A study in fractional diffusion: Fractured rocks produced through horizontal wells with multiple, hydraulic fractures

Abstract

The spatiotemporal evolution of transients in fractured rocks often displays unusual characteristics and is traced to multifaceted origins such as micro-discontinuity in rock properties, rock fragmentation, long-range connectivity and complex flow paths. A physics-based model that incorporates transient propagation wherein the mean square displacement of the diffusion front follows a nonlinear scaling with time is proposed. This model is based on fractional diffusion. The motivation for fractional flux laws follows from the existence of long-range connectivity that results in the mean square displacement of fronts moving faster than predicted by classical models; correspondingly, obstructions and discontinuities, local flow reversals, intercalations, etc. produce the opposite effect with fronts moving at a slower rate than classical predictions. The interplay of these two competing behaviors is quantified. We simulate transient production in a porous rock at the Theis scale as a result of production through a horizontal well consisting of multiple hydraulic fractures. Asymptotic solutions are derived and computations verified. The practical potential of this model is described through an example and the movement of fronts under the constraints of this model is demonstrated through the new expressions developed in this work. We demonstrate that this model offers a potential avenue to explain other behaviors noted in the literature. Though this work is developed in the context of applications to the earth sciences (production of hydrocarbons, extraction of geothermal resources, sequestration of radioactive waste and other fluids, groundwater flow), a minimal change in the Nomenclature permits application to other contexts. Ideas proposed here are particularly useful in the context of superdiffusion in bounded systems which until now, in many ways, has been considered to be an open problem.

Space-time fractional diffusion: transient flow to a line source

“…One immediate advantage of this work is that it provides an understanding of long term behaviors of the flow of fluids produced through complex wellbores (fractured wells, horizontal wells), for pseudoradial flow will ultimately prevail in all situations. For example, if a well produces through a finite-conductivity fracture, then exponents of ðb þ 1 À aÞ=½2ðb þ 1Þ; ðb þ 1 À aÞ=ðb þ 1Þ and 1 À a corresponding to the bilinear, linear and pseudoradial flow-periods, respectively, should be evident on the derivative trace whenever all three flow regimes are evident; see Raghavan and Chen (2020). Responses to a line-source well are presented both analytically and graphically.…”

“…Considered individually, the exponent, a, represents the reciprocal of the anomalous diffusion coefficient (random walk dimension), d w , (Metzler et al, 1994) for diffusion on fractal objects, < r 2 >$ t 2=dw , and the exponent, b, represents the Hurst index, H, for Levy-stable diffusion; H is 1/(b + 1) or <r 2 > ~tH . Elaborations of these ideas with respect to transient, anomalous diffusion in porous solids are given in Raghavan and Chen (2020). For a = b = 1, the fractional equation reverts to the classical form with alternating sequences of fixed-velocity runs and reorientations.…”