### abstract

- Copyright © 2018, Society of Petroleum Engineers. Double-porosity/naturally-fractured reservoir (NFRs) models have traditionally been used to represent the flow and pressure behavior for highly fractured carbonate reservoirs. Given that unconventional reservoirs such as shale oil/gas reservoirs may or may not be considered to be multi-porosity media, the use of the traditional/classical "double-porosity" models may not be adequate (or appropriate). The recent development of anomalous diffusion models has opened the possibility of adapting double-porosity models to estimate reservoir (and related) parameters for unconventional reservoirs. The primary objective of this work is to develop and demonstrate analytical reservoir models that provide (possible) physical explanations for the anomalous diffusion phenomenon. The models considering anomalous diffusion in reservoirs with Euclidean shape are developed using a convolved (i.e., time-dependent) version of Darcy's law. The use of these models can yield a power-law (straight-line) behavior for the pressure and/or rate performance — similar to the fractal reservoir models. The main advantage of using anomalous diffusion models compared to models considering fractal geometry is the reduction from two parameters (i.e., the Fractal Dimension and the Conductivity Index) to only one parameter (i.e., the anomalous diffusion exponent). However, the anomalous diffusion exponent does not provide information about the geometry or spatial distribution of the reservoir properties. To provide an alternative explanation for the anomalous diffusion phenomenon in petroleum reservoirs, we have developed double-porosity models considering matrix blocks with fractal geometry and fracture networks with either radial or fractal fracture networks. The flows inside the matrix blocks and the fractal fracture network assume that Darcy's Law is valid in its space-dependent (fractal) form, whereas the classical version of Darcy's Law is assumed for the radial fracture network case. The transient interporosity transfer is modeled using the classical convolution schemes given in the literature. For the system composed by a radial fracture network and fractal matrix blocks, we have investigated three cases by changing the producing conditions for the well and the flowing conditions for the matrix blocks: These three cases are: 1. A well producing at a constant-rate and closed matrix blocks, 2. A well producing at a constant-rate and "infinite-acting" matrix blocks, and 3. A well producing at a variable-rate (time-dependent inner boundary condition) and "infinite-acting" matrix blocks. We have defined the matrix blocks to be "infinite-acting" in order to represent the nano/micro permeability of shale reservoirs. For the system defined by a fractal fracture network and infinite-acting fractal matrix blocks, we have investigated the influence of the fractal parameters (both matrix and fracture network) in the pressure- and rate-transient performance behaviors. We have defined the periods of flow that can be observed in these sorts of systems and we have developed analytical solutions for pressure-transient analysis. We demonstrate that the use of the convolved version of Darcy's Law results in a model very similar to the diffusivity equation for double-porosity systems (which incorporates transient interporosity flow). In performing this work, we establish the following obervations/conclusions using our new solutions: 1. We have found that the assumption of a well producing at variable-rate (time-dependent inner boundary condition) has a more significant impact on the pressure (and derivative) functions and hinders the effects of the properties of the reservoir. 2. We have demonstrated that the anomalous diffusion phenomena in unconventional reservoirs can be related to their multi-porosity nature. 3. The pressure and pressure derivative responses may be used in the diagnosis of flow periods and the evaluation of reservoir parameters in unconventional reservoirs.