Hussein Mustapha

A New Approach to Simulating Flow in Discrete Fracture Networks with an Optimized Mesh

Natural fractured media are highly unpredictable because of existing complex structures at the fracture and at the network levels. Fractures are by themselves heterogeneous objects of broadly distributed sizes, shapes, and orientations that are interconnected in large correlated networks. With little field data and evidence, numerical modeling can provide important information on the underground hydraulic phenomena. However, it must overcome several barriers. First, the complex network structure produces a structure difficult to mesh. Second, the absence of a priori homogenization scale, along with the double fracture and network heterogeneity levels, requires the calculation of large but finely resolved fracture networks resulting in very large simulation domains. To tackle these two related issues, we reduce the highly complex geometry of the fractures by applying a local transformation that suppresses the cumbersome meshing configurations while keeping the networks fundamental, geological, and geometrical characteristics. We show that the flow properties are marginally affected while the problem complexity (i.e., memory capacity and resolution time) can be divided by orders of magnitude. The goal of this article is to propose a method of resolution which takes into account the geometrical complexity met in the networks and which makes it possible to treat a few thousand fractures. The principal aim of this article is to present a tool to slowly modify the structures of the fracture networks to have a good quality mesh with a marginal loss in precision.

A new approach for geological pattern recognition using high-order spatial cumulants

Spatially distributed natural phenomena represent complex non-linear and non-Gaussian systems. Currently, their spatial distributions are typically studied using second-order spatial statistical models, which are limiting considering the spatial complexity of natural phenomena such as geological applications. High-order geostatistics is a new area of research based on higher-order spatial connectivity measures, especially spatial cumulants as suitable for non-Gaussian and non-linear phenomena. This paper presents HOSC or High-order spatial cumulants, an algorithm for calculating spatial cumulants, including anisotropic experimental cumulants based on spatial templates. High-order cumulants are calculated on two- and three-dimensional synthetic training images so as to elaborate on their characteristics. Spatial cumulants up to and including the fifth-order are found to be efficient in characterizing patterns on both binary and continuous images. The behaviour of spatial cumulants is shown to characterize well the behaviour of the spatial architecture of geological data, including the degree of homogeneity and connectivity. The high-order cumulants are found to be relatively insensitive to the number of data used, and relatively small data sets are sufficient to provide cumulant maps. HOSC has been coded in FORTAN 90 and is easily integrated to the S-GeMS open source platform.

Dimensional Reduction of Pattern-Based Simulation Using Wavelet Analysis

A pattern-based simulation technique using wavelet analysis is proposed for the simulation (wavesim) of categorical and continuous variables. Patterns are extracted by scanning a training image with a template and then storing them in a pattern database. The dimension reduction of patterns in the pattern database is performed by wavelet decomposition at certain scale and the approximate sub-band is used for pattern database classification. The pattern database classification is performed by the k-means clustering algorithm and classes are represented by a class prototype. For the simulation of categorical variables, the conditional cumulative density function (ccdf) for each class is generated based on the frequency of the individual categories at the central node of the template. During the simulation process, the similarity of the conditioning data event with the class prototypes is measured using the L2-norm. When simulating categorical variables, the ccdf of the best matched class is used to draw a pattern from a class. When continuous variables are simulated, a random pattern is drawn from the best matched class. Several examples of conditional and unconditional simulation with two- and three- dimensional data sets show that the spatial continuity of geometric features and shapes is well reproduced. A comparative study with the filtersim algorithm shows that the wavesim performs better than filtersim in all examples. A full-field case study at the Olympic Dam base metals deposit, South Australia, simulates the lithological rock-type units as categorical variables. Results show that the proportions of various rock-type units in the hard data are well reproduced when similar to those in the training image; when rock-type proportions between the training image and hard data differ, the results show a compromise between the two.

HOSIM: A high-order stochastic simulation algorithm for generating three-dimensional complex geological patterns

Abstract The three-dimensional high-order simulation algorithm HOSIM is developed to simulate complex non-linear and non-Gaussian systems. HOSIM is an alternative to the current MP approaches and it is based upon new high-order spatial connectivity measures, termed high-order spatial cumulants. The HOSIM algorithm implements a sequential simulation process, where local conditional distributions are generated using weighted orthonormal Legendre polynomials, which in turn define the so-called Legendre cumulants. The latter are high-order conditional spatial cumulants inferred from both the available data and training images. This approach is data-driven and reconstructs both high and lower-order spatial complexity in simulated realizations, while it only borrows from training images information that is not available in the data used. However, the three-dimensional implementation of the algorithm is computationally very intensive. To address his topic, the contribution of high-order conditional spatial cumulants is assessed in this paper through the number of Legendre cumulants with respect to the order of approximation used to estimate a conditional distribution and the number of data used within the respective neighbourhood. This leads to discarding the terms of Legendre cumulants with negligible contributions and allows an efficient simulation algorithm to be developed. The current version of the HOSIM algorithm is several orders of magnitude faster than the original version of the algorithm. Application and comparisons in a controlled environment show the excellent performance and efficiency of the HOSIM algorithm.

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We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an

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-Version Discontinuous Galerkin Method for Integro-Differential Equations of Parabolic Type

-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal

CDFSIM: Efficient Stochastic Simulation Through Decomposition of Cumulative Distribution Functions of Transformed Spatial Patterns

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A Gabriel-Delaunay triangulation of 2D complex fractured media for multiphase flow simulations

-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near

Generalized Laguerre expansions of multivariate probability densities with moments

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