## 1. Introduction

[2] Over the past several decades there have been significant advances in the capabilities of reservoir simulators. Numerical simulators can account for thermal effects as well as phase changes [*O'Sullivan et al.*, 2001]. Many varieties of coupled simulators have been developed. For example, there are simulators which account for reactive transport and radioactive decay [*Yeh and Tripathi*, 1991; *Xu and Pruess*, 2001; *Xu et al.*, 2001], simulators which model coupled mechanical deformation and fluid flow [*Noorishad and Tsang*, 1996; *Rutqvist et al.*, 2002], and simulators which incorporate biological phenomena [*Rittmann and VanBriesen*, 1996]. Numerical simulators such as these represent a significant investment in time and resources. As such, they are valuable tools for tackling the forward problem of reservoir simulation.

[3] In solving the inverse problem of reservoir characterization we should take full advantage of the sophistication of current numerical simulators. That is, we should develop methods which operate directly on the output of a simulator run. This is already done in techniques that numerically difference simulator runs to calculate the components of the gradient vector for use in an iterative inversion [*Finsterle and Pruess*, 1995]. Also, a stochastic method, such as simulated annealing [*Datta-Gupta et al.*, 1995], operates directly on the output of a numerical simulator in order to update a prior model. Unfortunately, both of these approaches can require significant computation for large three-dimensional models. Alternative approaches, such as the adjoint-state and sensitivity equation methods are more efficient but require significant additional coding to form and solve the adjoint or coefficient equations [*Yeh*, 1986]. There is a need for inverse techniques that require the same level of computation as the solution of the forward problem and yet are easy to implement.

[4] In this paper we introduce a simple yet efficient approach based upon asymptotic solutions to the equations of transient flow and tracer transport. All quantities necessary for taking a step in our inversion algorithm, head and solute concentration histories, are produced by a conventional numerical simulator. In what follows we use the integral finite difference simulator TOUGH2 [*Pruess et al.*, 1999]. We merely post-process the results of a single simulation in order to take a step in our inversion algorithm. The asymptotic solutions are defined along trajectories through the model, much like optical or seismic rays. Thus the solution and model parameter sensitivities only depend on quantities defined along the trajectories. Because of this, the asymptotic approach scales quite well with problem size. The trajectories are the fundamental elements in our inversion technique, and defining them is an important aspect of our approach. The computer routine for accomplishing this is quite brief, involving of the order of one hundred lines of code. In the asymptotic methodology the data-fitting problem partitions into a kinematic, or travel time matching problem, and a dynamic, or amplitude-matching problem. This provides additional flexibility in the inversion. There is increasing recognition of the utility and advantages of arrival times [*Rubin and Dagan*, 1992; *Harvey and Gorelick*, 1995; *Woodbury and Rubin*, 2000; *Feyen et al.*, 2003]. For example, solute travel times are independent of the amount of tracer injected and less affected by pore-scale dispersivity [*Feyen et al.*, 2003]. Our experience indicates that travel time matching is quite robust and provides a prior model with which to fit the amplitudes [*Vasco and Datta-Gupta*, 1999; *Vasco et al.*, 2000].

[5] The asymptotic approach to inversion was introduced by *Vasco and Datta-Gupta* [1999] for tracer transport and extended to transient head by *Vasco et al.* [2000]. Asymptotic techniques have also proven useful in the inversion of two-phase flow data [*Vasco and Datta-Gupta*, 2001]. In those papers the trajectories required to construct an asymptotic solution were computed using either ray-tracing or streamline simulation. Here we describe how to compute the trajectories directly from the output of a numerical simulator. Note that, for transient flow, the trajectories can differ from streamlines [*Datta-Gupta and King*, 1995; *Crane and Blunt*, 1999]. In this paper we also extend the tracer transport results of *Vasco and Datta-Gupta* [1999] to include general dispersion and hydraulic conductivity tensors. We find that the trajectories for tracer transport depend on the flow field, similar to the situation for streamlines. In fact, the asymptotic methodology provides a mathematical framework for streamline modeling of tracer transport and multiphase flow. Utilzing this framework we can link streamline simulation to ray methods from geometrical optics and geophysics. Our treatment of transient head and tracer transport by no means exhausts the possible applications of asymptotic methods. Asymptotic and related techniques are useful for modeling a wide variety of phenomena such as spiral and toroidal scroll waves, multiple reaction fronts, solitons, fingering [*Grindrod*, 1996] and traveling wave convection [*Knobloch and De Luca*, 1990].