flux approximation
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Modern biomedical and tribological systems are increasingly deploying combinations of nanofluids and bioconvecting microorganisms which enable improved control of thermal management. Motivated by these developments, in this study, a new mathematical model is developed for the combined nanofluid bioconvection axisymmetric squeezing flow between rotating circular plates (an important configuration in, for example, rotating bioreactors and lubrication systems). The Buongiorno two-component nanoscale model is deployed, and swimming gyrotactic microorganisms are considered which do not interact with the nanoparticles. Thermal radiation is also included, and a Rosseland diffusion flux approximation is utilized. Appropriate similarity transformations are implemented to transform the nonlinear, coupled partial differential conservation equations for mass, momentum, energy, nanoparticle species and motile microorganism species under suitable boundary conditions from a cylindrical coordinate system into a dimensionless nonlinear ordinary differential boundary value problem. An efficient scheme known as differential transform method (DTM) combined with Padé-approximations is then applied to solve the emerging nonlinear similarity equations. The impact of different non-dimensional parameters i.e. squeezing Reynolds number, rotational Reynolds number, Prandtl number, thermophoresis parameter, Brownian dynamics parameter, thermal radiation parameter, Schmidt number, bioconvection number and Péclet number on velocity, temperature, nanoparticle concentration and motile gyrotactic microorganism density number distributions is computed and visualized graphically. The torque effects on both plates, i.e. the lower and the upper plate, are also determined. From the graphical results, it is seen that momentum in the squeezing regime is suppressed clearly as the upper disk approaches the lower disk. This inhibits the axial flow and produces axial flow retardation. Similarly, by enhancing the value of squeezing Reynolds number, the tangential velocity distribution also decreases. More rigorous squeezing clearly therefore also inhibits tangential momentum development in the regime and leads to tangential flow deceleration. Tables are also provided for multiple values of flow parameters. The numerical values obtained by DTM-Padé computation show very good agreement with shooting quadrature. DTM-Padé is shown to be a precise and stable semi-numerical methodology for studying rotating multi-physical flow problems. Radiative heat transfer has an important influence on the transport characteristics. When radiation is neglected, different results are obtained. It is important therefore to include radiative flux in models of rotating bioreactors and squeezing lubrication dual disk damper technologies since high temperatures associated with radiative flux can impact significantly on combined nanofluid bioconvection which enables more accurate prediction of actual thermofluidic characteristics. Corrosion and surface degradation effects may therefore be mitigated in designs.

Abstract The process of coupled flow and mechanics occurs in various environmental and energy applications, including conventional and unconventional fractured reservoirs. This work establishes a new formulation for modeling hydro-mechanical coupling in fractured reservoirs. The discrete-fracture model (DFM), in which the porous matrix and fractures are represented explicitly in the form of unstructured grid, has been widely used to describe fluid flow in fractured formations. In this work, we extend the DFM approach for modeling coupled flow-mechanics process, in which flow problems are solved using the multipoint flux approximation (MPFA) method, and mechanics problems are solved using the multipoint stress approximation (MPSA) method. The coupled flow-mechanics problems share the same computational grid to avoid projection issues and allow for convenient exchange between them. We model the fracture mechanical behavior as a two-surface contact problem. The resulting coupled system of nonlinear equations is solved in a fully-implicit manner. The accuracy and generality of the numerical implementation are accessed using cases with analytical solutions, which shows an excellent match. We then apply the methodology to more complex cases to demonstrate its general applicability. We also investigate the geomechanical influence on fracture permeability change using 2D rock fractures. This work introduces a novel formulation for modeling the coupled flow-mechanics process in fractured reservoirs, and can be readily implemented in reservoir characterization workflow.

Abstract An unstructured grid generation method is presented that automates control-volume boundary alignment to geological objects and control point alignment to complex wells. The grid generation method is coupled with an iterative acute mesh reconstruction technique, to construct essentially acute triangulations, while satisfying quite general geometric constraints. For well aligned grids control points are constrained to the well trajectory and protection circles are used, whereas for boundary aligned grids halo construction is performed. Unstructured Delaunay triangulations (DT) have the desirable locally orthogonal perpendicular bisectional (PEBI) property, required by the industry standard two-point flux approximation for consistency for isotropic fields. The PEBI property can only be exploited if such grids are comprised of acute simplexes, with circumcentres located inside their respective elements. The method presented enables acute DT layered mesh generation while honoring internal boundaries and wells in a two dimensional space. A dual (Voronoi) grid derived from a feature honored simplicial mesh is then projected in the vertical direction generating 2.5D PEBI grids. Effectiveness of the method to construct acute PEBI grids honoring geological objects and complex wells is demonstrated by meshing representative reservoir geometries. Numerical results are presented that verify consistency of the two-point flux on the resulting boundary-aligned acute PEBI grids. Development of an unstructured grid generation method which 1) Automates interior boundary alignment, 2) Honors features with respect to control point and/or control volume, and 3) Generates acute PEBI grids for reservoir geometries is presented. A unique workflow is presented to generate boundary aligned acute PEBI grids for complex geometries. Development of boundary aligned grids that honor both geological objects and multilateral complex wells, together with mesh reconstruction to ensure circumcenter containment is presented. Further, 3D PEBI grid generation method which can limit refinement to well perforations and geological objects is also described.

Abstract Solving the equations governing multiphase flow in geological formations involves the generation of a mesh that faithfully represents the structure of the porous medium. This challenging mesh generation task can be greatly simplified by the use of unstructured (tetrahedral) grids that conform to the complex geometric features present in the subsurface. However, running a million-cell simulation problem using an unstructured grid on a real, faulted field case remains a challenge for two main reasons. First, the workflow typically used to construct and run the simulation problems has been developed for structured grids and needs to be adapted to the unstructured case. Second, the use of unstructured grids that do not satisfy the K-orthogonality property may require advanced numerical schemes that preserve the accuracy of the results and reduce potential grid orientation effects. These two challenges are at the center of the present paper. We describe in detail the steps of our workflow to prepare and run a large-scale unstructured simulation of a real field case with faults. We perform the simulation using four different discretization schemes, including the cell-centered Two-Point and Multi-Point Flux Approximation (respectively, TPFA and MPFA) schemes, the cell- and vertex-centered Vertex Approximate Gradient (VAG) scheme, and the cell- and face-centered hybrid Mimetic Finite Difference (MFD) scheme. We compare the results in terms of accuracy, robustness, and computational cost to determine which scheme offers the best compromise for the test case considered here.

Abstract A novel higher resolution spectral volume method coupled with a control-volume distributed multi-Point flux approximation (CVD-MPFA) is presented on unstructured triangular grids for subsurface reservoir simulation. The flow equations involve an essentially hyperbolic convection equation coupled with an elliptic pressure equation resulting from Darcy’s law together with mass conservation. The spectral volume (SV) method is a locally conservative, efficient high-order finite volume method for convective flow. In 2D geometry, the triangular cell is subdivided into sub-cells, and the average state variables in the sub-cells are used to reconstruct a high-order polynomial in the triangular cell. The focus here is on an efficient strategy for reconstruction of both a higher resolution approximation of the convective transport flux and Darcy-flux approximation on sub-cell interfaces, which is also coupled with a discrete fracture model. The strategy involves coupling of the SV method and reconstructed CVD-MPFA fluxes at the faces of the spectral volume, to obtain an efficient finer scale higher resolution finite-volume method which solves for both the saturation and pressure. A limiting procedure based on a Barth-Jespersen type limiter is used to prevent non-physical oscillations on unstructured grids. The fine scale saturation/concentration field is then updated via the reconstructed finite volume approximation over the sub-cell control-volumes. Performance comparisons are presented for two phase flow problems on 2D unstructured meshes including fractures. The results demonstrate that the spectral-volume method achieves further enhanced resolution of flow and fronts in addition to that of achieved by the standard higher resolution method over first order upwind, while improving upon efficiency.

Abstract Combining the Perpendicular Bisector (PEBI) grids with the Two Point Flux Approximation (TPFA) scheme demonstrates a potential to accurately model on unstructured grids, conforming to the geological and engineering features of real grids. However, with the increased complexity and resolution of the grids, the PEBI conditions will inevitably be violated in some cells and the approximation properties will be compromised. The objective is to develop accurate and practical grid quality measures that quantify such errors. We critically evaluated the existing grid quality measures and found them lacking predictive power in several areas. The available k-orthogonality measures predict error for flow along the strata, although TPFA provides an accurate approximation. The false-positive results are not only misleading but can overwhelm further analysis. We developed the so-called "truncation error" grid measure which is probably the most accurate measure for flow through a plane face and accurately measures the error along the strata. We also quantified the error due to the face curvature. Curved faces are bound to exist in any real grid. The impact of the quality of the 2-D Delaunay triangulation on TPFA approximation properties is usually not taken into account. We investigate the impact of the size of the smallest angles that can cause considerable increase of the condition number of the matrix and an eventual loss of accuracy, demonstrated with simple examples. Based on the analysis, we provide recommendations. We also show how the size of the largest angles impacts the approximation quality of TPFA. Furthermore, we discuss the impact of the change of the permeability on the TPFA approximation. Finally, we present simple tools that reservoir engineers can use to incorporate the above-mentioned grid quality measures into a workflow. The grid quality measures discussed up to now are static. We also sketch the further extension to dynamic measures, that is, how the static measures can be used to detect change in the flow behavior, potentially leading to increased error. We investigate a comprehensive set of methods, several of them new, to measure the static grid quality of TPFA on PEBI grids and possible extension to dynamic measures. All measures can be easily implemented in production reservoir simulators and examined using the suggested tools in a workflow.

Abstract Mechanical deformation induced by injection and withdrawal of fluids from the subsurface can significantly alter the flow paths in naturally fractured reservoirs. Modeling coupled fluid-flow and mechanical deformation in fractured reservoirs relies on either sophisticated gridding techniques, or enhancing the variables (degrees-of-freedom) that represent the physics in order to describe the behavior of fractured formation accurately. The objective of this study is to develop a spatial discretization scheme that cuts the "matrix" grid with fracture planes and utilizes traditional formulations for fluid flow and geomechanics. The flow model uses the standard low-order finite-volume method with the Compartmental Embedded fracture Model (cEDFM). Due to the presence of non-standard polyhedra in the grid after cutting/splitting, we utilize numerical harmonic shape functions within a Polyhedral finite-element (PFE) formulation for mechanical deformation. In order to enforce fracture-contact constraints, we use a penalty approach. We provide a series of comparisons between the approach that uses conforming Unstructured grids and a Discrete Fracture Model (Unstructured DFM) with the new cut-cell PFE formulation. The manuscript analyzes the convergence of both methods for linear elastic, single-fracture slip, and Mandel’s problems with tetrahedral, Cartesian, and PEBI-grids. Finally, the paper presents a fully-coupled 3D simulation with multiple inclined intersecting faults activated in shear by fluid injection, which caused an increase in effective reservoir permeability. Our approach allows for great reduction in the complexity of the (gridded) model construction while retaining the solution accuracy together with great saving in the computational cost compared with UDFM. The flexibility of our model with respect to the types of grid polyhedra allows us to eliminate mesh artifacts in the solution of the transport equations typically observed when using tetrahedral grids and two-point flux approximation.

Abstract We develop a collocated Finite Volume Method (FVM) to study induced seismicity as a result of pore pressure fluctuations. A discrete system is obtained based on a fully-implicit coupled description of flow, elastic deformation, and contact mechanics at fault surfaces on a fully unstructured mesh. The cell-centered collocated scheme leads to convenient integration of the different physical equations, as the unknowns share the same discrete locations on the mesh. Additionally, a multi-point flux approximation is formulated in a general procedure to treat heterogeneity, anisotropy, and cross-derivative terms for both flow and mechanics equations. The resulting system, though flexible and accurate, can lead to excessive computational costs for field-relevant applications. To resolve this limitation, a scalable parallel solution algorithm is developed and presented. Several proof-of-concept numerical tests, including benchmark studies with analytical solutions, are investigated. It is found that the presented method is indeed accurate, stable and efficient; and as such promising for accurate and efficient simulation of induced seismicity.

AbstractIn this paper, we introduce a special kind of finite volume method called Multi-Point Flux Approximation method (MPFA) to price European and American options in two dimensional domain. We focus on the L-MPFA method for space discretization of the diffusion term of Black–Scholes operator. The degeneracy of the Black-Scholes operator is tackled using the fitted finite volume method. This combination of fitted finite volume method and L-MPFA method coupled to upwind methods gives us a novel scheme, called the fitted L-MPFA method. Numerical experiments show the accuracy of the novel fitted L-MPFA method comparing to well known schemes for pricing options.

Summary A class of monotone cell-centered nonlinear finite-volume methods has been proposed in the past decade to solve the anisotropic diffusion equation. The nonlinear two-point flux approximation (TPFA) (NTPFA) method preserves the nonnegativity of the solution values but can violate the discrete maximum/minimum principle (DMP). To enforce DMP, the nonlinear multipoint flux approximation (NMPFA) method ought to be used. In this work, we propose a novel NTPFA method that can reduce the severity of DMP violations significantly compared with the standard NTPFA method. The new formulation uses conormal decomposition for the construction of the one-sided fluxes. To define the unique flux through a connection between two cells, we choose a convex combination of the two one-sided fluxes such that the absolute differences of the magnitudes of the two transmissibility terms associated with the two neighboring cells are minimized, thus bringing the discrete coefficient matrix closer to having the zero row-sum property. Numerical experiments are conducted to test the performance of the new NTPFA method. The results demonstrate that the new scheme has comparable convergence order for both the solution and the flux compared with the standard NTPFA method or the classical multi-point flux approximation (MPFA-O) method. Moreover, the new NTPFA formulation shows marked improvements over the standard NTPFA in terms of reducing DMP violations. However, depending on the specific problem configuration, our new NTPFA formulation can lead to a system of nonlinear equations that is more difficult to solve.