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ECMOR XIII - 13th European Conference on the Mathematics of Oil Recovery
- Conference date: 10 Sep 2012 - 13 Sep 2012
- Location: Biarritz, France
- ISBN: 978-90-73834-30-9
- Published: 10 September 2012
61 - 80 of 114 results
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Comparison of a Finite Element Method and a Finite Volume Method for Flow on General Grids in 3D
Authors H. Hægland, I. Aavatsmark, C. Guichard, R. Masson and R. KaufmannWe compare the recently developed Vertex Approximate Gradient (VAG) scheme developed in [R. Eymard et al., ESAIM: Mathematical Modelling and Numerical Analysis, 46(2), 2012] and the multipoint flux approximations (MPFA) O- and L-methods on 3D irregular meshes. It is found that the VAG scheme converges for a wider range of problems than the MPFA methods, however when the MPFA-methods converge, the convergence rate in flux is better than for the VAG method.
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A Monotone Non-linear Finite Volume Method for Advection-diffusion Equations and Multiphase Flows
Authors K. Nikitin and Y. VassilevskiWe present a new nonlinear monotone finite volume method for diffusion and convection-diffusion equations and its application to two-phase black oil models. We consider full anisotropic discontinuous diffusion/permeability tensors and discontinuous velocity fields on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional 7-point stencil for cubic meshes and diagonal tensors. The approximation of the advective flux is based on the second-order upwind method with the specially designed minimal nonlinear correction. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water-breakthrough time. We compare the new nonlinear two-point flux discretization with the conventional linear two-point scheme. The new nonlinear scheme has a number of important advantages over the traditional linear discretization. First, it demonstrates low sensitivity to grid distortions. Second, it provides appropriate approximation in the case of full anisotropic permeability tensor. For non-orthogonal grids or full anisotropic permeability tensors the conventional linear scheme provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional dicretization, yet it is rather competitive.
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Voronoi Grids Conformal to 3D Structural Features
Authors R. Merland, B. Lévy and G. CaumonWhen simulating flow in a reservoir, errors due to upscaling can have a significant impact on the quality of results. To reduce these errors, the cells of the simulation grid should be as homogeneous as possible, hence conform to horizons and faults. In this paper, we optimize the coordinates of the 3D Voronoi seeds so that cell facets honor the structural features. These features are modeled by piecewise linear complex (PLC). The optimization consists in minimizing a function made of two parts: • A barycentric function, called Centroidal Voronoi Tessellation (CVT) function, which ensures that the cells will be of good quality by maximizing their compactness. • A conformal function, which measures the proportion of cells that is on the "wrong side" of the structural features (if the cell is cut in two by a structural feature, the "good side" contains the Voronoi seed). The novelty in this paper concerns the method of cutting cells by structural features which are locally approximated inside the Voronoi cells. These methods used jointly with an adaptive gradient solver allow dealing with complex 3D geological cases, presented in the paper.
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Adaptive Fully Implicit Multi-scale Meshless Multi-point Flux Method for Fluid Flow in Heterogeneous Porous Media
By A. LukyanovA sequential fully implicit multi-scale meshless multi-point flux method (MS-MMPFA) for nonlinear hyperbolic partial differential equations of fluid flow in heterogeneous porous media is described in this paper. The method extends the recently proposed the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media [Lukyanov, “Meshless Upscaling Method and its Application to a Fluid Flow in Porous Media”, Proceeding ECMOR XII, 2010] by utilizing advantages of the existing multi-scale finite volume (MSFV) schemes. The MMPFA is based on a gradient approximation commonly used in meshless method and combined with the mixed corrections which ensure linear completeness. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The MMPFA method was successfully tested for a number of problems where it was clearly shown that the MMPFA gives a good agreement with analytical solutions for a given number of particles. However, the level of detail and range of property variability included in reservoir characterization models leads to a large number of particles to be considered in MMPFA method. In this paper this problem is resolved using a sequential fully implicit MS-MMPFA method. The results are presented, discussed.
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CVD-MPFA Based Multiscale Formulation on Structured and Unstructured Grids
Authors E.T. Parramore, M. G. Edwards, M. Pal and S. LamineSubsurface reservoirs generally have complex geological and geometrical features, such as faults fractures, pinchouts, shales and layers defined on varying length scales. In addition the effect of heterogeneity leads to further multiscale features that cannot be modelled with desired precision on relatively coarse meshes. This has lead to development of multiscale methods over the last decade. This paper focuses on methods for fine scale modelling and presents development of multiscale methods in an unstructured grid framework with particular emphasis on the numerical flux approximation. 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 pressure and rock properties sharing the same location in a given control-volume and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale formulation applicable to both structured and unstructured grids in two-dimensions. Performance of the Darcy-flux approximations are compared in the multiscale modelling environment on a range of grid types resulting from both structured and unstructured grids. The methods are applied to domains with homogeneous and heterogeneous permeability fields involving a range of test cases. The effects of quadrature range of the schemes is tested. Boundary condition constraints and consequences of basis function formulation, together with implications of scheme and grid type are presented. The development of a CVD-MPFA based multiscale formulation leads to a novel approach for fine scale modelling. The results demonstrate the benefits of the new formulation.
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Multiscale Method for Two and Three-phase Flow Simulation in Subsurface Petroleum Reservoirs
Authors M. Pal, S. Lamine, K.A. Lie and S. KrogstadMultiscale simulation is a new and promising approach that enables simulation of detailed geological model and the retention of level of detail and heterogeneity that would not be possible via conventional upscaling methods. Most multiscale methods are developed from a sequential formulation, in which flow (pressure-flux) and transport (saturation) equations are solved in separate steps. The flow equation is solved using a set of special multiscale basis functions that attempt to incorporate the effects of sub-grid geological heterogeneity into a global flow equation formulated on a coarsened grid. The multiscale basis functions are computed numerically by solving local flow problems, and can be used to construct conservative fluxes on the coarsened as well as the original fine grid. Herein, we consider one particular multiscale method, the multiscale mixed finite-element method, and discuss how it can be extended to account for capillary pressure effects. The method is evaluated for computational efficiency and accuracy on a series of models with a high degree of realism, including spatially dependent relative permeability and capillary effects, gravity, and highly heterogeneous rock properties specified on representative corner-point grids.
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A Framework for Hybrid Simulations of Two-phase Flow in Porous Media
Authors I. Lunati, P. Tomin, A. Ferrari and R. KuenzeIn the last decade multiscale methods have proven efficient in solving large reservoir-scale problems with satisfactory accuracy. Computational efficiency is achieved by splitting the original problem into a set of local problems coupled through a global coarse problem. Although these techniques are usually employed for problems in which the fine-scale processes are described by Darcy’s law, they can also be applied to pore-scale simulations and used as a mathematical framework for hybrid methods that couples a Darcy and pore scales. In this work, we consider a pore-scale description of fine-scale processes. The Navier-Stokes equations are numerically solved in the pore geometry to compute the velocity field and obtain generalized permeabilities. In the case of two-phase flow, the dynamics of the phase interface is described by the volume of fluid method with the continuum surface force model. The MsFV method is employed to construct an algorithm that couples a Darcy macro-scale description with a pore-scale description at the fine scale. The hybrid simulations results presented are in good agreement with the fine-scale reference solutions. As the reconstruction of the fine-scale details can be done adaptively, the presented method offers a flexible framework for hybrid modeling.
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An Unconditionally Stable Splitting Method Using Reordering for Simulating Polymer Injection
Authors H. M. Nilsen, K.A. Lie, A.F. Rasmussen and X. RaynaudWe present an unconditionally stable algorithm for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a two-phase system with rock compressibility and equal fluid compressibilities. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the time step. The resulting nonlinear system of discrete transport equations can, in the absence of gravity and capillary forces, be permute to lower triangular form by using a simple topological sorting method from graph theory. This gives a nonlinear Gauss--Seidel method that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of scalar nonlinear equations that can easily be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in runtimes and increases the feasible time-step sizes. Hence, sequential splitting combined with standard upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.
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Algebraic Multiscale Linear Solver for Heterogeneous Elliptic Problems
Authors Y. Wang, H. Hajibeygi and H. TchelepiAn Algebraic Multiscale Solver (AMS) for the pressure system of equations arising from incompressible flow in heterogeneous porous media is developed. The algorithm allows for several independent preconditioning stages to deal with the full spectrum of errors. In addition to the fine-scale system of equations, AMS requires information about the superimposed (dual) coarse grid to construct a wirebasket reordered system. The primal coarse grid is used in the construction of a conservative coarse-scale operator and in the reconstruction of a conservative fine-scale velocity field. The convergence properties of AMS are studied for various combinations including (1) the MultiScale Finite-Element (MSFE) method, (2) the MultiScale Finite-Volume (MSFV) method, (3) Correction Functions (CF), (4) Block Incomplete LU factorization with zero fill-in (BILU), and (5) point-wise Incomplete LU factorization with zero fill-in (ILU). The reduced-problem boundary condition, which is used for localization, is investigated. For a wide range of test cases, the performance of the different preconditioning options is analyzed. It is found that the best overall performance is obtained by combining MSFE and ILU as the global and local preconditioners, respectively. Comparison between AMS and the widely used SAMG solver illustrates that they are comparable, especially for very large heterogeneous problems.
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How Fast Is Your Newton-Like Nonlinear Solver?
Authors R.M. Younis and H.A. TchelepiThis work answers the question for any Newton-like solver that is applied to nonlinear residual systems arising during the course of implicit Reservoir Simulations. We start by developing a mathematical foundation that characterizes the asymptotic convergence rate of infinite dimensional Newton methods applied to continuous form reservoir simulation problems. Using the fact that finite dimensional (discretized) methods are related to their infinite dimensional counterparts through the approximation accuracy of the underlying numerical discretization scheme, we translate the infinite dimensional characterizations to the finite dimensional world. The analysis reveals the asymptotic scaling relations between nonlinear convergence rate and time-step and mesh size. In particular, we show a constant scaling relation for elliptic problems, a set of super-linear relations for hyperbolic situations, and for mixed parabolic problems. Numerical examples are used to illustrate the theoretical results, and we compare the direct convergence results from this work to those obtained using existing convergence monitoring methods. This work should be of interest to any simulation practitioner or developer who previously relied on text-book quadratic local convergence rate characterizations that did not hold in simulation practice and that perhaps are never even observed. The practical applications of this work are in time-step selection for convergence, generalizing single cell safeguarding tactics, and building insight into asymptotic acceleration methods.
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Fast Linear Solver for Pressure Computation in Layered Domains
Authors P. van Slingerland and C. VuikAccurate simulation of fluid pressures in layered reservoirs with strong permeability contrasts is a challenging problem. For this purpose, the Discontinuous Galerkin (DG) method has become increasingly popular. Unfortunately, standard linear solvers are usually too inefficient for the aforementioned application. To increase the efficiency of the Conjugate Gradient (CG) method for linear systems resulting from Symmetric Interior Penalty (discontinuous) Galerkin (SIPG) discretizations, we have cast an existing two-level preconditioner into the deflation framework. The main idea is to use coarse corrections based on the DG solution with polynomial degree p=0. This paper provides a numerical comparison of the performance of both two-level methods in terms of scalability and overall efficiency. Furthermore, it studies the influence of the SIPG penalty parameter, the smoother, damping of the smoother, and the strategy for solving the coarse systems. We have found that the penalty parameter can best be chosen diffusion-dependent. In that case, both two-level methods yield fast and scalable convergence. Whether preconditioning or deflation is to be favored depends on the choice for the smoother and on the damping of the smoother. Altogether, both two-level methods can contribute to faster and more accurate fluid pressure simulations.
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Coupled Geomechanics and Flow in Fractured Porous Media
Authors T. T. Garipov, K.A. Levonyan, M. Karimi-Fard and H.A. TchelepiThe effects of geomechanics on the reservoir response can be important, and this is especially true for naturally fractured formations. Modeling the mechanical deformation of naturally fractured formations poses significant numerical challenges, and accurate coupling between mechanical deformation and flow adds to the challenge. We describe a simulation framework for coupled mechanics and flow based on a Discrete Fracture Model (DFM). An important aspect is that the mechanics and flow problems share the same unstructured DFM grid. The geomechanical model is based on the classical Biot theory. The Barton-Bandis model is used to describe the fracture mechanical response. For the flow problem, we use Darcy’s law and mass conservation for slightly compressible fluids. The fractured formation is discretized using DFM, which leads to complex unstructured grids. Three standard elements (hexahedrons, tetrahedrons and wedges) are used to represent the volumes of the matrix, and the fractures are represented using lower dimensional objects (triangles or quadrangles). The Galerkin finite-element method is used for the mechanics, and a DFM finite-volume method is used the flow equations. Two different coupling strategies are considered: the fully implicit method and the fixed-stress sequential-implicit scheme. Several examples of fractured porous media are used to illustrate our methodology.
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Coupled Flow-deformation Simulations of Realistic Hydraulic Fractured Systems
Authors A.A. Rodriguez, H. Florez and J. MonteagudoAccurate modeling of fractures growth / propagation and their induced perturbation in the stress field suggests the need for coupled flow and fracture mechanics simulations. In order to tackle these challenges, an integrated workflow that considers multiple complex non-planar fractures within a coupled simulation framework will be presented here. A symmetric Galerkin Boundary Element Method (SGBEM) developed by Rungamornrat et al. (SPE 96968), which treats the elasticity problems arising from the presence of a fracture in an unbounded domain, is used to simulate fracture evolution. Fractures generated by the SGBEM are gridded using a triangular mesh and embeded inside a box where boundary conditions for both flow and mechanics are imposed. Using the surface mesh and a triangulation of the box are used as constraints to the volume discretization. In this work we perform calculations of the fracture stress shadow using a FEM approach along the volume tetrahedral grid described above. This is done by spliting the nodes that lie on the fracture an imposing the corresponding displacement boundary conditions in agreement with the results obtained from the SGBEM code. Flow calculations are performed using a control volume finite element approach which allows the incorporation of discrete fractures.
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Stress Dependent Anisotropy of Relative Permeabilities in Naturally Fractured Reservoirs
Authors P. Lang, S. Steinecker, S. Bazr Afkan and S.K. MatthäiRelative permeabilities of fracture networks as used in dual-continua simulations determine predicted producer behavior and ultimately a field’s achievable recovery. We present numerically derived ensemble (upscaled) relative permeability curves as obtained from discrete fracture and matrix (DFM) imbibition simulations. Our flow simulations are based on unstructured finite element grids and fully capable to account for capillary forces which determine the fluid transfer between fractures and adjacent matrix. Joint aperture distributions are obtained for various trends of maximum horizontal stress using finite element analysis assuming a matrix obeying linear-elasticity and accounting for fracture dilation due to normal stress and displacement. Results obtained from two-phase flow simulations show that relative permeability curves for the case of dominant fracture flow and medium to high flow rates cannot be matched by conventional analytic relationships. A strong anisotropy of relative permeability curve is found - not only as a result of fracture set orientation and degree of percolation, but very much due to the stress dependent ratio between matrix and fracture flow. This result reflects the ability of displacing phase to invade small fractures dependent on stress induced opening/closing. Fracture surface area where capillary transfer processes take place hence strongly depends on stress orientation.
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Numerical Convergence Study of Iterative Coupling for Coupled Flow and Geomechanics
Authors M.F. Wheeler, A.M. Mikelić and B.W. WangIn this paper we consider algorithms that will enable scientists and engineers to readily model complex processes in porous media taking into account fluid motion and the accompanying solid deformations. Numerous field applications would benefit from a better understanding and integration of porous flow and solid deformation. Important applications in environmental engineering and petroleum engineering include carbon sequestration, surface subsidence, pore collapse, cavity generation, hydraulic fracturing, thermal fracturing, wellbore collapse, sand production, fault activation, and waste disposal, while similar issues arise in biosciences and chemical sciences as well. Here we consider solving iteratively the coupling of flow and mechanics. We employ mixed finite element method for flow and a continuous Galerkin method for elasticity. For single phase flow, we demonstrate the convergence and convergence rates for two widely used schemes, the undrained split and the fixed stress split. We discuss the extension of the fixed stress iterative coupling scheme to an equation of state compositional flow model coupled with elasticity and a single phase poroelasticity model on general hexahedral grids. Computational results are presented which include parallel simulation of carbon sequestration in saline aquifer, and single phase poroelasticity examples on an unstructured wellbore grid and an unstructured reservoir grid.
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A Numerical Method for Chemical Equilibrium Calculations in Multiphase Systems
Authors A.M.M. Leal, M.J. Blunt and T.C. LaForceWe present a method for calculating chemical equilibria of general multiphase systems. The method is based on a stoichiometric approach, which uses Newton's method to solve a system of mass-balance and mass-action equations. A stabilisation procedure is developed to promote convergence of the calculation when a presupposed phase in the chemical system is absent in the equilibrium state. The formulation of the chemical equilibrium problem is developed by presuming no specific details of the involved phases and species. As a consequence, the method is flexible and general enough so that the calculation can be customised with a combination of thermodynamic models that are appropriate for the problem of interest. Finally, we show the use of the method to solve relevant geochemical equilibrium problems found in modelling of carbon storage in highly saline aquifers.
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Simulation of Near-Well Pressure Build-up in Models of CO2 Injection
Authors G.E. Pickup, M. Jin and E.J. MackayReservoir simulation plays an important role in predicting the outcome of a CO2 storage project, although it is challenging to simulate all the processes that arise. In particular, we need to predict the build-up of pressure in the near well region to be able to estimate the optimum injection rate whilst ensuring that the formation and overlying caprock are not fractured. In this work, we compare simulations of horizontal homogeneous models, with both 1D radial and 2D Cartesian grids, with analytical calculations of pressure build-up. Our results show that several inaccuracies arise when using too coarse a grid, due to the inability to resolve the shock fronts adequately. In a coarse cell, the amount of dissolution is over-estimated and the gas saturation builds up slowly. The presence of a large cell with intermediate gas saturation gives rise to a peak in the pressure build-up curve (due to low mobility). The pressure eventually reduces to the “correct” value when the dry-out region forms. However, if injection ceases before this time, the final pressure will be over-estimated. As the grid size is reduced, these effects become less severe.
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Upscaled Models for CO2 Migration in Geological Formations with Structural Heterogeneity
Authors S.E. Gasda, H.M. Nilsen and H.K. DahleGeological carbon sequestration involves large-scale CO2 migration and immobilization within geometrically heterogeneous storage formations. Recent modeling studies have shown that structural features along the upper boundary of a storage formation can significantly decrease updip CO2 migration speed and increase structural trapping. This impact depends on caprock roughness, which can be present at different spatial scales--from seismic-resolution features such as domes, traps, and spill points to centimeter-scale rugosity observed at outcrops. The ability to resolve all relevant features within large-scale domains is not always practical, and thus upscaled modeling approaches may be required. We propose an alternative modeling approach, the VE model, which is based on the vertical equilibrium assumption. This type of simulator is well suited for modeling CO2 migration in gravity-dominated systems. The Utsira Formation is one such system due to the strong buoyancy effects are observed in the seismic data. We use 4D seismic data and our VE modeling tool to understand the physical parameters that control CO2 migration in the Utsira. Given the uncertainty in some important parameters--CO2 density, porosity, and topography of the top Utsira--we determine the range of uncertainty in CO2 and rock properties that is supported by the data.
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Mixed Multiscale Methods for Compressible Flow
Authors K.-A. Lie, S. Krogstad and B. SkaflestadMultiscale methods are a robust and accurate alternative to traditional upscaling methods. Multiscale methods solve local problems to numerically construct a set of basis functions that later can be used to compute global solutions that describe the flow on both the coarse computational scale and the underlying fine parameter scale. This way, one is able to account for both effective coarse-scale properties and sub-scale variations. The methods are particularly efficient when the flow field must be updated repeatedly. Because temporal changes in the flow equations are moderate compared to the spatial variability, it is seldom necessary to recompute basis functions each time the global flow field is recomputed. Herein, we discuss and compare two ways of extending a multiscale mixed method that was originally developed for incompressible flow to compressible flow. The first approach is based upon a mixed residual formulation with a fine-scale domain-decomposition corrector. The second approach is to associate more than one basis function for each coarse face and coarse cell and use bootstrapping to dynamically build a basis function dictionary that spans the evolving flow patterns. We present and discuss several numerical examples, from simplified 1D cases to 3D cases with realistic reservoir geometries.
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GAMPACK (GPU Accelerated Algebraic Multigrid Package)
Authors K.P. Esler, V. Natoli and A. SamardzicIn reservoir simulation, the elliptic character of the pressure subsystem and the inhomogeneous permeability field result in extremely slow convergence for conventional iterative solvers. Algebraic multigrid (AMG) methods address this challenge by constructing a multilevel hierarchy of matrices that naturally adapts to the permeability channels of the underlying geology. Preconditioning with AMG allows difficult cases with millions of unknowns to be solved in just a few iterations. In just a few years, graphical processing units (GPUs) have progressed from a research curiosity to a productivity workhorse by reducing time-to-solution and overall hardware cost. The highly irregular computation patterns of AMG, however, require new approaches to adapt to the many-core paradigm. The construction of the coarse matrix hierarchy and grid transfer operators poses a particular challenge for GPU acceleration. We show that by carefully selecting algorithms with sufficient fine-grained parallelism, and implementing them with novel approaches, it is possible to substantially accelerate both the setup and solve stages. We present GAMPACK, a library for accelerated AMG, and show that on a single GPU it can typically reduce the total setup and solve time by a factor of over 5, when compared to a widely-used AMG solver running on 8 Xeon cores.
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