History matching and rate forecasting in unconventional oil reservoirs with 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 vs. time. This production signature (half-slope) is identical to that expected from a 1D 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. On the basis of 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 use of 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 (PDEs) for the double-porosity model into a system of ordinary-differential equations (ODEs). After which, we developed a function that gives the relationship between the average pressures in the highand the low-permeability regions. With this relationship, the system of ODEs was solved and used to obtain a rate/time function that one can use 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. After the model was validated and tested, we applied it to field-production data by partially history matching and forecasting the expected ultimate recovery (EUR). 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.