Water Resources Research

An asymptotic solution for two-phase flow in the presence of capillary forces

Authors


Abstract

[1] 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.

1. Introduction

[2] 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].

[3] Recently, Vasco and Datta-Gupta [1999] 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 [1999], 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].

[4] 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. Methodology

2.1. Governing Equations for Two-Phase Flow

[5] 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]

equation image

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

equation image

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,

equation image

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:

equation image

and F1 (S) and F2(S) are specified functions of saturation. In particular, F1(S) is given by

equation image

Similarly, the function F2(S) is given by

equation image

and the capillary pressure is denoted by

equation image

The quantity μd denotes the ratio of viscosities μwn and Γ is the difference

equation image

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.

2.2. An Asymptotic Solution for Two-Phase Flow

[6] Asymptotic solutions are extremely valuable for modeling a wide range of physical phenomena [Jeffrey and Taniuti, 1964; Jeffrey and Kawahara, 1982; Anile et al., 1993]. Asymptotic methods underlie a wealth of practical techniques for modeling electromagnetic and seismic wave propagation [Kline and Kay, 1979; Bouche et al., 1997; Aki and Richards, 1980], as well as for medical imaging [Arridge, 1999].

[7] 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 Ll. 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:

equation image

where α is a rational number.

[8] The quantity α is chosen such that the nonlinearity balances the dispersion and dissipation [Taniuti and Nishihara, 1983; Anile et al., 1993]. The formula determining α, which is based upon dimensional arguments [Anile et al., 1993, p. 105] is

equation image

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 [1960] and generalized by Taniuti and Wei [1968]. 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.

[9] 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)

equation image

[10] Formally, an asymptotic solution of equation (3) is a power series representation of the saturation distribution, in terms of the scale parameter ε

equation image

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.

[11] 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

equation image

where

equation image

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

equation image

where

equation image
equation image

(see equation (A21)).

[12] 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

[13] If the right-hand side of equation (12a) vanishes, it reduces to a differential equation for the travel time, ϕ,

equation image

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

equation image
equation image

[Courant and Hilbert, 1962, p. 70]. For a coordinate system with one axis oriented along U I may write (17) as

equation image

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

equation image

where Σ is the trajectory from the injection well to the observation well.

[14] 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

[15] 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

equation image

a nonlinear differential equation for the saturation amplitude s1. I rewrite equation (21) in characteristic coordinates defined by equations (17) and (18)

equation image

where ∇ · U is a damping term due to gravitational forces and spatial variations in relative permeability parameters. Next, I define the variable

equation image

and rewrite equation (22) in terms of τ

equation image

where I have defined

equation image
equation image

Owing to the presence of the variable coefficients Ψ(τ) and Φ(τ), equation (24) is a generalization of Burgers' equation describing the evolution of a nonlinear diffusive wave [Burgers, 1948, 1974]. Burgers' equation has been used to model water flow in soils and one-dimensional drainage [Clothier et al., 1981; Hills and Warrick, 1993; Warrick and Parkin, 1995]. Because equation (24) is a scalar differential equation which only depends on a single space-like variable, it may be solved efficiently using a numerical scheme such as a total variation diminishing (TVD) algorithm [Datta-Gupta et al., 1991].

2.4.1. Similarity Solution

[16] 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 [1979] and Scott [1981], 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.

[17] Consider the similarity solutions of equation (24) subject to the initial conditions

equation image

where u2 and u1 are the limiting values of saturation as θ → ±∞. Similarity solutions are of the form

equation image

where A and γ are specific constants [Barenblatt, 1996]. Similarity solutions are related to traveling wave solutions

equation image

if one makes the following change of variables [Sachdev, 2000]

equation image
equation image
equation image
equation image

[18] 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.

[19] 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 [1981]. The stability analysis of Scott [1981] followed upon the derivation of a self-similar solution of the cylindrical Burgers' equation by Chong and Sirovich [1973] and Rudenko and Soluyan [1977]. Crighton and Scott [1979] 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

equation image

Substituting this expression into equation (24), with Φ(τ) = 0, results in the ordinary differential equation

equation image

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 [1981] with regard to the self-similar behavior of the solution. For completeness, I briefly discuss each case in succession.

2.4.2. Cylindrical Case: equation image → β, a Constant

[20] 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 [1977] and is given implicitly by

equation image
equation image

where

equation image

In this case, diffusive decay and nonlinear steepening balance each other, resulting in a self-similar front profile.

2.4.3. Subcylindrical Case: equation image → 0

[21] 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 [1981] proved that the asymptotic form is of an “expansion front” type and corresponds to ζ0→ −equation image in equation (37). For u2 > 0, Scott [1981] conjectures, but does not prove, that the similarity solution converges to the “Taylor shock” solution [Taylor, 1910].

[22] 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

equation image

to convert (24) to the heat equation

equation image

[Sachdev, 1987]. This equation may be solved exactly and the explicit expression for the saturation is

equation image

where equation image is the ratio of the integrals

equation image

and equation image is the average

equation image

[Whitham, 1974, p. 102; Sachdev, 2000, p. 132]. In the limit as τ → ∞ the ratio equation image converges to 1. Thus, for large τ

equation image

[Whitham, 1974; Sachdev, 2000]. Note that, for Ψ(τ) identically zero, I have the similarity solution of Buckley and Leverett [1942].

2.4.4. Supercylindrical Case: equation image → ∞

[23] In this case the nonlinear term is dominated by the diffusive term and the solution is that of the linearized version of equation (24). The solution is in the form of an error function

equation image

where z is the integral

equation image

[Rudenko and Soluyan, 1977]. Thus the front decays strongly as a function of distance.

3. Numerical Illustration

[24] 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.

[25] 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 [1977]

equation image
equation image

where

equation image

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 [1980] is used for this illustration

equation image

where

equation image

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.

Figure 1.

The results of a simulation using the TOUGH2 integral finite difference code [Pruess et al., 1999]. The reservoir model is a single layer, characterized by a uniform distribution of properties, a porosity of 0.18, and a conductivity of 2.0 × 10−13 m2. Initially, the layer contains 95% air and 5% water. Water is injected into a borehole in the lower left quadrant of the model at a rate of 0.097 kg/s. The water saturation at three times is indicated by the gray scale in each plot. The star in this plot indicates the location of an “observation” point, at which I compute the breakthrough of the aqueous phase.

Figure 2.

The arrival time of the injected water in days. The arrival time is defined as the time associated with the occurrence of the maximum time derivative of the aqueous phase fraction observed in each grid block. The solid circle denotes the injection well; the open circles and the star indicate observation wells. I compute the breakthrough of the aqueous phase for the well, denoted by the star.

[26] 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 [2004]. 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)

equation image

The (i+1)th step direction is computed using the average of the gradients at the ith point Xi and the intermediate point equation imagei

equation image

Figure 3 displays the resulting trajectory, which is a straight line in this case.

Figure 3.

Steady state pressure variations induced by the injection of the water. The solid line corresponds to the trajectory X(r) extending from the pumping well to the observation well (star), computed using equations (50) and (51).

[27] 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.

Figure 4.

Plot of Ψ(τ)/τ, where Ψ(τ) is the coefficient of the Burger's equation (24), as a function of distance along the trajectory shown in Figure 3. The quantity Ψ(τ) is defined in equations (25a), (14), and (15). The variable τ is defined by equation (23). As discussed in section 2.4 on saturation amplitude, the ratio Ψ(τ)/τ determines the nature of the solution in the far field, i.e., away from the pumping well. The ratio rapidly approaches a constant value of approximately 3.0 × 10−12 with distance along the trajectory. Thus the solution should be well approximated by the cylindrical case (equations (35), (36), and (37)).

Figure 5.

Calculated water saturation at the observation well, which is indicated by a star in Figure 3. The saturation history output by the TOUGH2 numerical simulator [Pruess et al., 1999] is shown by squares. The result of a numerical integration of the generalized Burgers' equation (equation (24)) using a total variation diminishing (TVD) algorithm [Datta-Gupta et al., 1991] is shown by crosses). An analytic expression corresponding to equations (35), (36), and (37) is shown by circles.

Figure 6.

Absolute (circles) and relative (squares) errors associated with the analytic solution. The errors were computed with respect to the numerical simulator estimates.

[28] 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].

Figure 7.

Calculated water saturation at the observation well as a function of time. Each curve corresponds to a different, constant value of Ψ in equation (24). Larger values of Ψ indicate the increasing significance of capillary forces in the two-phase flow.

[29] 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.

Figure 8.

Hydraulic conductivity variation used for numerical trajectory computations in a heterogeneous medium. The shading indicates logarithmic multipliers which are applied to the uniform model used for the first synthetic case. High conductivity is indicated by the darker tones. The injection (solid circle) and observation wells (open circles and star) are identical to those used for the uniform test (Figure 1).

Figure 9.

Steady state pressure variation in the heterogeneous medium. The trajectory is denoted by a solid line and extends from the injection well (solid circle) to the observation point (star). The trajectory is computed from the pressure field using equations (50) and (51). Owing to the heterogeneity the trajectory deviates from the straight line path in Figure 3.

Figure 10.

Plot of Ψ(τ)/τ as a function of distance along the trajectory shown in Figure 9. Ratio Ψ(τ)/τ determines the nature of the solution in the far field. The ratio approaches zero with distance along the trajectory. Thus the solution may be approximated by the subcylindrical case (equations (40) and (41)).

Figure 11.

Calculated water saturation at the observation well, which is indicated by a star in Figure 9. The saturation history output by the TOUGH2 numerical simulator [Pruess et al., 1999] is shown by circles. The result of a numerical integration of the generalized Burgers' equation (equation (24)) using a total variation diminishing (TVD) algorithm [Datta-Gupta et al., 1991] is shown by squares. An analytic expression corresponding to the subcylindrical solution (equations (40) and (41)) is shown by crosses.

Figure 12.

Absolute (circles) and relative (squares) errors associated with the analytic solution for the heterogeneous test case. The errors were computed with respect to the numerical simulator estimates.

4. Discussion

[30] 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 [1999].

[31] 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.

[32] 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. [2000] and Vasco and Finsterle [2004]. 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

equation image

where

equation image
equation image

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

equation image

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)

equation image

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

equation image

or θ = equation imageτ. Using the definition of τ, provided by equation (23), I arrive at the condition

equation image

or, using equation (20),

equation image

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 equation image, 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.

[33] 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].

5. Conclusions

[34] 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.

[35] 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

[36] 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

equation image

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.

[37] 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

equation image

Thus the spatial derivatives contained in the gradient operator applied to S(X, T) take the form

equation image

For quantities, such as K(x), which only depend on spatial variables, the derivative is

equation image

Higher-order spatial derivatives are obtained by successive applications of (A3).

[38] 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

equation image

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).

[39] Now I expand the expression (A5) and consider terms of the lowest two orders in ε. Multiplying out the factors in (A5) gives

equation image

Neglecting terms of order ε2 or greater I arrive at the equation

equation image

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

equation image

where O2) 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

equation image
equation image

Substituting the power series expansions into equation (A7) and retaining only terms up to order ε2 results in the expression

equation image

A2. Terms of Order ε: An Equation for the Travel Time

[40] Consider terms of order ε in expression (A11)

equation image

Or, regrouping terms and dividing out ε,

equation image

Defining the vector U

equation image

where the derivatives are evaluated with respect to the background saturation s0, I may write (A13) more compactly as

equation image

[41] 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

equation image

A3. Terms of Order ε2: An Equation for the Saturation Amplitude

[42] Terms of order ε2 are

equation image

[43] If I regroup them and write (A17) in terms of U, defined above, I have

equation image

Defining the coefficients

equation image
equation image

allows one to write equation (A18) as

equation image

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 s1s1/∂θ and equation (A21) is reminiscent of the nonlinear diffusion equation.

Acknowledgments

[44] 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.

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