## 1. Introduction

[2] Nonisothermal flow is a common occurrence in actual field settings. That is, there are variations in the temperature of fluids in the subsurface, for example due to the Earth's thermal gradient or due to seasonal effects [*Grifoll et al.*, 2005]. These temperature variations are usually small enough that their effect on fluid properties can be safely ignored. In certain activities, such as geothermal production [*Brownell et al.*, 1977; *Mercer and Faust*, 1979] or high-temperature water flooding [*Boberg*, 1988], the differences in fluid temperature are so large that they must be accommodated when modeling such flow.

[3] Fluid flow under nonisothermal conditions is typically more complicated than is isothermal fluid flow [*Brownell et al.*, 1977; *Bear and Corapcioglu*, 1981; *Noorishad et al.*, 1984; *McTigue*, 1986]. Nonisothermal flow, even under the simplest conditions in which matrix deformation is neglected, is governed by two coupled equations. Furthermore, two forms of nonlinearity are present in the governing equations. First, there are nonlinearities due to the temperature dependence of the coefficients in the governing equations. Second, there are explicit nonlinear terms in the mass and energy equations. These nonlinearities take the form of the scalar products of the pressure and temperature gradients. Due to the complexities of such coupled nonlinear equations with spatially varying parameters, most investigators have turned to numerical methods in order to model nonisothermal flow [*Noorishad et al.*, 1984]. Previous analytic or semianalytic studies have been limited to rather restrictive situations, such as a homogeneous [*Booker and Savvidou*, 1984] or one-dimensional media [*O'Sullivan*, 1981; *Natale and Salusti*, 1996; *Doughty and Pruess*, 1992], or linearized versions of the governing equations [*McTigue*, 1986].

[4] In this paper I present a semianalytic approach for modeling nonisothermal flow in a heterogeneous medium. The approach is based upon an asymptotic technique that has proven useful in modeling coupled linear processes [*Vasco*, 2009], as well as nonlinear processes such as two-phase flow [*Vasco*, 2004]. The asymptotic procedure assumes that the length scale of the heterogeneity is greater than the length scale of the propagating pressure and temperature disturbances. Here, I extend this approach to coupled, nonlinear processes in a three-dimensional heterogeneous medium. Away from layer boundaries and faults, the heterogeneity is assumed to vary in a smooth fashion but with an arbitrary magnitude. Specifically, the length scale of the heterogeneity is assumed to be longer than the length scale of the propagating pressure and temperature disturbance. One use of such modeling is for the efficient inversion of flow and temperature data. The techniques developed here are intended to be used in conjunction with a numerical simulator, both to aid in the interpretation of results and as a method for the inverse modeling of observations. For example, the asymptotic formulation provides explicit semianalytic expressions for model parameter sensitivities [*Vasco et al.*, 2000; *Vasco*, 2008]. Though the specific formulation of *Noorishad et al.* [1984], modified along the lines of the passive reservoir model of *Brownell et al.* [1977], is used here, and the fluid is assumed to be incompressible, the approach is general and applicable to more comprehensive formulations. That is, the techniques can be applied to general systems of nonlinear, as well as linear, governing equations.