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ECMOR XII - 12th European Conference on the Mathematics of Oil Recovery
- Conference date: 06 Sep 2010 - 09 Sep 2010
- Location: Oxford, UK
- ISBN: 978-90-73781-89-4
- Published: 06 September 2010
21 - 40 of 117 results
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Analysis of Grid Resolution for Simulations of CO2 Storage in Deep Saline Aquifers
Authors G.E. Pickup, M. Kiatsakulphan and J.R. MillsThe simulation of CO2 injection into deep saline aquifers provides challenges to reservoir modellers. Estimates of storage and trapping of CO2 must be made for large-scale aquifers for planning purposes, and these simulations are generally undertaken on coarse models with grid cells of several 100m in length. However, some of the physical processes which arise during CO2 storage can only be represented on finer grids. In this work, we used a variety of 2D models to study the effect of grid resolution on a range of processes which take place when CO2 is injected into a saline aquifer. A reasonably fine grid is required to model the buoyant rise of the CO2 plume and the migration under the caprock. Cell sizes of several meters in the horizontal and approximately 1m in the vertical are adequate. However, an even finer grid is required to limit the effects of numerical dispersion and correctly simulate the dissolution of CO2 in brine. When CO2 dissolves in brine, the brine becomes more dense and convection may take place, enhancing the dissolution. Estimates of time to onset of convection and critical wavelength have been determined by a number of people with variable results, but for a typical aquifer, a grid resolution of less than 1m may be required. If the cell size is too large, the amount of convection will be underestimated, and therefore the amount of dissolution will be underestimated. However, our results showed that the error in the amount of CO2 dissolved was greater at the end of injection than at the end of the simulation (100 years after injection ceased), due to numerical dispersion. On the other hand, our simulations showed that the grid resolution had little effect on the build-up of pressure, which indicates that coarse grids may be sufficient for initial assessments of storage potential.
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Accurate Discretization of Vertically-averaged Models of CO2 Plume Migration
Authors K.A. Lie, I.S. Ligaarden and H.M. NilsenWhen CO2 is injected into a deep formation, it will migrate as a plume that moves progressively higher in the formation, displacing the resident brine. The invasion front is driven by gravity, and the upward movement of the plume is limited by a low-permeable caprock. Several authors have recently proposed to make a sharp-interface assumption and only describe the plume migration in a vertically-averaged sense. For inhomogeneous permeability, the plume migration is then described by a system of conservation laws with spatially discontinuous flux. If one disregards dissolution and residual trapping, the system reduces to a scalar conservation law with a spatially dependent flux function, which may exhibit different solutions depending on the entropy condition that is enforced to pick a unique solution. We propose a certain set of assumptions that lead to the so-called minimum-jump condition and derive the corresponding solutions to the Riemann problem. Solutions to this problem are fundamental when developing accurate Godunov-type schemes. Here, we take a slightly different approach and present an unconditionally stable front-tracking method, which is optimal for this type of problem. Moreover, we verify the well-known observation that a standard upstream mobility discretization can give wrong solutions in certain cases.
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Effects of Non-Darcy Flow in CO2 Injection into Saline Aquifers
Authors A. Mijic and T.C. LaForceOne of the most important aspects of flow in porous media is the flow velocity. It defines the convective part of the transport and usually is assumed to be given by Darcy law. However, for the analysis of near-wellbore phenomena, such as salt precipitation during CO2 injection, it is necessary to include nonlinear flow behaviour in the zone of interest. This paper presents the modification of existing solutions for the immiscible non-Darcy displacement in order to account for mutual solubility of fluids and salt precipitation. The flow is governed by the Forchheimer type equation for the two phase flow. The analytical solution is implemented in CO2 injection modelling to investigate the effects of a nonlinear flow regime on phase saturation profiles. The results enable the determination of a drying front and hence the zone where the salt precipitation will occur. By implementing the approximate solution for the solid salt saturation it is shown that non Darcy displacement has a significant influence on obtained saturation values and the skin factor than previously estimated. Finally, the non-Darcy displacement dependency on injection rates further contributes to the modification of salt precipitation pattern.
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Multicomponent Multiphase Transport Solutions for Application to CO2 Storage
Authors A.L. Goater and T. LaForceReliable prediction of subsurface flow will be vital as Carbon Dioxide (CO2) storage grows into a global-scale industry. Therefore, it is essential to achieve fundamental objectives such as identifying and numerically modelling the physical solutions to three-phase transport problems. In this paper we present a one-dimensional (1D) conservation law for a three-phase, three-component transport problem of interest for CO2 storage in an oil reservoir. An example analytic solution for the purely advective problem is proposed. The componentwise Essentially Non-Oscillatory (ENO) method is used to minimise numerical error as compared with single-point upstream weighting (SPU). The simulated solutions both converge to the proposed dispersion-free analytic solution. However, the ENO approach exhibits only first order convergence, which is the same as SPU. Fine-grid ENO simulations of the advection/dispersion equation in three-phase fluid flow with non-negligible capillary pressures are also considered. It is demonstrated that the physically-realistic dispersion caused by capillary pressure alters the simulated solution in a fundamentally different way than numerical effects, dispersing some shocks while having little impact on others.
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Multidimensional and Higher Order Edge Based Upwind Schemes for Hyperbolic Systems Including Gravity in Porous Media on Structured and Unstructured Grids
Authors S. Lamine and M.G. EdwardsStandard 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 higher-order multidimensional upwind schemes are presented for convective flow approximation in porous media that reduce both directional and crosswind diffusion. The schemes are coupled with full-tensor Darcy flux approximations and handle general flow conditions including counter current gravity driven flows and systems of hyperbolic equations. Characteristic vector upwind approximations are proposed and compared with the simulation standard single-point upstream weighting schemes. When dealing with systems of hyperbolic equations, upwind characteristic wave decomposition is used in combination with different limiting strategies involving conservative, primitive and characteristic variables. Alternate wave vector tracing approximations are proposed based on phase velocities and characteristic velocities and comparisons are presented. The higher order multidimensional schemes are designed for general grids. Conditions for a local discrete maximum principle are presented that ensure solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for gravity segregated flow and polymer flood systems. The test cases involve a range of unstructured grid types with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher order multidimensional formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts.
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Monotone Multi-dimensional Upstream Weighting on General Grids
Authors E. Keilegavlen, J. Kozdon and B. MallisonThe governing equations for multi-phase flow in porous media often have a mixed elliptic and (nearly) hyperbolic character. The total flux for each phase consists of two parts: a geometry and rock dependent term that resembles a single-phase flux and a mobility term representing fluid properties and rock-fluid interactions. The geometric term is commonly discretized by two or multi point flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, e.g. gas injection, SPU yields grid orientation effects. For these physical processes, the governing equations are unstable on the scale at which they are discretized, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results and to improve the overall performance prediction of enhanced oil recovery processes. In this work, we present a framework for multi-dimensional upstream weighting for multi-phase flow with gravity on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multi-dimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for treatment of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multi-dimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multi-dimensional upstream weighting is used.
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On the Upstream Mobility Finite Difference Scheme
Authors A. Adimurthi, J. Jaffré, S. Mishra and G.D. Veerappa GowdaIncompressible two-phase flow in porous media when capillarity effects are neglected can be modeled as a nonlinear scalar conservation with unknown one of the phase saturations. The standard scheme used in reservoir simulation or in hydrogeology uses the upstream mobility flux. The solution calculated with such a scheme was proven to be converging to the entropy solution. This scheme is used in the homogeneous case as well as in the case of a change in rock types when the mobilities functions are changing from one rock type to the other. However in this latter case, while the scheme is giving the right solution most of the time, we will exhibit some examples of mobility functions for which the scheme does not give the right solution. An alternative scheme is to use an extension of the Godunov flux for which we give a simple formula which can be used when a change of rock types occurs. Thus the solution given by this scheme can be proven to be correct even in cases where the upstream mobility scheme fails.
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Finite Volume Schemes for Multiphase Flow Simulation on Near Well Grids
Authors C. Guichard, J. Brac, R. Eymard and R. MassonIn reservoir engineering a proper well modeling requires to simulate accurately multiphase flow taking into account the singular pressure distribution in the well vicinity and the large difference of scales between the wellbore radius and the reservoir dimension. This paper investigates the numerical simulation of a 3D near-well model using a radial mesh exponentially refined down to the well boundary. The radial and the reservoir CPG grids are matched using either hexahedra or both tetrahedra and pyramids. Various multipoint finite volume methods are compared such as the O and L schemes, and the newer G and GradCell schemes which are briefly described. Our first objective is to introduce a new scheme which includes the advantages of compact stencil, symmetry and convergence. Our second objective is to study the behavior of these schemes on the above two types of meshes. First, their convergence behavior are compared for the deviated well single-phase flow analytical solution described in the literature with an homogeneous anisotropic permeability tensor. Second, the influence of the type of mesh is studied on a CO2 injection test case in an aquifer with solubility of CO2 in the water phase and anisotropic permeability fields.
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Horizontal Simulation Grids as Alternative to Structure-based Grids for Thin Oil-zone Problems
By O. PettersenAs a general rule, the layering in reservoir simulation grids is based on the geology, e.g. structure tops. In this paper we investigate the alternative of using horizontal layers, where the link to the geology model is by the representation of the petrophysics alone. The obvious drawback is the failure to honor the structure in the grid geometry. On the other hand a horizontal grid will honor the initial fluid contacts perfectly, and horizontal wells can also be accurately represented. Both these issues are vital in thin oil-zone problems, where horizontal grids may hence be a viable alternative. To investigate this question, a number of equivalent simulation models were built for a segment of the Troll Field, both geology-based and horizontal, and various combinations of these. In the paper it is demonstrated that the horizontal grid is able to capture the essentials of fluid flow with the same degree of accuracy as the geology-based grid, and near-well flow is considerably more accurate. For grids of comparable resolution, more reliable results were obtained by a horizontal grid than a geo-grid. A geo-grid with local grid refinement and a horizontal grid produced almost identical results, but the ratio of computing times was more than 20 in favor of the horizontal grid. In the one-phase regions of the reservoir, relatively coarse cells can be used without significant loss of accuracy.
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A Model for Conductive Faults in Heterogeneous Media for Non-matching Grids
Authors X. Tunc, I. Faille, T. Gallouët, M.C. Cacas and P. HavéIn the standard Corner Point Grid approach, a fault zone is supposed to be sufficiently thin to be represented as a surface with non-matching interfaces, due to the throw, across the fault. Following this point of view, we study an approach where fluid flow along the fault is modelized by a lower dimensionnal model, the fault thickness becoming a model parameter.A fault zone is represented by two sets of faces, corresponding to the fault surface, each set being conforme with its neighbouring matrix grid-blocks. Fault zone properties are assigned to each face. Finite Volume Discretization of fluid flow equations leads to an additional mass balance equations in each fault face, with three types of fluxes. The first one represents mass exchange between fault faces along each set. The second one is between neighbouring non-conforme fault faces across the fault and finally, the last one, is between fault faces and their neighbouring matrix grid-blocs. In this paper, we mainly focus on how to handle the non-matching grids and therefore consider only one-phase fluid flow. However, the extension of this approach to complex multiphase flow is straightforward as it enters into the Finite Volume framework.
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Modification of a Reservoir Grid to Achieve Monotonicity of the Control Volume Method
Authors R.A. Yorgova and I. AavatsmarkSpurious oscillations may occur for nonmonotone discrete methods. Therefore, monotonicity is a desired property for control-volume methods. Control-volume MPFA methods are only conditionally monotone. Sufficient criteria for monotonicity of the MPFA O-method on quadrilateral grids in 2D for the pressure equation has previously been developed. Tests on rough quadrilateral grids show that the monotonicity conditions are often violated. In this work we present modification of rough quadrilateral grids on homogenous and on heterogeneous media obtaining grids where the MPFA O-method is monotone. The reconstruction is based on short moves of the grid nodes. The moves are defined by weighted gradients on functions based on the monotonicity criteria. The main steps of the reconstruction algorithm are described. Examples of rough quadrilateral grids and their modified grids are included in order to show the efficiency of the presented algorithm.
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Practical Gridding Algorithms for Discrete Fracture Modeling Workflows
Authors B.T. Mallison, M.H. Hui and W. NarrThis paper describes grid generation algorithms for simulating flow and transport in fractured porous media. The methods are designed to only capture details of the fracture network geometry larger than the specified grid resolution. The final grids honor fractures approximately while maintaining good cell quality. Improved representations of the input geometry can be obtained by generating a new grid with finer resolution. Several numerical examples are presented in two and three dimensions that demonstrate that the algorithms are robust and practical for industrial applications.
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Accurate Locally Conservative Discretizations for Modeling Multiphase Flow in Porous Media on General Hexahedra Grids
Authors M.F. Wheeler and G. XueFor many years there have been formulations considered for modeling single phase ow on general hexahedra grids. These include the extended mixed nite element method, and families of mimetic nite difference methods. In most of these schemes either no rate of convergence of the algorithm has been demonstrated both theoret- ically and computationally or a more complicated saddle point system needs to be solved for an accurate solution. Here we describe a multipoint ux mixed nite element (MFMFE) method [5, 2, 3]. This method is motivated from the multipoint ux approximation (MPFA) method [1]. The MFMFE method is locally conservative with continuous ux approximations and is a cell-centered scheme for the pressure. Compared to the MPFA method, the MFMFE has a variational formulation, since it can be viewed as a mixed nite element with special approximating spaces and quadrature rules. The framework allows han- dling of hexahedral grids with non-planar faces by applying trilinear mappings from physical elements to reference cubic elements. In addition, there are several multi- scale and multiphysics extensions such as the mortar mixed nite element method that allows the treatment of non-matching grids [4]. Extensions to the two-phase oil-water ow are considered. We reformulate the two- phase model in terms of total velocity, capillary velocity, water pressure, and water saturation. We choose water pressure and water saturation as primary variables. The total velocity is driven by the gradient of the water pressure and total mobility. Iterative coupling scheme is employed for the coupled system. This scheme allows treatments of different time scales for the water pressure and water saturation. In each time step, we first solve the pressure equation using the MFMFE method; we then Center for Subsurface Modeling, The University of Texas at Austin, Austin, TX 78712; [email protected]. yCenter for Subsurface Modeling, The University of Texas at Austin, Austin, TX 78712; [email protected]. 1 solve the saturation using discontinuous Galerkin (DG) method by taking multiple small time steps within the large time step. In addition, the MFMFE method allows effcient computations of the total and capillary velocity since the method gives the local velocity approximation in terms of surrounding pressure degrees of freedom. Both theoretical and computational results are discussed and presented. Exten- sions to advection-diffusion equations and non-Newtonian polymer ooding [6] are also discussed.
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Quasi-positive Continuous Darcy-flux Approximation on Cell-centred Triangular Grids
Authors M.G. Edwards and H.A. FriisA new family of cell-centered finite-volume schemes is presented for solving the general full-tensor pressure equation on arbitrary unstructured triangulations. The new schemes are flux continuous and have full pressure support (FPS) over each subcell with continuous pressure imposed across each control-volume sub-interface, in contrast to earlier formulations. The earlier methods are point-wise continuous in pressure and flux with triangle-pressure-support (TPS) leading to a limited quadrature range. An M-matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle. Positive definite conditions are also given. Comparisons show that the earlier pointwise TPS methods can induce strong spurious oscillations in solutions for problems involving strong full-tensor anisotropy where the M-matrix conditions are violated, leading to unstable solutions in such cases. The earlier unstructured cell-centered TPS schemes are investigated in terms of stability and are shown to be decoupled at high anisotropy ratio. In contrast to TPS, the new FPS formulation leads to well resolved stable solutions that are essentially free of spurious oscillations. A substantial degree of improved convergence behavior, for both pressure and velocity, is observed in all convergence tests.
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Convergence of the MPFA L-method – Strengths and Difficulties
More LessThe convergence of the Multi Point Flux Approximation (MPFA) O-method on general grids has recently been proved by rewriting the method in a weak form using quadratures. We will examine in detail the quadrature of the MPFA L-method in 2D. To obtain the correct quadrature one must introduce auxiliary variables that impose the continuity of the pressure over an interface – a characteristic of the L-method. The auxiliary variables can successively be eliminated from the quadrature. Compared with the O-method, the L-method obtains a stronger diagonalization of the quadrature matrix. This is important as the convergence proof of the method relies on the assumption that the symmetric part of the quadrature matrix has positive eigenvalues. On the other hand, the diagonalization also means that the L-method has a stronger inter-element coupling, as pressure continuity is imposed along element interfaces and not within the partial elements themselves. We will examine and compare the assumption for convergence for the O-method and the L-method in 2D aiming for a better understanding of the merits of the two methods and how anisotropy and grid deformation influence their behaviour.
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On Rough Grids – Convergence and Reproduction of Uniform Flow
Authors R.A. Klausen and A.F. StephansenIn reservoir simulation the flow driven by pressure differences and gravity is usually approximated by means of the empirically derived Darcy's law. The elliptic pressure equation that results is preferably solved with a mass conservative method, for which several candidates are available. We distinguish between full field methods and raw field methods. On the one hand we have continuous full field methods which approximate the velocity with a field that is defined inside the element itself. On the other hand we have discrete methods that approximate the flux, and for which no velocity field is defined inside the elements. We denote this raw field. We will examine some mass-conservative methods on the basis of two different characteristics: uniform flow reproduction and convergence on rough grids. Analysis apart, we propose an interpolation on polygons that can reproduce uniform flow by using edge basis functions constructed by means of barycentric functions. This permits us to obtain an H(div) field starting from discrete fluxes evaluated at the interfaces of the elements. We also show that a similar interpolation can be constructed to obtain an H(curl) field on polygons in 2D.
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Locally Adaptive Timestepping in Reservoir Simulators
Authors H. Mc Namara, G. Bowen and P.J. DellarAn algorithm for locally adapting the step-size for large scale finite volume simulations of multi- phase flow in petroleum reservoirs is suggested which allows for an “all-in-one” implicit calculation of behaviour over a very large time scale. Some numerical results for simple two-phase flow in one space dimension illustrate the promise of the algorithm, which has also been applied to very simple 3D cases. A description of the algorithm is presented here along with early results. Further development of the technique is hoped to facilitate useful scaling properties.
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Improved Gravity Splitting for Streamline and Reordering Methods
Authors H.M. Nilsen and J.R. NatvigFor heterogeneous reservoirs, fast, stable and accurate methods are hard to obtain. Large changes in the velocity field leads to severe time step restrictions in explicit schemes or expensive time steps in implicit schemes. In the absence of gravity, the exact velocity field will be loop free in the sense that there are no closed integral curves. For streamline methods, the absence of closed integral curves ensures that all streamlines have endpoints in wells. Likewise, this property implies that the Jacobian matrix of an implicit scheme with an upwind flux approximation can be reduced to a triangular matrix by a permutation, or reordering, of the unknowns. For the above methods the effect of gravity is usually handled by operator splitting in the transport equation. Gravity will introduce rotation in the total velocity, which may yield closed integral curves. Even when the effect of gravity is small, it can limit the efficiency and the robustness of streamline and reorder methods. To overcome this problem, we split the total velocity field in a loop-free part without the effect of gravity, and a part driven by gravity only. This is achieved by solving the pressure equation two times with different righthand sides.
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A Robust Streamline Tracing Method for Systems with Non–neighbor Connections
By D. KachumaThe use of complex corner point systems in field flow reservoir simulators allows the modelling of a wider variety of cell geometries and hence a better representation of faulted reservoirs. If faults are involved, then a cell face may be next to two or more cell faces. This phenomenon can also be observed in situations where the cells in a particular region have been refined to improve accuracy. This creates complex connections between cell faces which require special attention and representation within the simulator. Conventionally, this is handled by the use of the so-called non-neighbour connections (NNC) whereby the simulator includes, on top of the regular neighbour fluxes, the fluxes between the extra cell connections. We present a robust technique which is applicable regardless of the flow configuration. Our technique involves subdividing each cell with non-neighbour fluxes into a local regular subgrid. Using this grid the local streamfunction is constructed numerically. By reducing the size of the local subgrid, the accuracy of the traced streamlines improves. We demonstrate this technique first on a detailed 2D synthetic example that is designed to show the robustness of the technique. The practical utility of our algorithm is then demonstrated in a structurally complex and heavily faulted full model of a North Sea field which includes cells with several non-neighbour configurations in different faces. This treatment is contrasted with the usual approach and also other techniques designed to trace streamlines in heavily faulted systems.
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Streamline Tracing on Irregular Geometries
More LessThe accurate and efficient tracing of streamlines is fundamental to any streamline-based simulation method. For grids with irregular cell geometries, such as corner-point grids with faults or Voronoi-diagram (pebi) grids, most efforts to trace streamlines have been focused on subdividing irregular cells into sets of simpler subcells, typically hexahedra or simplices. Then one proceeds by reconstructing the velocity field on, and tracing through, the sets by a more basic algorithm. One such basic algorithm applies to incompressible flow on simplices. In that case there is a cell-wise constant velocity that is consistent with given face fluxes, as long as those face fluxes sum to zero for the cell. We give an efficient and simple formulation of this algorithm using barycentric coordinates. Another approach to irregular cell tracing is computing the streamline directly on the complex cell geometry. We give a new method based on generalized barycentric coordinates for direct tracing on arbitrary convex polygons, which generalizes the corner velocity interpolation method of Hægland et al (2007). The method generalizes to convex polytopes in 3D, with a restriction on the polytope topology near corners that is shown to be satisfied by several popular grid types.
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