A Note On Pressure-Transient Analysis Of Fractal Reservoirs

Abstract

Abstract In a recent paper1 we introduced a novel approach for the identification and description of a fractal reservoir using concepts from fractal geometry. However, the analysis presented in Ref. 1 was unduly restrictive and suffers from two drawbacks: First, it is based on the use of a generalized diffusivity equation, the validity of which has been the subject of recent debate.2 In that formulation, the diffusivity was taken to be position-dependent. This reflects the long memory associated with random walks (diffusion) on fractal objects. Alternative approaches in anamalous diffusion have also been used, e.g. with time-dependent diffusivity, or with the use of an integro-differential equation (see Ref. 3 and references therein). Second, as in all formalisms, the underlying assumption is that of a single source (well) in an unbounded medium. As a result, the equation cannot be directly extended to many wells and variable rates as is the case with normal diffusion. Although generalized diffusivity equations have also been used in other fields involving fractals4 it appears that alternative approaches based on Green's functions are more appropriate. It is the purpose of this note to provide such an alternative and to clarify certain issues related to the description of single-phase, slightly compressible fluid flow in fractal reservoirs. We emphasize from the outset that as in Ref. 1, finite effects are not considered in this note. This presupposes a reservoir which is fractal over a wide range of scales, such that upper and lower cutoffs are greatly separated. The implication is that effects of boundaries are absent. Our recent numerical results5 have demonstrated that finite size, namely boundary, effects affect significantly the late-time behavior and must be carefully differentiated from the purely fractal response. Nonetheless, developing the proper formalism for an unbounded (theoretical) fractal is still desirable to serve as a guide for the more realistic finite size cases. In fact, it is only outside of boundary effects that a fractal response can be clearly identified. In theory, fractals obey power-law scaling around an arbitrary point. In particular, properties, such as permeability, are scale-dependent with the same exponent at any location. Consider then the pressure response in such a fractal reservoir at location x due to n wells located at x_i(i =1, 2, …, n), and producing at rate qi(t) (where qi may also be negative). The solution of this problem can be obtained in terms of the free space Green's function g(ξ_−ξ_′, t−t′), which represents the response at point ξ_ and time t due to an instantaneous point source at point ξ_′ and time t′. Because of the linearity of the process, a simple superposition gives the result