Accurate Multi-Phase Flow Simulation in Faulted Reservoirs Using Mimetic Finite Difference Methods on Polyhedral Cells

Faulted reservoirs are commonly modeled by corner-point grids. Since the two-point flux approximation (TPFA) method is not consistent on non-orthogonal grids, multi-phase flow simulation using TPFA on corner-point grids may have significant discretization errors if grids are not K-orthogonal. To improve the simulation accuracy, we developed a novel method where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular orthogonal grid is first constructed,and then fault planes are added by dividing cells at fault planes. Most cells remain orthogonal while irregular non-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to part of the domain while the TPFA method is applied to rest of the domain. We compared flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid while the same problem is solved by TPFA and MFD methods on a grid with distorted cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault while fluxes computed using MFD are still as accurate as the reference solution. We also compared saturation accuracy of two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when pressure equation is solved by the TPFA method. Since the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions match the reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.

Unified analysis for variational time discretizations of higher order and higher regularity applied to non-stiff ODEs

We present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin–Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results.

Solution of the Black-Scholes Equation: LDM and SDM

The main aim of this paper is to discuss a new way of a non-discretization method for the solution of the Black-Scholes equation. Black-Scholes is a mathematical model based on a partial differential equation. The solution of the model is of utmost importance in financial mathematics to estimate option pricing. Several analytical, numerical, and non-discretization methods are existing in the literature to solve the model. Two decomposition methods namely the Laplace decomposition method (LDM) and Sumudu decomposition method (SDM) are adopted for the present study. The results of the present techniques have closed an agreement with an approximate solution which has been obtained with the help of the Adomian Decomposition Method (ADM).

Toward Accurate Reservoir Simulations on Unstructured Grids: Design of Simple Error Estimators and Critical Benchmarking of Consistent Discretization Methods for Practical Implementation

Summary In this paper, we aim to identify discretization errors caused by non-K-orthogonal grids upfront through simple preprocessing tools and perform a comparative study of a set of representative, state-of-the-art, consistent discretizations [multipoint flux approximation (MPFA-O), mimetic finite difference (MFD), nonlinear two-pointflux approximation (NTPFA, TPFA), and average multipoint flux approximation (AvgMPFA)] to select the method most suited for inclusion in a commercial reservoir simulator. To predict the potential impact of discretization errors, we propose two types of error indicators. Static indicators measure the degree of nonconsistency of the two-point method at a cell level, and dynamic indicators measure how local discretization errors affect flow paths. The latter are computed using a series of idealized tracer simulations. By changing monitoring and injection points, one can mimic the reservoir-development strategy and thus focus on the errors introduced on quantities of real interest. To assess the practical usability of various consistent methods and validate our new error indicators, we use a set of representative grid models generated by contemporary commercial tools, for which we discuss static error indicators and compare tracer responses for the various discretization methods. We also compare degrees of freedom, sparsity, and the condition number of the alternative methods and discuss challenges related to their practical implementation. Our results indicate that tracer simulations constitute an efficient tool to identify and classify discretization errors and quantify their potential impact. We observe distinctively different behavior with the inconsistent two-point method and the consistent methods, which agree closely in terms of accuracy of the response. We also note a deficiency in the commercial realization of so-called Depogrids, which can result in unnecessarily complicated polytopal cells with hundreds of faces. Our overall conclusion is that NTPFA and AvgMPFA are the most viable solutions for integration into a commercial simulator, with the linear AvgMPFA method being the least invasive.

Effect of grey-level discretization on texture feature on different weighted MRI images of diverse disease groups

Purpose Many studies of MRI radiomics do not include the discretization method used for the analyses, which might indicate that the discretization methods used are considered irrelevant. Our goals were to compare three frequently used discretization methods (lesion relative resampling (LRR), lesion absolute resampling (LAR) and absolute resampling (AR)) applied to the same data set, along with two different lesion segmentation approaches. Methods We analyzed the effects of altering bin widths or bin numbers for the three different sampling methods using 40 texture indices (TIs). The impact was evaluated on brain MRI studies obtained for 71 patients divided into three different disease groups: multiple sclerosis (MS, N = 22), ischemic stroke (IS, N = 22), cancer patients (N = 27). Two different MRI acquisition protocols were considered for all patients, a T2- and a post-contrast 3D T1-weighted MRI sequence. Elliptical and manually drawn VOIs were employed for both imaging series. Three different types of gray-level discretization methods were used: LRR, LAR and AR. Hypothesis tests were done among all diseased and control areas to compare the TI values in these areas. We also did correlation analyses between TI values and lesion volumes. Results In general, no significant differences were reported in the results when employing the AR and LAR discretization methods. It was found that employing 38 TIs introduced variation in the results when the number of bin parameters was altered, suggesting that both the degree and direction of monotonicity between each TI value and binning parameters were characteristic for each TI. Furthermore, while TIs were changing with altering binning values, no changes correlated to neither disease nor the MRI sequence. We found that most indices correlated weakly with the volume, while the correlation coefficients were independent of both diseases analyzed and MR contrast. Several cooccurrence-matrix based texture parameters show a definite higher correlation when employing the LRR discretization method However, with the best correlations obtained for the manually drawn VOI. Hypothesis tests among all disease and control areas (co-lateral hemisphere) revealed that the AR or LAR discretization techniques provide more suitable texture features than LRR. In addition, the manually drawn segmentation gave fewer significantly different TIs than the ellipsoid segmentations. In addition, the amount of TIs with significant differences was increasing with increasing the number of bins, or decreasing bin widths. Conclusion Our findings indicate that the AR discretization method may offer the best texture analysis in MR image assessments. Employing too many bins or too large bin widths might reduce the selection of TIs that can be used for differential diagnosis. In general, more statistically different TIs were observed for elliptical segmentations when compared to the manually drawn VOIs. In the texture analysis of MR studies, studies and publications should report on all important parameters and methods related to data collection, corrections, normalization, discretization, and segmentation.