## 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].