Publications
30 results found
Clare MCA, Percival JR, Angeloudis A, et al., 2021, Hydro-morphodynamics 2D modelling using a discontinuous Galerkin discretisation, Computers and Geosciences, Vol: 146, Pages: 1-13, ISSN: 0098-3004
The development of morphodynamic models to simulate sediment transport accurately is a challenging process that is becoming ever more important because of our increasing exploitation of the coastal zone, as well as sea-level rise and the potential increase in strength and frequency of storms due to a changing climate. Morphodynamic models are highly complex given the non-linear and coupled nature of the sediment transport problem. Here we implement a new depth-averaged coupled hydrodynamic and sediment transport model within the coastal ocean model Thetis, built using the code generating framework Firedrake which facilitates code flexibility and optimisation benefits. To the best of our knowledge, this represents the first full morphodynamic model including both bedload and suspended sediment transport which uses a discontinuous Galerkin based finite element discretisation. We implement new functionalities within Thetis extending its existing capacity to model scalar transport to modelling suspended sediment transport, incorporating within Thetis options to model bedload transport and bedlevel changes. We apply our model to problems with non-cohesive sediment and account for effects of gravity and helical flow by adding slope gradient terms and parametrising secondary currents. For validation purposes and in demonstrating model capability, we present results from test cases of a migrating trench and a meandering channel comparing against experimental data and the widely-used model Telemac-Mascaret.
Kramer S, Wilson C, Davies R, et al., 2020, FluidityProject/fluidity: New test cases "Analytical solutions for mantle flow in cylindrical and spherical shells"
This release adds new test cases described in the GMD paper "Analytical solutions for mantle flow in cylindrical and spherical shells"
Nunez Rattia JM, Percival J, Neethling S, et al., 2018, Modelling local scour near structures with combined mesh movement and mesh optimisation, Journal of Computational Physics, Vol: 375, Pages: 1220-1237, ISSN: 0021-9991
This paper develops a new implementation coupling optimisation-based anisotropic mesh adaptivity algorithms to a moving mesh numerical scour model, considering both turbulent suspended and bedload sediment transport. The significant flexibility over mesh structure and resolution, in space and time, that the coupling of these approaches provides makes this framework highly suitable for resolving individual marine structure scales with larger scale ocean dynamics. The use of mesh optimisation addresses the issue of poor mesh quality and/or inappropriate resolution that have compromised existing modelling approaches that apply mesh movement strategies alone, especially in the case of extreme scour. Discontinuous Galerkin finite element-based discretisation methods and a Reynolds Averaged Navier–Stokes-based turbulent modelling approach are used for the hydrodynamic fluid flow. In this work the model is verified in two dimensions for current-dominated scour near a horizontal pipeline. Combined adaptive mesh movement and anisotropic mesh optimisation is found to maintain both the quality and validity of the mesh in response to morphological bed evolution changes, even in the case where it is severely constrained by nearby structures.
McManus TM, Percival JR, Yeager BA, et al., 2017, Moving mesh methods in Fluidity and Firedrake, Archer report eCSE06-1
Adam A, Pavlidis D, Percival JR, et al., 2017, Dynamic Mesh Adaptivity for Immiscible Viscous Fingering, Pages: 788-802
The unstable displacement of one fluid by another in a porous medium occurs frequently in various branches of enhanced oil recovery. It is now well known that when the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive, particularly in 3D. We use IC-FERST, a novel control volume finite element (CVFE) code that uses dynamic mesh adaptivity on unstructured meshes to simulate 2D and 3D viscous fingering with higher accuracy and lower computational cost than conventional methods. We provide evidence that these unstructured mesh simulations in fact yield better results that are less influenced by grid orientation error than their structured counterparts. We also include the effect of capillary pressure and show three examples that are very challenging to simulate using more conventional approaches.
Xie Z, Pavlidis D, Salinas P, et al., 2016, A balanced-force control volume finite element method for interfacial flows with surface tension using adaptive anisotropic unstructured meshes, Computers and Fluids, Vol: 138, Pages: 38-50, ISSN: 0045-7930
A balanced-force control volume finite element method is presented for three-dimensional interfacial flows with surface tension on adaptive anisotropic unstructured meshes. A new balanced-force algorithm for the continuum surface tension model on unstructured meshes is proposed within an interface capturing framework based on the volume of fluid method, which ensures that the surface tension force and the resulting pressure gradient are exactly balanced. Two approaches are developed for accurate curvature approximation based on the volume fraction on unstructured meshes. The numerical framework also features an anisotropic adaptive mesh algorithm, which can modify unstructured meshes to better represent the underlying physics of interfacial problems and reduce computational effort without sacrificing accuracy. The numerical framework is validated with several benchmark problems for interface advection, surface tension test for equilibrium droplet, and dynamic fluid flow problems (fluid films, bubbles and droplets) in two and three dimensions.
Gomes JLMA, Pavlidis D, Salinas P, et al., 2016, A force-balanced control volume finite element method for multi-phase porous media flow modelling, International Journal for Numerical Methods in Fluids, Vol: 83, Pages: 431-445, ISSN: 1097-0363
A novel method for simulating multi-phase flow in porous media is presented. The approach is based on acontrol volume finite element mixed formulation and new force-balanced finite element pairs. The novelty ofthe method lies in: (a) permitting both continuous and discontinuous description of pressure and saturationbetween elements; (b) the use of arbitrarily high-order polynomial representation for pressure and velocityand (c) the use of high-order flux-limited methods in space and to time avoid introducing non-physicaloscillations while achieving high-order accuracy where and when possible. The model is initially validatedfor two-phase flow. Results are in good agreement with analytically obtained solutions and experimentalresults. The potential of this method is demonstrated by simulating flow in a realistic geometry composed ofhighly permeable meandering channels.
Savre J, Percival J, Herzog M, et al., 2016, Two-Dimensional Evaluation of ATHAM-Fluidity, a Nonhydrostatic Atmospheric Model Using Mixed Continuous/Discontinuous Finite Elements and Anisotropic Grid Optimization, Monthly Weather Review, Vol: 144, Pages: 4349-4372, ISSN: 0027-0644
This paper presents the first attempt to apply the compressible nonhydrostatic ATHAM-Fluidity solver to a series of idealized atmospheric test cases. ATHAM-Fluidity uses a hybrid finite-element discretization where pressure is solved on a continuous 2nd order grid while momentum and scalars are computed on a 1st order discontinuous grid (also known as 1DG–2). ATHAM-Fluidity operates on two- and three-dimensional unstructured meshes, using triangular or tetrahedral elements respectively, with the possibility to employ an anisotropic mesh optimization algorithm for automatic grid refinement and coarsening during run-time. The solver is evaluated using two-dimensional only dry idealized test cases covering a wide range of atmospheric applications. The first three cases, representative of atmospheric convection, reveal the ability of ATHAM-Fluidity to accurately simulate the evolution of large scale flow features in neutral atmospheres at rest. Grid convergence without adaptivity as well as the performances of the Hermite-WENO slope limiter are discussed. These cases are also used to test the grid optimisation algorithm implemented in ATHAM-Fluidity. Adaptivity can result in up to a six-fold decrease in computational time and a five-fold decrease in total element number for the same finest resolution. However, substantial discrepancies are found between the uniform and adapted grid results, thus suggesting the necessity to improve the reliability of the approach. In the last three cases, corresponding to atmospheric gravity waves with and without orography, the model ability to capture the amplitude and propagation of weak stationary waves is demonstrated. This work constitutes the first step towards the development of a new comprehensive limited area atmospheric model.
Nunez Rattia JM, Percival JR, Yeager B, et al., 2016, Numerical simulation of scour below pipelines using flexible mesh methods, The 8th International Conference on Scour and Erosion, Pages: 101-108
Evaluating bed morphological structure and evolution (specifically the scoured bed level) accurately using numerical models is critical for analyses of the stability of many marine structures. This paper discusses the performance of an implementation within Fluidity, an open source, general purpose, Computational Fluid Dynamics (CFD) code, capable of handling arbitrary multi-scale unstructured tetrahedral meshes and including algorithms to perform dynamic anisotropic mesh adaptivity. The flexibility over mesh structure and resolution that these capabilities provide makes it potentially highly suitable for coupling the structural scale with larger scale ocean dynamics. In this very preliminary study the solver approach is demonstrated for an idealised scenario. Discontinuous Galerkin finite-element (DG-FEM) based discretisation methods have been used for the hydrodynamics and morphological calculations, and automatic mesh deformation has been utilised to account for bed evolution changes while preserving the validity and quality of the mesh. In future work, the solver will be used in three-dimensional impinging jet and other industrial and environmental scour studies.
Adam A, Pavlidis D, Percival J, et al., 2016, Higher-order conservative interpolation between control-volume meshes: Application to advection and multiphase flow problems with dynamic mesh adaptivity, Journal of Computational Physics, Vol: 321, Pages: 512-531, ISSN: 1090-2716
A general, higher-order, conservative and bounded interpolation for the dynamic and adaptive meshing of control-volume fields dual to continuous and discontinuous finite element representations is presented. Existing techniques such as node-wise interpolation are not conservative and do not readily generalise to discontinuous fields, whilst conservative methods such as Grandy interpolation are often too diffusive. The new method uses control-volume Galerkin projection to interpolate between control-volume fields. Bounded solutions are ensured by using a post-interpolation diffusive correction. Example applications of the method to interface capturing during advection and also to the modelling of multiphase porous media flow are presented to demonstrate the generality and robustness of the approach.
Pavlidis D, Gomes JLMA, Xie Z, et al., 2016, Compressive advection and multi-component methods for interface-capturing, International Journal for Numerical Methods in Fluids, Vol: 80, Pages: 256-282, ISSN: 0271-2091
This paper develops methods for interface-capturing in multiphase flows. The main novelties of these methods are as follows: (a) multi-component modelling that embeds interface structures into the continuity equation; (b) a new family of triangle/tetrahedron finite elements, in particular, the P1DG-P2(linear discontinuous between elements velocity and quadratic continuous pressure); (c) an interface-capturing scheme based on compressive control volume advection methods and high-order finite element interpolation methods; (d) a time stepping method that allows use of relatively large time step sizes; and (e) application of anisotropic mesh adaptivity to focus the numerical resolution around the interfaces and other areas of important dynamics. This modelling approach is applied to a series of pure advection problems with interfaces as well as to the simulation of the standard computational fluid dynamics benchmark test cases of a collapsing water column under gravitational forces (in two and three dimensions) and sloshing water in a tank. Two more test cases are undertaken in order to demonstrate the many-material and compressibility modelling capabilities of the approach. Numerical simulations are performed on coarse unstructured meshes to demonstrate the potential of the methods described here to capture complex dynamics in multiphase flows.
Adam AG, Pavlidis D, Percival JR, et al., 2016, Simulation of immiscible viscous fingering using adaptive unstructured meshes and controlvolume galerkin interpolation
Displacement of one fluid by another in porous media occurs in various settings including hydrocarbon recovery, CO2 storage and water purification. When the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive. We demonstrate here the use of a novel control volume - finite element (CVFE) method along with dynamic unstructured mesh adaptivity to simulate viscous fingering with higher accuracy and lower computational cost than conventional methods. Our CVFE method employs a discontinuous representation for both pressure and velocity, allowing the use of smaller control volumes (CVs). This yields higher resolution of the saturation field which is represented CV-wise. Moreover, dynamic mesh adaptivity allows high mesh resolution to be employed where it is required to resolve the fingers and lower resolution elsewhere. We use our results to re-examine the existing criteria that have been proposed to govern the onset of instability. Mesh adaptivity requires the mapping of data from one mesh to another. Conventional methods such as collocation interpolation do not readily generalise to discontinuous fields and are non-conservative. We further contribute a general framework for interpolation of CV fields by Galerkin projection. The method is conservative, higher order and yields improved results, particularly with higher order or discontinuous elements where existing approaches are often excessively diffusive.
Adam AG, Pavlidis D, Percival JR, et al., 2016, Simulation of immiscible viscous fingering using adaptive unstructured meshes and controlvolume galerkin interpolation
Displacement of one fluid by another in porous media occurs in various settings including hydrocarbon recovery, CO2 storage and water purification. When the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive. We demonstrate here the use of a novel control volume - finite element (CVFE) method along with dynamic unstructured mesh adaptivity to simulate viscous fingering with higher accuracy and lower computational cost than conventional methods. Our CVFE method employs a discontinuous representation for both pressure and velocity, allowing the use of smaller control volumes (CVs). This yields higher resolution of the saturation field which is represented CV-wise. Moreover, dynamic mesh adaptivity allows high mesh resolution to be employed where it is required to resolve the fingers and lower resolution elsewhere. We use our results to re-examine the existing criteria that have been proposed to govern the onset of instability. Mesh adaptivity requires the mapping of data from one mesh to another. Conventional methods such as collocation interpolation do not readily generalise to discontinuous fields and are non-conservative. We further contribute a general framework for interpolation of CV fields by Galerkin projection. The method is conservative, higher order and yields improved results, particularly with higher order or discontinuous elements where existing approaches are often excessively diffusive.
Jackson MD, Percival JR, Mostaghiml P, et al., 2015, Reservoir modeling for flow simulation by use of surfaces, adaptive unstructured meshes, and an overlapping-control-volume finite-element method, SPE Reservoir Evaluation and Engineering, Vol: 18, Pages: 115-132, ISSN: 1094-6470
We present new approaches to reservoir modeling and flow simulation that dispose of the pillar-grid concept that has persisted since reservoir simulation began. This results in significant improvements to the representation of multiscale geologic heterogeneity and the prediction of flow through that heterogeneity. The research builds on more than 20 years of development of innovative numerical methods in geophysical fluid mechanics, refined and modified to deal with the unique challenges associated with reservoir simulation.Geologic heterogeneities, whether structural, stratigraphic, sedimentologic, or diagenetic in origin, are represented as discrete volumes bounded by surfaces, without reference to a predefined grid. Petrophysical properties are uniform within the geologically defined rock volumes, rather than within grid cells. The resulting model is discretized for flow simulation by use of an unstructured, tetrahedral mesh that honors the architecture of the surfaces. This approach allows heterogeneity over multiple length-scales to be explicitly captured by use of fewer cells than conventional corner-point or unstructured grids.Multiphase flow is simulated by use of a novel mixed finite-element formulation centered on a new family of tetrahedral element types, PN(DG)–PN+1, which has a discontinuous Nth-order polynomial representation for velocity and a continuous (order N +1) representation for pressure. This method exactly represents Darcy-force balances on unstructured meshes and thus accurately calculates pressure, velocity, and saturation fields throughout the domain. Computational costs are reduced through dynamic adaptive-mesh optimization and efficient parallelization. Within each rock volume, the mesh coarsens and refines to capture key flow processes during a simulation, and also preserves the surface-based representation of geologic heterogeneity. Computational effort is thus focused on regions of the model where it is most required.After valid
Mostaghimi P, Percival JR, Pavlidis D, et al., 2015, Anisotropic Mesh Adaptivity and Control Volume Finite Element Methods for Numerical Simulation of Multiphase Flow in Porous Media, MATHEMATICAL GEOSCIENCES, Vol: 47, Pages: 417-440, ISSN: 1874-8961
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- Citations: 35
Salinas P, Percival J, Pavlidis D, et al., 2015, A discontinuous overlapping control volume finite element method for multi-phase porous media flow using dynamic unstructured mesh optimization, SPE Reservoir Simulation Symposium
Su K, Latham J-P, Pavlidis D, et al., 2015, Multiphase flow simulation through porous media with explicitly resolved fractures, Geofluids, Vol: 15, Pages: 592-607, ISSN: 1468-8123
Accurate simulation of multiphase flow in fractured porous media remains a challenge. An important problem is the representation of the discontinuous or near discontinuous behaviour of saturation in real geological formations. In the classical continuum approach, a refined mesh is required at the interface between fracture and porous media to capture the steep gradients in saturation and saturation-dependent transport properties. This dramatically increases the computational load when large numbers of fractures are present in the numerical model. A discontinuous finite element method is reported here to model flow in fractured porous media. The governing multiphase porous media flow equations are solved in the adaptive mesh computational fluid dynamics code IC-FERST on unstructured meshes. The method is based on a mixed control volume – discontinuous finite element formulation. This is combined with the PN+1DG-PNDG element pair, which has discontinuous (order N+1) representation for velocity and discontinuous (order N) representation for pressure. A number of test cases are used to evaluate the method's ability to model fracture flow. The first is used to verify the performance of the element pair on structured and unstructured meshes of different resolution. Multiphase flow is then modelled in a range of idealised and simple fracture patterns. Solutions with sharp saturation fronts and computational economy in terms of mesh size are illustrated.
Salinas P, Percival JR, Pavlidis D, et al., 2015, A new approach to reservoir modeling and simulation using boundary representation, adaptive unstructured meshes and the discontinuous overlapping control volume finite element method, Pages: 241-245
We present a new, high-order, control-volume-finite-element (CVFE) method with discontinuous Nthorder representation for pressure and (N+1)th-order for velocity. The method conserves mass and ensures that the extended Darcy equations for multi-phase flow are exactly enforced, but does not require the use of control volumes (CVs) that span domain boundaries. We demonstrate that the approach, amongst other features, accurately preserves sharp saturation changes associated with high aspect ratio geologic features such as fractures and mudstones, allowing efficient simulation of flow in highly heterogeneous models. Moreover, in conjunction with dynamic mesh optimization, in which the mesh adapts in space and time to key solution fields such as pressure, velocity or saturation whilst honoring a surface-based representation of the underlying geologic heterogeneity, accurate solutions are obtained at significantly lower computational cost than an equivalent fine, fixed mesh and conventional CVFE methods. The work presented is significant for two reasons. First, it resolves a longstanding problem associated with the use of classical CVFE methods to model flow in highly heterogeneous porous media; second, it reduces computational cost/increases solution accuracy through the use of dynamic mesh optimization without compromising parallelization.
Xie Z, Pavlidis D, Percival JR, et al., 2014, Adaptive unstructured mesh modelling of multiphase flows, International Journal of Multiphase Flow, Vol: 67, Pages: 104-110, ISSN: 0301-9322
© 2013 Elsevier Ltd. Multiphase flows are often found in industrial and practical engineering applications, including bubbles, droplets, liquid film and waves. An adaptive unstructured mesh modelling framework is employed here to study interfacial flow problems, which can modify and adapt anisotropic unstructured meshes to better represent the underlying physics of multiphase problems and reduce computational effort without sacrificing accuracy. The numerical framework consists of a mixed control volume and finite element formulation, a 'volume of fluid'-type method for the interface capturing based on a compressive control volume advection method and second-order finite element methods. The framework also features a force-balanced algorithm for the surface tension implementation, minimising the spurious velocities often found in such flows. Numerical examples of the Rayleigh-Taylor instability and a rising bubble are presented to show the ability of this adaptive unstructured mesh modelling framework to capture complex interface geometries and also to increase the efficiency in multiphase flow simulations.
Xie Z, Pavlidis D, Percival JR, et al., 2014, Adaptive unstructured mesh modelling of multiphase flows, International Journal of Multiphase Flow, Vol: 67, Pages: 104-110, ISSN: 0301-9322
Multiphase flows are often found in industrial and practical engineering applications, including bubbles, droplets, liquid film and waves. An adaptive unstructured mesh modelling framework is employed here to study interfacial flow problems, which can modify and adapt anisotropic unstructured meshes to better represent the underlying physics of multiphase problems and reduce computational effort without sacrificing accuracy. The numerical framework consists of a mixed control volume and finite element formulation, a 'volume of fluid'-type method for the interface capturing based on a compressive control volume advection method and second-order finite element methods. The framework also features a force-balanced algorithm for the surface tension implementation, minimising the spurious velocities often found in such flows. Numerical examples of the Rayleigh-Taylor instability and a rising bubble are presented to show the ability of this adaptive unstructured mesh modelling framework to capture complex interface geometries and also to increase the efficiency in multiphase flow simulations.
Percival JR, Pavlidis D, Xie Z, et al., 2014, Control volume finite element modelling of segregation of sand and granular flows in fluidized beds, INTERNATIONAL JOURNAL OF MULTIPHASE FLOW, Vol: 67, Pages: 191-199, ISSN: 0301-9322
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- Citations: 5
Pavlidis D, Xie Z, Percival JR, et al., 2014, Two- and three-phase horizontal slug flow simulations using an interface-capturing compositional approach, INTERNATIONAL JOURNAL OF MULTIPHASE FLOW, Vol: 67, Pages: 85-91, ISSN: 0301-9322
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- Citations: 26
Che Z, Fang F, Percival J, et al., 2014, An ensemble method for sensor optimisation applied to falling liquid films, International Journal of Multiphase Flow, Vol: 67, Pages: 153-161, ISSN: 1879-3533
Multiphase flow problems are often extremely complex due to their strong nonlinearity. To study multiphase flow, it is important to simulate or measure key parameters accurately, such as pressure drops and flow rates. Therefore, it is essential to place the sensors at the locations with high impact, and to avoid locations with low impact, where impact is determined by a function such as one of the key variables like pressure drop or flow rate. In this paper, an ensemble method is used to optimise sensor locations for falling film problems based on an importance map. The importance map can identify the important regions according to a target function. The sensor locations are selected based on the importance map, the variation of the variables, and the costs of performing the measurements. We demonstrate the approach by applying data assimilation and show that the optimised sensor locations can significantly improve the data assimilation results. Through sensitivity analysis, sensor optimisation, and data assimilation, this study, for the first time, provides a systematic linkage between the experiments and the models for falling film problems. It also presents a new goal or target based method for sensor placement. This method can be extended to other complex multiphase flow problems.
Sakai M, Abe M, Shigeto Y, et al., 2014, Verification and validation of a coarse grain model of the DEM in a bubbling fluidized bed, CHEMICAL ENGINEERING JOURNAL, Vol: 244, Pages: 33-43, ISSN: 1385-8947
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- Citations: 177
Xie Z, Pavlidis D, Percival JR, et al., 2014, Adaptive unstructured mesh modelling of multiphase flows, International Journal of Multiphase Flow, ISSN: 0301-9322
Multiphase flows are often found in industrial and practical engineering applications, including bubbles, droplets, liquid film and waves. An adaptive unstructured mesh modelling framework is employed here to study interfacial flow problems, which can modify and adapt anisotropic unstructured meshes to better represent the underlying physics of multiphase problems and reduce computational effort without sacrificing accuracy. The numerical framework consists of a mixed control volume and finite element formulation, a 'volume of fluid'-type method for the interface capturing based on a compressive control volume advection method and second-order finite element methods. The framework also features a force-balanced algorithm for the surface tension implementation, minimising the spurious velocities often found in such flows. Numerical examples of the Rayleigh-Taylor instability and a rising bubble are presented to show the ability of this adaptive unstructured mesh modelling framework to capture complex interface geometries and also to increase the efficiency in multiphase flow simulations.
Jackson MD, Gomes JLMA, Mostaghimi P, et al., 2013, Reservoir modeling for flow simulation using surfaces, adaptive unstructured meshes and control-volume-finite-element methods, Pages: 774-792
We present new approaches to reservoir modeling and flow simulation that dispose of the pillar-grid concept that has persisted since reservoir simulation began. This results in significant improvements to the representation of multi-scale geological heterogeneity and the prediction of flow through that heterogeneity. The research builds on 20+ years of development of innovative numerical methods in geophysical fluid mechanics, refined and modified to deal with the unique challenges associated with reservoir simulation. Geological heterogeneities, whether structural, stratigraphic, sedimentologic or diagenetic in origin, are represented as discrete volumes bounded by surfaces, without reference to a pre-defined grid. Petrophysical properties are uniform within the geologically-defined rock volumes, rather than within grid-cells. The resulting model is discretized for flow simulation using an unstructured, tetrahedral mesh that honors the architecture of the surfaces. This approach allows heterogeneity over multiple length-scales to be explicitly captured using fewer cells than conventional corner-point or unstructured grids. Multiphase flow is simulated using a novel mixed finite element formulation centered on a new family of tetrahedral element types, PN(DG)-PN+1, which has a discontinuous Nth-order polynomial representation for velocity and a continuous (order N+1) representation for pressure. This method exactly represents Darcy force balances on unstructured meshes and thus accurately calculates pressure, velocity and saturation fields throughout the domain. Computational costs are reduced through (i) efficient parallelization and (ii) automatic mesh adaptivity in time and space. Within each rock volume, the mesh coarsens and refines to capture key flow processes, whilst preserving the surface-based representation of geological heterogeneity. Computational effort is thus focused on regions of the model where it is most required. Having validated the approach aga
Holm DD, Ivanov RI, Percival JR, 2012, <i>G</i>-Strands, JOURNAL OF NONLINEAR SCIENCE, Vol: 22, Pages: 517-551, ISSN: 0938-8974
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- Citations: 11
Cotter CJ, Holm DD, Ivanov RI, et al., 2011, Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 44, ISSN: 1751-8113
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- Citations: 13
Cotter CJ, Holm DD, Percival JR, 2010, The square root depth wave equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 466, Pages: 3621-3633, ISSN: 1364-5021
We introduce a set of coupled equations for multi-layer water waves that removes the ill-posedness of the multi-layer Green–Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear, they retain the same travelling wave solutions as MGN. We call the new model the square root depth (Inline Formula) equations from the modified form of their kinetic energy of vertical motion. Our numerical results show how the Inline Formula equations model the effects of multi-layer wave propagation and interaction, with and without shear.
Percival JR, Cotter CJ, Holm DD, 2008, A Euler–Poincaré framework for the multilayer Green–Nagdhi equations, Meeting held in Honor of Darryl D Holms on Geometry and Analysis in Physical Systems, Publisher: IOP Publishing, Pages: 344018-344031, ISSN: 1751-8113
The Green–Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler–Poincaré theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.
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