Abstract

Foam has been widely used as a mobility control agent for Improved Oil Recover (IOR), gas blocking and acid diversion during matrix stimulation. The prediction of foam performance relies on macroscopic modeling. Foam modeling approaches include fractional flow theories and population balance models.

Traditionally, fractional foam models assume implicitly that foam is incompressible and do not account directly for the evolution of bubble population. The population balance models, instead, rely on the idea that foam mobility depends on bubble density and are more comprehensive. Yet, population balance models did not gain full acceptance thus far, because of their perceived complexity, with parameters that are hard to obtain experimentally.

This paper presents an improved foam model based on a simpler but realistic foam rheology and stochastic bubble generation ideas. Physical ideas in agreement with pictures emerging from recent foam studies using X-ray computed tomography form the basis for the new model. First, we provide the conservation equations for foam motion in porous media. Then we present their analytical treatment considering several cases that are likely to exist in the laboratory and in the field. We present an analysis of quasi-incompressible foam, reconciling for the first time the population balance and fractional flow ideas. We demonstrate why fingering is likely to occur during liquid injection following foam. Then we provide a solution for the coupling of liquid drainage through foam and viscous fingering.

Introduction

Foam motion in granular porous media is a phenomenon of common experience with applications in various fields, including oil and gas recovery,[1–6] environmental remediation[1,4–6] and water purification.[1] The description of foam behavior in porous media relies on phenomenological modeling. The available foam models aim to capture the drastic lowering of gas mobility associated with foam development. Nevertheless, they differ on their approaches to accomplish this task. With some simplification, current foam models can be grouped in (semi-) empirical,[7] fractional flow,[8,9] population balance[10–13] and percolation or network[14–16] approaches. The emphasis of this work is on fractional flow and population balance methods.

The modeling of foam using fractional flow ideas was advocated by Rossen and co-workers.[2,8,9] These authors identified foam states in so-called time-distance diagrams computed from core flow experimental data. They argued that the computation is simplified when done near the critical capillary pressure.[17] Foam fractional flow theory is based on the assumption that foam is incompressible and, therefore, is valid for cases where pressure variations remain small compared with the reference pressure. However, current fractional flow models do not account for the evolution of bubble population explicitly and therefore may lack accuracy when tackling transient foam motion.

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