Pressure-Transient Behavior of Double-Porosity Reservoirs with Transient Interporosity Transfer with Fractal Matrix Blocks

Abstract

Summary Double-porosity/naturally fractured reservoir 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 might not be considered to be multiporosity media, the use of the traditional/classical “double-porosity” models might 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 with 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 regarding 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. We have defined the matrix blocks to be “infinite-acting” to represent the nano/micropermeability of shale reservoir. 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 flow periods 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 observations/conclusions derived from our new solutions: We find that the assumption of a well producing at variable rate (time-dependent inner-boundary condition) has a more-significant effect on the pressure (and derivative) functions and obscures the effects of the properties of the reservoir. We demonstrate that the anomalous-diffusion-phenomena model proposed for unconventional reservoirs can be directly related to the multiporosity concept model. Pressure and pressure-derivative responses can be used in the diagnosis of flow periods and in the evaluation/estimation of reservoir parameters in unconventional reservoirs.