arXiv subjects

Inga Berre

Publications and source records attributed to Inga Berre.

46 records · Page 3Linked to original sources

arXiv2017

PorePy: An Open-Source Simulation Tool for Flow and Transport in Deformable Fractured Rocks

Fractures are ubiquitous in the subsurface and strongly affect flow and deformation. The physical shape of the fractures, they are long and thin objects, puts strong limitations on how the effect of this dynamics can be incorporated into standard reservoir simulation tools. This paper reports the development of an open-source software framework, termed PorePy, which is aimed at simulation of flow and transport in three-dimensional fractured reservoirs, as well as deformation of the reservoir due to shearing along fracture and fault planes. Starting from a description of fractures as polygons embedded in a 3D domain, PorePy provides semi-automatic gridding to construct a discrete-fracture-matrix model, which forms the basis for subsequent simulations. PorePy allows for flow and transport in all lower-dimensional objects, including planes (2D) representing fractures, and lines (1D) and points (0D), representing fracture intersections. Interaction between processes in neighboring domains of different dimension is implemented as a sequence of couplings of objects one dimension apart. This readily allows for handling of complex fracture geometries compared to capabilities of existing software. In addition to flow and transport, PorePy provides models for rock mechanics, poro-elasticity and coupling with fracture deformation models. The software is fully open, and can serve as a framework for transparency and reproducibility of simulations. We describe the design principles of PorePy from a user perspective, with focus on possibilities within gridding, covered physical processes and available discretizations. The power of the framework is illustrated with two sets of simulations; involving respectively coupled flow and transport in a fractured porous medium, and low-pressure stimulation of a geothermal reservoir.

cs.CE↗

arXiv2017

Three-Dimensional Numerical Modeling of Shear Stimulation of Naturally Fractured Reservoirs

Shear dilation based hydraulic stimulations enable exploitation of geothermal energy from reservoirs with inadequate initial permeability. While contributing to enhancing the reservoir's permeability, hydraulic stimulation processes may lead to undesired seismic activity. Here, we present a three dimensional numerical model aiming to increase understanding of this mechanism and its consequences. The fractured reservoir is modeled as a network of explicitly represented large scale fractures immersed in a permeable rock matrix. The numerical formulation is constructed by coupling three physical processes: fluid flow, fracture deformation, and rock matrix deformation. For flow simulations, the discrete fracture matrix model is used, which allows the fluid transport from high permeable conductive fractures to the rock matrix and vice versa. The mechanical behavior of the fractures is modeled using a hyperbolic model with reversible and irreversible deformations. Linear elasticity is assumed for the mechanical deformation and stress alteration of the rock matrix. Fractures are modeled as lower dimensional surfaces embodied in the domain, subjected to specific governing equations for their deformation along the tangential and normal directions. Both the fluid flow and momentum balance equations are approximated by finite volume discretizations. The new numerical model is demonstrated considering a three dimensional fractured formation with a network of 20 explicitly represented fractures. The effects of fluid exchange between fractures and rock matrix on the permeability evolution and the generated seismicity are examined for test cases resembling realistic reservoir conditions.

physics.geo-ph↗

arXiv2017

Reactivation of Fractures in Subsurface Reservoirs - a Numerical Approach using a Static-Dynamic Friction Model

Fluid-induced slip of fractures is characterized by strong multiphysics couplings. Three physical processes are considered: Flow, rock deformation and fracture deformation. The fractures are represented as lower-dimensional objects embedded in a three-dimensional domain. Fluid is modeled as slightly compressible, and flow in both fractures and matrix is accounted for. The deformation of rock is inherently different from the deformation of fractures; thus, two different models are needed to describe the mechanical deformation of the rock. The medium surrounding the fractures is modeled as a linear elastic material, while the slip of fractures is modeled as a contact problem, governed by a static-dynamic friction model. We present an iterative scheme for solving the non-linear set of equations that arise from the models, and suggest how the step parameter in this scheme should depend on the shear modulus and mesh size.

physics.geo-ph↗

arXiv2017

Post-injection normal closure of fractures as a mechanism for induced seismicity

Understanding the controlling mechanisms underlying injection-induced seismicity is important for optimizing reservoir productivity and addressing seismicity-related concerns related to hydraulic stimulation in Enhanced Geothermal Systems. Hydraulic stimulation enhances permeability through elevated pressures, which cause normal deformations, and the shear slip of pre-existing fractures. Previous experiments indicate that fracture deformation in the normal direction reverses as the pressure decreases, e.g., at the end of stimulation. We hypothesize that this normal closure of fractures enhances pressure propagation away from the injection region and significantly increases the potential for post-injection seismicity. To test this hypothesis, hydraulic stimulation is modeled by numerically coupling fracture deformation, pressure diffusion and stress alterations for a synthetic geothermal reservoir in which the flow and mechanics are strongly affected by a complex three-dimensional fracture network. The role of the normal closure of fractures is verified by comparing simulations conducted with and without the normal closure effect.

physics.geo-ph↗

arXiv2017

High-accuracy phase-field models for brittle fracture based on a new family of degradation functions

Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients, which in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations, one enforcing stress equilibrium and another governing phase-field evolution. The two equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature.

cond-mat.mtrl-sci↗

arXiv2017

Heterogeneity Preserving Upscaling for Heat Transport in Fractured Geothermal Reservoirs

In simulation of fluid injection in fractured geothermal reservoirs, the characteristics of the physical processes are severely affected by the local occurence of connected fractures. To resolve these structurally dominated processes, there is a need to develop discretization strategies that also limit computational effort. In this paper we present an upscaling methodology for geothermal heat transport with fractures represented explicitly in the computational grid. The heat transport is modeled by an advection-conduction equation for the temperature, and solved on a highly irregular coarse grid that preserves the fracture heterogeneity. The upscaling is based on different strategies for the advective term and the conductive term, respectively. The coarse scale advective term is constructed from sums of fine scale fluxes, whereas the coarse scale conductive term is constructed based on numerically computed basis functions. The method naturally incorporates a coupling between the matrix and the fractures via the discretization, so that explicit transfer terms that couple solution variables in the fractures and the matrix are avoided. Numerical results show that the upscaling methodology performs well, in particular for large upscaling ratios, and that it is applicable also to highly complex fracture networks.

cs.CE↗

arXiv2017

Benchmarks for single-phase flow in fractured porous media

This paper presents several test cases intended to be benchmarks for numerical schemes for single-phase fluid flow in fractured porous media. A number of solution strategies are compared, including a vertex and a cell-centered finite volume method, a non-conforming embedded discrete fracture model, a primal and a dual extended finite element formulation, and a mortar discrete fracture model. The proposed benchmarks test the schemes by increasing the difficulties in terms of network geometry, e.g. intersecting fractures, and physical parameters, e.g. low and high fracture-matrix permeability ratio as well as heterogeneous fracture permeabilities. For each problem, the results presented by the participants are the number of unknowns, the approximation errors in the porous matrix and in the fractures with respect to a reference solution, and the sparsity and condition number of the discretized linear system. All data and meshes used in this study are publicly available for further comparisons.

math.NA↗

arXiv2012

Thermal Fracturing of Geothermal Wells and the Effects of Borehole Orientation

An enhanced geothermal system (EGS) expands the potential of geothermal energy by enabling the exploitation of regions that lack conventional hydrothermal resources. The EGS subsurface system is created by engineering enhanced flow paths between injection and production wells. Hydraulic stimulation of existing fracture networks has been successfully achieved for unconventional geothermal resources. More recently proposed concepts increase the use of drilled wellbores in hard rock to connect the injection and production wells. The present work investigates the long-term thermal effects of deviated geothermal wellbores and studies how the cooling of the borehole wall results in thermally induced tensile fractures. The results show that induced fractures are created by a combination of in situ and thermal stresses, and that the extent to which thermally induced tensile wall fractures are created largely depends on how the wellbores are oriented with respect to the pre-existing stresses of the reservoir. If the system is not optimized with respect to in situ stresses, the risk of wellbore instability becomes severe within less than a year of production. In contrast, if the orientation of the wellbores is optimized, thermally induced instabilities can be completely excluded as potential risks for the operational lifetime of the system. Furthermore, our results show that the thermal failure process strongly depends on the temperature of the injected water but is only weakly affected by the injection rate.

physics.geo-ph↗

arXiv2012

Efficient Simulation of Geothermal Processes in Heterogeneous Porous Media based on the Exponential Rosenbrock-Euler and Rosenbrock-type Methods

Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock--Euler and Rosenbrock-type methods with control-volume (two-point flux approximation) space discretizations leads to efficient numerical techniques for simulating geothermal systems. In terms of efficiency and accuracy, the exponential Rosenbrock--Euler time integrator has advantages over standard time-dicretization schemes, which suffer from time-step restrictions or excessive numerical diffusion when advection processes are dominating. Based on linearization of the equation at each time step, we make use of matrix exponentials of the Jacobian from the spatial discretization, which provide the exact solution in time for the linearized equations. This is at the expense of computing the matrix exponentials of the stiff Jacobian matrix, together with propagating a linearized system. However, using a Krylov subspace or Leja points techniques make these computations efficient. The Rosenbrock-type methods use the appropriate rational functions of the Jacobian from the spatial discretization. The parameters in these schemes are found in consistency with the required order of convergence in time. As a result, these schemes are A-stable and only a few linear systems are solved at each time step. The efficiency of the methods compared to standard time-discretization techniques are demonstrated in numerical examples.

math.NA↗

arXiv2012

Interaction between Injection Points during Hydraulic Fracturing

We present a model of the hydraulic fracturing of heterogeneous poroelastic media. The formalism is an effective continuum model that captures the coupled dynamics of the fluid pressure and the fractured rock matrix and models both the tensile and shear failure of the rock. As an application of the formalism, we study the geomechanical stress interaction between two injection points during hydraulic fracturing (hydrofracking) and how this interaction influences the fracturing process. For injection points that are separated by less than a critical correlation length, we find that the fracturing process around each point is strongly correlated with the position of the neighboring point. The magnitude of the correlation length depends on the degree of heterogeneity of the rock and is on the order of 30-45 m for rocks with low permeabilities. In the strongly correlated regime, we predict a novel effective fracture-force that attracts the fractures toward the neighboring injection point.

cond-mat.mtrl-sci↗