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Reservoir Simulation and the Mathematics of Oil Recovery

The following is a brief and timely reflection from the author of Reservoir Simulation: Mathematical Techniques in Oil Recovery, which was published by SIAM in 2007 as part of the CBMS-NSF Regional Conference Series in Applied Mathematics. The text addresses classical reservoir engineering and basic reservoir simulation methods. It also presents an overview of various types of flows and includes a detailed glossary of petroleum engineering terms.

Reservoir Simulation: Mathematical Techniques in Oil Recovery evolved from a series of lectures that I delivered at the 2006 NSF-CBMS Regional Research Conferences in the Mathematical Sciences, where I spoke about multiphase flows in porous media and reservoir simulation. The text is appropriate for senior undergraduate students and first-year graduate students in geology, petroleum engineering, and applied mathematics. It can also serve as a reference book for geologists, petroleum engineers, applied mathematicians, and scientists in petroleum reservoirs. Employees in the petroleum industry can even use Reservoir Simulation as a handbook for modeling and simulation.

The book’s 10 chapters correspond to my 10 respective lectures at the NSF-CBMS conference. Chapter one offers an introduction to classical reservoir engineering and basic reservoir simulation. In chapter two, I review a glossary of terms that are common in the area of reservoir simulation. With the exception of chapter four, chapters three through nine present the governing differential equations and their numerical solutions for single-phase, two-phase, black oil (three-phase), single-phase with multi-components, compositional, and thermal flows. For each of these flows, the text (i) gives the basic flow and transport equations; (ii) states the corresponding rock and fluid properties; (iii) discusses peculiar features of the equations; (iv) diligently describes the procedure for obtaining their numerical solutions; and (v) addresses difficulties and practical issues in the solutions. Specifically, the book studies the treatments of rock, fluid, and rock/fluid properties at the internal boundaries of gridblocks in great depth.

Visual representation of the integration of reservoir modeling, big data, and virtual reality. Photo courtesy of Zhangxin Chen.
Chapter four describes the well constraints that contribute to the numerical simulations of each flow, and chapter ten summarizes several practices in reservoir simulation — including data gathering and analysis, selection of a simulation model, history matching, and reservoir performance prediction. This final chapter also provides examples for each numerical solution that carry out the discretization procedure in detail for the finite difference method; other work presents the finite volume and finite element methods in a similar fashion [1].

Previous researchers have widely used mathematical models to predict, understand, and optimize complex physical processes for the modeling and simulation of multiphase fluid flows in petroleum reservoirs. These models are important when one seeks to understand the fate and transport of chemical species and heat. Armed with this knowledge, practitioners in the petroleum industry then apply the models in order to design enhanced oil recovery strategies.

Reservoir Simulation focuses on the modeling and simulation of multiphase flow in porous media, the derivation of mathematical models, the use of numerical methods to solve these models, and applications to petroleum recovery. One can even extend the presented mathematical techniques for conventional oil and gas reservoirs to unconventional oil and gas reservoirs, such as tight and shale oil and gas, coalbed methane, and gas hydrate.

Reservoir simulation has become a standard predictive tool in the petroleum industry, as it can obtain accurate performance predictions for hydrocarbon reservoirs under different conditions. A single hydrocarbon recovery project involves a capital investment of hundreds of millions of dollars, and companies must assess and minimize the risk that is associated with its selected development and production strategies. This risk includes important factors — like the complexity of a petroleum reservoir and the fluids that fill it, an understanding of hydrocarbon recovery mechanisms, and the applicability of selected strategies. Researchers can account for these important factors in reservoir simulation by inputting data into a simulation model.

Determining the simulation process involves three major interrelated stages. First, one must develop a set of mathematical model equations that incorporate the necessary physics and chemistry to describe the essential features of flow and transport processes in petroleum reservoirs. Second, well-posed equations with clearly expressed solutions enable the establishment of robust numerical models. These models must be stable, accurate, and able to reliably generate solutions that demonstrate basic physical and chemical features without introducing spurious phenomena. Third, researchers must design computer algorithms and codes to efficiently solve the large-scale systems of algebraic equations that arise from the numerical models. As consumers around the world look to greener, cleaner, and more efficient sources of fuel in response to climate change, reservoir simulation will play an increasingly important role in the petroleum industry.

My hope is that the readers of Reservoir Simulation: Mathematical Techniques in Oil Recovery will join me in exploring and understanding the mathematical and computational foundations that comprise reservoir simulation.

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References
[1] Chen, Z., Huan, G., & Ma, Y. (2006). Computational methods for multiphase flows in porous media. In Computational science and engineering (Vol. 2). Philadelphia, PA: Society for Industrial and Applied Mathematics.

Zhangxing Chen holds two research chairs at the University of Calgary: Alberta Innovates Industrial Chair in Reservoir Modeling and NSERC/Energi Simulation Senior Industrial Research Chair in Reservoir Simulation. He is a Fellow of the Royal Society of Canada, the Canadian Academy of Engineering, and the Engineering Institute of Canada, as well as an Academician of the Chinese Academy of Engineering. Chen has received numerous awards, including the Killam Professor Award, CAIMS-Fields Industrial Mathematics Prize, and Gerald J. Ford Research Fellowship.