Production analysis of oil production from unconventional reservoirs using bottom hole pressures entirely in the Laplace space

Laplace transforms are a powerful mathematical tool to solve many problems that describe fluid flow in unconventional reservoirs. However, for the solutions to be useful in applications, for instance history matching, they must be converted from the Laplace space into the real-time domain. A common practice is to numerically invert the transformed Laplace solution. However, we find substantial benefits if the data sets are handled entirely in the Laplace domain, and fitted to models presented in Laplace space rather than in the time domain. The data set used in this work is oil production rate and bottom hole pressure (BHP) from a liquid-rich shale play in North America, which we study to understand the decline of production from a tight formation produced by a fractured horizontal well. Since the BHP is relatively constant in the long run, a constant BHP solution is appropriate to analyze inflow performance analysis for most wells. However in some cases, as a result of operational changes to some wells, mainly periodic shut-ins, the production rate experiences isolated pressure build-ups. Both the production rate and BHP are transformed into the Laplace domain and accounted for in the model. Ours is the first analysis that combines rate and BHP entirely in the Laplace domain. There is no need for a Laplace transform inversion. Two models whose Laplace solutions are readily available are studied side-by-side, a single-compartment model versus a dual-compartment model. We fit the transformed production data of hundreds of wells to the Laplace models. The algorithm to transform data is fairly simple and computationally inexpensive. Since Laplace transformation smoothes the data, the fits are consistently good. Both models yield realistic and similar estimates of ultimate recovery. In most cases the effect of the second compartment in the dual-compartment model can be ignored, i.e., neglecting the fracture-well interaction. The single-compartment model seems adequate for modeling unconventional reservoirs performance. The knowledge of the reservoir model parameters provides estimation of the drainage volume and forecast future production. One of the main advantages of this novel history matching method is its ability to eliminating noise from data scatter without losing important information. As a result, we can match data more easily. Moreover, real-time solutions to many fluid flow problems in porous media often cannot be obtained analytically but rather via numerical computation. Our current method eliminates the need of inverting to real-time solutions. Additionally, these solutions often assume simple closed forms in Laplace domain even for very complex geometry (higher number of compartments), facilitating the task of history matching.