Diffusive Mass Transfer and Gaussian Pressure Transient Solutions for Porous Media

Ruud Weijermars

doi:10.3390/fluids6110379

Abstract

This study revisits the mathematical equations for diffusive mass transport in 1D, 2D and 3D space and highlights a widespread misconception about the meaning of the regular and cumulative probability of random-walk solutions for diffusive mass transport. Next, the regular probability solution for molecular diffusion is applied to pressure diffusion in porous media. The pressure drop (by fluid extraction) or increase (by fluid injection) due to the production system may start with a simple pressure step function. The pressure perturbation imposed by the step function (representing the engineering intervention) will instantaneously diffuse into the reservoir at a rate that is controlled by the hydraulic diffusivity. Traditionally, the advance of the pressure transient in porous media such as geological reservoirs is modeled by two distinct approaches: (1) scalar equations for well performance testing that do not attempt to solve for the spatial change or the position of the pressure transient without reference to a well rate; (2) advanced reservoir models based on numerical solution methods. The Gaussian pressure transient solution method presented in this study can compute the spatial pressure depletion in the reservoir at arbitrary times and is based on analytical expressions that give spatial resolution without gridding-meaning solutions that have infinite resolution. The Gaussian solution is efficient for quantifying the advance of the pressure transient and associated pressure depletion around single wells, multiple wells and hydraulic fractures. This work lays the basis for the development of advanced reservoir simulations based on the superposition of analytical pressure transient solutions.

Highlights

A key step made here is that we show how the diffusion equation can be adapted for the rapid generation of transient pressure profiles, due to certain engineering interventions in porous media that begin with the introduction of instantaneous pressure drops
The present study argues that the advance of pressure transients, the gradient of which drives transport that has been traditionally labeled as advection, can be described by Gaussian solutions, similar to Fourier descriptions of molecular random walk diffusion with net mass transport
Generic solutions are given for the Gaussian probability of diffusive mass transfer due to a molecular random walk, taking into account the specific geometric position of the source in unbound 1D, 2D and 3D space

Summary

Introduction

This article shows how the advection and diffusion terms in the equation for convective mass transport by Fokker–Planck can be solved for use in certain types of pressuredriven flows, typically caused by engineering intervention in geological reservoirs. The result is an expression with exponential terms for the probability position of particles transported by the convective process, both for advection driven by migrating pressure gradients and for the random walk diffusion. Before setting out to derive the computational solutions, it may be useful to describe some typical examples of situations where advection and diffusive mass transport may occur jointly, as summarized in the principle sketches of Figure 1. The examples of mass transport systems are ranked from relatively simple (mass transport only) to more complex cases (mass and heat transport) based on the number of scaling parameters involved

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Discussion

Conclusion