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Quasi K-orthogonal grid generation is presented, to improve grid quality and method stability with respect to flux approximation in the presence of strongly anisotropic full-tensor permeability fields.K-orthogonal grid generation is only possible for low anisotropy ratios. Quasi K-orthogonal grid generation involves satisfying the K-orthogonal condition approximately, resulting in grids that place less demand on an approximation with respect to stability conditions, and therefore improve grid quality with respect to flux approximation in the presence of anisotropic permeability fields. The method employed enables Delaunay grid generation principles to be employed in a locally transformed system according to local permeability tensor variation. The resulting method has great flexibility for handling complex geometries and can handle jumps in permeability tensor principal axes orientation and jumps in coefficients and details will be presented. Results are presented that demonstrate the benefit of a quasi K-orthogonal grid. Highly challenging cases involving strong full-tensor permeability fields where control-volume distributed multi-point flux approximation (CVD-MPFA) schemes exceed their stability limits and yield solutions with spurious oscillations when using conventional grids, are solved using the new grid generation method. CVD-MPFA schemes are still required as the grids are only approximately K-orthogonal in such cases, however the schemes retain a discrete maximum principle on the new quasi-K-orthogonal grids and yield well resolved solutions that are free of spurious oscillations. While the two-point flux approximation (TPFA) requires strict K-orthogonality, results using both CVD-MPFA and TPFA will be presented. New Quasi K-orthogonal grid generation methods are presented that satisfy the K-orthogonal condition approximately, resulting in practical grids that restore a discrete maximum principle (stability) for the CVD-MPFA schemes when applied to cases involving general full-tensor permeability fields. Results are presented for a variety of test cases that confirm the validity of the grids.
A novel discrete fracture model (DFM) approximation is presented and coupled with the control-volume distributed multi-point flux approximation (CVD-MPFA) formulation. The reduced-dimensional discrete discontinuous pressure model for intersecting fractures is extended to two-phase flow, including gravity and discontinuous capillary pressure. A novel higher resolution hybrid upwind method provides improved flow resolution on unstructured grids.
Novel discontinuous fracture models together with appropriate interface conditions, essential for application to cases involving continuous and discontinuous capillary pressure, and for fractures with permeability barrier effects are presented. The CVD-MPFA based discontinuous DFM models are coupled with higher resolution methods on unstructured grids, including an extended higher resolution hybrid upwind method for gravity driven flow and a novel higher resolution capillary flux approximation. A special DFM approximation is presented for fracture intersection cells located in flow fields where viscous and gravity forces interact.
Performance comparisons are presented for tracer and two-phase flow and fracture problems involving discontinuous capillary pressure and gravity on unstructured meshes. The results demonstrate the importance of the discontinuous DFM model to resolve flow problems including oil trapping in fractures. In addition comparison between the standard lower order method and the higher resolution hybrid upwind scheme shows that the higher resolution method yields significantly improved flow resolution in gravity driven flow fields.
A novel DFM approximation is presented and coupled with the CVD-MPFA formulation on unstructured grids. The method includes a discontinuous discrete fracture model with appropriate interface conditions for application to discontinuous capillary pressure fields, and a new method for treatment of intersecting fractures is also introduced for viscous-gravity driven flow. The method is also coupled with a higher resolution hybrid upwind scheme which yields improved flow resolution.
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. 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 the Barth-Jespersen 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. The method is also coupled with a discrete fracture model. Performance comparisons are presented for tracer and 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.
Three-dimensional unstructured grid generation for reservoirs with geological layers, faults, pinchouts, fractures and wells is presented. Grids are generated for example cases, and pressure fields and flow fields computed by the cell-centered and vertex-centered control-volume distributed multi-point flux approximation (CVD-MPFA) schemes are compared and contrasted together with the methods. Grid generation for reservoir simulation, must honour classical key geological features and multilateral wells. The geological features are classified into two groups; 1) involving layers, faults, pinchouts and fractures, and 2) involving well distributions. In the former, control-volume boundary aligned grids (BAGs) are required, while in the latter, control-point well aligned grids (WAGs) are required. In reservoir simulation a choice of grid type and consequent control-volume type is made, i.e. either primal or dual-cells are selected as control-volumes. The control-point is defined as the centroid of the control-volume for any grid type. Three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to wells, yielding essentially perpendicular bisector (PEBI) meshes either with respect to primal or dual-cells depending on grid type. Both primal and dual-cell boundary aligned grid generators use primal-cells (tetrahedra, pyramids, prisms and hexahedra) as grid elements. Dual-cell feature aligned grids are derived from underlying primal-meshes, such that features are recovered, with control-volume faces aligned with interior feature boundaries. The grids generated enable a comparative performance study of cell- vertex versus cell-centered CVD-MPFA finite-volume formulations using equivalent degrees of freedom. The benefits of both types of approximation are presented in terms of flow resolution relative to the respective degrees of freedom employed. Stability limits of the methods are also explored. For a given mesh the cell-vertex method uses approximately a fifth of the unknowns used by a cell-centered method and proves to be the most beneficial with respect to accuracy and efficiency, which is verified by flow computation. Novel techniques for generating three-dimensional unstructured hybrid essentially PEBI-grids, honouring geological features are presented. Geological boundary aligned grid generation is performed for primal and dual-cell grid types. Flow results show that vertex-centered CVD-MPFA methods outperform cell-centered CVD-MPFA methods.
A three-dimensional symmetric positive definite (SPD) cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) is presented for porous media flow simulation on tetrahedral grids. The scheme depends on a single degree of freedom per control-volume and is derived in physical space where the continuous fluxes are resolved directly along the face normals of the tetrahedra. We believe this is the first and possibly only general three-dimensional full-tensor finite-volume scheme that is flux-continuous and SPD in physical space, while depending on a single degree of freedom per control-volume. The equivalent general two-dimensional CVD-MPFA scheme that is SPD in physical space is presented in (Friis et al., 2008).
Wu, Minglu (China University of Petroleum, QingDao) | Ding, Mingcai (China University of Petroleum, QingDao) | Yao, Jun (China University of Petroleum, QingDao) | Li, Chenfeng (Swansea University) | Huang, Zhaoqin (China University of Petroleum, QingDao) | Xu, Sinan (China University of Petroleum, QingDao)
Summary A shale-gas reservoir with a multiple-fractured horizontal well (MFHW) is divided into two regions: The inner region is defined as stimulated reservoir volume (SRV), which is interconnected by the fracture network after fracturing, while the outer region is called unstimulated reservoir volume (USRV), which has not been stimulated by fracturing. Considering an arbitrary interface boundary between SRV and USRV, a composite model is presented for MFHWs in shale-gas reservoirs, which is based on multiple flow mechanisms, including adsorption/desorption, viscous flow, diffusive flow, and stress sensitivity of natural fractures. The boundary-element method (BEM) is applied to solve the production of MFHWs in shale-gas reservoirs. The accuracy of this model is validated by comparing its production solution with the result derived from an analytical method and the reservoir simulator. Furthermore, the practicability of this model is validated by matching the production history of the MFHW in a shale-gas reservoir. The result shows that the model in this work is reliable and practicable. The effects of relevant parameters on production curves are analyzed, including Langmuir volume, Langmuir pressure, hydraulic-fracture width, hydraulic-fracture permeability, natural-fracture permeability, matrix permeability, diffusion coefficient, stress-sensitivity coefficient, and the shape of the SRV. The model presented here can be used for production analysis for shale-gas-reservoir development.
In this paper, we investigate potential changes to hydrodynamics of a proposed tidal stream turbine energy generation site in the Inner Sound channel of Pentland Firth, located between mainland Scotland and Orkney Islands, using a state of the art coastal area model. The modelling results suggest that the selected site is very suitable for tidal stream energy harvesting however, some morphological changes should be anticipated in sedimentary deposits located close the proposed turbine array, which may have implications on benthic ecology of the area.
Marine renewable energy has attracted the interest of renewable energy developers worldwide. The British Government has pledged that a significant proportion of the United Kingdom's energy will be developed using renewable energy sources by 2050. Among many forms of marine renewables, tidal stream energy has been identified as the most reliable and sustainable energy resource (European Ocean Energy Association, 2012). As a result, the tidal stream energy sector is a rapidly growing industry and is moving fast towards deployment of large scale turbine arrays in numerous part of the UK waters.
Tidal stream energy harvesting is not without some implications on the environment. Especially, when energy harvesting is done at very large scale, changes may be expected to hydrodynamics of the far field areas as well as in and around energy farms. This may have some implications on the sedimentary and ecological environment, depending on the site conditions, the level of energy extraction and the sensitivity of the site to potential changes to the hydrodynamics (Shields, et al., 2009). Particular concerns are the extent to which the alterations of the tidal current regime following tidal stream energy extraction on sediment transport and sea bed morphodynamics. It is therefore important to investigate hydrodynamic climate, the energy resource and implication of energy extraction at sites potentially suitable for large scale turbine arrays, before design and implementation of large scale tidal energy harvesting schemes.
Ahmed, R. (Swansea University) | Edwards, M.G. (Swansea University) | Lamine, S. (Shell Global Solutions International B.V.) | Huisman, B.A.H. (Shell Global Solutions International B.V.) | Pal, M. (Maersk Oil and Gas A/S)
A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture-matrix simulations in three-dimensions. The grid is aligned with the fractures and barriers which are then modelled as lower-dimensional interfaces located between the matrix cells in the physical domain. The three-dimensional(3D) pressure equation is solved in the matrix domain coupled with a two-dimensional(2D) pressure equation solved over fracture networks via a 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 must be added to the lower-dimensional flow equation (called the transfer function). An additional transmission condition is used between matrix cells adjacent to low permeable fractures to link the velocity and pressure jump across the fractures. Numerical tests serve to assess the convergence and accuracy of the lower-dimensional fracture model for highly anisotropic fractures having different apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approach 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. Highly conductive fractures are modelled as lower-dimensional entities with continuous pressure across these leading to reduced local degrees of freedom for the cluster of cells. Moreover, we present 3D simulations involving geologically representative complex fracture networks.
A new robust metaheuristic optimization method, namely modified cuckoo search (MCS), is presented in this paper. MCS is inspired by breeding behavior of cuckoo birds and combined with Levy flight approach to efficiently search for optimal solutions. MCS is coupled with a filtering technique to provide the ability to handle nonlinear constraints. The filter-based MCS is efficient insofar as it provides a bias toward exploration during early generations allowing for global search and then shifts that bias toward exploitation at final generations allowing to search promising areas of the solution. This helps in finding feasible solutions at earlier search stages and consequently improves convergence rate.
Two example cases involving two-dimensional synthetic reservoir models are presented. The first case compares the performance of MCS to that of genetic algorithm (GA) to maximize oil recovery by optimizing the location of four injection wells. It is shown that MCS outperforms GA in terms of the optimal solution as well as the rate of convergence. The second case entails the use of filter-based MCS to maximize NPV under maximum water cut constraint at the production well. The results indicate the superior performance of the filter-based MCS as it was able to quickly find feasible solutions even though all previous initial solutions were infeasible. The incorporation of filtering technique allows to assess the sensitivity of the objective function to the constraint violation. This provides additional insights that can lead to better future planning.