Stochastic Modeling of Wormhole Growth in Carbonate Acidizing With Biased Randomness

Abstract

SPE Members Abstract "Wormholes", which are characteristic shapes resulting from the acidizing of carbonate formations have been considered in the past as fractals. In previous publications, analytical relationships of the area penetrated by wormholes, the wormhole porosity and the fractal dimension have been presented. Local mineral compositional heterogeneities and structures result in uneven reaction profiles when acid reacts with carbonate rocks. This, coupled with permeability heterogeneities, leads to microscopic flow instabilities which may evolve into macroscopic wormhole patterns. The understanding of the physics of acidizing is becoming a serious issue with the emergence of horizontal wells, where massive volumes of acid may be needed for effective stimulation. The stochastic nature of the wormholing process has been a limiting factor for a physical interpretation. The simulation of this unstable growth process is the purpose of this paper. The impacts of permeability anisotropy, heterogeneous distribution of the properties of the formation, such as microfractures and zones of different permeabilities, are investigated. A simulation model, the permeability driven fingering model (PDF) is presented. This technique is a new approach to diffusion limited growth, which traditionally has been simulated with diffusion limited aggregation models (DLA). The randomness of fractal growth is changed by introducing a bias representing the permeability anisotropy and the preferential reaction kinetics of lithologic heterogeneities. Introduction The formation of wormholes in carbonate acidizing is driven by a dynamic fluid instability which is conceptually similar to what is known as viscous finger instability. The physical difference of wormhole growth is that the instability is caused by a discontinuous jump in the permeability between the untreated matrix and the highly conductive paths of the wormhole network. In viscous fingering it is the difference between the viscosity of the displacing and displaced fluids which causes the growth of small perturbations to the two-fluid interface. In the case of the wormhole instability (later referred to as permeability fingering, in contrast to viscous fingering) the viscosity ratio between injection fluid and the reservoir fluids is close to one, especially if an acid pre-flush is considered. Therefore, viscosity will not be the primary driving force for the instability in the injection front which leads to wormhole growth. The striking similarity between how a viscosity ratio and permeability ratio cause a fingering pattern to develop has encouraged us to apply a similar model based on stochastic growth. In this paper we will describe the permeability driven fingering model (PDF) which is an extension of the dielectric breakdown model with tunable noise. P. 413^