By Matthias Albert Augustin
This monograph specializes in the numerical tools wanted within the context of constructing a competent simulation software to advertise using renewable power. One very promising resource of strength is the warmth kept within the Earth’s crust, that's harnessed by way of so-called geothermal amenities. Scientists from fields like geology, geo-engineering, geophysics and particularly geomathematics are known as upon to aid make geothermics a competent and secure strength creation strategy. one of many demanding situations they face contains modeling the mechanical stresses at paintings in a reservoir.
The goal of this thesis is to enhance a numerical resolution scheme by way of which the fluid strain and rock stresses in a geothermal reservoir should be decided sooner than good drilling and through construction. For this goal, the tactic may still (i) comprise poroelastic results, (ii) offer a way of together with thermoelastic results, (iii) be reasonably cheap when it comes to reminiscence and computational strength, and (iv) be versatile with reference to the destinations of knowledge points.
After introducing the fundamental equations and their kin to extra known ones (the warmth equation, Stokes equations, Cauchy-Navier equation), the “method of primary strategies” and its strength price touching on our activity are mentioned. in keeping with the homes of the basic ideas, theoretical effects are validated and numerical examples of tension box simulations are awarded to evaluate the method’s functionality. The first-ever 3D pics calculated for those subject matters, which neither requiring meshing of the area nor regarding a time-stepping scheme, make this a pioneering volume.
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Extra resources for A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs
The existence and uniqueness of solutions for this case is shown in . 51). 49b). It can be considered as part of the right-hand side term h. Moreover, we assume homogeneous Dirichlet boundary conditions. Inhomogeneous Dirichlet boundary values can be treated similarly. ˝/, respectively. We assume u 2 V, p 2 Q. Please note that throughout this section, v is used to denote test functions which are always in V and should neither be confused with the deformation velocity vs of an elastic solid, nor with the fluid velocity vf which we used in the previous section.
A; b/ 1 1 open interval and t be a C -regular motion. a; b// and B ˝ be open with a piecewise C1 -regular boundary @B. 81) is the so-called material time derivative. 1] and [11, page 3]. 50, and a reference configuration B is crucial here. If, for example, in the settings of fluid mechanics, the fluid is allowed to leave the volume Bt , the velocity that has to be used in the Transport Theorem may differ from the velocity of the fluid. 5 Differential Equations Assume we have an open bounded domain ˝ k 1 k Rn F W Rn ::: Rn R Rn , n 2 N, n > 1, and a map ˝ !
When there is a function of x as coefficient of the time derivative term. , they are of one of these types on certain subdomains of ˝. 36 2 Preliminaries (ii) Not all linear second order PDEs are of one of the above classes for n > 2. Indeed, the system of equations which we discuss in this thesis is of neither of the above types, but has similarities with some elliptic and parabolic equations. We are interested in finding solutions of differential equations. Usually, we have some more information than just the differential equation itself in the form of initial and boundary conditions.