In the prevailing context of the 21st century, characterized by a predominant reliance on oil and gas, or in the promising future where green energy shapes a human society committed to net-zero emissions, the role of underground fractured formations in energy production and storage remains pivotal and irreplaceable. Geological faults typically act as non-permeable sealing boundaries for reservoirs used in storage, including those for hydrogen and carbon dioxide. In contrast, artificial fractures can serve as highly permeable conduits for fluid flow into wellbores, particularly in applications such as enhanced geothermal reservoirs. In the past decade, the hazardous consequences of failing to predict the geomechanics behaviors of fractured formations have led to a pronounced focus on developing simulation strategies that are both accurate and efficient for fractured formations.
It is challenging to understand formations riddled with fractures. From a computational perspective, the complex fracture networks typically demand a way finer unstructured grid. However, using such an unstructured grid is impractical for real-world applications due to their high computational load. Conversely, coarser grids paired with strategies such as homogenization could result in loss of crucial details. Heterogeneous properties of geological formations that span on large length sizes require the simulation strategies to be scalable, in order to be relevant.
This thesis proposes a novel approach named as multiscale extended finite element method (MS-XFEM) to tackle these challenges. The challenges related to discretization are resolved by applying the extended finite element method (XFEM) which allows for the use of structured grids. This simplified mesh, however, leads to an augmented matrix size due to extra degrees of freedom (DOFs) introduced by enrichments. A multiscale approach is therefore combined with XFEM. The computational process is operated on the larger yet sparser coarse grids and then the coarse scale mesh solutions are interpolated back to fine scale mesh. The novelty of this work is to involve the fractures into basis functions only, thus the coarse scale system is constructed based on a finite element method. More importantly, this construction of basis functions is fully algebraic and can be updated locally and adaptively for the simulation of propagating fractures.
This method has been implemented and tested to prove its efficiency and accuracy. All tests results prove the good qualities of solutions computed from MS-XFEM when compared to fine scale XFEM solutions. Basis functions are constructed successfully with the algebraic method. These tests reveal the potential of the MS-XFEM in simulating real-world subsurface fractured formations.@en

In fractured geological formations, as a result of the in-situ stress changes, fractures can propagate or slide. This phenomenon can be efficiently modeled by the extended finite element method (XFEM) when there are only a few fractures present. However, geological reservoirs contain many fractures which can also cross and are densely populated. Therefore, the classical XFEM is too expensive to be applied for the simulation of propagating fractures in geological formations. To reduce the costs, typically, homogenization or upscaling is used. However, they result in inaccurate solutions, since no separation of scales exists in this process. To resolve this challenge, in this work, a multiscale XFEM (MS-XFEM) for propagating fractures is developed and presented. In each time step, given the current geometries of the fractures, local XFEM-based basis functions are constructed or adaptively updated. The adaptive update takes place in certain regions where fracture geometries are changed due to propagation. Using these basis functions, a very efficient FEM-based coarse-scale system is developed since it has no extra degrees of freedom (DOFs). Once the coarse-scale system is solved, its solution is prolonged to the fine-scale original resolution using the basis functions. This approximate fine-scale solution is then used to estimate the group of growing fractures tips and their growing angles. This allows for exploiting the locality of the propagation process fully while solving a global system. To control the error, an iterative procedure is also developed. Proof-of-concept test cases are presented to study the developed MS-XFEM algorithm. It is shown that MS-XFEM results are capable of predicting the propagating paths for complicated fracture patterns. As such, MS-XFEM casts a promising method for field-scale applications.

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Altering the state of the stress of the subsurface reservoirs can lead fractures to slip and extend their lengths (i.e., to propagate). This process can even be engineered, in many applications, e.g., enhanced geothermal systems. As such, accurate and efficient simulation of the mechanical deformation of the subsurface geological reservoirs, allowing for fracture propagation, is at the core of many geoscientific operational designs. Subsurface reservoirs entail many fractures at multiple scales. Implementation of 3D complex grids on these complex fractured systems, for mechanical deformation analyses, is extremely challenging. An alternative approach can be developed by using extended finite element methods (XFEM). XFEM allows for capturing the fractures effects on a conveniently-generated structured matrix mesh. The cracks are introduced by extra degrees of freedom (DOFs) on the nodes of the matrix rock mesh. For geoscientific applications, however, XFEM results in too many DOFs which are beyond the scope of simulators. Additionally, for propagating fractures, these DOFs need to be updated in response to the dynamic extension of the fractures in the domain. The propagation process not only adds to the sensitivity of the outputs to the accuracy of the estimated stress field, but also increases the size of the linear systems. In addition to these, matrix rocks are often highly heterogeneous, at high resolutions. In this work, we present a novel multiscale procedure for propagating fractures in heterogeneous geological reservoirs. For the first time in the community, we present the highly fractured systems at coarser resolutions via XFEM-based basis functions, which also account for the propagating effects. Fractures are allowed to extend their scale and the enriched basis functions are locally updated. Using these bases, the coarse scale system is obtained in which no extra DOFs due to fractures exist. This significantly reduces the computational complexity. As a significant step forward compared with our recently-published journal paper [Xu, Hajibeygi, Sluys, Journal of Computational Physics, 2021], in this conference contribution we allow the fractures to propagate. Specially, we introduce a local-global-based approach, in which fracture propagation is treated only at local stage; while the stress and deformation are modelled at global scale. In the search of convenient implementation, the procedure is presented algebraically. Through several test cases, we demonstrate the applicability of the method for complex fractured media. Specially we demonstrate that propagation can be modeled at local scale, while accurate stress and deformation fields are obtained at global scale.

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Deformable fractured porous media appear in many geoscience applications. While the extended finite element method (XFEM) has been successfully developed within the computational mechanics community for accurate modeling of deformation, its application in geoscientific applications is not straightforward. This is mainly due to the fact that subsurface formations are heterogeneous and span large length scales with many fractures at different scales. To resolve this limitation, in this work, we propose a novel multiscale formulation for XFEM, based on locally computed enriched basis functions. The local multiscale basis functions capture heterogeneity of th e porous rock properties, and discontinuities introduced by the fractures. In order to preserve accuracy of these basis functions, reduced-dimensional boundary conditions are set as localization condition. Using these multiscale bases, a multiscale coarse-scale system is then governed algebraically and solved. The coarse scale system entails no enrichment due to the fractures. Such formulation allows for significant computational cost reduction, at the same time, it preserves the accuracy of the discrete displacement vector space. The coarse-scale solution is finally interpolated back to the fine scale system, using the same multiscale basis functions. The proposed multiscale XFEM (MS-XFEM) is also integrated within a two-stage algebraic iterative solver, through which error reduction to any desired level can be achieved. Several proof-of-concept numerical tests are presented to assess the performance of the developed method. It is shown that the MS-XFEM is accurate, when compared with the fine-scale reference XFEM solutions. At the same time, it is significantly more efficient than the XFEM on fine-scale resolution, as it significantly reduces the size of the linear systems. As such, it develops a promising scalable XFEM method for large-scale heavily fractured porous media.

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We present the first multiscale extended finite element (MS-XFEM) method. MS-XFEM develops local multiscale basis functions which allow for constructing accurate coarse-scale systems. These basis functions are computed locally using XFEM. Both enrichment functions along the fracture nodes and tips are considered in the basis function calculation. These functions are then used to construct and solve the coarse-scale system. Note that no enrichment exists on the coarse level. This makes our MS-XFEM procedure computationally efficient, yet accurate, for highly-fractured real-field applications. For several test cases, the MS-XFEM performance is investigated compared with the classical XFEM method. MS-XFEM casts a promising approach for real-field analyses of mechanical deformation influenced by the reservoir pore-pressure change.

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