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Magdalena Toda

Publications and source records attributed to Magdalena Toda.

29 records · Page 2Linked to original sources
arXiv2017

Fracture Model Reduction and Optimization for Forchheimer Flows in Reservoir

In this study, we analyze the flow filtration process of slightly compressible fluids in fractured porous media. We model the coupled fractured porous media system, where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. Flow in the fracture is modeled as a reduced low dimensional BVP which is coupled with an equation in the reservoir. We prove that the solution of the reduced model can serve very accurately to approximate the solution of the actual high-dimensional flow in reservoir fracture system, because the thickness of the fracture is small. In the analysis we consider two types of Forchhemer flows in the fracture: isotropic and anisotropic, which are different in their nature. Using method of reduction, we developed a formulation for an optimal design of the fracture, which maximizes the capacity of the fracture in the reservoir with fixed geometry. Our method, which is based on a set point control algorithm, explores the coupled impact of the fracture geometry and beta-Forchheimer coefficient.

math.AP↗
arXiv2013

Geometric framework for modeling nonlinear flows in porous media, and its applications in engineering

This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an important tool for evaluating technological parameters in reservoir engineering.

math.DG↗
arXiv2013

Geometric Methods in the Analysis on Non-linear Flows in Porous Media (Preliminary Version)

Over the past few years, we developed a mathematically rigorous method to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. We have proved the existence of a geometric transformation which relates constant mean curvature surfaces and time-invariant pressure distribution graphs constrained by the Darcy-Forchheimer law. We therein established a direct relationship between the CMC graph equation and a certain family of equations which we call $g$-Forchheimer equations. The corresponding results, on fast flows and their geometric interpretation, can be used as analytical tools in evaluating important technological parameters in reservoir engineering.

math.DG↗
arXiv2013

Weingarten surfaces with moving frames -- a tribute to S.S. Chern and C.L. Terng -- and a duality result

The techniques used in this paper are based on the exterior calculus of Maurer-Cartan forms, and Weingarten surfaces are used to illustrate the methods that apply to quadratic exterior equations with constant coefficients. Isothermic {\it surfaces of constant astigmatism} (non-linear Weingarten surfaces whose difference of principal curvatures is a constant) are shown to represent dual surfaces of isothermic surfaces which satisfy the relation $H + αK = 0$.

math.DG↗
arXiv2013

Constructing isothermal curvature line coordinates on surfaces which admit them

Isothermic parameterizations} are synonyms of isothermal curvature line parameterizations, for surfaces immersed in Euclidean spaces. We provide a method of constructing isothermic coordinate charts on surfaces which admit them, starting from an arbitrary chart. One of the primary applications of this work consists of numerical algorithms for surface visualization.

math.DG↗
arXiv2005

Initial Value Problems of the Sine-Gordon Equation and Geometric Solutions

Recent results using inverse scattering techniques interpret every solution $ϕ(x,y)$ of the sine-Gordon equation as a non-linear superposition of solutions along the axes $x=0$ and $y=0$. Here we provide a geometric method of integration, as well as a geometric interpretation. Specifically, every weakly regular surface of Gauss curvature $K=-1$, in arc length asymptotic line parametrization, is uniquely determined by the values $ϕ(x,0)$ and $ϕ(0,y)$ of its coordinate angle along the axes. Based on a generalized Weierstrass pair that depends only on these values, we prove that to each such unconstrained pair of differentiable functions, there corresponds uniquely an associated family of pseudospherical immersions; we construct these immersions explicitely.

math.DG↗
arXiv2004

The Hyperbolic Geometry of the Sinh-Gordon Equation

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different perspective: the equivalence of their Weierstrass representations (normalized potentials). This work also presents a construction algorithm for the moving frame, the adjusted frame, their Maurer-Cartan forms, and ultimately the CMC immersion.

math.DG↗
arXiv2004

Timelike Minimal Surfaces via Loop Groups

We study manifolds with split-complex structure and apply some general results to the study of Lorentz surfaces. In particular, we apply our results to timelike minimal immersions. The conformal realization of these surfaces is obtained using a representation based on loop groups. The classical Weierstrass representation (integral formula) is recovered as a byproduct of this general setting.

math.DG↗
arXiv2003

Weierstrass-type Representation of Weakly Regular Pseudospherical Surfaces in Euclidean Space

The paper presents a generalized Weierstrass representation for pseudospherical surfaces in terms of 3x3 matrices, using moving frames and loop group decompositions. The construction of all such surfaces, starting from a given representation, constitutes the subject of a separate, ulterior report. The appendix of the present work gives an elementary proof of the Birkhoff decomposition theorem for "loop-groups" of spectral parameter based on the real line, instead of the unit circle.

math.DG↗