillary effects and hydrodynamic dispersion. Thus, all the physical mechanisms that account for solute transport and mixing in a homogeneous two-phase system are taken into account. The solution is obtained from two main ideas. First, we note that, for cases where Sw and qw are known in equation (9) either from analytical or numerical solutions, the problem of solving equations (1)–(6) reduces to solving an advection-dispersion equation (ADE). The Sw and qw are fully determined by equations (1) together with equations (4) and (5). The highly nonlinear term because of capillary forces in equation (4) poses a main mathematical difficulty for deriving analytical solutions and, thus, only few exact solutions are known. We will capitalize on the one derived by McWhorter and Sunada  for reasons explained below. Although this significantly reduces the complexity of the problem, the ADE (equation (9)) still has time- and space-dependent coefficients, and no analytical solutions are known. Second, to derive a solution for it, we use a general analytical approximation that separates the two physical transport mechanisms in equation (6), i.e., the advective motion because of viscous and capillary forces and dispersive mixing. The advective part is solved for exactly by two different approaches: First, we use the physical notion that if dispersion can be ignored, i.e., in equation (6), C is a function of Sw only, and an explicit expression for the location of the solute front can be derived. Second, we use the method of characteristics. Both approaches yield the same result, which furthermore gives the mathematically rigorous justification that for the dispersion-free limit, C is a function of Sw only. We show that, if qw and Sw are described by the McWhorter and Sunada problem, the location of the solute front can be determined graphically by a modified Welge tangent [Welge, 1952]. To the best of our knowledge, this is the first analytical solution that accounts for capillary effects on tracer transport.
 Next, the effect of hydrodynamic dispersion is superimposed on the advective motion via a singular perturbation expansion around the advective front of the solute. Singular perturbation techniques have been used previously for describing dispersion in unsteady flow fields of a single phase [Gelhar and Collins, 1971; Dagan, 1971; Eldor and Dagan, 1972; Nachabe et al., 1995; Wilson and Gelhar, 1981, 1974]. We show that if the dispersion is small compared to a characteristic length of the system, very good agreement between our analytical approximation and a numerical reference solution is achieved. While we are mainly concerned with the combined effects of capillary, viscous, and dispersive processes in this paper, the equations derived for the characteristics and the hydrodynamic dispersion are valid for any given flow field. For illustration, we also combine them with the solution for the capillary-free limit, i.e., the Buckley-Leverett problem [Buckley and Leverett, 1942], which is the classical solution for the case where external driving forces become large and capillary effects become negligible. To the best of our knowledge, this is the first analytical solution that fully describes the complex dependence of the hydrodynamic dispersion on the simultaneous and unsteady flow of the two phases. From these analytical expressions, we finally obtain equations for the growth rate of the dispersive zone both for the case where capillary pressure is considered and for the viscous limit.
2.2. Semianalytical Solutions for Immiscible Two-Phase Flow
 Since the derivatives in the conservation equation for the solutes can be written out because of the product rule, for known qw and Sw, the problem of solving for simultaneous flow and transport, and thus the whole set of conservation equations, reduces to solving one advection-dispersion equation for highly nonlinear, but known, coefficients. Hence, we first give a short overview over analytical solutions for qw and Sw that satisfy equations (1)–(5). The strong nonlinearity of the capillary drive in equation (4) makes the determination of exact solutions difficult, and only a few analytical solutions are known. Two approaches exist: either closed-form solutions are determined at the cost of restricting the capillary-hydraulic properties krw(Sw), krn(Sw), and Pc(Sw) to very particular nonlinearities, or more general nonlinearities are chosen and the resulting exact analytical expressions are mostly nonlinear expressions that generally need to be solved numerically. Examples for the first approach are the solution given by [Fokas and Yortsos, 1982; Chen, 1988; van Duijn and de Neef, 1998; Philip, 1960; Kashchiev and Firoozabadi, 2002]. For all these solutions, the specific form of the nonlinearities excludes the adequate study of flow for different porous media and often also the possibility to study both cocurrent and countercurrent flows. Examples for the second approach can be found by Chen et al. , McWhorter and Sunada , and van Duijn and Peletier . We choose the ones derived by McWhorter and Sunada  since they allow for both general capillary-hydraulic properties and the consideration of cocurrent and countercurrent flows. The only time we make specific use of the special form of these solutions, however, is for the explicit determination of the saturation level at which the solute advective front breaks through, and in the examples given in section 5. The nonlinear expressions derived for the characteristics and the hydrodynamic dispersion are valid for any flow and saturation field known, either from numerical solutions such as streamline simulations [Blunt et al., 1996; King and Datta-Gupta, 1998; Datta-Gupta and King, 1995] or analytical considerations.