We characterize the pore-scale fluid distributions, phase connectivity, and local capillary pressures during three-phase flow in a water-wet Berea sandstone sample. In this investigation, we use a set of x-ray micro-tomography images acquired during three-phase core-flooding experiments conducted on a miniature core sample. We use several image analysis techniques to analyze the pore-scale fluid occupancy maps and use this information to develop several insights related to pore occupancy, oil and gas cluster distribution, and interfacial curvature during the gas injection process. The results of our investigation show that the large-, intermediate-, and small-sized pores are mostly occupied with gas, oil, and brine, respectively, which is consistent with the wetting order of the fluids (i.e., gas, oil, and brine are the nonwetting, intermediate wetting, and wetting phases, respectively). In addition, the connectivity analysis reveals that a significant amount of the gas phase was in the form of disconnected ganglia separated from the connected invading cluster. The presence of these trapped nonwetting phase clusters during the drainage process is presumably attributed to Roof snap-off and Haines jump events, as well as the anti-ripening phenomenon. Moreover, the average local oil-water capillary pressures are found to be greater than the gas-oil counterparts. This observation is then related to the relative location of the interfaces in the pore space and the threshold capillary pressures at which the various displacement events take place.
We propose a simple post-processing technique for linear and high order continuous Galerkin Finite Element Methods (CGFEMs) to obtain locally conservative flux field. The post-processing technique requires solving an auxiliary problem on each element independently which results in solving a linear algebra system whose size is 1 2 (k + 1)(k + 2) for k th order CGFEM. The post-processing could have been done directly from the finite element solution that results in locally conservative flux on the element. However, the normal flux is not continuous at the element's boundary. To construct locally conservative flux field whose normal component is also continuous, we propose to do the post-processing on the nodal-centered control volumes which are constructed from the original finite element mesh. We show that the post-processed solution converges in an optimal fashion to the true solution in an H 1 semi-norm. We present various numerical examples to demonstrate the performance of the post-processing technique. Keywords CGFEM; FVEM; conservative flux; post-processing However, the current understanding and implementation of higher order FVEMs are still at its infancy and are not as satisfactory as linear FVEM. For one-dimensional elliptic equations, high order FVEMs have been developed in [29].Other relevant high order FVEM work can be found in [26,25,9,11,12].As mentioned, FVEM produces locally conservative fluxes while, due to the global formulation, CGFEMs do not.Robustness of the CGFEMs for any order has been established through extensive and rigorous error analysis, while this is not the case for FVEM. Development of linear algebra solvers for CGFEMs has reached an advanced stage, mainly driven from a solid understanding of the variational formulations and their properties, such as coercivity (and symmetry) of the bilinear form in the Galerkin formulation. On the other hand, the resulting linear algebra systems derived from FVEMs, especially high order FVEMs, are not that easy to solve. Typically, the matrices resulting from FVEMs are not symmetric even if the original boundary value problem is. Furthermore, at most FVEM discretization with linear finite element basis yields M-matrix, while with quadratic finite element basis it is not (see [26]).Preservation of numerical local conservation property of approximate solutions are imperative in simulations of many physical problems especially those that are derived from law of conservation. In order to maintain the advantages of CGFEM as well as to obtain locally conservative fluxes, post-processing techniques are developed; see [1,6,7,13,14,15,16,19,21,23,24,27,28,30,31,32,33,34,36]. The post-processing techniques proposed in the aforementioned references are mainly techniques for post-processing linear finite element related methods and they include finite element methods for solving pure elliptic equations, advection diffusion equations, advection dominated diffusion equations, elasticity problems, Stokes problem, etc. Among them, some of the proposed post-process...
In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm.
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