#### 2.1. Equations Governing Two-Phase Flow

[7] To begin, I consider the set of simultaneous partial differential equations describing the flow of a wetting phase and a nonwetting phase [*Bear*, 1972; *Peaceman*, 1977; *de Marsily*, 1986]:

where *S*_{w} and *S*_{n} denote the saturation of the wetting and nonwetting phases, respectively. The relative permeabilities of the wetting and nonwetting phases, which are functions of the saturations, are represented by *k*_{rw} and *k*_{rn}, while the absolute permeability is given by *k*(**x**). The respective densities are and , the gravitational constant is *g*, and the porosity is . The pressure associated with the wetting phase is *P*_{w}(**x**, *t*) while the nonwetting phase pressure is *P*_{n}(**x**, *t*); the fluid viscosities are and . The two equations are coupled because the two fluids are assumed to fill the available pore space, and thus, their saturations sum to unity:

[8] I also assume that the phases are incompressible so that their densities are constant. I define the saturation-dependent component of the fluid mobility [*Peaceman*, 1977, p. 18] by the ratios

[9] I shall assume that the relative permeability properties are constant for a given formation. Thus, within a given heterogeneous layer I shall assume that the relative permeabilities are only functions of the fluid saturations. Because the saturations sum to unity, I can write the governing equations (1) in terms of one of the saturations, say

and hence, the system of equations reduces to two equations in three unknowns (*S*, *P*_{w}, and *P*_{n}).

[10] To reduce the system (1) to two equations in two unknowns, I invoke the assumption that the fluid pressure difference, the capillary pressure *P*_{c}, in the pores is a function of the fluid saturation [*Bear*, 1972]; thus,

[11] As was done for the relative permeabilities, I shall assume that the capillary pressure function only varies across a layer boundary and does not depend explicitly upon the spatial location within a particular formation. Rather, in a given formation, the capillary pressure function is only a function of the pressure in one fluid phase and the saturation distribution within the layer. Denoting the fluid pressure in the wetting phase by *P*,

and writing the fluid pressure for the nonwetting phase as

[12] I can reduce the system of equations (1) to two equations in two unknowns. First, because of equation (8), I write the gradient of the fluid pressure of the nonwetting phase in terms of the gradients of the pressure and saturation of the wetting phase:

[13] As done by *de Marsily* [1986], I assume linear elastic behavior for the porous matrix to arrive at a relationship between a change in fluid pressure and a change in matrix porosity. The exact relationship is

where is a proportionality coefficient that depends upon the compressibilities of the fluids and the solid and on the porosity. Making the various substitutions described above, the original system of equations (1) reduces to two equations in two unknowns:

where is a vector in the direction of the gravitational attraction and

[14] Carrying out the differentiations associated with the outer divergence operator, I can write equations (11) and (12) as the pair of equations

where

[15] Equations (14) and (15) are the governing equations and serve as the starting point for my application of the method of multiple scales, an asymptotic technique described in section 2.2.

#### 2.2. An Asymptotic Analysis of the Governing Equations

[16] The governing equations (14) and (15) are rather complicated as they are nonlinear, of mixed character, and coupled partial differential equations with spatially varying coefficients. Without some manner of simplification an analytic solution is certainly not possible. Because one goal of this work is to develop techniques to solve inverse problems, for example, using the saturation front arrival time to infer the flow properties of the medium, retaining the heterogeneity is essential. However, because of the limited resolution of most inverse methods, in which a finite number of data are used to estimate a field of properties, one typically seeks models with smoothly varying heterogeneity. Thus, I am most interested in two-phase flow in a model with smoothly varying properties. I should note that sharp boundaries, in the form of layering, are allowed as explicit boundary conditions.

[17] I can build the assumption of smoothly varying heterogeneity into the modeling through a technique known as the method of multiple scales [*Anile et al.*, 1993, p. 49]. This approach is suited to the construction of asymptotic solutions for a porous medium with heterogeneous, yet smoothly varying, flow properties. The measure of smoothness is with respect to the scale length of the two-phase front. In order to define this formally, I first denote the scale length of the two-phase front, the distance over which the saturation changes from the background value to the value behind the front, by *l*. In addition, let *L* denote the scale length of the heterogeneity within the medium. The smoothness of the medium is stipulated by the requirement that . An asymptotic solution can be formulated in terms of the ratio of scale lengths . To this end, I will define the slow spatial coordinates

the scale over which many of the quantities of interest, such as the travel time, will vary. Similarly, I can define a slow time:

[18] An asymptotic solution is a power series representation of the dependent variables, that is, the saturation and pressure. The power series is in terms of the scale variable ɛ. For example, the saturation is represented as

where *S*_{b} is the background saturation that may be a function of space and time, is a phase function that is related to the propagation time of the saturation front, and *S*_{i}(**X**, *T*) is the *i*th contribution to the saturation amplitude. The integral appears in the representation because I will be considering a step function source, rather than a pulse-like source. The saturation contains both an explicit and an implicit dependence upon the spatial and temporal coordinates. The implicit dependence is through the phase function . The representation (22) is in the form of a traveling front, a propagating change in the saturation with respect to the background saturation.

[19] The increase in pressure across the two-phase front has a similar representation:

[20] Note that the phase function for the pressure can differ from the saturation phase, meaning that the saturation and pressure can move with different speeds. This allows the jump in pressure to propagate much faster than the saturation change, for example. In section 2.3 I will examine the situation in which .

[21] The governing equations (14) and (15) can be rewritten in terms of the slow coordinates. In order to do this I first express the partial derivatives in terms of *X*_{i}, *T*, and . In doing so I make use of the relationships (20) and (21) between the fast and slow coordinates and the explicit and implicit dependence upon the independent variables. Thus, I can write the partial derivative with respect to time as

because, from the definition of *T* in equation (21), . Similarly, I can express the derivative with respect to *x*_{i} as

and thus, the gradient in terms of the **X** coordinates is given by

where the subscripts *X* and *x* indicate that the derivatives are with respect to the **X** and **x** coordinates, respectively.

#### 2.3. Expression Governing the Evolution of the Two-Phase Front

[24] However, before delving into a detailed derivation, I need to discuss an important issue regarding the nature of the propagating front. As noted, it is assumed that the leading edge of the front is defined by a rapid change in saturation and pressure. Ahead of the front, the saturation and pressure are at their background values; behind the front the saturation and/or pressure assume new values, different from the background values. The concept of a propagating front has proven extremely useful in a wide variety of fields, such as electromagnetics [*Kline and Kay*, 1965; *Luneburg*, 1966], and is central to treatments of nonlinear wave propagation [*Whitham*, 1974; *Maslov and Omel'yanov*, 2001]. Many of the coefficients, for example, , , and , in equations (27) and (28) are functions of the saturation and pressure. Thus, one may ask, What values of saturation and pressure should be used in determining the coefficients? Because I am interested in the arrival time of the leading edge of the front, which evolves according to the saturation and pressure encountered before the jump to new values, I will use the background conditions to compute the coefficients in (27) and (28). Following *Anile et al.* [1993], a more formal mathematical approach may be taken, on the basis of an expansion of the coefficients in powers of ɛ. The expansion follows from the representation of the pressure as an asymptotic series in ɛ (see equation (23)). Consider, for example, , which may be expanded as

where

in the case of a step function source. To order one finds that . The moment at which the coupled front arrives at an observation point **X**, which I denote by *T*_{arrival}, the quantity is zero, and hence, .

[25] Fixing the coefficients in equations (27) and (28) to their background values, the next task involves estimating the slowness of the propagating front. If the governing equations were linear differential equations, then the zeroth-order terms would form a linear system, and one could use the condition that the linear system has a nontrivial solution to find the admissible slowness values [*Kline and Kay*, 1965; *Kravtsov and Orlov*, 1990]. Equations (27) and (28) are not linear in and ; rather, they comprise two quadratic equations. A formal approach, similar to that used for linear systems of equations [*Noble and Daniel*, 1977], may be based on techniques from algebraic geometry [*Cox et al.*, 1998]. The condition under which the two polynomial equations (27) and (28) have common zeros is the vanishing of the resultant [*Cox et al.*, 1998; *Sturmfels*, 2002]. The resultant is a polynomial equation in *s*, *p* with coefficients that depend upon either or and the properties of the medium.

[26] Here I take a direct approach, first solving equation (27) for the product term,

where I have used the background value *S*_{b} for the saturation. Substituting this expression for into equation (28) and grouping terms according to their degrees in *s* and *p* gives

an equation for the slownesses.

[27] In the most general situation, in which no assumptions are made regarding **s** and **p**, there are more unknowns than equations. However, I am primarily interested in the propagation of a two-phase front in which the change in saturation and the change in pressure are coupled. That is, the jumps in saturation and pressure occur simultaneously as the front passes. Thus, and , the phase terms associated with the saturation and pressure changes, are equal to , , and *s* = *p*. In that case, equation (36) reduces to a single quadratic equation in *p*:

[28] Defining the ratio

and the coefficients

#### 2.5. Zeroth-Order Solution for the Saturation and Pressure Changes

[36] Armed with expressions for the trajectory of the propagating front and the phase function , one can construct a low-order representation of the saturation and pressure fields using the series solutions (22) and (23). Because I am interested in solutions for a model with smoothly varying flow properties, ɛ is assumed to be small, and thus, the first few terms of the series dominate. Here I consider a zeroth-order solution, taking only the first term of each series. The expression for the saturation change with respect to the background value *S*_{b}(**X**, *T*) is

and similarly for the pressure change, it is

[38] One can gain some physical insight into the meaning of the phase terms and following a line of reasoning first suggested by *Virieux et al.* [1994]. I discuss this approach in some detail in Appendix C. Note that each of the semianalytic expressions (51) and (52) contain temporal integrals of an exponential of the phase function multiplied by an amplitude function. The exponential of the phase function is always a positive number, and the amplitude function is typically of one sign for a passing front. For example, the amplitude function *S*_{0} will either describe a decrease or an increase in the saturation of the aqueous phase, depending on the nature of the passing coupled front. Thus, the integrals are typically piecewise monotonic if the coupled multiphase front is a result of the injection of a particular fluid component.

[39] As noted by *Vasco et al.* [2000] and *Vasco and Finsterle* [2004], for a step function source, the transient, wave-like nature of a solution to the diffusion equation is emphasized by taking the derivative of the head or pressure with respect to time. Thus, I shall be interested in the time derivatives of the saturation and pressure changes, which are of the form

[40] For a separable phase function, as given by equation (50), the saturation and pressure resemble the product of a Gaussian function and the time-varying amplitude function. In Appendix C I derive a relationship between the phase function and the arrival time of the pressure and saturation fronts *T*_{peak} for a separable phase function with a power law time dependence:

where is given by expression (40).