Rami Mustafa Younis
Published: 2011
Total Pages: 206
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As conventional hydrocarbon resources dwindle, and environmentally-driven markets start to form and mature, investments are expected to shift into the development of novel emerging subsurface process technologies. While these processes are characterized by a high commercial potential, they are also typically associated with high technical risk. The time-to-market along comparable development pipelines, such as for Enhanced Oil Recovery (EOR) methods in the Oil and Gas sector, is on the order of tens of years. It is anticipated that in the near future, there will be much value in developing simulation tools that can shorten time-to-market cycles, making investment shifts more attractive. There are two forces however that may debilitate us from delivering simulation as a scientific discovery tool. The first force is the growing nonlinearity of the problem base. The second force is the flip-side of a double edged sword; a rapidly evolving computer architecture scene. The first part of this work concerns the formulation and linearization of nonlinear simultaneous equations; the archetypal inflexible component of all large scale simulators. The proposed solution is an algorithmic framework and library of data-types called the Automatically Differentiable Expression Templates Library (ADETL). The ADETL provides generic representations of variables and discretized expressions on a simulation grid, and the data-types provide algorithms employed behind the scenes to automatically compute the sparse analytical Jacobian. Using the library, large-scale simulators can be developed rapidly by simply writing the residual equations, and without any hand differentiation, hand crafted performance tuning loops, or any other low-level constructs. A key challenge that is addressed is in enabling this level of abstraction and programming ease while making it easy to develop code that runs fast. Faster than any of several existing automatic differentiation packages, faster than any purely Object Oriented implementation, and at least in the order of the execution speed of code delivered by a development team with hand-optimized residuals, analytical derivatives, and Jacobian assembly routines. A second challenge is in providing a generic multi-layered software framework that incorporates plug-in low-level constructs tuned to emerging architectures. The inception of the ADETL spurred an effort to develop the new generation AD-GPRS simulator, which we use to demonstrate the powers of the ADETL. We conclude with a thought towards a future where simulators can write themselves. The second part of this work develops nonlinear methods that can exploit the nature of the underlying physics to deal with the current and upcoming challenges in physical nonlinearity. The Fully Implicit Method offers unconditional stability of the discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations that are solved at each timestep. Current reservoir simulators apply safe-guarded variants of Newton's method that can neither guarantee convergence, nor provide estimates of the relation between convergence rate and timestep size. In practice, timestep chops become necessary, and they are guided heuristically. With growing complexity, convergence difficulties can lead to substantial losses in computational effort and prohibitively small timesteps. We establish an alternate class of nonlinear iteration that converges and that associates a timestep to each iteration. Moreover, the linear solution process within each iteration is performed locally. Several challenging examples are presented, and the results demonstrate the robustness and computational efficiency of the proposed class of methods. We conclude with thoughts to unify timestepping and iterative nonlinear methods.
