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ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery
- Conference date: 29 Aug 2016 - 01 Sep 2016
- Location: Amsterdam, Netherlands
- ISBN: 978-94-6282-193-4
- Published: 29 August 2016
41 - 60 of 163 results
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The Interplay of Capillary and Viscous Forces Driving Flow through Layered Porous Media
Authors Y. Debbabi, M.D. Jackson, G.J. Hampson, P.J.R. Fitch and P. SalinasWe examine the impact of viscous and capillary forces on immiscible, two-phase flow parallel and perpendicular to continuous layers of contrasting material properties. We consider layers of contrasting porosity and relative permeability, in addition to the contrasts in absolute permeability investigated previously. We define a set of dimensionless numbers which characterize flow. Some of these are common to flow both parallel and perpendicular to layering, such as the longitudinal permeability ratio σ and the ratio Rs of the moveable pore volumes (MPV) in each layer. Others are specific to a given flow direction, such as the dimensionless capillary to viscous ratio Ncv, and the effective aspect ratio RL that quantifies crossflow for layer-parallel flow. We examine how variations in the dimensionless numbers affect the trapping/recovery efficiency, defined as the fraction of the model MPV occupied by the injected phase after 1 MPV injected, and which is numerically equivalent to the fraction of the displaced phase recovered from the model after 1 MPV injected. The results are directly applicable to geological carbon storage and hydrocarbon production. We find that the trapping efficiency is clearly controlled by the dimensionless numbers. When flow is perpendicular to layering, heterogeneity only influences flow when capillary forces are significant (Ncv>0). As Ncv is increased, a larger fraction of the non-wetting phase is trapped if the layers have contrasting capillary pressure curves. When flow is parallel to layering, both viscous and capillary forces are important. In the viscous limit (Ncv=0), heterogeneity reduces trapping efficiency if σ≠Rs. As capillary forces become more significant (Ncv increases) and if crossflow between layers can occur (RL>0), the trapping efficiency also increases in response to capillary crossflow and reaches a maximum at a given Ncv. At higher Ncv, the benefit of crossflow is outweighed by along layer diffusion of the injected phase.
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Low Salinity Carbonated Waterflooding
Authors T. Blom, A.C. Alvarez, W.J. Lambert, D. Marchesin and J. BruiningIt has been shown in the literature that a secondary low salinity waterflood can improve the oil recovery by 5-20%. A possible mechanism is that the low salinity causes desorption of organic material, which may increase water-wetness and lead to more favorable relative permeability behavior. A less well-known mechanism is enhanced solvent (e.g., carbonated water) recovery as low salinity enhances the aqueous solubility of neutral components, which after injection will be transferred from the aqueous phase to oleic phase thus decreasing the oil concentration in the oleic phase and diluting the residual oil. By way of example we consider a low salinity carbonated waterflood into a reservoir containing oil equilibrated with high salinity carbonated water. For a given pH, the CO2 equilibrium concentration in low salinity injection water is higher than in the high salinity initial water. PHREEQC, a geochemical aqueous equilibrium programme, can be extended to obtain the accurate partition coefficient of neutral species that are soluble both in the oleic and the aqueous phase. For this we use the Krichevsky-Ilinskaya extension of Henry’s law for solubility of gases in liquids. Gibbs phase rule shows that the phase behavior only depends on the pH and the chloride concentration. In PHREEQC, we use Pitzer’s activity coefficients to extend the validity up to 6M. The output of PHREEQC can only be successfully incorporated in multiphase flow simulation programmes, e.g. COMSOL(TM), after applying a smoothing procedure for which we choose symbolic regression (EUREQA(TM)). An optimal formulation avoids spurious broadening of the concentration profiles in contact “discontinuities”. We obtain the saturation, composition and the total Darcy velocity profiles. The significant new insight is that by changing the salinity at constant pH the oil recovery by carbonated water flooding can be enhanced. This insight can be applied to optimize enhanced oil recovery with a low salinity waterflood.
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Analyses of Geo-statistical Modelling, in the Wavenumber Domain, for Multi-dimensional Models
By J. LeguijtThe probabilistic seismic inversion program, Promise, has been equipped with a module that is able to account for lateral continuity. In this module, the prior probability density function (pdf) is generated using two point statistics. Bayes rule is used to account for the observations; the result is a posterior pdf. The observations consist of seismic traces and arrival times of interpreted horizons. An iterative algorithm is deployed to sample the posterior pdf. This visits all the locations in its model, numerous times in succession and effectively creates a Metropolis algorithm. Unexpected results are sometimes generated by this. Consequently, a closer examination of the algorithm and its theoretical background has been partaken. The models that are used in Promise are multi-dimensional; a one to one mapping between the model parameters and the observations does not always exist. This makes the analyses more complicated. The uncertainty in the observations is represented by the seismic noise. For convenience, it is often assumed that the seismic noise at different locations is statistically independent. This is often an erroneous assertion and causes the modeling results to be adversely affected. To make the analyses comprehensible, they are carried out in the wavenumber domain, for models with a linear relation between the model parameters and the observations. Some of the learnings will be discussed for models that are based on a 1D grid.
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Oil Flow Rate History Reconstruction using Downhole Transient Temperature Data with Wavelet Transform
More LessFlow rate history is important for pressure-transient analysis and history matching, but due to high cost of metering equipment, the accurate real-time flow rate history for individual well is not always available. This paper presents a method of reconstructing unknown flow rate history from downhole transient temperature data with wavelet transform (WT). Firstly, the time of flow rate change was identified from transient temperature data with the Haar wavelet. As reservoir temperature changes dramatically at the time of flow rate change due to the Joule-Thomson effect, and the time of flow rate change can be identified in the WT detailed signal of the transient temperature data. Then, the proportional relationship between the amplitude of WT coefficients and flow rate change was discovered and theoretically proved. Based on this relationship, the flow rate for each flow event was allocated back from the cumulative production. Finally, with the identified time and calculated flow rate for each flow event, the unknown flow rate history was reconstructed from downhole transient temperature data. This method works well in the oil reservoirs with constant reservoir and fluid properties, and the reliability of this method was demonstrated with a field case study.
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Discrete Fracture Method for Petroleum Reservoir Simulation Using an Element-based Finite Volume Method
Authors C.R. Maliska, B.T. do Vale and F. MarcondesAny porous media flow is inherently complex to model due to the impossibility of giving the real flow geometry to the simulation model. If heterogeneous media is involved, as is the case of naturally fractured petroleum reservoirs, the difficulty increases even more, since large fractures can be seen as discontinuities, having as background the porous matrix, in which many smaller sized fractures are present. The porous matrix can be treated by a stochastic procedure and are, normally, large deposits of oil, while the large fractures are better solved through a deterministic treatment. A network of connected large fractures linked to the porous matrix may be the most important flow path for oil production. In the other hand, depending on the physical properties, capillarity and permeability, the fractures and porous matrix combination may lead to a undesirable secondary oil recovery, leaving considerable amount of oil in the porous matrix. Therefore, the prediction of this combined flow (fractures +porous matrix) is of utmost importance for the oil industry. There are several approaches to solve this combined flow, all of them based, of course, on a idealized fracture configuration, which gets more and more realistic as the characterization methods evolves, due to the specialization of well-logging, 4D seismic and other methods. The final goal would be to solve the local flow for any single fracture, irrespective its size, nowadays an impossible task due to the lack of characterization methods and computer capacity. However, as computational power and characterization techniques evolve, methods able to solve the details of the flow should be devised. This paper follows this route and presents a DFM (Discrete Fracture Method) in the framework of an Element-based Finite Volume Method (EbFVM) using unstructured grids. The EbFVM is per se a multi-point flux approximation, avoiding the usual two-point approach, which is conceptually wrong, since the errors do not vanish as the grid is refined. The EbFVM also avoids the need of more complex MPFA algorithms for having correct flux evaluation. Additionally, the EbFVM framework allows the use of truly directional upwind and higher order schemes with no extra efforts. The 2D oil-water flow using the DFM method with superposition for connecting the fractures and the porous matrix is solved. The fractures are assumed to be 1D considering its real thickness. Several aspects of the model are investigated, as the capillarity effects, especially in the situation in which imbibition of the porous matrix occurs and the anisotropy of the coefficients of the linear system resulting from the superposition of the equations. Since IMPES method is used, comments on the time step adaption is also given.
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Simulation of Densely Fractured Tight Oil Reservoirs Using a New Simulator with Automatic Differentiation
Authors W.C. Fang, H.Q. Jiang, J. Killough, J.J. Li, W.C. Teng, L.K. Li and L. ZhaoDevelopment of tight oil reservoirs mainly relies on massive hydraulic fracturing, which can generate complex fracture networks in the reservoirs. These highly discrete fracture networks bring great challenges for reservoir simulation. An efficient model applying an unstructured discretization method and automatic differentiation is proposed. The flexibility in unstructured control volume shapes enables the gridding of complex fracture systems. By introducing the concept of half-transmissibility for each grid, transmissibility list including connections of matrix-matrix, matrix-fracture, fracture-fracture is established. Nonlinear flow and transport equation system is solved by a modified Newton’s method, in which the Jacobian matrix is computed by automatic differentiation (AD). Accuracy of the model was validated by performing simulations using a commercial simulator. We implemented our model in several cases with a uniform physical domain (radial model with diameter=1 km) but different fracture properties. Results show that fracture configuration and property have significant impacts on the production. Moreover, our model shows a high efficiency in the densely fractured system with fracture density up to 1018/km2. The novelty of the model is in the ability to represent complex fracture systems individually and explicitly, and in the application of automatic differentiation, which greatly facilitates the model establishment and improves computational efficiency.
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Mixed Hybrid Finite-element Formulation for General Purpose Reservoir Simulation
Authors A.S. Abushaikha, D.V. Voskov and H.A. TchelepiWe present a mixed hybrid finite-element (FE) formulation for modeling subsurface flow and transport for general-purpose compositional reservoir simulation. The formulation is fully implicit in time and employs a hybrid FE method for the spatial discretization of the conservation equations. The hybrid FE formulation is implemented in the Automatic Differentiation General Purpose Research Simulator (ADGPRS); consequently, the new FE-based methodology inherits all the `physics’ capabilities of ADGPRS, including compositional EOR models. The high-order mixed hybrid FE discretization scheme works for many types of finite elements and can handle highly anisotropic material properties. The formulation is locally conservative. The momentum and mass balance equations are solved simultaneously, including Lagrange multipliers on element interfaces. The fully implicit scheme uses the automatic differentiation capability to construct the Jacobian matrix. The hybrid FE approach accommodates unstructured grids, which are needed for honouring the complex geometry of the subsurface, in a straightforward manner. We present compositional test cases with full permeability tensors, and we discuss the accuracy and computational efficiency of the formulation. We also compare the performance of the hybrid FE-based scheme with finite-volume based Multi-Point Flux Approximation (MPFA) methods.
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3D Geological Feature Honored Cell-centered and Vertex-centered Unstructured Grid Generation, and CVD-MPFA Performance
Authors S. Manzoor, M.G. Edwards, A.H. Dogru and T.M. Al-ShaalanGrid generation for reservoir simulation, must honor classical key geological features and multilateral wells. For the purpose of grid generation, the geological features are classified into two groups; 1) involving layers, faults, pinchouts and fractures, and 2) involving well distributions. In the former, control-volume boundary aligned grids(BAGs) are required, while in the latter, control-point well aligned grids(WAGs) are required. In reservoir simulation a choice of grid type and consequent control-volume type is made, i.e. either primal or dual-cells are selected as control-volumes. Regardless of control-volume type, the control-point is defined as the centroid of the control-volume. Three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to wells, yielding essentially PEBI-meshes either with respect to primal or dual-cells depending on grid type. In the grid generation methods presented, for both primal and dual-cell feature based meshes, both frameworks use primal-cells (tetrahedra, pyramids, prisms and hexahedra) as grid elements. Dual-cell feature honored grids are derived from underlying primal-meshes such that features are recovered in the dual-setting. Geological features are honored by using the idea of protection spheres, and protection halos around key geological features. Halo construction requires the use of prisms and/or hexahedra. Pyramids are used as transition elements providing interfaces between quad faces of the halo elements and triangular faces of the main tetra-mesh. A novel method for constructing pyramids as transition elements in an unstructured mesh together with a novel technique for ensuring fully constrained recovery of geological features is proposed. The grids generated are employed to study comparative performance of cell-vertex versus cell-centred CVD-MPFA finite-volume formulations using equivalent degrees of freedom. The benefits of both types of approximation are presented in terms of flow resolution relative to the respective degrees of freedom employed. The cell vertex method proves to be the most beneficial with respect to accuracy and efficiency.
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Immiscible Two-phase Darcy Flow in Fractured Porous Media - New Robust Formulation and Application to the Tight Gas Recovery
Authors K. Brenner, M. Groza, L. Jeannin, R. Masson and J. PellerinNumerical simulations of two-phase Darcy flows in heterogeneous porous media requires choosing an appropriate set of primary unknowns, which may be challenging, especially when dealing with very flat capillary pressure curves, dry regions or saturation jumps at the rock type interfaces. The classical approaches fail to cope with all of those difficulties. In particular the two-pressure formulation allows handling saturation jumps, but beaks down if the capillary pressure doesn’t depend on saturation. It also lacks robustness when dealing with nearly residual water saturations. On the other hand, for homogeneous medium, the pressure – saturation formulation is known to be robust when dealing with dry media and can handle vanishing or constant capillary pressure curves. Unfortunately it is not always possible to extend it to the case of discontinuous capillary pressure curves. In this paper, a new formulation based on parametrization techniques for the capillary pressure monotone graph extension is proposed which handles all the above mentioned difficulties while still using only two unknowns by degree of freedom. We illustrate the efficiency of our approach by numerous numerical experiments dealing with water gas flow in fractured tight gas reservoirs using the data set presented in [2]. Following [1], the fractures are modelled as interfaces of codimension one with continuous pressure at the matrix fracture interfaces. During the injection phase of the simulation, water penetrates only a few tens of centimetres deep in the matrix rock. Therefore, in order to obtain an accurate numerical approximation, an anisotropic refinement of the mesh is used in the neighbourhood of the fractures using prismatic elements. The connection with the surrounding tetrahedral mesh in the matrix domain is achieved using pyramids. Following [1], the model is discretized using the Vertex Approximate Gradient scheme which allows for general polyhedral cells. The numerical performance of the new approach is evaluated for various choices of capillary pressures curves. The comparison with classical formulations shows that the new approach is more efficient both in terms of Newton iterations and CPU time. [1] K.Brenner, M.Groza, C.Guichard, R.Masson: Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media, M2AN, pp.49 2, 303-330, 2015. [2] D.Y.Ding, H.Langouet, L.Jeannin: Simulation of Fracturing Induced Formation Damage and Gas from Fractured Wells in Tight Gas Reservoirs, SPE 153255, 2012.
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Enhanced Nonlinear Finite Volume Scheme for Multiphase Flows
Authors K. Nikitin, V. Kramarenko and Y. VassilevskiWe present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest enhancement of the nonlinear method takes into account the nonlinear (e.g. logarithmic) singularity of the pressure in the near-well region and introduces the nonlinear correction to improve accuracy of the pressure and the flux calculation. The new method is generalized for anisotropic media, polyhedral grids and nontrivial wells cases. Numerical experiments show the noticeable reduction of the numerical errors compared to the original monotone nonlinear FV scheme with the conventional Peaceman well model and even with the given analytical well rate.
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A Topological Approach for Automated Unstructured Meshing of Complex Reservoir
Authors V. Gauthier, A. Arnould, H. Belhaouari, S. Horna, M. Perrin, M. Poudret and J.F. RainaudEstimations of petroleum reserves rest on finite volumes computational simulations of the reservoir fluid dynamics. These simulations are operated on 3D meshed reservoir models produced by a complex and poorly automated chain of operations. This paper proposes a mesh building methodology, which uses geological rules for building reservoir meshes in a more automated way. We start from a surface structural model and from a description of the stratigraphy both packed thanks to the industry standard RESQML. We construct a volume structural framework based on generalized map topological structures. These structures include topological boundary relations between the represented geological objects (horizons, faults, units) and some dedicated data attached to the topological cells (vertices, faces, volumes, etc.), such as geometry or geological labels (e.g. names, relative ages, deposit methods). In particular, on a single topological representation, we can attach two different geometric representations respectively describing the present day layer geometry (“folded model”) and the original positions of the various layers in their “deposition space” ("unfolded model"). Thanks to a dedicated rule-based language, we deduce from the geological interpretation, a set of topological and geometric operations that allow an automated building of the structural framework, on which the reservoir meshes will be implemented. This language allows a fast prototyping of complex operations (boolean operations for instance) and it guarantees the geological and topological consistency of the model. Using this consistent and fully informed structural framework, we can create in an automated way various conformal unstructured 3D meshes organized in layers. These meshes agree both with the topology induced by the succession of deposition, erosion and tectonic events that constitute the local geological history and with the peculiarities of the used fluid flow simulators. A use-case is presented to demonstrate the feasibility of our method.
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Numerical Simulation of Three-dimensional Complex Fracture Geometries Using an Unstructured Voronoi Mesh
Authors Y. Wang and S.A. AryanaTo date, most numerical simulators developed to simulate production from unconventional gas reservoirs focus on two-dimensional (2d) representations of geological properties. Fracture geometry is, however, highly uncertain and three-dimensional models would enable more robust representations while considering gravity effects. In this paper, a Discrete Fracture Model (DFM) of 3d complex fracture geometries is developed where unstructured PEBI grids are used to discretize flow equations. The mesh generation algorithm, implemented in 3d space, is based on the equilibrium state of forces in a truss system. Each fracture is modeled as a 2d plane with an arbitrary orientation and shape. Nodes along the fracture plane are populated uniformly while the proposed dynamic force-based model is applied in the matrix region. A distance function is developed to assist node population in the matrix region. Fractures are regarded as bounded plane-constraints and the distance function is expressed in terms of geometrical information from nodes in the matrix and fracture planes in 3d space. Assuming isotropic media, the generated mesh guarantees local orthogonality, and two-point flux approximations (TPFA) are used in the numerical discretization scheme. Hydraulic and induced fractures and their intersections are accounted for explicitly without invoking transformations. A sensitivity study is performed to investigate the effect of grid size in regards to embedded fractures and optimal node density to balance accuracy and efficiency. The fully 3d model is validated against results from a commercial simulator using a synthetic case. Inclusion of gravity is shown to have a significant impact on gas and water production rates. In summary, the proposed approach enables modeling complex fractures in fully 3d space using an unstructured Voronoi mesh with gravity effects and provides a better understanding of two-phase systems representing recovery from fractured unconventional gas reservoirs.
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C1-PPU Schemes for Efficient Simulation of Coupled Flow and Transport with Gravity
Authors J.M. Jiang and R.M. YounisIt is revealed by recent studies that in the presence of counter-current flow due to buoyancy, nonlinear convergence problems may be pronounced when the popular PPU scheme is used to approximate the numerical flux. The PPU numerical flux is non-differentiable across the co-current/counter-current flow regimes and thus may lead to oscillations or even divergence in the Newton iterations. Recently proposed methods address improved smoothness of the numerical flux. In this paper we devise and analyze an alternative numerical flux scheme called C1-PPU that allows a smooth variation between the co-current/counter-current flow regimes as well as an optimal balance between the scalar nonlinearity and accuracy of the flux function. The C1-PPU scheme involves a novel use of the flux limiter concept from the context of high-resolution methods. Numerical examples including 1D scalar transport problem and 2D heterogeneous problem with fully-coupled flow and transport are presented. The results indicate that in addition to smoothness, nonlinearity may also be critical for convergence behavior and thus needs to be considered in the design of an efficient numerical flux scheme. Moreover, the results show that our C1-PPU scheme exhibits superior convergence properties for large time steps compared to the other alternatives.
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Simulation of Immiscible Viscous Fingering Using Adaptive Unstructured Meshes and Control-volume Galerkin Interpolation
Authors A.G. Adam, D. Pavlidis, J.R. Percival, P. Salinas, Z. Xie, C.C. Pain, A.H. Muggeridge and M.D. JacksonDisplacement 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.
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Cell-centred Higher Resolution Finite-volume Total Velocity Vt and Va Formulations on Structured and Unstructured grids
Authors Y. Xie and M.G. EdwardsY.Xie [email protected], [email protected] Novel cell-centred finite-volume formulations are presented for two-phase flow with gravity and capillary pressure on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity and capillary pressure flux operators. Two formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both the Vt and the Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However when gravity is present, flow regimes are identified where the Va method becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first order single point upstream weighting.
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Compositional Dual Mesh Method for Single Phase Flow in Heterogeneous Porous Media - Application to CO2 Storage
Authors D. Guerillot and J. BruyelleThe geological static models of realistic contexts are described with high resolution meshes (HRM) and cannot be directly used as input for fluid flow reservoir simulators due to memory and/or running time constraints. The pragmatic approach consists in averaging the high resolution petrophysical values to assign to a low resolution mesh (LRM) used to perform reservoir simulations. Hence, predictions made with these coarser meshes are inevitably less accurate than those that would have been obtained on HRM. For compositional modelling, the loss of accuracy due to upscaling processes will come not only for the component displacements but also from the solution of the thermodynamic and/or geochemical equilibrium equations. For example, a chemical reaction of an acid on carbonated rock may highly depends on its concentration. Therefore, our main motivation here is to keep an HRM for calculating those chemical equilibriums. We propose to name this innovative approach “Compositional Dual Mesh Method” (CDMM). The CDMM is a formulation with two different meshes: The pressure equation is solved on a LRM using upscaled properties and the transport equation and chemical equilibrium are solved on a HRM.
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Modelling Near-well Flow Performance for Horizontal Wells in Anisotropic Media
Authors J. Cao, L.A. James and T.E. JohansenThis paper presents a novel methodology to model flow performance in an anisotropic reservoir in the near-well region with an arbitrary well trajectory. It is based on an analytical productivity model describing coupled axial reservoir flow and radial well inflow. To apply this model in an anisotropic reservoir, the permeability field relative to the radial direction perpendicular to the well trajectory and the axial direction along the well trajectory must first be determined. A classical transformation is used to obtain a virtual isotropic model. The transformation preserves the volumes and average pressures. It is applied in the near-well region without modifying the outer boundary conditions. The use of this virtual isotropic model requires the Dietz shape factor for an ellipse, which is determined numerically. For example, in a circular-cylindrical near-well region, this transformation method maps the anisotropic reservoir onto an equivalent virtual isotropic media which is an elliptical cylinder. The coupled axial and radial productivity model is implemented in a numerical simulator incorporating formation anisotropy and wellbore hydraulics. The specific productivity index along the well trajectory is generated using the virtual configuration. Numerical results for different anisotropy ratios and also incorporating frictional losses in the well are presented.
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FV-MHMM Methods for Reservoir Modelling
Authors J. Franc, L. Jeannin, R. Masson, P. Egermann and G. DebenestThe present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing to define accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called Multiscale Hybrid-Mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach [1]. We extend here the MH-Mixed Method to a Finite Volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the Lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively with respect to a posteriori error estimators. The method is illustrated by the application of single phase (well-testing) and multiphase flow in heterogeneous porous media at the field scale. [1] R. Araya, C. harder, D. Parades, F. Valentin, Multiscale Hybrid-Mixed Method, SIAM J. Numer. Anal. 51(6), 3505-3531, 2013.
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Modelling of the Waterflooding Process in the Presence of Discontinuities in the Reservoir
Authors E.V. Andriyanova, V.I. Astafev and A.E. KasatkinThe knowledge of the nature of the fluid motion in the reservoir allows us to optimize the system of oilfield development. Thus, the study of the filtration process in reservoirs with discontinuities, such as fractures, has a great importance for the oilfield development. For instance, the hydraulic fracturing is one of the most common recovery methods for the unconventional reserves. But the modern level of geophysics can show that mostly reservoirs have the tectonic faults with various permeability, and that has a great impact on well productivity. This article will show the impact of inclusions of different permeability in the reservoir on the waterflooding process. The steady-state flow process of incompressible fluid to the production well in a reservoir of constant height and permeability is considered. There is a thin area in the reservoir with constant permeability, which might be a highly permeable crack or low permeable barrier. The production and injection wells are placed inside the reservoir’s external boundary. The characteristics of waterflooding process are studied for various permeability values and different locations of the fracture and a pair of wells. Finally, flow lines of the fluid flow will be analyzed for every considered case.
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Analysis of Sparse Matrix-vector Multiply for Large Sparse Linear Systems
Authors M. Grossman, M. Araya-Polo, F.O. Alpak, F. Frank, J. Limbeck and V. SarkarDiscretization of the partial differential equations that govern the physics of multi-phase multi-component fluid flow and transport gives rise to large sparse linear systems for practical pore-scale simulation. In this work, we focus on a linear system arising from the discretization of the Cahn-Hilliard equation that governs the separation of a two-component mixture in the pore space. The discretization is performed using the discontinuous Galerkin method. The resulting nonlinear system is solved by use of the Newton's method, which entails multiple large sparse linear systems over Newton iterations course. The sparse linear systems are solved by use of an iterative linear solver. Iterative linear solvers approach the solution process by the computation of sparse matrix-vector (SpMV) products. SpMV products are computational bottlenecks for the simulation of large problems, since they are extremely memory bound. In this work, we contribute with a quantitative and qualitative evaluation of techniques for performing SpMV on large matrices. We evaluate different SpMV software implementations (frameworks and kernels) across a range of state-of-the-art hardware platforms. For example, we find that for a 5GB matrix wrt to a naive multi-threaded x86 baseline the highest performing GPU kernel runs 1.75x faster, and the highest performing x86 kernel runs 1.29x faster.
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