History Matching and Rate Forecasting in Unconventional Oil Reservoirs Using an Approximate Analytical Solution to the Double Porosity Model

Production data from most fractured-horizontal wells in gas and liquid-rich unconventional reservoirs plot as straight lines with a one half slope on a log-log plot of rate versus time. This production signature (half slope) is identical to that expected from a one-dimensional linear flow from reservoir matrix to the fracture face, when production occurs at constant-bottomhole pressure. In addition, microseismic data obtained around these fractured wells suggest that an area of enhanced permeability is developed around the horizontal well, and outside this region is an undisturbed part of the reservoir with low permeability. Based on these observations geoscientists have, in general, adopted the conceptual double-porosity model in modeling production from fractured horizontal wells in unconventional reservoirs. The analytical solution to this mathematical model exists in Laplace space but it cannot be inverted back to real-time space without using a numerical inversion algorithm. We present a new approximate analytical solution to the double-porosity model in real-time space and its use in modeling and forecasting production from unconventional-oil reservoirs. The first step in developing the approximate solution was to convert the systems of partial differential equations for the dual-porosity model into a system of ordinary-differential equations. After which we developed a function that gives the relationship between the average pressures in the high-and the low-permeability regions. Using this relationship, the system of ordinary differential equations was solved and used to obtain a rate/time function that can be used to predict oil production from unconventional reservoirs. The approximate solution was validated with numerical reservoir simulation. We then performed a sensitivity analysis on the model parameters to understand how the model behaves. Once the model was validated and tested, we applied it to field production data by partially history matching and forecasting the expected ultimate recovery. The rate/time function fits production data and also yields realistic estimates of ultimate oil recovery. We also investigated the existence of any correlation between the model-derived parameters and available reservoir and well completion parameters. Introduction Many studies have been published that focus on the solution of the double porosity model for flow in hydraulically fractured horizontal wells. Barenblatt and Zheltov (1960) presented the first formulation of the double-porosity model. Warren and Root (1962) presented the first application of the double-porosity model to flow problems in the petroleum industry. Since then many authors (de Swaan, 1976; Mayerhofer, 2006; Carlson and Mercer, 1989; El-Banbi, 1998; Ozkan et al., 1987) have presented applications of the model. All the analytical solutions presented have all been in Laplace space and have had to be numerically transformed to real time space using some form of inversion algorithm of which the Stehfest algorithm (Stehfest 1970) is the most popular. More recently, Bello and Wattenbarger (2008) presented the solution to the double porosity model for linear flow in which they were able to obtain closed form analytical solutions for certain ranges of time. To do this they broke their Laplace space solution in to smaller bits using special properties of the solution which they could invert to real-time space. This piece-wise solution would have to be applied sequentially. Samandarli et al. (2011) presented the application of this solution to history matching and forecasting the performance of shale gas wells. Song (2014) presented a finite-difference solution to this problem and its application to oil production from hydraulically fractured wells. In this study, we present an approximate analytical solution to the double-porosity model in real-time space that is valid across all time domains, that is, it is a continuous function that is valid during the transient and late time flow from the fracture and matrix. We validate our solution against numerical simulation and also show that our solution reproduces the production behavior obtained from the inverted Laplace space solution. We also present example applications of our solution to field data.