Determining optimal locations and operation parameters for wells in oil and gas reservoirs has a potentially high economic impact. Finding these optima depends on a complex combination of geological, petrophysical, flow regimen, and economical parameters that are hard to grasp intuitively. On the other hand, automatic approaches have in the past been hampered by the overwhelming computational cost of running thousands of potential cases using reservoir simulators, given that each of these runs can take on the order of hours. Therefore, the key issue to such automatic optimization is the development of algorithms that find good solutions with a minimum number of function evaluations. In this work, we compare and analyze the efficiency, effectiveness, and reliability of several optimization algorithms for the well placement problem. In particular, we consider the simultaneous perturbation stochastic approximation (SPSA), finite difference gradient (FDG), and very fast simulated annealing (VFSA) algorithms. None of these algorithms guarantees to find the optimal solution, but we show that both SPSA and VFSA are very efficient in finding nearly optimal solutions with a high probability. We illustrate this with a set of numerical experiments based on real data for single and multiple well placement problems.

fax 01-972-952-9435. AbstractA primary challenge for a new generation of reservoir simulators is the accurate description of multiphase flow in highly heterogeneous media and very complex geometries. However, many initiatives in this direction have encountered difficulties in that current solver technology is still insufficient to account for the increasing complexity of coupled linear systems arising in fully implicit formulations. In this respect, a few works have made particular progress in partially exploiting the physics of the problem in the form of two-stage preconditioners.

a b s t r a c tThe aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments. The governing equations are based on Darcy's law with stochastic permeability. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product Hermite polynomials is used in the stochastic dimensions. Error analysis is performed and experimentally verified with numerical simulations. Computational results include incompressible and slightly compressible single and two-phase flow.

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The solution of the linear system of equations for a large scale reservoir simulation has several challenges. Preconditioners are used to speed up the convergence rate of the solution of such systems. In theory, a preconditioner defines a matrix M that can be inexpensively inverted and represents a good approximation of a given matrix A. In this work, two-stage preconditioners consisting of the approximated inverses M 1 and M 2 are investigated for multiphase flow in porous media. The first-stage preconditioner, M 1 , is approximated from A using four different solution methods: (1) constrained pressure residuals (CPR), (2) lower block Gauss-Seidel, (3) upper block Gauss-Seidel, and (4) one iteration of block Gauss-Seidel. The pressure block solution in each of these different schemes is calculated using the Algebraic Multi Grid (AMG) method. The inverse of the saturation (or more generally, the nonpressure) blocks are approximated using Line Successive Over Relaxation (LSOR). The second stage preconditioner, M 2 , is a global preconditioner based on LSOR iterations for the matrix A, that captures part of the original interaction of different coefficient blocks. Several techniques are also employed to weaken the coupling between the pressure block and the nonpressure blocks. Effective decoupling is achieved by: (1) an Implicit Pressure Explicit Saturation (IMPES) like approach designed to preserve the integrity of pressure coefficients, (2) Householder transformations, (3) the alternate block factorization (ABF), and (4) the BDS based on least squares. The fourth method is a new technique developed in this work. The aforementioned preconditioning techniques were implemented in a parallel reservoir simulation environment, and tested for large-scale two-phase and three-phase black oil simulation models. This study demonstrates that a two-stage preconditioner based on BDS or ABF combined with Gauss-Seidel sweeps, that also incorporate nonpressure solutions for M, delivers both the fastest convergence rate and the most robust option overall without compromising parallel scalability.

Abstract. The adequate location of wells in oil and environmental applications has a significant economic impact on reservoir management. However, the determination of optimal well locations is both challenging and computationally expensive. The overall goal of this research is to use the emerging Grid infrastructure to realize an autonomic self-optimizing reservoir framework. In this paper, we present a policy-driven peer-to-peer Grid middleware substrate to enable the use of the Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm, coupled with the Integrated Parallel Accurate Reservoir Simulator (IPARS) and an economic model to find the optimal solution for the well placement problem.

The forthcoming generation of many-core architectures suggests a strong paradigm shift in the way algorithms have been designed to achieve maximum performance in reservoir simulations. In this work, we propose a novel poly-algorithmic solver approach to develop hybrid CPU multicore and GPU computations for solving large sparse linear systems arising in realistic black oil and compositional flow scenarios. The GPU implementation exploits data parallelism through the simultaneous deployment of thousands of threads while reducing memory overhead per floating point operations involved in most BLAS kernels and in a suite of preconditioner options such as BILU(k), BILUT and multicoloring SSOR. On the other hand, multicore CPU computations are used to exploit functional parallelism to perform system partitioning and reordering, algebraic multigrid preconditioning, sparsification and model reduction operations in order to accelerate and reduce the number of GCR iterations. The efficient orchestration of these operations relies on carefully designing the heuristics depending on the timestep evolution, degree of nonlinearity and current system properties. Hence, we also propose several criteria to automatically decide the type of solver configuration to be employed at every time step of the simulation. To illustrate the potentials of the proposed solver approach, we perform numerical computations on state-of-the-art multicore CPU and GPU platforms. Computational experiments on a wide range of highly complex reservoir cases reveal that the solver approach yields significant speedups with respect to conventional CPU multicore solver implementations. The solver performance gain is of the order of 3x which impacts in about 2x the overall compositional simulation turnaround time. These results demonstrate the potential that many core solvers have to offer in improving the performance of near future reservoir simulations.

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