Straubhaar, Julien

2014-7, Lochbühler, Tobias, Pirot, Guillaume, Straubhaar, Julien, Linde, Niklas

Geophysical tomography captures the spatial distribution of the underlying geophysical property at a relatively high resolution, but the tomographic images tend to be blurred representations of reality and generally fail to reproduce sharp interfaces. Such models may cause significant bias when taken as a basis for predictive flow and transport modeling and are unsuitable for uncertainty assessment. We present a methodology in which tomograms are used to condition multiple-point statistics (MPS) simulations. A large set of geologically reasonable facies realizations and their corresponding synthetically calculated cross-hole radar tomograms are used as a training image. The training image is scanned with a direct sampling algorithm for patterns in the conditioning tomogram, while accounting for the spatially varying resolution of the tomograms. In a post-processing step, only those conditional simulations that predicted the radar traveltimes within the expected data error levels are accepted. The methodology is demonstrated on a two-facies example featuring channels and an aquifer analog of alluvial sedimentary structures with five facies. For both cases, MPS simulations exhibit the sharp interfaces and the geological patterns found in the training image. Compared to unconditioned MPS simulations, the uncertainty in transport predictions is markedly decreased for simulations conditioned to tomograms. As an improvement to other approaches relying on classical smoothness-constrained geophysical tomography, the proposed method allows for: (1) reproduction of sharp interfaces, (2) incorporation of realistic geological constraints and (3) generation of multiple realizations that enables uncertainty assessment.

2007-1, Straubhaar, Julien

This paper is devoted to the study of some preconditioners for the conjugate gradient algorithm used to solve large sparse linear and symmetric positive definite systems. The construction of a preconditioner based on the Gram-Schmidt orthogonalization process and the least squares method is presented. Some results on the condition number of the preconditioned system are provided. Finally, numerical comparisons are given for different preconditioners.

2014-2, Straubhaar, Julien, Malinverni, Duccio

Multiple-point statistics (MPS) allows simulations reproducing structures of a conceptual model given by a training image (TI) to be generated within a stochastic framework. In classical implementations, fixed search templates are used to retrieve the patterns from the TI. A multiple grid approach allows the large-scale structures present in the TI to be captured, while keeping the search template small. The technique consists in decomposing the simulation grid into several grid levels: One grid level is composed of each second node of the grid level one rank finer. Then each grid level is successively simulated by using the corresponding rescaled search template from the coarse level to the fine level (the simulation grid itself). For a conditional simulation, a basic method (as in snesim) to honor the hard data consists in assigning the data to the closest nodes of the current grid level before simulating it. In this paper, another method (implemented in impala) that consists in assigning the hard data to the closest nodes of the simulation grid (fine level), and then in spreading them up to the coarse grid by using simulations based on the MPS inferred from the TI is presented in detail. We study the effect of conditioning and show that the first method leads to systematic biases depending on the location of the conditioning data relative to the grid levels, whereas the second method allows for properly dealing with conditional simulations and a multiple grid approach.

2008-10, Straubhaar, Julien

This paper is devoted to the study of some preconditioned conjugate gradient algorithms on parallel computers. The considered preconditioners (presented in [J. Straubhaar, Preconditioners; for the conjugate gradient algorithm using Gram-Schmidt and least squares methods, Int. J. Comput. Math. 84 (1) (2007) 89-108]) are based on incomplete Gram-Schmidt orthogonalization and least squares methods. The construction of the preconditioner and the resolution are treated separately. Numerical tests are performed and speed-up curves are presented in order to evaluate the performance of the algorithms.