Introduction

Diffusive processes are often assumed to have very little in common with solutions of the hyperbolic partial differential equations governing wave-like propagation. Indeed, solutions of the diffusion equation decay in an exponential fashion with distance from the source of a disturbance. Thus, to propagate an observable disturbance away from a source it is often necessary to continuously supply mass or energy. For example, in order to propagate an observable pressure disturbance from an injection point to a distant location it is typically necessary to introduce fluid at a constant rate into the porous medium for a sustained period of time. However, as indicated in Chapter 1, transient solutions to the diffusion equation do indeed propagate, and, as will be shown in this chapter, one may derive an expression governing the phase of such a propagating disturbance. The governing equation for the phase is a Hamilton–Jacobi equation, identical in form to the eikonal equation encountered earlier. However, the interpretation and the effect of the phase function is significantly different from that of a propagating wave, in keeping with the very different nature of diffusion. In this chapter we shall consider one approach for developing a trajectory-based solution for diffusion equation, a high-frequency asymptotic expansion. We shall illustrate the approach and its utility with a few applications, including diffusion tomography.

A high-frequency asymptotic solution

Here we derive a high-frequency asymptotic solution for the diffusion equation. The phrase ‘high-frequency’ is not commonly associated with diffusive processes. However, a high-frequency component is present in the early rapid variations of pressure due to the onset of fluid injection. Also, the notion of what constitutes a high frequency is relative to the temporal variation of the background field, which can be gradual. Admittedly, the approximation does not have the long reach associated with processes such as elastic wave propagation and is often restricted to a region surrounding the source. Still, the range over which we can utilize a transient pressure disturbance is far enough to provide a powerful tool for imaging flow properties between a given source and receiver. Furthermore, our asymptotic analysis allows us to draw a connection between the well developed modeling of wave propagation and transient solutions to the diffusion equation. Such a connection is useful in understanding coupled processes and equations of a mixed nature: those that are neither entirely wave-like nor completely diffusive in character.