Horacio Florez

Applications and comparison of model-order reduction methods based on wavelets and POD

We present a wavelet-based model-order reduction method (MOR) that provides an alternative subspace when Proper Orthogonal Decomposition (POD) is not a choice. We thus compare the wavelet- and POD-based approaches for reducing high-dimensional nonlinear transient and steady-state continuation problems. We also propose a line-search regularized Petrov-Galerkin (PG) Gauss-Newton (GN) algorithm that includes a regularization procedure and a globalization strategy. Numerical results included herein indicate that wavelet-based method is competitive with POD for compression ratios below 25% while POD achieves up to 90%. Full-order-model (FOM) results demonstrate that the proposed PGGN algorithm outperforms the standard GN method.

Spline-based reservoir’s geometry reconstruction and mesh generation for coupled flow and mechanics simulation

In this paper, the geometry of oil reservoirs is reconstructed by using B-splines surfaces. The technique exploits the reservoir’s static model’s simplicity to build a robust piecewise continuous geometrical representation by means of Bèzier bicubic patches. Interpolation surfaces can manage the reservoir’s topology while translational surfaces allow extrapolating it towards its sideburdens. After that, transfinite interpolation (TFI) can be applied to generate decent hexahedral meshes. In order to test the procedure, several open-to-the-public oil reservoir datasets are reconstructed and hexahedral meshes around them are generated. This reconstruction workflow also allows having different meshes for flow and mechanics by computing a projection operator in order to map pressures from the original flow mesh to the generated reference mechanics mesh. As an update respect to a previous version of this research, we already incorporate blending functions to the TFI procedure in order to attract the mesh towards the reservoir, which allows grading the hexahedral meshes in the appropriate manner. Finally, field scale reservoir compaction and subsidence computations are carried out by using continuous Galerkin FEM for both flow and mechanics in order to demonstrate the applicability of the proposed algorithm.

Upscaling of Hydraulic Properties of Fractured Porous Media: Full Permeability Tensor and Continuum Scale Simulations

The simulation of flow and transport phenomena in fractured media is a challenging problem. Despite existing advances in computer capabilities, the fact that fractures can occur over a wide range of scales within porous media compromises the development of detailed flow simulations. Current discrete approaches are limited to systems that contain a small number of fractures. Alternatively, continuum approaches require the input of effective parameters that must be obtained as accurately as possible, based on the actual fracture network or its statistical description. In this work, a novel method based on the utilization of the Delta-Y transformation is introduced for obtaining the effective permeability tensor of a 2D fracture network. This approach entails a detailed description of the fracture network, where each fracture is represented as a segment with a given length, orientation and permeability value. A fine rectangular grid is then superimposed on the network, and the fractures are discretized so that each one of them is represented as a connected sequence of bonds on the grid with a hydraulic conductivity proportional to the ratio of effective permeability over fracture discretization length. The next step consists of the selection of a coarser rectangular grid on which the continuum simulation is performed. In order to obtain the permeability tensor for each one of the resulting blocks, the Delta-Y method is used. Finally, the resulting continuum permeability tensor is used to simulate the steady-state flow problem, and the results are compared with the actual flow pattern yielded by the fracture network simulation. The results obtained with both methods follow a similar flux pattern across the reservoir system. This shows that the proposed approach allows for efficient perform upscaling of hydraulic properties by honoring both the underlying physics and details of fracture network connectivity.

A Mortar Method Based on NURBS for Curve Interfaces

The Mortar Finite Element Method (MFEM) has been demonstrated to be a powerful technique in order to formulate a weak continuity condition at the interface of sub-domains in which different meshes, i.e. non-conforming or hybrid, and / or variational approximations are used. This is particularly suitable when coupling different physics on different domains, such as elasticity and poro-elasticity, for example, in the context of coupled flow and geomechanics. In this area precisely, geometrical aspects play also a role. It is very expensive, from the computational standpoint, having the same mesh for flow and mechanics. Tensor product meshes are usually propagated from the reservoir in a conforming way into its surroundings, which makes non-conforming discretizations a highly attractive option for these cases. In order to tackle these general sub-domains problems, a MFEM scheme on curve interfaces based on Non-Uniform Rational B-Splines (NURBS) curves and surfaces is presented in this paper. The goal is having a more robust geometrical representation for mortar spaces which allows gluing non-conforming interfaces on realistic three-dimensional geometries. The resulting mortar saddle point problem will be decoupled by means of standard Domain Decomposition techniques such as Dirichlet-Neumann and Neumann-Neumann, in order to exploit current parallel machine architectures. Three-dimensional examples ranging from near-wellbore applications to field level subsidence computations show that the proposed scheme can handle problems of practical interest. In order to facilitate the implementation of complex workflows, an advanced Python wrapper interface that allows programming capabilities have been implemented. Extensions to couple elasticity and plasticity, which seems very promising in order to speed up computations involving poroplasticity, will be also discussed.

A model reduction for highly non-linear problems using wavelets and the Gauss-Newton method

A global regularized Gauss-Newton method is proposed to obtain a zero residual for square nonlinear problems on an affine subspace. The affine subspace is characterized by using wavelets which enable us to solve the problem without making simulations before solving it. We pose the problem as a zero-overdetermined nonlinear composite function where the inside function provided the solution we are seeking. A Gauss-Newton method is presented together with its standard Newtons assumptions that guarantee to retain the q-quadratic rate of convergence. To avoid the singularity and the high-nonlinearity a regularized strategy is presented which preserves the fast rate of convergence. A line-search method is included for global convergence. We rediscover that the Petrov-Galerkin (PG) inexact directions for the Newton method are the Gauss-Newton (GN) directions for the composite function. The results obtained in a set of large-scale problems show the capability of the method for reproducing their essential features while reducing the computational cost associated with high-dimensional problems by a substantial order of magnitude.

Towards predictions of large dynamic systems' behavior using reduced-order modeling and interval computations

The ability to conduct fast and reliable simulations of dynamic systems is of special interest to many fields of operations. Such simulations can be very complex and, to be thorough, involve millions of variables, making it prohibitive in CPU time to run repeatedly for many different configurations. Reduced-Order Modeling (ROM) provides a concrete way to handle such complex simulations using a realistic amount of resources. However, uncertainty is hardly taken into account. Changes in the definition of a model, for instance, could have dramatic effects on the outcome of simulations. Therefore, neither reduced models nor initial conclusions could be 100% relied upon. In this research, Interval Constraint Solving Techniques (ICST) are employed to handle and quantify uncertainty. The goal is to identify key features of a given dynamical phenomenon in order to be able to propagate the characteristics of the model forward and predict its future behavior to obtain 100% guaranteed results. This is specifically important in applications, as a reliable understanding of a developing situation could allow for preventative or palliative measures before a situation aggravates.