Simulation of fracture slip and propagation in hydraulic stimulation of geothermal reservoirs

Rollen til hydraulisk stimulering i å øke produksjonen fra geotermiske reservoarer, og muliggjøre kommersiell utnyttelse av et større spekter av geotermiske ressurser, har fått økt oppmerksomhet de siste tiårene. Under stimulering kan eksisterende sprekker sideforskyves, forplante seg og koble seg til andre sprekker og der igjennom øke permeabiliteten i reservoaret. Prosessene er preget av sterke hydromekaniske interaksjoner, som vi har begrensede muligheter til å overvåke. Numeriske simuleringer er derfor et viktig verktøy for å hjelpe oss til å bedre forstå mekanismene som er i spill. Avhandlingen tar sikte på å utvikle en omfattende matematisk modell og en numerisk tilnærming for å analysere bruddmekanismer og undersøke koblede hydromekaniske prosesser som forekommer i oppsprukne porøse medier. Den foreslåtte modellen benytter en blandet-dimensjonal konseptuell modell, som inkluderer porelastisitet i det porøse mediet og kontaktmekanikk for sprekkene. Modellen tillater også forplantning og koalescens av eksisterende sprekker. Et nytt diskretiseringsskjema for å løse den foreslåtte matematiske modellen presenteres. Den foreslåtte metoden bruker en to-nivå simuleringstilnærming, kategorisert i grove og fine nivåer, for å redusere beregningskostnader og sikre nøyaktighet. En endelig volummetode kombineres med en aktiv-sett løsningsstrategi for å diskretisere porelastisitet og bruddkontaktmekanikk på det grove nivået. Sprekkeforplantning betraktes på et fint nivå, der en endelig elementmetode kombineres med kollapsede kvartpunktselementer for å approksimere singulariteten i spenningen ved enden av sprekkene. Adaptiv gitring basert på en feilestimator og Laplace-glatting av gitteret introduseres på begge nivåer for effektivt å håndtere sprekkepropagering og koalescens. Simuleringene utført i denne avhandlingen forbedrer vår forståelse av hydraulisk stimulering og dens effekt på forbedring av sprekkepermeabilitet og konnektivitet i geotermiske reservoarer.The role of hydraulic stimulation in enhancing geothermal reservoir production and allowing for commercial exploitation of a larger range of geothermal resources has attracted attention from researchers in recent decades. During stimulation, preexisting fractures may slip, propagate, and connect to other fractures to enhance permeability. The processes are characterized by strong hydromechanical interactions, which have limited monitoring opportunities. Therefore, numerical simulations provide a powerful tool to help us better understand the mechanisms. This thesis aims to develop a comprehensive mathematical model and a numerical approach to analyze fracture mechanisms, and to investigate the coupled hydromechanical processes occurring in fractured porous media. The proposed model will employ a mixed-dimensional conceptual model, incorporating the concepts of poroelasticity and fracture contact mechanics. The model will also allow for the growth and coalescence of preexisting fractures. A novel discretization scheme for solving the proposed mathematical model is presented. The proposed scheme employs a two-level simulation approach, categorized into coarse and fine levels, to reduce the computational costs and ensure accuracy. A finite volume method is combined with an active set strategy to discretize poroelasticity and fracture contact mechanics on the coarse level. Fracture propagation is considered on a fine level, in which a finite element method is combined with collapsed quarter-point elements to capture the stress singularity at the fracture tips. Adaptive remeshing based on an error estimator and Laplacian smoothing is introduced on both levels to effectively capture fracture propagation and coalescence in the computational grid. The simulations conducted in this thesis improve our understanding of hydraulic stimulation and its effect on enhancing fracture permeability and connectivity in geothermal reservoirs.Doktorgradsavhandlin

Existence for noncoercive nonlinear elliptic equations with two lower-order terms

This paper considers a class of noncoercive nonlinear elliptic problems with coefficients defined in Marcinkiewicz and Lorentz spaces. We prove the existence of a solution for the corresponding Dirichlet problem and investigate the higher integrability properties of the solution.Comment: 13 page

Modelling of mixed-mechanism stimulation for the enhancement of geothermal reservoirs

Hydraulic stimulation is a critical process for increasing the permeability of fractured geothermal reservoirs. This technique relies on coupled hydromechanical processes induced through pressurized fluid injection into the rock formation. The injection of fluids causes poromechanical stress changes that can lead to fracture slip and shear dilation, as well as tensile fracture opening and propagation, so-called mixed-mechanism stimulation. The effective permeability of the rock is particularly enhanced when new fractures connect with pre-existing fractures. While hydraulic stimulation can significantly improve the productivity of fractured geothermal reservoirs, the process is also related to induced seismicity. Hence, understanding the coupled physics is central, for both reservoir engineering and seismic risk mitigation. This article presents a modelling approach for simulating the deformation, propagation and coalescence of fractures in porous media under the influence of anisotropic stress and fluid injection. It uses a coupled hydromechanical model for poroelastic, fractured media. Fractures are governed by contact mechanics and a fracture propagation model. For numerical solutions, we employ a two-level approach, combining a finite volume method for poroelasticity with a finite element method for fracture propagation. The study investigates the impact of injection rate, matrix permeability and stress anisotropy on stimulation outcomes.publishedVersio

Multiscale simulation of injection-induced fracture slip and wing-crack propagation in poroelastic media

In fractured poroelastic media under high differential stress, the shearing of fractures and faults and the corresponding propagation of wing cracks can be induced by fluid injection. Focusing on low-pressure stimulation with fluid pressures below the minimum principal stress but above the threshold required to overcome the fracture's frictional resistance to slip, this paper presents a mathematical model and a numerical solution approach for coupling fluid flow with fracture shearing and propagation. Numerical challenges are related to the strong coupling between hydraulic and mechanical processes, the material discontinuity the fractures represent in the medium, the wide range of spatial scales involved, and the strong effect that fracture deformation and propagation have on the physical processes. The solution approach is based on a multiscale strategy. In the macroscale model, flow in and poroelastic deformation of the matrix are coupled with the flow in the fractures and fracture contact mechanics, allowing fractures to frictionally slide. Fracture propagation is handled at the microscale, where the maximum tangential stress criterion triggers the propagation of fractures, and Paris' law governs the fracture growth processes. Simulations show how the shearing of a fracture due to fluid injection is linked to fracture propagation, including cases with hydraulically and mechanically interacting fractures

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) for dynamic response of laminated composite plate subjected to blast loading

The paper investigates the dynamic response of laminated composite plate under the effect of blast loading. The cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) and the equivalent layer theory (ELT) is used to model the behavior of the laminated composite plate. The blast loading is simulated by a time diagram of rectangular, triangular, exponential, or sinusoidal shape. The effects of the number of layers and fiber orientation to the displacement and stress fields of the laminated composite plate are discussed. Numerical results demonstrated the accuracy and reliability of the present method compared with previous published methods

Numerical modeling of wing crack propagation accounting for fracture contact mechanics

As a consequence of shearing, wing cracks can emerge from pre-existing fractures. The process involves the interaction of sliding of the existing fracture surfaces and the tensile material failure that creates wing cracks. This work devises a numerical model to investigate how wing cracks emerge, propagate and connect pre-existing fractures under shear processes. A mathematical and numerical model for wing crack propagation based on linear elastic fracture mechanics that also accounts for fracture contact mechanics is presented. Computational efficiency is ensured by an adaptive remeshing technique. The numerical model is verified and validated through a comparison of the analytical and experimental results. Additional numerical examples illustrate the performance of the method for complex test cases where wing-cracks develop for multiple pre-existing and interacting fractures.publishedVersio

Analyses of stiffened plates resting on the viscoelastic foundation subjected to a moving vehicle by a cell-based smoothed triangular plate element

Recently, a cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the firstorder shear deformation theory (FSDT) was proposed for static and free vibration analyses of Mindlin plates. The CS-FEM-DSG3 uses three-node triangular elements that can be easily generated automatically for arbitrary complicated geometric domains. This paper further extends the CS-FEMDSG3 for static, free vibration, and dynamic response of the stiffened plate resting on viscoelastic foundation subjected to a moving vehicle. The viscoelastic foundation is modeled by discrete springs and dampers whereas the stiffened plate can be considered as the combination between the Mindlin plate and the Timoshenko beam elements. The moving vehicle is transformed into one concentrated load at its central point. Some numerical examples are investigated and numerical results show that the CS-FEMDSG3 overcomes shear-locking phenomena and has a fast convergence. The results also illustrate the good agreement of the CS-FEM-DSG3 for static and free vibration analyses of un-stiffened plate compared with the previous published methods. In addition, the numerical results for dynamic analysis of stiffened plates by the CS-FEM-DSG3 also show the expected property in which the deflection of the stiffened plate is much smaller than those of the un-stiffened plate

Static and free vibration analyses of laminated composite shells by cell-based smoothed discrete shear gap method (CS-DSG3) using three-node triangular elements

A cell-based smoothed discrete shear gap method (CS-DSG3) using three-node triangular elements was recently proposed to improve the performance of the discrete shear gap method (DSG3) for static and free vibration analyses of isotropic Reissner-Mindlin plates and shells. In this paper, the CS-DSG3 is further extended for static and free vibration analyses of laminated composite shells. In the present method, the first-order shear deformation theory (FSDT) is used in the formulation due to the simplicity and computational efficiency. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of others available numerical results

Numerical modeling of wing crack propagation accounting for fracture contact mechanics

As a consequence of shearing, wing cracks can emerge from pre-existing fractures. The process involves the interaction of sliding of the existing fracture surfaces and the tensile material failure that creates wing cracks. This work devises a numerical model to investigate how wing cracks emerge, propagate and connect pre-existing fractures under shear processes. A mathematical and numerical model for wing crack propagation based on linear elastic fracture mechanics that also accounts for fracture contact mechanics is presented. Computational efficiency is ensured by an adaptive remeshing technique. The numerical model is verified and validated through a comparison of the analytical and experimental results. Additional numerical examples illustrate the performance of the method for complex test cases where wing-cracks develop for multiple pre-existing and interacting fractures