Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Grid generation for reservoir simulation must honor classical key constraints and ensure boundary alignment such that control-volume boundaries are aligned with geological features including layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. Novel unstructured grid generation methods are proposed that automate control-volume and/or control point boundary alignment and yield perpendicular-bisector (PEBI) meshes both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles that contain segments of key geological boundaries, while in order to generate a dual-cell feature aligned grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex flux-continuous control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom and thus ensure application of the most efficient methods. The CVD-MPFA formulation (c.f. Edwards et al.) in cell-centred and cell-vertex modes is somewhat analogous and requires switching control-volume from primal to dual or vice versa, together with appropriate data structures and boundary conditions, however dual-cells are generated after primal grid generation. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are contrasted in terms of flow resolution and degrees of freedom required

Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model

A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture–matrix simulations on unstructured grids in three-dimensions (3D). The grid is aligned with fractures and barriers which are then modelled as lower-dimensional surface interfaces located between the matrix cells in the physical domain. The three-dimensional pressure equation is solved in the matrix domain coupled with a two-dimensional (2D) surface pressure equation solved over fracture networks via a novel surface CVD-MPFA formulation. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix–fracture fluxes are expressed in terms of matrix and fracture pressures and define the transfer function, which is added to the lower-dimensional flow equation and couples the three-dimensional and surface systems. An additional transmission condition is used between matrix cells adjacent to low permeable fractures to couple the velocity and pressure jump across the fractures. Convergence and accuracy of the lower-dimensional fracture model is assessed for highly anisotropic fractures having a range of apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approximation for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretisation. Lower-dimensional fracture model results are compared with equi-dimensional model results. Fractures and barriers are efficiently modelled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Pressure continuity is built into the model across highly conductive fractures, leading to reduced local degrees of freedom in the CVD-MPFA approximation. The formulation is applied to geologically complex fracture networks in three-dimensions. The effects of the fracture permeability, aperture and grid resolution are also assessed with respect to convergence and computational cost

Multidimensional upwind schemes and higher resolution methods for three-component two-phase systems including gravity driven flow in porous media on unstructured grids

Standard reservoir simulation schemes employ single-point upstream weighting for approximation of the convective fluxes when multiple phases or components are present. These schemes introduce both coordinate-line numerical diffusion and crosswind diffusion into the solution that is grid and geometry dependent.Families of locally conservative multidimensional upwind schemes are presented for essentially hyperbolic three-component two-phase flow systems of conservation laws in porous media including counter current gravity flow on unstructured grids. The multidimensional methods employ cell-based tracing, which involves tracing characteristic wave directions over each control-volume subquadrant. The multidimensional methods reduce crosswind diffusion inherent in standard methods for convective flow approximation in porous media. The schemes are coupled with continuous Darcy-flux approximations resulting from the elliptic pressure equation on unstructured grids.Characteristic upwind approximations are proposed and compared with the classical upstream weighting schemes for cases including gravity segregated flow. When dealing with systems of hyperbolic equations, upwind characteristic wave decomposition is used for wave tracing. The multidimensional upwind cell-based tracing formulations are designed for unstructured grids (and include structured grids by default) and are stable subject to conditions on the tracing direction and CFL number and satisfy a local maximum principle that ensures solutions are free of spurious oscillations.Benefits of the resulting schemes are demonstrated for two-phase flow and a three-component two-phase flow system including gravity segregated flow. The multidimensional cell based schemes are shown to reduce crosswind diffusion induced by standard upwind methods, and prove to be particularly effective when flow is strongly non-aligned with the grid, leading to improved resolution of numerical saturation and concentration fronts. Extension of higher order schemes to a three-component two-phase flow systems of conservation laws on unstructured grids is also presented, which provides a significant improvement in flow resolution for the system cases. Comparison is drawn between the methods

Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model

A novel cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture-(rock)matrix flow simulations. The grid is aligned with the fractures and barriers which are then modeled by lower-dimensional interfaces located between rock matrix cells in the physical domain. The n D (n-dimension) pressure equation in the rock matrix is coupled with the (n−1)D pressure equation in the fractures, leading to the first reduced dimensional flux-continuous CVD-MPFA formulation. This formulation naturally handles fractures efficiently on unstructured grids. Matrix-fracture fluxes are expressed in terms of matrix and fracture pressures, resulting in a transfer function, which is added to the lower-dimensional flow equation. An additional transmission condition is used between matrix cells separated by low permeable fractures to couple the velocity and pressure jump across the fractures. Numerical tests serve to assess the convergence and accuracy of the lower-dimensional fracture model for lower anisotropic fractures having a range of apertures and permeability tensors. A tracer flow transport equation is solved for problems with single and intersecting fractures. A lower-dimensional mass balance for intersecting fracture cells circumvents the more restrictive CFL condition corresponding to standard equi-dimensional approximation with explicit time discretization. Lower-dimensional fracture model results are compared with hybrid-grid and equi-dimensional model results. Fractures and barriers are efficiently modeled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Highly conductive fractures are modeled as lower-dimensional entities without the use of locally refined grids that are required by the equi-dimensional model, while pressure continuity across fractures is built into the model, without depending on the extra degrees of freedom which must be added locally by the hybrid-grid method. The lower-dimensional fracture model also yields improved results when compared to those of the hybrid-grid model for fractures with low-permeability in the normal direction to the fracture where pressure is discontinuous. In addition, transient pressure simulation involving geologically representative complex fracture networks is presented

Multiscale Finite-Volume CVD-MPFA Formulations on Structured and Unstructured Grids

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values

A study of methods of measurement of the electric charge on a rocket and of ambient electric fields using probe techniques

A summary is given of literature on causes of rocket charge, ambient fields, and on probe techniques. A variety of techniques are available for determining rocket charge, represented by a potential relative to space of 0 to 10 volts. A cylindrical probe characteristic, analysed by the older Engel and Steenback methods, was obtained for a pulse discharge with a low electron concentration of 2 x 10(^7) electrons/c.c, in the author’s experiments. The probe sheath was collision free, and conditions for such sheaths in the upper atmosphere are given. The predicted ambient field is about .02 mV/cm, fields measured being .60 mV/cm. (by a probe technique) and 200 V/m (by field meters). The author’s tentative opinion is that the ambient field might be measured by a simultaneous determination of potential at two points using probes. Laboratory simulation seems to require a low voltage gradient discharge, (preferably with ions and electrons in thermal equilibrium,) because two probes cannot be placed as close as a Debye length. The contract specifically excluded development of circuitry so a precise answer cannot be given, but definite proposals are made. In the experiment, the measured voltage gradient was about 30 V/cm, which seems abnormally high at a discharge current of about 10(^-4) amps when compared with recent results for steady glow discharges. The explanation may reside in the existence of striations. From the work recently reported in a Czechoslovakian journal it might be argued that the probe is responsible foe the striations. No material is included oh re-entry physics, nor on the time and space variations of electric fields. A study of the latter should be related to a precise knowledge of the rocket motion. Tine research represents a branch of Atmospheric Electricity which had not previously been studied at Durham