Inga Berre

An efficient multi-point flux approximation method for Discrete Fracture-Matrix simulations

We consider a control volume discretization with a multi-point flux approximation to model Discrete Fracture-Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions. Inspired by a recently introduced approach based on a two-point flux approximation, we explicitly account for the fractures by representing them as hybrid cells between the matrix cells. As well as simplifying the grid generation, our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time-step restrictions associated with small cells. Excluding the small cells also reduces the condition number of the discretization matrix. For examples involving realistic anisotropy ratios in the permeability, numerical results show significant improvement compared to existing methods based on two-point flux approximations. We also investigate the hybrid method by studying the convergence rates for different apertures and fracture/matrix permeability ratios. Finally, the effect of removing the cells in the intersections of the fractures are studied. Together, these examples demonstrate the efficiency, flexibility and robustness of our new approach.

Combined Adaptive Multiscale and Level-Set Parameter Estimation

We propose a solution strategy for parameter estimation, where we combine adaptive multiscale estimation (AME) and level-set estimation (LSE). The approach is applied to the nonlinear inverse probl...

A 3D computational study of effective medium methods applied to fractured media

This work evaluates and improves upon existing effective medium methods for permeability upscaling in fractured media. Specifically, we are concerned with the asymmetric self-consistent, symmetric self-consistent, and differential methods. In effective medium theory, inhomogeneity is modeled as ellipsoidal inclusions embedded in the rock matrix. Fractured media correspond to the limiting case of flat ellipsoids, for which we derive a novel set of simplified formulas. The new formulas have improved numerical stability properties, and require a smaller number of input parameters. To assess their accuracy, we compare the analytical permeability predictions with three-dimensional finite-element simulations. We also compare the results with a semi-analytical method based on percolation theory and curve-fitting, which represents an alternative upscaling approach. A large number of cases is considered, with varying fracture aperture, density, matrix/fracture permeability contrast, orientation, shape, and number of fracture sets. The differential method is seen to be the best choice for sealed fractures and thin open fractures. For high-permeable, connected fractures, the semi-analytical method provides the best fit to the numerical data, whereas the differential method breaks down. The two self-consistent methods can be used for both unconnected and connected fractures, although the asymmetric method is somewhat unreliable for sealed fractures. For open fractures, the symmetric method is generally the more accurate for moderate fracture densities, but only the asymmetric method is seen to have correct asymptotic behavior. The asymmetric method is also surprisingly accurate at predicting percolation thresholds.

Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods

Abstract Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock–Euler method 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-discretization 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 of the ODEs resulting 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.

Upscaling of Non-isothermal Reactive Porous Media Flow with Changing Porosity

Motivated by rock–fluid interactions occurring in a geothermal reservoir, we present a two-dimensional pore scale model of a periodic porous medium consisting of void space and grains, with fluid flow through the void space. The ions in the fluid are allowed to precipitate onto the grains, while minerals in the grains are allowed to dissolve into the fluid, and we take into account the possible change in pore geometry that these two processes cause, resulting in a problem with a free boundary at the pore scale. We include temperature dependence and possible effects of the temperature both in fluid properties and in the mineral precipitation and dissolution reactions. For the pore scale model equations, we perform a formal homogenization procedure to obtain upscaled equations. A pore scale model consisting of circular grains is presented as a special case of the porous medium.

Anisotropic effective conductivity in fractured rocks by explicit effective medium methods

In this work, we assess the use of explicit methods for estimating the effective conductivity of anisotropic fractured media. Explicit methods are faster and simpler to use than implicit methods but may have a more limited range of validity. Five explicit methods are considered: the Maxwell approximation, the T-matrix method, the symmetric and asymmetric weakly self-consistent methods, and the weakly differential method, where the two latter methods are novelly constructed in this paper. For each method, we develop simplified expressions applicable to flat spheroidal “penny-shaped” inclusions. The simplified expressions are accurate to the first order in the ratio of fracture thickness to fracture diameter. Our analysis shows that the conductivity predictions of the methods fall within known upper and lower bounds, except for the T-matrix method at high fracture densities and the symmetric weakly self-consistent method when applied to very thin fractures. Comparisons with numerical results show that all the methods give reliable estimates for small fracture densities. For high fracture densities, the weakly differential method is the most accurate if the fracture geometry is non-percolating or the fracture/matrix conductivity contrast is small. For percolating conductive fracture networks, we have developed a scaling relation that can be applied to the weakly self-consistent methods to give conductivity estimates that are close to the results from numerical simulations.

A model for non-isothermal flow and mineral precipitation and dissolution in a thin strip

Motivated by porosity changes due to chemical reactions caused by injection of cold water in a geothermal reservoir, we propose a two-dimensional pore scale model of a thin strip. The pore scale model includes fluid flow, heat transport and reactive transport where changes in aperture is taken into account. The thin strip consists of void space and grains, where ions are transported in the fluid in the void space. At the interface between void and grain, ions are allowed to precipitate and become part of the grain, or conversely, minerals in the grain can dissolve and become part of the fluid flow, and we honor the possible change in aperture these two processes cause. We include temperature dependence and possible effects of the temperature in both fluid properties and in the mineral precipitation and dissolution reactions. For the pore scale model equations, we investigate the limit as the width of the thin strip approaches zero, deriving upscaled one-dimensional effective equations.

Linear and nonlinear convection in porous media between coaxial cylinders

We uncover novel features of three-dimensional natural convection in porous media by investigating convection in an annular porous cavity contained between two vertical coaxial cylinders. The investigations are made using a linear stability analysis, together with high-order numerical simulations using pseudospectral methods to model the nonlinear regime. The onset of convection cells and their preferred planform are studied, and the stability of the modes with respect to different types of perturbation is investigated. We also examine how variations in the Rayleigh number affect the convection modes and their stability regimes. Compared with previously published data, we show how the problem inherits an increased complexity regarding which modes will be obtained. Some stable secondary modes or mixed modes have been identified and some overlapping stability regions for different convective modes are determined.

Postinjection 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 preexisting 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 postinjection seismicity. To test this hypothesis, hydraulic stimulation is modeled by numerically coupling flow in the fractures and matrix, fracture deformation, and matrix deformation for a synthetic 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.

Robust inversion of controlled source electromagnetic data for production monitoring

Monitoring of the flow pattern during reservoir production is a potential new application for Controlled Source Electromagnetic (CSEM) data. However, it is a challenging problem due to a restricted resolution power in the data and possible high signal to noise ratio. We present a novel 3-D inversion algorithm for estimating changes in the electric conductivity based on time-lapse CSEM data. The inversion is based on a reduced representation of the unknown parameter function, where the degree of freedom in the estimation is determined by the information content in the data. The chosen representation facilitates the estimation of flow patterns with varying structure and also varying degree of smoothness in the transition between the oil and water saturated regions.