Under the assumption of smoothly varying background properties I derive an asymptotic solution for two-phase flow. This formulation partitions the modeling of two-phase flow into two subproblems: an arrival time calculation and a saturation amplitude computation. The asymptotic solution itself is defined along a trajectory though the model. If gravitational forces are not important and the flow field is independent of changes in the background saturation, the trajectory may be identified with a streamline, and the asymptotic approach provides a mathematical basis for streamline simulation. The evolution of the saturation amplitude is governed by a generalization of Burgers' equation, defined along the trajectory. If the capillary properties are uniform and gravitational forces are negligible, one can derive self-similar solutions for the three limiting behaviors of a two-phase front. Asymptotic solutions were found to deviate by less than 5% from saturations calculated using a numerical simulator. The numerical results indicate that changes in capillary properties can introduce significant variations in the arrival time of a specific aqueous fraction. A more robust definition of arrival time appears to be the time associated with the peak rate of change in aqueous phase fraction, the greatest slope of the breakthrough curve.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 An adequate representation of subsurface variability typically requires a large-scale model of flow properties. Accurate characterization of such large-scale models must be based upon an large set of observations. Increasingly, such data sets are becoming available for characterizing flow properties. For example, extensive networks of multilevel samplers are now used to measure tracer concentrations. New methods, such as hydraulic tomography, provide numerous sets of pressure observations [Karasaki et al., 2000; Yeh and Liu, 2000; Vasco and Karasaki, 2001]. Laboratory techniques such as X-ray computer tomography (CT) can image fluid content changes at high resolution [Vinegar and Wellington, 1987; Clausnitzer and Hopmans, 2000]. Utilizing such large data sets for inverse modeling can be a computational challenge. This is particularly true for two-phase flow observations, because of the computational burden of numerical simulation [Finsterle and Pruess, 1995].
 Recently, Vasco and Datta-Gupta  introduced a trajectory-based approach for inverse modeling. The technique, which is similar to ray methods for medical and geophysical imaging, makes it possible to consider large-scale, three-dimensional models and extensive sets of observations, such as provided by multilevel samplers [Yoon et al., 2001; Datta-Gupta et al., 2002]. The initial work, Vasco and Datta-Gupta , was concerned with purely advective tracer transport. The methodology was subsequently extended to diffusive pressure propagation and the inversion of transient pressure data [Vasco et al., 2000]. The asymptotic approach also forms the basis for an efficient inversion of two-phase flow observations [Vasco et al., 1999; Vasco and Datta-Gupta, 2001]. The methodology has been shown to be applicable to the general equations governing transient flow and tracer transport and may be implemented using an existing numerical simulator [Vasco and Finsterle, 2004].
 In our previous work on the inversion of two-phase flow observations [Vasco et al., 1999; Vasco and Datta-Gupta, 2001] we neglected capillary effects. In this paper I demonstrate that the asymptotic methodology is also applicable to modeling two-phase flow in the presence of capillary forces. When capillary effects are included in the analysis, the resulting equation governing the evolution of the saturation amplitude along a trajectory is a generalization of Burger's equation [Burgers, 1948]. Burger's equation governs nonlinear diffusive wave propagation and is used in numerous applications [Burgers, 1974; Sachdev, 1987]. Thus the asymptotic formulation reveals a link between two-phase flow and nonlinear wave propagation [Whitham, 1974].
2.1. Governing Equations for Two-Phase Flow
 My starting point is the set of simultaneous partial differential equations describing the flow of an aqueous (wetting) phase and a nonaqueous (nonwetting) phase [Bear, 1972; Peaceman, 1977; de Marsily, 1986]
where Sw and Sn denote the saturations of the aqueous and nonaqueous phases respectively. The relative permeabilities of the aqueous and nonaqueous phases, which are functions of the saturations, are represented by krw and krn while the hydraulic conductivity is given by K(x). The respective densities are ρw and ρn, the gravitational constant is g and the porosity is ϕ(x). The pressure associated with the aqueous phase is Pw(x, t) while the pressure for the nonaqueous phase is Pn(x, t), the viscosities are μw and μn. The flow equations are coupled because the saturations are assumed to sum to unity
I also assume that the phases are incompressible. Making use of the fact that Sn = 1 − Sw one may derive a single equation describing the evolution of the aqueous phase saturation, which I shall denote by S(x, t) in all that follows,
where z is the unit vector in the direction of the gravity field, q denotes total velocity, the sum of the velocities of the aqueous (qw) and nonaqueous (qn) phases:
and F1 (S) and F2(S) are specified functions of saturation. In particular, F1(S) is given by
Similarly, the function F2(S) is given by
and the capillary pressure is denoted by
The quantity μd denotes the ratio of viscosities μw/μn and Γ is the difference
In general, equation (3) must be solved numerically, using a method such as integral finite differences [Narasimhan and Witherspoon, 1976; Pruess et al., 1999]. Here I derive an approximate asymptotic solution which is valid when the background saturation and flow properties are smoothly varying in a sense made more precise below.
 There are two assumptions invoked in the derivation of an asymptotic solution for two-phase flow. First, it is assumed that the front or boundary separating the injected phase from existing fluid in the aquifer is the result of a balance between dispersive and diffusive effects and the nonlinearity of two-phase propagation. Second, it is assumed that the background saturation and flow properties vary smoothly between known boundaries. That is, there may be discontinuities, such as layering or faults, which are modeled as boundary conditions, and smoothly varying properties between these interfaces. In addition, there will be one or more saturation fronts due to the injection of fluid. The fronts define relatively sharp variations in saturation when compared to the smoothly varying background. I can represent the time and space scale of the front saturation variation by l. Similarly, the background variations are over a time and space scale L, where L ≫ l. I represent the ratio l/L by a dimensionless parameter ε and require that 0 < ε ≪ 1. I may define slow variables, in both space and time, in terms of the ratio ε:
where p is the highest order of the derivatives in the governing equations [Anile et al., 1993]. The argument is that, for a small perturbation, the most significant nonlinear term is of order ε2/L, assuming nondegeneracy. The effect of dispersion and dissipation is of order ε/Lp. For dispersion and dissipation to balance nonlinearity one must have L ∼ ε−(α−1). The extra ε enters because, in addition to the balance between nonlinearity and dispersion and dissipation, I would like the front properties to vary smoothly as a function of distance. For equation (3)p equals 2 and the corresponding value of α is 2. This generalization in scaling, which allows one to apply asymptotic methods to nonlinear, dispersive, and diffusive phenomena, was introduced by Gardner and Morikawa  and generalized by Taniuti and Wei . I should note that the exact form of the scaling is not unique. Other scalings are valid and will lead to somewhat different formulations. See Korsunsky [1997, p. 17–21] for an example of two possible choices of scaling in modeling ion acoustic waves.
 Much of the formalism developed in the asymptotic approach [Anile et al., 1993], is based upon concepts associated with a propagating front. For example, I shall consider properties such as the amplitude and travel time of the moving front [Whitham, 1974]. The local travel time θ of the front is a rapidly varying quantity which is defined in terms of a smoothly varying function, ϕ(X, T)
 Formally, an asymptotic solution of equation (3) is a power series representation of the saturation distribution, in terms of the scale parameter ε
where s0(X, T) is the background saturation distribution, which is assumed to vary smoothly in both space and time. The unknown quantities in equation (11), the functions θ(X, T) and sn(X, T), are found by substituting the series into the governing equation for saturation (3) and examining terms of various orders in ε. The low-order components in ε are of special interest for they dominate for a relatively sharp saturation front, e.g., for ε = l/L ≪ 1.
 As shown in Appendix A, the substitution of the asymptotic expansion (11) into equation (3) results in a series of equations associated with varying orders in ε. The equation of lowest order in ε is
and where the partial derivatives are evaluated with respect to the background saturation s0 (see equation (A16)). The equation of next highest order in ε is
 It is possible to treat equation (12a), for the travel time, and equation (13), for the amplitude, in a general setting. For example, by considering the propagation of a front with a variable velocity that depends on the saturation. Here I consider the case in which a two-phase front is propagating into saturation conditions such that the term on the right-hand side of equation (12a) vanishes. This may be due to a relatively uniform initial capillary pressure or an initial saturation distribution such that either F1(s0) or ∇Pc(s0) is negligible.
2.3. Saturation Travel Time
 If the right-hand side of equation (12a) vanishes, it reduces to a differential equation for the travel time, ϕ,
assuming that ∂s1/∂θ does not vanish. I may solve equation (16) directly, using the method of characteristics [Courant and Hilbert, 1962, p. 70]. In the method of characteristics, solutions are developed along particular trajectories, the characteristic curves, which are denoted by X(l), where l is a parameter signifying position along the curve. The equations for the characteristic curves are a set of four ordinary differential equations
where U = ∣U∣ and r denotes the distance along the axis aligned with U. Combining equations (18) and (19), I can write the travel time as an integral
where Σ is the trajectory from the injection well to the observation well.
 Note that, in specific instances, one may associate the trajectories with streamlines used to model tracer transport and multiphase flow [Datta-Gupta and King, 1995; King and Datta-Gupta, 1998; Crane and Blunt, 1999]. In particular, when the vector U, defined in equation (12b), is primarily dependent on q the trajectories coincide with streamlines. However, if q depends significantly on saturation then the trajectories will deviate from streamlines. Similarly, if gravitational forces are important, the path X(l) will deviate from a streamline.
2.4. Saturation Amplitude
 Now consider variations of the saturation amplitude associated with the passage of the two-phase front. The evolution of the saturation amplitude is governed by equation (13). In view of equation (16), it is clear that equation (13) reduces to
a nonlinear differential equation for the saturation amplitude s1. I rewrite equation (21) in characteristic coordinates defined by equations (17) and (18)
where ∇ · U is a damping term due to gravitational forces and spatial variations in relative permeability parameters. Next, I define the variable
 An analytic solution of equation (24) is not generally possible. However, if gravitational forces are negligible and the capillary properties of medium are uniform, then Φ(τ) in equation (24) vanishes and it is possible to develop similarity solutions for the saturation amplitude. In particular, as noted by Crighton and Scott  and Scott , depending on the behavior of the ratio Ψ(τ)/τ as τ increases, there are three classes of solutions. These three families of solutions, the cylindrical, the subcylindrical, and the supercylindrical, are described in this section. I should emphasize that these solutions are valid in the presence of arbitrarily large, but smoothly varying, conductivity and porosity heterogeneity.
 Consider the similarity solutions of equation (24) subject to the initial conditions
where u2 and u1 are the limiting values of saturation as θ → ±∞. Similarity solutions are of the form
where A and γ are specific constants [Barenblatt, 1996]. Similarity solutions are related to traveling wave solutions
 Self-similar and traveling wave solutions form a vast subject of investigation [Sachdev, 2000]. Such solutions have been grouped into two categories: self-similar solutions of the first and second kind [Barenblatt, 1979]. Similarity solutions of the first kind may be completely specified using dimensional analysis [Sachdev, 1987]. Similarity solutions of the second kind, also known as intermediate asymptotics, are solutions which are stable over a wide range of propagation distances [Barenblatt, 1979]. The exponent γ cannot be determined by dimensional considerations alone. Rather, γ is found by solving an eigenvalue problem which depends on the boundary conditions [Barenblatt, 1996]. The constant A is usually determined by matching the self-similar solution to a numerical solution.
 When Φ(τ) is negligible, equation (24) reduces to a generalized Burgers' equation. The generalized Burgers' equation, which arose in nonlinear acoustics [Lighthill, 1956], appears in many other contexts and is a canonical equation, describing dispersive and diffusive nonlinear wave propagation in heterogeneous media [Sachdev, 1987; Jeffrey, 1989; Anile et al., 1993]. The long-time asymptotic solutions of the generalized Burgers' equation were classified by Scott . The stability analysis of Scott  followed upon the derivation of a self-similar solution of the cylindrical Burgers' equation by Chong and Sirovich  and Rudenko and Soluyan . Crighton and Scott  treated the cylindrical and spherical generalized Burgers' equations in some detail. The significance of the ratio Ψ(τ)/τ is clear if one assumes a solution of the form
Substituting this expression into equation (24), with Φ(τ) = 0, results in the ordinary differential equation
where η = θ/τ. The nature of this differential equation depends on the behavior of the ratio Ψ(τ)/τ as τ increases. With increasing τ the ratio may either converge to a constant value, vanish, or increase without bound. These three limiting cases were examined by Scott  with regard to the self-similar behavior of the solution. For completeness, I briefly discuss each case in succession.
2.4.2. Cylindrical Case: → β, a Constant
 This is the most general case in which the ratio of Ψ(τ) to τ approaches a limiting value. In a sense, the other two solutions are just special cases of this one. The similarity solution for this case was derived by Rudenko and Soluyan  and is given implicitly by
In this case, diffusive decay and nonlinear steepening balance each other, resulting in a self-similar front profile.
2.4.3. Subcylindrical Case: → 0
 In this case the nonlinear behavior is the most significant. Equation (34) has two solutions, depending on the sign of u2. For u2 ≤ 0, Scott  proved that the asymptotic form is of an “expansion front” type and corresponds to ζ0→ − in equation (37). For u2 > 0, Scott  conjectures, but does not prove, that the similarity solution converges to the “Taylor shock” solution [Taylor, 1910].
 For the special case in which Ψ(τ) = Ψ0, a constant, equation (24) reduces to the well-known Burgers' equation [Burgers, 1974]. Burgers' equation can be solved exactly using the Cole-Hopf transformation
 In this section I illustrate the construction of the trajectory-based asymptotic solution. I compare a purely numerical solution of equation (24) to the analytical solutions given above. The asymptotic solutions are subsequently compared with the output of the numerical simulator TOUGH2 [Pruess et al., 1999]. In these examples I shall not consider gravitational effects. Rather, I will focus on the role of capillary forces on the solutions.
 The first model consists of a uniform, 4 m thick layer, with a reference porosity of 0.18 and hydraulic conductivity of 2.0 × 10−13 m2. Initially, the layer contains 95% air and 5% water. Water is injected into a grid block within the layer at a rate of 0.097 kg/s. The relative permeability functions for gas (krg) and liquid (krl) are those of Grant 
Slr = 9.6 × 10−4 is the residual liquid saturation and Sgr = 0.01 is the residual gas saturation. The capillary function of van Genuchten  is used for this illustration
with λ = 0.75, P0 = 1.49 × 105, and Sls = 1.0. The water saturations, calculated by the numerical simulator TOUGH2, are shown for three different times (0.5, 1.5, and 2.0 days) in Figure 1. The arrival time of the water, shown in Figure 2, is defined as the time at which the derivative of the water saturation is a maximum. The motivation for this definition is provided in section 4.
 The first step in the construction of an asymptotic solution is to define the trajectory, X(l), from the observation point to the injection well, based upon equation (17). This may be done using a numerical simulator, as discussed by Vasco and Finsterle . The vector field U is computed using a TOUGH2 simulation. Figure 3 shows the steady state pressure field produced by the simulator. The trajectory is obtained numerically, by starting at the observation well and marching in the direction of U. The formal procedure I use, a second-order Runge-Kutta technique, is known as Heun's method. Heun's method is quite simple and can be implemented in a few tens of lines of computer code. In essence, Heun's method improves upon an Euler iteration by computing U at an intermediate point. That is, after the ith step along the trajectory an intermediate step is taken, based upon equation (17)
The (i+1)th step direction is computed using the average of the gradients at the ith point Xi and the intermediate point i
Figure 3 displays the resulting trajectory, which is a straight line in this case.
 The second step in computing the asymptotic solution entails solving the differential equation (24). Generally, equation (24) must be solved numerically using a technique such as finite differences. However, when the capillary properties are uniform and gravitational forces are not significant, I may derive analytic, self-similar solutions. As noted above, the nature of the self-similar solutions depends upon the behavior of the ratio Ψ(τ)/τ as τ increases. In Figure 4 I plot this ratio as a function of the distance along the trajectory. The ratio Ψ(τ)/τ rapidly approaches a nonzero constant value as the aqueous phase propagates from the injection well. Thus it is valid to use the cylindrical solution, equation (37), to compute the saturation at the observation well. The solution of the generalized Burgers' equation is also obtained numerically, using a total variation diminishing (TVD) scheme [Datta-Gupta et al., 1991] to solve equation (24). In Figure 5 I compare the analytic expression (ANALYTIC), the numerical solution of the generalized Burgers' equation (TVD), and the saturation history output by the numerical reservoir simulator TOUGH2 (TOUGH2). The saturation history produced by analytic expression and the TOUGH2 simulation are essentially identical. The TVD results differ slightly from the TOUGH2 calculations, perhaps because of numerical dispersion or discretization error. Figure 6 displays both the absolute and relative error between the analytic solution and the TOUGH2 estimates. The peak absolute error is approximately 2% and takes the form of a long-period oscillation over the entire breakthrough curve. The relative error peaks at 29%, near the initial portion of the breakthrough curve. Near this portion of the curve the relative error is strongly dependent on the initial water saturation (5%). Note that errors in the numerical simulation may also contribute to the disagreement. For example, there will be numerical dispersion and discretization errors in the TOUGH2 estimates.
 In Figure 7 one notes the effect of increasing the coefficient Ψ(τ) in equation (24). In Figure 7 I plot the observed water breakthroughs for three values of Ψ(τ), 0.001, 0.010, and 0.100. Increasing the coefficient broadens the front and results in a more gradual increase in S with time. Note that, if the arrival time of the aqueous phase is measured by the observation of some arbitrary fraction, say 0.05, increasing the capillary forces will significantly modify the arrival time. Thus, for the same porosity and hydraulic conductivity, the arrival time can vary significantly, depending on the strength of capillary forces. A better measure of arrival time appears to be the time associated with the greatest rate of increase of the water with time, i.e., the greatest slope of the breakthrough curve. This measure of arrival time, is similar to the definition for a propagating pressure front [Vasco et al., 2000; Vasco and Finsterle, 2004].
 Now consider two-phase flow in a heterogeneous environment. Specifically, hydraulic conductivity modifiers are applied to the uniform background model. Figure 8 displays the distribution of conductivity modifiers. A high-conductivity zone extends from the south central region to the northeast. Generally higher permeabilities are also found in the southwest quadrant of the model. The construction of the trajectory from the injection well to the observation well proceeds as in the homogeneous case. The output of a TOUGH2 numerical simulation is used to calculate the vector field U and Huen's method is used to solve equation (17). The resulting trajectory for the heterogeneous case is shown in Figure 9, along with the steady state pressure field output by TOUGH2. Note how the trajectory bends in response to variations in hydraulic conductivity. As before, I compute the ratio Ψ(τ)/τ for increasing τ (Figure 10) in order to determine the nature of the solution to the generalized Burgers' equation. From Figure 10 one sees that the ratio Ψ(τ)/τ approaches zero as τ increases. Thus the solution is in the form of a shock front, equation (40). Figure 11 indicates the agreement between the analytic, the generalized Burgers (TVD), and the TOUGH2 solutions. Figure 12 displays both the absolute and relative differences between the numerical simulation and the analytic solution. For the heterogeneous case the largest absolute error is roughly 4%, in the form of an oscillation. The peak relative error is just above 12%, occurring near the onset of the breakthrough of the aqueous phase.
 I have presented an asymptotic solution which is valid for smoothly varying heterogeneous media. The approach is appropriate for arbitrary conductivity contrasts as long at the scale length of the heterogeneity is significantly longer than the width of the two-phase front. Thus the method could break down in the presence of a sharp jump in conductivity which is not accounted for as a boundary condition. The assumption of smoothly varying properties is in accord with the use of the asymptotic solution for inverse modeling. Typically, it is not possible to resolve small-scale variations in reservoir properties based upon multiphase flow data. Thus one often seeks the smoothest model which is consistent with the observations. In this context, the asymptotic solution may be used to invert multiphase flow data for smoothly varying heterogeneity, as discussed by Vasco and Datta-Gupta .
 As noted above, the asymptotic solution is defined along a trajectory or curve through the model. As a result, for a two- or three-dimensional model, the computational complexity associated with the asymptotic forward and inverse modeling grows in proportion to the greatest dimension of the model. Thus it is possible to apply this modeling technique to large-scale, models [Vasco and Datta-Gupta, 2001; Datta-Gupta et al., 2002]. The asymptotic approach shares characteristics with streamline-based modeling schemes [Datta-Gupta and King, 1995; King and Datta-Gupta, 1998; Datta-Gupta et al., 2002]. In a sense, the asymptotic approach provides a mathematical framework for streamline modeling. However, the asymptotic approach is applicable to the fully general two-phase flow equations including gravity and capillary effects. Furthermore, one can implement the trajectory-based asymptotic techniques using a purely numerical simulator [Vasco and Finsterle, 2004]. It is only necessary to postprocess the output of the numerical simulator to determine the components of U in order to compute the trajectories and to define the travel times. Depending on the limiting nature of the coefficient Ψ(τ), there are three regimes in which analytic solutions are possible. One can take advantage of the analytic expressions for saturation amplitudes in order to derive sensitivity functions for the inverse modeling of the saturation amplitudes. This provides an efficient method for matching saturation history data [Vasco and Datta-Gupta, 2001]. This will be the topic of a future investigation.
 The numerical results indicate that capillary forces can dramatically alter the breakthrough time (Figure 7). That is, for a breakthrough time defined with respect to an observed fluid fraction, I find significant variation, depending on the strength of the capillary forces. Thus care is required when defining the breakthrough time, if one is to obtain a measure that is directly sensitive to porosity and conductivity. Numerical results indicate that the time corresponding to the observation of the peak slope of the breakthrough curve may be a better measure of arrival time. The use of the peak slope of the breakthrough curve is similar to the definition of the arrival time of a transient pressure front as discussed by Vasco et al.  and Vasco and Finsterle . I can motivate the use of the peak of the derivative of the saturation curve as a measure of the arrival time by considering a special case of the cylindrical solution (35), (36), and (37). Specifically, I consider the subcylindrical case in the limit at τ → ∞. The solution is given by the expression (42), which I write as
The variable θ functions as a time-like variable, while τ is a measure of reduced distance from the injection well. I shall consider s(θ, τ) as a function of θ and fix τ. Because I am interested in the slope of the breakthrough curve, consider the derivative of (52) with respect to θ
where the prime denotes the derivative with respect to θ. The quantity s′(θ, τ) is an extremum when its derivative vanishes. Thus I differentiate the expression (54)
The derivative is a minimum or maximum when the expression (55) is zero. The right-hand side of (55) vanishes, for a finite value of θ, when
or θ = τ. Using the definition of τ, provided by equation (23), I arrive at the condition
From the definition of Ω(s0), given by (15), I find that the value of θ associated with the observation of the largest slope, only depends on flow properties as contained in ϕ and U, the mean of the limiting saturations , and the relative permeability parameters contained in Ω(s0). Most importantly, this definition of arrival time does not depend on capillary forces within the medium. The independence of the arrival time of the steepest slope, with respect to changes in Ψ(τ), was observed in the numerical calculations.
 The techniques I have discussed may be extended in several respects. For example, asymptotic methods may be applied to coupled dispersive and dissipative systems [Korsunsky, 1997]. Thus, using an asymptotic approach it is possible to formally consider the coupled system for saturation and pressure. I should note that Burgers' equation also arises when one considers dynamic capillary pressure [Cuesta et al., 2000]. Thus it may be possible to extend the asymptotic approach to include dynamic capillary pressure. In this paper I have only considered the situation in which the capillary pressure variation prior to injection of the second phase is relatively constant. However, this in not a fundamental requirement of the approach taken here. In fact, asymptotic methods are applicable to coupled systems exhibiting fingering [Grindrod, 1996] and systems with curvature-dependent propagation velocities [Sethian, 1999].
 In this paper I derive an asymptotic method for modeling two-phase flow in the presence of capillary forces. The technique is well suited for efficient inverse modeling and provides a flexible tool for aquifer characterization. In particular, the asymptotic solution partitions into two subproblems: an arrival time calculation and an amplitude calculation. The governing equation for the arrival time provides an analytic relationship between the breakthrough time and the flow properties of the aquifer. The amplitude subproblem reduces to the solution of a one-dimensional generalization of Burgers' equation, an equation governing dispersive and diffusive nonlinear wave propagation in heterogeneous media [Sachdev, 1987; Anile et al., 1993]. Because the solution is defined along a trajectory through the reservoir model, the level of computation is proportional to the largest single dimension of the model. The equation for the saturation amplitude is solved efficiently using a numerical approach, the total variation diminishing (TVD) scheme [Datta-Gupta et al., 1991]. In the situations considered here it was found that self-similar solutions were appropriate for modeling two-phase flow. Such solutions provide semianalytic expressions for saturation amplitude variations which may be used as a basis for an efficient inversion scheme.
 The asymptotic approach compares well with the output of the numerical simulator TOUGH2 [Pruess et al., 1999] for both homogeneous and heterogeneous conductivity models. The asymptotic approach was found to agree to within 5% of numerical estimates of saturation. The approach is valid for smoothly varying flow properties and initial saturation, relative to the scale length of the two-phase front. Sharp interfaces, such as an air-water contact, must be modeled as explicit boundary conditions. It is appropriate to comment on the computation time associated with each of the three approaches for calculating the breakthrough of the aqueous phase. The numerical simulator TOUGH2 took roughly 200 CPU s to calculate the water breakthrough. The TVD approach required 3.5 s to solve equation (24) numerically. The semianalytical solution of equations (35), (36), and (37) was completed in 0.1 s. While these times are indicative of the relative efficiencies of each algorithm, the real advantage of the asymptotic approach is in the solution of the inverse problem. Specifically, the asymptotic methodology provides semianalytic expressions for the two-phase front arrival time and amplitude. Such expressions form the basis for semianalytic sensitivity calculations and extremely efficient inverse modeling algorithms [Vasco and Datta-Gupta, 1999; Vasco et al., 1999]. Thus the true value of the expressions derived here will be in their use in inverse modeling.
Appendix A:: Terms of the Asymptotic Expansion
A1. Asymptotic Solution
 In this appendix I derive an asymptotic solution to equation (3). Formally, an asymptotic solution of equation (3) is a power series representation of the saturation distribution, in terms of the scale parameter ε
where s0(X, T) is the background saturation distribution, which is assumed to vary smoothly in both space and time. The unknown quantities in equation (A1), the coefficients θ(X, T) and sn(X, T), are found by substituting the series into the governing equation for saturation and examining terms of various orders in ε. The low-order components in ε are of special interest for they dominate for a rapidly varying saturation front because ε ≪ 1.
 Before substituting the series (A1) into equation (3) note that the partial derivative operators in (3) may be represented in terms of derivatives with respect to the slow variables X and T, see equation (8), and the travel time θ
Thus the spatial derivatives contained in the gradient operator applied to S(X, T) take the form
For quantities, such as K(x), which only depend on spatial variables, the derivative is
Higher-order spatial derivatives are obtained by successive applications of (A3).
 The asymptotic representation (A1) depends upon the travel time function θ and the successive saturation amplitude corrections si(X, T, θ), i = 1, 2, …∞. In order to determine these quantities I write the space and time derivatives in (3) in terms of the operators (A2). The result is the equation
where I have divided both sides by ε. Equations for the travel time and amplitude functions are obtained by considering terms of successive orders in ε in equation (A5).
 Now I expand the expression (A5) and consider terms of the lowest two orders in ε. Multiplying out the factors in (A5) gives
Neglecting terms of order ε2 or greater I arrive at the equation
The quantities F1(S), F2(S), and Pc(S) depend strongly on saturation S and possibly on x. For conciseness I shall suppress the explicit dependence on x, but the following should be kept in mind. In constructing the asymptotic representation I substitute the expansion (A1) for S(x, t) into the terms F1(S), F2(S), and Pc(S) of equation (3). For example, consider Pc(S) which may be represented as a power series in S. The expansion is given by
where O(ε2) denotes terms of order ε2 and higher. Note that the derivatives in the expansion are evaluated with respect to the smoothly varying background saturation, s0(x, t). Similarly, the power series representations of F1(S) and F2 (S) are given by
Substituting the power series expansions into equation (A7) and retaining only terms up to order ε2 results in the expression
A2. Terms of Order ε: An Equation for the Travel Time
 Consider terms of order ε in expression (A11)
Or, regrouping terms and dividing out ε,
Defining the vector U
where the derivatives are evaluated with respect to the background saturation s0, I may write (A13) more compactly as
 If I add the term −∇ · [K(x)F1(s0)∇Pc(s0)] to each side of equation (A15) and note that the background saturation distribution s0 satisfies equation (3), I arrive at
A3. Terms of Order ε2: An Equation for the Saturation Amplitude
 Terms of order ε2 are
 If I regroup them and write (A17) in terms of U, defined above, I have
a nonlinear partial differential equation for the saturation amplitudes s1 and s2. Note that the coefficients of the differential equation depend on the flow properties, the background saturation distribution, and the travel time. The nonlinearity takes the specific form s1∂s1/∂θ and equation (A21) is reminiscent of the nonlinear diffusion equation.
 This work was supported by the Assistant Secretary, Office of Basic Energy Sciences of the U.S. Department of Energy under contract DE-AC03-76SF00098. All computations were carried out at the Center for Computational Seismology, LBL.