## 1. Introduction

[2] In a wide range of activities, including environmental remediation, the geological sequestration of carbon dioxide, and geothermal energy development, it is important to correctly model the flow of fluids within the subsurface. To this end, one must adequately characterize the flow properties at depth. This is typically accomplished through the solution of an inverse problem in which observations are used to constrain medium parameters, such as the formation permeability [*Sun*, 1994]. The practice of inverse modeling has advanced in recent years through improved field methods and the development of flexible modeling techniques. For example, there are networks of multilevel samplers and cross-well configurations of transducers capable of generating a dense array of observations [*Hsieh et al.*, 1985; *Butler et al.*, 1999; *Karasaki et al.*, 2000; *Yeh and Liu*, 2000; *Vesselinov et al.*, 2001; *Datta-Gupta et al.*, 2002]. In addition, geophysical measurements have been used to augment hydrological data in order to characterize flow properties in the subsurface [*Paillet*, 1993; *Schmidt and Bürgmann*, 2003; *Vasco et al.*, 2004; *Vasco*, 2004b; *Kowalsky et al.*, 2004, 2005; *Bell et al.*, 2008; *Vasco et al.*, 2010; *Rucci et al.*, 2010].

[3] Given the wide variety of hydrologic and geophysical data, it is important to have access to flexible and efficient approaches for modeling and inversion. Purely numerical methods provide the most comprehensive solutions to multiphase flow problems. However, numerical approaches tend to be computationally intensive and provide less insight because such solutions do not produce explicit expressions in terms of the model parameters. Analytic solutions can be efficient and do produce explicit expressions in terms of the parameters of a medium but are usually limited to fairly simple situations, such as linearized perturbations on a homogeneous background model. There are semianalytic techniques for modeling and inversion that display some of the efficiency and insight of analytic methods while extending to the more complicated situations that can be treated by numerical techniques. One class of semianalytic methods, the trajectory-based approaches described by *Cohen and Lewis* [1967], *Shen* [1983], *Vasco and Datta-Gupta* [1999], *Vasco et al.* [1999], and *Vasco et al.* [2000], has the additional flexibility of partitioning the inverse problem into a travel time–matching problem [*Vasco and Datta-Gupta*, 1999; *Vasco et al.*, 2000; *Brauchler et al.*, 2003] and an amplitude-matching problem [*Vasco*, 2008a]. As noted by *Cheng et al.* [2005], the travel time problem is quasi-linear and thus converges more readily than the highly nonlinear amplitude matching problem. Furthermore, inverting travel times is much more efficient than is amplitude inversion, and such inverse problems form the basis of medical imaging [*Arridge*, 1999] and geophysical tomography [*Iyer and Hirahara*, 1993].

[4] While asymptotic, trajectory-based solutions have been used to treat a number of inverse problems in hydrology [*Vasco*, 2008b], there are some limitations in current derivations. Specifically, while asymptotic techniques are applicable to nonlinear processes [*Whitham*, 1974; *Anile et al.*, 1993] and have been applied to two-phase flow [*Vasco et al.*, 1999; *Vasco*, 2004a], the applications have been limited in some respects. For example, capillary effects were neglected by *Vasco et al.* [1999]. Typically, when capillary effects are included, the background fields, such as the initial saturation and the capillary pressure, are assumed to be uniform [*Anile et al.*, 1993; *Vasco*, 2004a]. Furthermore, when the governing equation is written in terms of a distinct saturation equation, as in the work of *Vasco* [2004a], the resulting equation for the velocity of the saturation front is a complicated expression that contains an implicit dependence on the solution of the pressure equation.

[5] In this paper I present a new derivation of a trajectory-based solution for two-phase flow in the presence of capillary forces. The derivation is based upon a general approach that is applicable to any set of coupled nonlinear partial differential equations. The resulting expression for the phase velocity depends explicitly upon the saturation and pressure amplitude changes in a rather simple fashion. Because of the presence of the saturation and pressure terms, the phase velocity must be calculated in conjunction with the results from a numerical simulator. However, the expression provides insight into the way in which saturation and pressure changes control the propagation of a coupled two-phase front.