arXiv subjects
Publications and source records attributed to Eren Ucar.
We investigate and solve a special class of integrals involving associated Legendre functions, which can be regarded as generalized Mehler-Fock transformations. Some of the integrals appear naturally when dealing with the heat or resolvent equation of a wedge in the hyperbolic plane. As an application, we derive explicit formulas for the Green's function and the heat kernel of a wedge.
In this thesis we deal with spectral invariants for polygons and closed orbisurfaces of constant Gaussian curvature. In each case our method is to study the heat kernel and the asymptotic expansion of the heat trace. First, we investigate hyperbolic polygons, i.e. relatively compact domains in the hyperbolic plane with piecewise geodesic boundary. We compute the asymptotic expansion of the heat trace associated to the Dirichlet Laplacian of any hyperbolic polygon, and we obtain explicit formulas for all heat invariants. Analogous results for Euclidean and spherical polygons were known before. We unify these results and deduce the heat invariants for arbitrary polygons, i.e. for relatively compact domains with piecewise geodesic boundary contained in a complete Riemannian manifold of constant Gaussian curvature. It turns out that the heat invariants provide much information about a polygon, if the curvature does not vanish. For example, then the multiset of all real angles (i.e. those which are not equal to pi) and the Euler characteristic of a polygon are spectral invariants. Furthermore, we compute the asymptotic expansion of the heat trace for any closed Riemannian orbisurface of constant curvature, and obtain explicit formulas for all heat invariants. If the curvature does not vanish, then it is possible to detect interesting information about the topology and the singular set of an orbisurface from the heat invariants.
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.
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.
Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multipoint stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (faces in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems for which analytical solutions are available and more complex benchmark problems, including comparison with a finite-element discretization.