A major assumption incorporated into conventional pressure-transient models is that the conductive and storage properties of a formation are uniform in the unit of interest, i.e., the properties are invariant in space. Moreover, the majority of the existing pressure transient models assume Newtonian fluid flow in reservoirs. Although these assumptions have been shown to be viable in many situations, the geological complexity of some units and/or the non-Newtonian characteristics of certain fluids may make these assumptions of dubious value. The purpose of this study is to explore methods for the analysis of non-Newtonian fluid flow data in a class of geological settings that are not amenable to 'he conventional approach.
In this study, we present new analytical solutions for transient pressure behaviour during the radial flow of non-Newtonian power-law fluids in fractal reservoirs. The results are presented as pressure, pressure-derivative and pressure-derivative-to pressure- ratio solutions. The analytical solutions are compared with a more rigorous numerical solution. We consider both finite and line-source wells. It is shown that the present model reduces to the conventional (i.e.. for Newtonian fluids and homogeneous reservoirs) pressure-transient models as limiting cases. The results obtained in this study extend previous pressure-transient methods for single-phase Newtonian flow in 2-D cylndrical, single-porosity systems to reservoirs of arbitrary fractal dimensions and fluids of arbitrary flow behaviour indices.
In a few recent theoretical studies, several authors(1–6) have proposed the concept of fractal geometry for pressure transient modelling of Newtonian flow in various naturally fractured reservoirs. Chang and Vortsos(1) presented a mathematical formulation for a fractal fracture network embedded into a Euclidean matrix. The system pressure transient response was analyzed for flow only in the fracture network and for flow in [raclure with matrix participation. Beier(2) presented an extension of the work by Chang and Vortsos(1). Beier's model applies specifically to flow in a cylindrical symmetry reservoir (d = 2) containing a fractal permeable network, Beier expressed the pressure transient equations for a fradal reservoir in a form that required available estimates of "near wellbore porosity and permeabilily". Beier(2) also presented some field data from the Grayburg and San Andres formations in southeastern New Mexico that was quantitatively analyzed with his fractal reservoir model. In a subsequent study, Beier(3) presented a model to analyze the pressure response of a well with a vertical fracture in an infinite Iractal reservoir. The theoretical model of Chang and Yortsos(1) was used by Acuna et al.(4) to interpret the fractal characteristics of a naturally Iraclured geothermal field. In a more recent study, Acuna et al.(5) reviewed the theoretical background of fractal analysis. They also demonstrated the application of various diagnostic techniques for fractal pressure transient analysis as developed by Chang and Yortsos(1).
When a flow medium (e.g., a fracture network) is highly disordered and fractal, its geometric and transport properties differ in a non-trivial way from those for the corresponding Euclidean flow media(1,4,5). The theoretical, ideal response of perfect fractal objects are described as follows: