Bivariate Log-Normal Distribution of Stimulated Matrix Permeability and Block Size in Fractured Reservoirs: Proposing New Multilinear-Flow Regime for Transient-State Production
Authors: Al-Rbeawi Salam 
Published: June 2018

Summary
This paper investigates the impacts of varied stimulated matrix permeability and matrix-block size on pressure behaviors and flow regimes of hydraulically fractured reservoirs using bivariate log-normal distribution. The main objective is assembling the variance in these two parameters to the analytical models of pressure and pressure derivative considering different porous-media petrophysical properties, reservoir configurations, and hydraulic-fracture (HF) characteristics. The motivation is eliminating the long-run discretization treatment in the porous media required by applying analytical models to describe the variance in the previously discussed parameters with the distance between HFs.
Several analytical models for pressure response were generated in this study for hydraulically fractured reservoirs with rectangular-shaped drainage areas. These models take into account the change in stimulated matrix permeability from the maximum value close to the HF face to the minimum value at the so-called no-flow boundary between fractures. They also consider the change in the matrix-block size, corresponding to the change in the induced-fracture density (number of fractures per foot of length), from the minimum value close to the fracture face to the maximum value at the no-flow boundary. Bivariate log-normal distribution was used to describe the change in the stimulated matrix permeability and matrix-block size. The formations of interest are composed of stimulated reservoir volume (SRV), where the matrix is stimulated by the fracturing process, and unstimulated reservoir volume (USRV), where the stimulation process does not affect the matrix.
The outcomes of this study can be summarized as
Generating new analytical models for pressure and pressure derivative in hydraulically fractured reservoirs that consider the change in stimulated matrix-block size and permeability using bivariate log-normal distribution Understanding the effect of using the probability-density function (PDF) of matrix-block size and permeability in the pressure distribution of different reservoirs Observing the new multilinear-flow regime that develops at intermediate production time and represents several simultaneous linear-flow regimes inside HFs, SRV, and USRV Developing analytical models for the new multilinear-flow regime Studying the effects of petrophysical properties of HFs, induced fractures, and matrix as well as reservoir size and configuration on pressure behavior
The most interesting points in this study are
The applicability of bivariate log-normal distribution for describing the variance and nonuniform distribution of matrix-block size and permeability. The large variance in the matrix-block size and permeability causes significant decrease in wellbore-pressure drop. Small value of standard deviation of matrix-block size and permeability indicates the possibilities for significant decrease in wellbore pressure drop. The means of matrix-block size and permeability may not have significant effects on reservoir-pressure distribution. The new multilinear-flow regime is characterized by a one-eighth slope on the pressure-derivative curve and is seen always after HF linear flow and before boundary-dominated flow regime. Multilinear-flow regime develops to bilinear-flow regime with a one-quarter slope for uniform distribution of equal matrix-block size and permeability.