Julian Hennicker, Ph.D.Section de Mathématiques2-4 rue du Lièvre, CP 64 CH-1211 Genève Julian.Hennicker@unige.ch |
I am a mathematician, specialized in the theory and numerical analysis for partial differential equations.
Currently, I am a postdoc on Professor
Martin Gander's research team at the University of Geneva.
Currently, I work on the modelling of multiphase Darcy flow in fractured porous media. When one represents the fractures in all spatial dimensions, a numerical resolution of these models is not possible for cases of subsurface reservoir scale. In collaboration with Professor Roland Masson's research team, I develop so-called hybrid dimensional models (also called Discrete Fracture Matrix (DFM) models), for which the fractures are represented as interfaces of codimension one and the (d-1)-dimensional flow in the fractures is coupled with the d-dimensional flow in the surrounding rock matrix. The video on the right shows a comparison of the equi dimensional reference solution for oil saturation (upper left) and the solutions of three different hybrid dimensional models on a two phase flow fractured reservoir test case in 2D, where oil is injected in the bottom fracture. Formerly, I also worked on phase field models for non-isothermal Newtonian flow with regard to thermodynamical consistency and conducted an asymptotical analysis in order to derive the first order sharp interface limit. |
Current DFM models rely on ad hoc approximations in order to derive the matrix fracture coupling conditions. We work on a rigorous derivation of coupling conditions for DFM models of very general type, i.e. advection-diffusion-reaction in the fracture and even more general second order PDEs in the surrounding matrix domains. The derivation of coupling conditions relies on a Fourier transform of the physical unknowns in direction tangential to the fracture and, subsequently, on the elimination of the fracture unknowns’ Fourier coefficients by performing a continuous Schur complement. Reduced order coupling conditions can then be obtained up to a certain order of a given quantity (fracture width, conductivity, resistivity), by truncating the corresponding asymptotic expansions. The illustrations on the right hand side show solutions in the limit of vanishing fracture aperture for a given fixed resistivity (upper left), diffusive conductivity (upper right) and advective conductivity (lower figure). We compared the coupling conditions to a commonly used family of (diffusion) models from the literature and obtained correspondence for the coupling conditions truncated after the next-to-leading-order terms. In a next step, we give estimates for the error of the so derived approximate models and compare them to existing models in the literature. |
A complex model can only be solved efficiently by an adequate numerical scheme. The development of new models and of new numerical schemes goes hand in hand. For hybrid dimensional flow through fractured porous media for example, the scheme has to cope with anisotropic permeability tensors, with heterogeneous media, with polyhedral meshes and with the coupling of the fracture surface flow with the surrounding rock matrix. On the right hand side, you see the VAG fluxes, which are local to each cell/fracture face, for two different hybrid dimensional models. |
I think that numerical analysis is a great help in the conceptualization and understanding of numerical methods. In my research, I am concerned with the analysis of schemes in the general framework of the gradient discretization method, which is well suited to analyze the convergence of conforming and non conforming discretizations for linear and nonlinear second order diffusion and parabolic problems. The class of gradient discretizations contains various schemes such as Finite Element methods, Mixed and Mixed Hybrid Finite Element methods, some Finite Volume schemes like symmetric Multi Point Flux Approximation schemes, the Vertex Approximate Gradient (VAG) schemes and the Hybrid Mimetic Mixed schemes. The main advantage of this framework is to provide the convergence proof for all schemes satisfying some typical abstract conditions, such as coercivity, consistency, limit conformity, compactness. The image on the right hand side shows a test case, which demonstrates order one convergence for the VAG and HMM schemes in the discrete H^{1}-norm, a result for which we provide the first proof for a general fracture network in a three-dimensional surrounding matrix domain including fully, partially and non immersed fractures as well as fracture intersections. |
The concept of hierarchical matrix (H-matrix) compression relies on the possibility to partition a dense matrix into submatrices, which can be approximated by low rank matrices. It is well known that this approach is efficient for matrices that represent the inverse of a discretized differential operator with asymptotically smooth kernel, as it is the case for the Laplace operator. However, the method loses its efficiency for oscillatory kernels, as e.g. the kernel of the Helmholtz operator, which is not asymptotically smooth. In this case, it would be helpful to have an idea of the range of wave numbers, for which good results can be achieved by standard H-matrix techniques. I am currently working on estimtions for the rank of sub-blocks of the discretized Green's function for Helmholtz' equation in the high frequency domain. The picture on the right shows a matrix, which represents such a discretized Green's function. |