The primary goal of this work is to demonstrate that the fractal geometry can be used to accurately model the rate and pressure performance behavior of highly heterogeneous reservoirs such as shale oil/gas reservoirs. There are multi-scale heterogeneities that can hinder modelling and diagnostic analyses, and the use of large stimulation treatments (e.g., multi-fractured horizontal wells) can further complicate the modeling of heterogeneous reservoir systems. The primary objective of this work is to provide analytical reservoir models and diagnostic interpretation relations that can be used to estimate the well and reservoir parameters for the case of a hydraulic fracture intercepting a horizontal well within a fractal reservoir.

This work presents a semi-analytical solution for the pressure and rate transient behaviors of a horizontal well intercepting a single finite-conductivity fracture within a fractal reservoir considering either single or naturally-fractured/dual porosity reservoir conditions. The shape of the imposed hydraulic fracture can be either circular or rectangular. The media (the reservoir and the fracture) are "coupled" by discretizing the fracture, which defines a system of equations, the solution of which provides the pressure at any position inside the fracture. Naturally-fractured/dual porosity and anomalous diffusion effects are included by modifying the solution of the diffusivity equation for the reservoir in the Laplace domain.

The diagnostic signatures (i.e., the pressure derivative functions) for the proposed semi-analytical solution illustrate the following features:

  • Period 1 (Fracture flow): Radial or linear flow (depending on the geometry of the fracture) at very early times. As in classic studies for the case of a single finite-conductivity fracture, this period will never be observed in practice.

  • Period 2 (Fracture-reservoir interaction): "Radial-Fractal" or "Linear Fractal" at intermediate-transient times. This period can be subdivided into two sub-periods: (1) early-intermediate and (2) late-intermediate. The early-intermediate period is analogous to the bilinear flow regime for a finite-conductivity vertical fracture in an infinite-acting homogeneous reservoir, whereas the late-intermediate is analogous to the formation-linear flow regime observed at most times for a case with a very high conductivity vertical fracture and at later times for cases with a medium to high conductivity vertical fracture.

  • Period 3 (Reservoir dominated flow): "Pseudo-Fractal" flow. This flow period is dominated by the reservoir and yields power-law behavior (i.e., a straight line in the pressure drop and pressure derivative functions versus time on a log-log plot).

In this "mechanistic"-style of study, we found that the heterogeneities of the reservoir represented by the fractal parameters, and/or the characteristics of the fracture itself can distort the pressure response during early/very early-times. Specifically, this work shows that the interaction between the fracture and the fractal reservoir can exhibit power-law pressure drop and pressure drop derivative signatures which are different from the distinctive one-quarter slope expected for bilinear flow.

The following contributions are derived from this work:

  • Proposed solution for the pressure drop and pressure drop derivative behavior of a horizontal well intercepting a hydraulic fracture within a fractal reservoir that exhibits several "power-law" flow regimes.

  • Proposed features from our analytical solution for the characteristic early-time pressure drop and pressure drop derivative responses that may be of use in the diagnostic evaluation of fracturing treatments (e.g, the effectiveness of the fracture treatment in connection with existing/enhanced natural fractures).

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