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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
51 - 100 of 114 results
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Analytical Front Tracking in Numerical Modelling of Two-phase Flow in Porous Media
Authors I. Panfilova, J. Rihet and M. Panfilovuities in phase saturation are the obligatory attribute of any solution. Any numerical method should contain specific procedures capable to treat the discontinuities. We propose a specific two-scale numerical method which is based on replacing the saturation field by the field of discrete “elementary fronts”, whose movement is calculated on the basis of an algorithm similar to the dynamic invasion percolation. The pressure field is calculated within a macroscopic grid (scale l), while the movement of fronts is calculated inside each macroscopic cell, so that a step of the front motion h may be much lover than l. The equation of saturation transport becomes mono-dimensional within a cell and has the analytical solution. This solution gives the analytical relation for the front velocity. The time step for front motion is selected in such a way that the most rapid front would reach the limit of the corresponding macroscopic cell. Respectively the time step is variable and may be very small. When the elementary front reaches the inlet of the cell, the conditions of its penetration in the neighbouring cells are verified, including the connectivity of the displacing phase, the capillary counter-force, and so on. The connectivity of phase clusters is calculated on the basis of a special iterative algorithm developed by the group. The validity of such a method is proved theoretically: taking into account the very slow variation of the saturation far from the fronts, it is possible to replace the saturation field by a piece-wise constant approximation in the overall domain. Then the problem is reduced to the movement of the discontinuity surface. The advantage of the present method is its absolute physical and numerical stability, so that it can be applied to model the unstable displacement and analyse the fingering process. We illustrate the possibility of the method by simulating several examples of the unstable flow as (i) the gravity driven NAPL penetration in an aquifer (the Reyleigh-Taylor instability) and the (ii) displacement of heavy oil by gas (Saffman-Taylor instability), abd comparing them with experimental results.
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Trust-region Based Nonlinear Solver for Counter-current Two-phase Flow in Heterogeneous Porous Media
Authors X. Wang and H. TchelepiWe describe a new nonlinear solver for immiscible two-phase flow where viscous, buoyancy, and capillary forces are significant. The flux function is a nonlinear function of saturation and typically has inflection points and a unit-flux point. The non-convexity of flux function is a major source of convergence difficulty for nonlinear solvers. We describe a modified Newton solver that employs trust-regions of the flux function to guide Newton iterations and solution updating. The flux function is divided into saturation trust regions. The delineation of these regions is dictated by the inflection and unit-flux points. Newton update is performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, we "chop back" the saturation value at the appropriate trust-region boundary. This development is a significant generalization of the inflection-point approach of Jenny et al. (JCP, 2009) for viscous dominated flows. Mathematically we prove the global convergence of the trust-region based nonlinear solver. Numerically we test it for multiphase flow and transport in large-scale heterogeneous problem. Using our new nonlinear solver, we achieved significant reduction in the total Newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost.
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Bio-reactive Two-phase Transport and Population Dynamics in Underground Storage of Hydrogen: Natural Self-organisation
More LessTwo new research projects on hydrogen underground storage have been submitted in France (ANR - POWELTECH) and Germany (H2STORE), with collaboration of Kazakhstan National University.
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Pore-to-reservoir Modelling of Three-phase Flow Processes in Mixed-wet Carbonate Reservoirs
Authors M.I.J. van Dijke, A. Al-Dhahli and S. GeigerCarbonate reservoirs have structural heterogeneities at all length-scales (triple porosity: pore-vug-fracture) and tend to be mixed- to oil-wet. The interplay of structural and wettability heterogeneities impacts the sweep efficiency and oil recovery. The choice of an enhanced oil recovery process and the prediction of oil recovery require a sound understanding of the fundamental controls on fluid flow in mixed- to oil-wet carbonate rocks, as well as physically robust flow functions, i.e. relative permeability and capillary pressure functions. Obtaining these flow functions is a challenging task, especially when three fluid phases coexist, such as during water-alternating-gas injection (WAG). We have developed a new three-phase flow pore-network model, which comprises a novel thermodynamic criterion for formation and collapse of oil layers that strongly depends on the fluid spreading behaviour and the rock wettability. The criterion affects in particular the oil relative permeability at low oil saturations and the accurate prediction of residual oil saturations. Additionally, multiple displacement chains have been implemented, where injection of one phase at the inlet triggers a chain of interface displacements throughout the network. This allows accurate modelling of the mobilization of the many disconnected phase clusters that arise during higher order WAG cycles. Pore-networks extracted from pore-space reconstruction methods and CT images are used as input for the pore-scale simulations and the model comprises a constrained set of parameters that can be tuned to mimic the wetting state of a given reservoir. Three-phase flow functions generated from networks with carbonate pore geometries and connectivities have been used in a heterogeneous carbonate reservoir model and we demonstrate their impact on the sweep efficiency after gas injection and WAG for a range of realistic wettability scenarios. We also show that the network generated flow functions give distinctly different recovery curves compared to recoveries for traditional three-phase flow relative permeability functions, such as Stone’s.
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Modeling and Simulation of Shale Gas Production in Multi-Staged Hydraulic-Fractured Formations
More LessShale gas production is effectively enhanced by multi-staged hydraulic fracturing from horizontal wells. The characteristics of the generated fracture networks are crucial to estimating shale gas production rate and consequently determine the economics of shale gas projects. The location and geometry of hydraulic fractures are reasonably well known; whereas the secondary fractures, generated during the fracturing process, are numerous and can only be described by a stochastic framework. We thus propose three groups of fractures to be modeled: (1) hydraulic fractures whose location and geometry can be deterministically approximated, (2) smaller induced/natural fracture subset connected between hydraulic fractures, and (3) disconnected small scale (natural or induced) fractures. As the permeability contrast between fractures and micro or nano pores in shale is very large, the gas production rate will be controlled by the diffusion process that feeds gas from shale to fracture networks and by the pressure-drop propagation mechanism in the formation. The transport of gas from micro or nano pores to the fracture network comprises two mechanisms: (1) molecular (or density) diffusion and (2) convective flow due to gas compressibility. We derive a simple numerical solution for the advection/diffusion equation, coupled with statistical distribution of micro and nano pores.
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A Mathematical Model for Interpretation of Brine-Dependent Spontaneous Imbibition Experiments
Authors P.Ø. Andersen and S. EvjeIn this paper we consider a mathematical model that seeks to explain possible mechanisms for brine-dependent oil recovery in chalk. It is well documented through lab experiments that the brine composition has a strong impact on oil recovery. In particular, the role of the divalent ions (Ca2+, Mg2+ and SO24−) present in seawater have been extensively studied. Also the effect of salinity which is mainly controlled by the monovalent ions Na+ and Cl− has been carefully investigated. It has been observed that chemical reactions occur between rock and brine when seawater or seawater-like brines are injected or diffuse into chalk at high temperature. Different chemical mechanisms are involved like ion exchange, adsorption, and precipitation/dissolution of minerals such as calcite, magnesite and anhydrite. Hence, these experiments suggest that for spontaneous imbibition tests the produced oil is a result of an interplay between capillary forces and the imposed water-rock chemistry. We are interested in formulating a theory for this observed behavior based on a proper combination of geochemical and two-phase model components. The mathematical model we present couples geochemical reactive transport with the capillary forces trapping the oil. When a brine different from the formation brine enters pore space the water-rock chemistry induces changes on the rock surface. It is suggested that this leads to correspond- ing changes of the wetting state as represented by relative permeability and capillary pressure curves. Different hypothesis concerning the possible link between geochemical changes of the rock-surface and changes of wetting state are explored. Specifically, we employ the model to dis- cuss some previously published lab experiments where systematic variations in Ca2+ and SO24− in imbibing and initial brine were explored. The model suggests that at 70◦C neither dissolution nor precipitation are the main contributors for wettability alteration. Rather, a conceptual sulfate adsorption mechanism coupled to the surface activity of calcium readily explain how adding more sulfate and calcium to the system would increase oil recovery. Hence, we demonstrate how the model can be used as a tool for systematic investigations aiming at identifying key mechanisms important for mobilization of oil as a function of brine composition.
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Tie-simplex-based Nonlinear Solver for Mass-variables Compositional Formulation
Authors D.V. Voskov and H.A. Tchelepig flux function in parameterized compositional space is developed for general-purpose compositional simulation. This solver takes full advantage of the hyperbolic nature of the transport equations of compositional problem. Since compositional recovery processes evolve along a few ‘key’ tie-simplexes, the flux functions (fractional flow curves) parameterized along these tie-simplexes play a dominant role in the evolution of the solution. For a given nonlinear iteration, the flux functions associated with the parameterized tie-simplex are segmented into trust regions which includes appearance and disappearance of phases, changes in mobility of phases, and the inflection point of flux function. These regions are used to guide the evolution of the composition unknowns on nonlinear iteration since they delineate convex regions of the flux function, where convergence of the Newton iterations is guaranteed. Several challenging compositional problems are used to test the robustness and efficiency of this tie-simplex-based nonlinear solver. The convergence rate of the new nonlinear solver is always better than our standard safeguarded Newton method, which employs heuristics on maximum changes in the variables. We demonstrate that for aggressive time stepping, the new nonlinear solver converges within a fewer number of Newton iterations.
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Compositional Formulation Based on Piece-wise Linear Representation in Tie-simplex Space
Authors R. Zaydullin, D.V. Voskov and H.A. TchelepiCompositional formulations are necessary for numerical simulation of EOR (Enhanced Oil Recovery) processes, such as gas and steam injection. The coupling of the nonlinear conservation laws of multiphase flow and transport with the thermodynamic equilibrium relations poses significant challenges for compositional simulation. We describe a new framework, in which the thermodynamic phase behavior is cast in tie-simplex space as a function of composition, pressure and phase fraction. This parameter space is then used to specify the base nonlinear variables for fully-implicit compositional simulation. The compositional space is discretized using tie-lines. Thus, all the thermodynamic properties become piece-wise linear functions in this space. The numerical implementation employs multilinear interpolation of the phase behavior using adaptively constructed tie-line tables. The computation of the phase behavior in the course of a compositional simulation then becomes an iteration-free procedure and does not require any EoS (flashes or phase-stability tests) computations. The efficiency and accuracy of the method are demonstrated for several multidimensional compositional problems for both miscible and immiscible displacements. For the tested problems, the proposed method reduces the computational cost of the thermodynamic calculations significantly compared with standard EOS-based approaches. Moreover, the method shows better nonlinear convergence behavior for near-miscible gas injection displacements.
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Method of Negative Saturation and Interface Stabilization for Multiphase Compositional Flow in Porous Media
Authors M. Panfilov, M. Ghesmoune and A. AbadpourVarious EOR methods lead to the appearance of various zones with different number of phases and different thermodynamic state. They are separated by specific surfaces called the interfaces of phase transition. Consecutively, the flow equations are also different in various zones and cannot be deduced from each other by continuous degeneration, which imposes serious difficulties in numerical modelling. We suggest a new conceptual mathematical method based on the replacement of real single-phase fluid by an imaginary multiphase muticomponent continuum having fictitious properties. As the result, the fluid over all zones becomes three-phase and can be described by uniform three-phase hydro- and thermodynamic equations, which allows applying the direct numerical simulation. The equivalence principle determines the physical properties of the fictitious multiphase fluid, as well as the structure of the uniform multiphase equations. It also proves that the saturation of each phase may become negative in non-equilibrium zones, which becomes the efficient method of tracking the interface and the number of phases at any point. The method was developed by the authors for two-phase case. In the present paper the new version is developed for three-phase case. Several examples of simulation are presented.
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Vertex-centred Discretization of Multiphase Compositional Darcy Flows on General Meshes
Authors C. Guichard, R. Eymard, R. Herbin, R. Masson and P. SamierThis paper introduces a vertex centred discretization on general 3D meshes of multiphase Darcy flows in heterogeneous anisotropic porous media. The model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium. The conservative spatial discretization of the Darcy fluxes is based on the Vertex Approximate Gradient scheme (VAG) which is unconditionally coercive for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes. On tetrahedral meshes, the number of unknowns is considerably reduced, by typically a factor five, compared with usual cell-centred MultiPoint Fluxes Approximations, which is a key asset for multiphase flow simulations on unstructured meshes. An adaptive choice of the pore volume at the vertices ensures the accuracy of the discretization even for coarse meshes on highly heterogeneous media. This approach can easily be implemented on existing reservoir simulators using a graph of transmissibilities for the computation of the fluxes. The efficiency of our approach is exhibited on several two phase and three phase Darcy flow examples. In particular it includes the nearwell injection of miscible CO2 in a saline aquifer taking into account the precipitation of salt.
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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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Multi-core and GPU Parallelization of a General Purpose Reservoir Simulator
Authors Y. Zhou and H. TchelepiWe describe our multi-threading parallelization strategy of a general-purpose reservoir simulator (GPRS) based on a flexible Automatic Differentiation (AD) framework. Parallel Jacobian construction is achieved with a thread-safe extension of our AD library. For linear solution, we use a two-stage CPR (Constrained Pressure Residual) preconditioning strategy, combining the parallel multigrid solver XSAMG and the Block Jacobi technique with Block ILU(0) applied locally. The speedup of the full SPE 10 problem (1.1M cells) is about 5.0X on a dual quad-core Nehalem node. We then discuss the GPU parallelization of Nested Factorization (NF). The Massively Parallel NF (MPNF) algorithm was first introduced by Appleyard et al. (2011), where the 3D structured grid is divided into kernels, and each kernel is assigned a color such that no neighbouring kernels share the same color. Then, parallelism is exploited in the concurrent solution of all kernels with the same color. The most important aspects of our algorithm are: 1) coalesced memory access via special ordering of the matrix elements, and 2) application of the twisted factorization technique that further improves parallelism. With a 512-core Tesla M2090 GPU, the speedup of the full SPE10 problem is about 26X for single precision, and 19X for double precision.
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HPC-Based Optimal Well Placement
Authors A.M. Kuvichko and A.I. ErmolaevThis paper studies the mathematical aspects of well location and related optimization problems. These problems are formulated in terms or integer programming. Optimal solutions found are to formulate initial sets of cases and to improve the efficiency of oil and gas recovery. Described optimization algorithms are presented as high-scalable parallel programs making a vast majority of cases to be considered. Considered integer programming problems are extremely large-scale problems. The matrix structure, the number of feasible solutions, etc. was taken into account. A new fast algorithm for the generalized assignment (transportation) problem has been designed. Programs implementing this algorithm for CPU and GPU were tested and the results presented. A high scalability and good speedup achieved. Performed tests had also shown better timing results comparing to well-known common algorithms. It is reasonable to use the approach studied in the paper to design a set of appropriate initial cases for the small fields or fields with a complex geology. Presented workflow finds optimal well positions for a field or its part. A Brugge field has been taken as a test case. An improvement of production and NPV achieved the comparison between an initial and an optimal case presented.
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A Multilevel Multiscale Finite Volume Method
More LessThe Multiscale Finite Volume (MsFV) method has been developed to efficiently solve reservoir-scale problems while conserving fine-scale details. The method employs two grid levels: a fine grid and a coarse grid. The latter is used to calculate a coarse solution to the original problem, which is interpolated to the fine mesh. The coarse system is constructed from the fine-scale problem using restriction and prolongation operators that are obtained by introducing appropriate localization assumptions. Through a successive reconstruction step, the MsFV method is able to provide an approximate, but fully conservative fine-scale velocity field. For very large problems (e.g. one billion cell model), a two-level algorithm can remain computational expensive. Depending on the upscaling factor, the computational expense comes either from the costs associated with the solution of the coarse problem or from the construction of the local interpolators (basis functions). To ensure numerical efficiency in the former case, the MsFV concept can be reapplied to the coarse problem, leading to a new, coarser level of discretization. One challenge in the use of a multilevel MsFV technique is to find an efficient reconstruction step to obtain a conservative fine-scale velocity field. In this work, we introduce a three-level Multiscale Finite Volume method (MlMsFV) and give a detailed description of the reconstruction step. Complexity analyses of the original MsFV method and the new MlMsFV method are discussed, and their performances in terms of accuracy and efficiency are compared.
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The Gravitational Instability of a Diffusive Boundary Layer; Towards a Theoretical Minimum for Time of Onset of Convecti
More LessIn this paper we extend previous work in on the linearized analysis of gravitational instability of a diffusive boundary layer in a semi-infinite anisotropic homogenous porous medium. We express the time derivative of the square of the standard L^2-norm of a given perturbation as a time dependent quadratic form on an appropriate Hilbert space . Numerical analysis of the spectra of these quadratic forms give rise to results qualitatively similar to previous results in the litterature. We demonstrate that after the time of instability only perturbations having a non-zero projection onto a one-dimensional subspace of are unstable. We also find that the space of neutrally stable perturbations before onset of instability form a large subspace of the space of possible perturbations, where numerical analysis strongly indicate that this subspace is infinite dimensional. Error estimates for a certain part of the numerical analysis are not yet rigorous. In particular, estimating the spectrum of unbounded linear operators using finite matrix approximations still lacks a theoretical basis. However, the largest eigenvalues of larger and larger matrices approximating the operator converge quickly to well defined values, and it is conjectured that the given critical values are the correct ones for the problem at hand.
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Efficiency of Dissolution Trapping in Geological Carbon Storage
Authors M.T. Elenius, J.M. Nordbotten and H. KalischDuring geological storage of carbon dioxide (CO2), several mechanisms contribute to safe storage by immobilizing the CO2 in the injection formation. It has been shown that dissolution into resident brine can be one of the major contributors. The injected supercritical CO2 is buoyant, but dissolved CO2 increases brine density and therefore reduces the tendency for upward CO2 migration. The density increase with dissolved CO2 leads to convective mixing of the brine, thereby enabling more CO2 to dissolve. It is important to quantify the efficiency of CO2 dissolution, and therefore the efficiency of convective mixing. In previous work, we have shown that convective mixing can be considerably enhanced when taking into acccount the interaction between the two-phase region (supercritical CO2 and brine), and the single-phase brine region. Bounds on this impact were obtained for onset times, wavelengths of unstable fingers, and dissolution rates. The maximum increase in the dissolution rate was found to be large, when interaction with the plume was considered. In this paper, we use stability analysis to further study the dissolution in more detail. We make technical contributions to the field of stability analysis and in obtaining more reliable estimates of the efficiency of dissolution trapping.
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Evolution of Seismic Responses due to CO2 Injection in Carbonates Including Chemical Reactions and Rock-Physics Model
Authors L.G. Rodrigues, J.P. Nunes and D.R. GuérillotThe increase in worldwide activities related to CO2 injection in geological formations, for both EOR/CO2 and CCS projects, has pushed oil companies and universities to enhance the modeling of these processes for their better (1) designing and (2) monitoring. The objective of this paper is to describe new improvements for these two aspects through a multi-scale methodology of simulations from laboratory experiments to full-field modeling passing by studies around the wells including new rock-physics model. When injecting CO2 in carbonate rocks, one of the most critical aspects is to understand the complex chemical reactions occurring between the acidic fluid formed by the CO2, the in-situ water (connate and aquifer) and the carbonate matrix depending on its mineralogy. The mathematical formulation of the simultaneous thermodynamic equilibrium and the chemical reactions will be described completely. An original construction of the rock-physic model developed for this multi-phase flow based an effective medium theory will be described. A specific power law equation will be proposed to fit the relation between porosity and permeability obtained in the laboratory for carbonate rocks replacing the classically used Kozeny-Carman equation not valid in our case. To improve the quality of the forecasts at the entire reservoir level, simulations at different scales are performed and used sequentially. Results of sensitivity studies with various rock and mineralogy characteristics showing the impact on (a) the porosity and permeability field on the CO2 segregation, (b) the pH evolution in space and time, (c) the synthetic seismograms, will be described. This paper demonstrates the practicality of the modeling approach and software tools to address the design and monitoring of CO2 injected in a geological formation for CO2/EOR and or CCS processes. In particular, it helps the geophysicists and reservoir engineers based on the geological description of the reservoir to design injection plans for EOR or CCS processes defining plans for well tests and seismic campaigns around the wells and in between them (2D or 3D) whether such changes may be observable as a function of time. Technical contributions: 1. Multi-phase flows on realistic carbonate reservoirs with multi-phase thermodynamic equilibrium and geochemical reaction, 2. New rock-physic model for the evolution of the density and velocities used to construct surface seismic responses, 3. Methodology to improve the quality of the full-field forecast checking the results of the simulations results at the laboratory scale and generating synthetic seismograms to design seismic campaigns for monitoring.
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Classification of Digital Rocks by Machine Learning
Authors J. Ma, Z. Jiang, Q. Tian and G.D. CouplesThe availability of high-resolution 3D digital rocks in ever increasing quantities calls for intelligent Machine Learning (ML) techniques to classify them according to diverse characteristics of their pore structures. If stable classes could be identified, they would aid us to develop better models for rock typing, to gain sounder understanding of the links between the pore structures and the fluid flow behaviours and to develop predictive models of effective flow properties with many potential applications in the petroleum industry and beyond. We reported an approach that the authors developed for classifying digital samples. There, the pore structure is characterised by topological and geometrical attributes obtained from topology-preserved pore networks for each sample. Each attribute is then represented as a 1st-order tensor and normalised so that it is comparable for images sampled at different scales and resolutions. Machine learning techniques are then used to carry out actual classification from a training dataset containing labelled and unlabelled samples. The viability and extendibility of this approach are discussed. We show that this approach can be implemented to classify samples in progressive, recursive and regressive manners, and can be extended to develop correlation between the classes of samples and their fluid flow properties.
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A Flow and Transport Model in Porous Media for Microbial EOR Studies at Core Scale
Authors M. A. Diaz-Viera and J.R. Hernandez-PerezThe oil fields at their initial operation stage produce using basically its natural energy which is known as a primary recovery. As the reservoir loses energy in order to maintain the pressure it requires the injection of gas or water, which is called a secondary recovery. When the secondary recovery process becomes ineffective it is necessary to apply a more sophisticated approach such as steam injection, chemicals, etc. These are known as enhanced oil recovery methods. Some important oil fields in Mexico are entering the third stage. For the optimal design of oil recovery methods it is required to perform a variety of laboratory tests under controlled conditions to model the fundamental recovery mechanisms for a given recovery method in a specific reservoir. However, the laboratory tests commonly have a number of drawbacks, which include among others that they are very sophisticated, time consuming, expensive and always not enough to cover the whole range of field conditions involved. A proper modeling of the laboratory tests would be decisive in the interpretation, analysis and understanding of recovery mechanisms as well as in obtaining the relevant parameters for the subsequent implementation of recovery processes at the well and the reservoir scale. In this work we present a flow and transport model which was implemented using the finite element method to simulate, analyze and interpret MEOR processes at core scale under laboratory conditions. The flow model is biphasic and is based on the oil phase pressure and total velocity formulation given by Chen Z. et al. 2006, in which the capillary pressure, relative permeabilities, the effects of gravity and the dynamic porosity and permeability modification due to the clogging-declogging phenomena (adsorption-desorption of microorganisms) are taken in account. Whereas, the transport model consists of two phases (water-biofilm) and three components (microorganisms, nutrients and bioproducts). The transport model includes physical-chemical-biological phenomena such as advection, diffusion, dispersion, adsorption-desorption, growth and decay of microorganisms. Adsorption of nutrients is implemented through a linear adsorption isotherm. The effects of the bioproducts on the residual oil saturation are also included. From the methodological point of view, each stage of model development (conceptual, mathematical, numerical and computational) is described. Finally, the resulting coupled flow and transport model is numerically validated in a case study of oil displacement by the injection of water follows by the injection of water with microorganisms and nutrients. The oil recovery evaluation considering different scenarios is shown.
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Pore-scale Single and Two-phase Transport in Real Porous Medium
Authors I.I. Bogdanov, J. Kpahou and F. GuertonSince long time it has been recognized that the typical pore size is a fundamental scale in understanding of transport phenomena and determination of global transport properties of porous media. In a similar way like the Navier-Stokes equations may be used at certain limit to derive the Darcy law and define single phase transport properties, the modified Navier-Stokes model might be used to determine medium two-phase flow properties. Instead of using a regularization technique to capture the interface (cf. VOF or level-set functions approach), which may affect the modelling results in a non-trivial way, the diffuse interface method offers a thermodynamic treatment of phase “mixing” zone. As a result, it is a good choice for a numerical technique, handling the morphological changes of the interface which is of great importance for modelling of such a kind. Like zero-order approximation which is at the same time the classical theory assumptions case, the two-phase flow properties (e.g. phase relative permeabilities) are simply two ultimate single phase flow configurations, one per each phase. In both cases only volumes occupied by one fluid are considered so that wetting and capillary properties becomes very important, probably along with the process history as they all are responsible for particular fluid distribution in pore space. Taking advantage of recent advancements in X-ray computed micro-tomography (μCT), the reconstructed real porous medium samples (Bentheimer sandstone) are used for direct numerical simulations (DNS) of single and two-phase transport problems. Main model parameters - capillary, Reynolds, Cahn and Peclet numbers - are defined for each flow case. Emphasis is made on characterization of different steps and features of methodology based on μCT measurements, geometrical reconstruction, grid generation and computational models. The contribution of DNS to understanding of transport phenomena in real media becomes increasingly important factor of porous medium description efforts.
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Multi-scale Simulation of Permeability Fields and History-matching
Authors C. Gardet, M. Le Ravalec and E. GloaguenThe prediction of fluid flows within oil reservoirs or gas storage sites or aquifers requires the characterization of its petro-physical properties, i.e., facies, porosity, permeability, etc. This issue can be addressed through history-matching which calls for the determination of a three-dimensional model representing the studied reservoir. In a nutshell, a model is a grid populated by petrophysical properties. These ones have to be sequentially adjusted until the flow responses simulated for the resulting reservoir model reproduce the available dynamic data: pressures, flow rates, water cuts, 4D-seismic... A difficulty usually disregarded is that these data provide information about petrophysical properties at different scales. Referring to sequential simulation, we propose a method for generating multiscale realizations of both continuous or discrete random fields. These ones are then used to populate reservoir models with the required petrophysical properties. The integration of multiscale simulation within history-matching provides new facilities and makes it possible to incorporate dynamic data at different scales of resolution. When combined with geostatistical parameterization techniques as the gradual deformation method, it gives the essential ability to adjust the reservoir model at various scales. In addition, the overall history-matching process becomes more efficient as targeting the appropriate scale entails an economical parameterization of the model, i.e., the coarser the scale, the smaller the number of unknown parameters. Last, we present a numerical application case to highlight the advantages of the method for conditioning permeability models to dynamic data. For simplicity, we focus on two-scale processes. The coarse scale describes the variations in the mean while the fine scale characterizes local variations around the mean. We investigate the relationships between data resolution and parameterization.
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Upscaling of Vertically Heterogeneous Reservoirs
Authors A. Stovas and Y. RoganovAn accurate description of a reservoir is crucial to the management of production and efficiency of oil recovery. Reservoir modeling is an important step in a reservoir’s future performance, which is in direct proportion to reservoir management, risk analysis and making key economic decisions. Saturation and pressure changes, and porosity and permeability distributions are the most common parameters to estimate in the oil industry. In order to reduce the number of parameters in reservoir description, the different upscaling techniques have been used. At the rock physics level, the Gassmann and Hertz-Mindlin theories are applied in order to incorporate the fluid substitution and pressure changes, respectively. The most popular method at the elastic level is the Backus (1962) averaging. This method is based on the zero frequency limit of seismic wave field in a vertically heterogeneous structure. We extend the Backus averaging for the low-frequency regime by using the Baker-Campbell-Hausdorff series (Serre, 1965). That allows us to compute the frequency dependent effective medium parameters. These parameters can be used in seismic modeling and inversion with band-limited seismic wavelet.
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Homogenization of Relative Permeabilities Curves for Two-phase Flow in Porous Media Using an Optimization Method
Authors F. McKee, C. Preux and C. BerthonGrid coarsening remains essential in practical reservoir studies in order to get acceptable simulation time. This implies being able to upscale two-phase flow in particular the relative permeability. Upscaling can be divided in two stages: homogenization and mesh changing. Optimization gets involved here in the homogenization part. We proceed by identification between fine grid simulation on both representative heterogeneous regions and homogeneous equivalent region. We start with a mesh containing heterogeneous rock type. Each rock type has its own relative permeability curve and these curves are homogenized throughout the mesh with a unique relative permeability curve. In order to do this, the exit oil flow rates from the heterogeneous rock type case are considered as a reference solution. We then simulate the same flow except for the unique effective relative permeability curve. The exit oil flow rates from the two simulations are extracted to build a least squares objective function. The effective relative permeability curve (kr) is the main parameter of the optimization problem : the end points of a Brooks-Corey relative permeabilities model are used to look for a minimum objective function value.
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Mathematical Model of Horizontal Well Acidizing
Authors S.U. Zhuchkov and R.D. KanevskayaHorizontal wells are widely used to increase reservoir development efficiency. The most important factor of successful use of such wells is their capability to preserve reservoir properties in the vicinity of horizontal well. Acid treatment leads to dissolution of rock matrix and rock particles, which can plug flow channels. Therefore it is often used to recover permeability and intensify oil production. Acidizing of horizontal wells requires a special approach. The efficiency of stimulation depends on reagent distribution along the borehole, depth of acid penetration and reaction kinetics. To evaluate these characteristics it is necessary to work with adequate mathematical models giving the possibility to plan the acid treatments. These models should take into account the specific character of fluid flow close to horizontal well, pressure loss along the well, influence of gravity and heterogeneity of reservoir properties. The modeling of rock matrix dissolution should be carried out. Mathematical two-phase multicomponent model describing the flow of acid solution close to horizontal well is presented. The effects related to chemical reactions and fluid flow in well are taken into account. Calculation results for different cases are presented.
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Mixture Models for Sampling Conditional Facies Realizations from Multiple Training Images
Authors B. Jafarpour and M. KhodabakhshiMultiple point statistics (MPS) provides a systematic approach for pattern-based simulation of geologic objects from a conceptual training image (TI). The TI encodes the higher-order spatial statistics of the expected connectivity structures through stationary patterns representing the underlying geologic features. The pattern-imitating nature of MPS simulation implies that the simulated facies inherit the spatial structure of the general features in the TI. This property makes the MPS approach very sensitive to uncertainty in the prior TI. Since TIs are constructed using uncertain data and imperfect assumptions, multiple TIs may be necessary to account for the uncertainty and full range of structural variability in facies descriptions. We present a Bayesian mixture modeling approach for adaptively sampling conditional facies from multiple uncertain TIs using a probability conditioning method (PCM). Using the PCM, we invert the flow data to obtain a facies probability map for drawing conditional facies realizations from each TI. The number of samples drawn from each TI is proportional to the weight assigned to them. The TI weights are assigned based on the predictive performance of its corresponding conditional facies realizations. We demonstrate the suitability of the proposed method using numerical experiments in fluvial formations.
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Optimization of Dynamic 3D Hex-dominant Mesh Adapted for Basins Simulation Using the Smoothing Laplacian 2D
Authors B. Yahiaoui, H. Borouchaki, A. Benali and C. BennisTo improve a dynamic hex-dominant mesh for basins, a particular optimization in shape is proposed in this article. The aim is to return a mesh as regular as possible on $xy$ coordinates and align the $z$ coordinates. This optimization must complete an existing approach generate a hex-dominant mesh to improve these generated elements. To do this optimization for the $xy$ coordinate a Transformation called Smoothing Laplacian is applied. After, an iterative method which transforms some connections between layers in verticals. And it’s possible to conclude that this kind of optimization can be improved to have any shape wanted.
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Quantification of Uncertainty in Reservoir Connectivity for Field Test Evaluation
By H. OkanoProduction forecasts for petroleum reservoirs are essentially uncertain due to the lack of data. The unknown parameters are calibrated so that the simulated profile can match the observed data. A Bayesian framework has been applied to the evaluation of CO2 injection test in a tight oil reservoir. The observed data used for history-matching include the bottom-hole flowing pressure at the injector well and the gas composition at the wellhead of the producer wells. The key is starting with a simple model, because it is much quicker to adjust large-scale heterogeneity in a simple model than in a detailed model. The in-place volumes and connectivity between the wells have been calibrated in the simple models using a stochastic sampling method called the Neighbourhood Approximation algorithm. The aim of our study is to quantify uncertainty of reservoir connectivity. A Bayesian framework along with Markov Chain Monte Carlo and Neighbourhood Approximation in parameter space is used to calculate the posterior probability. We showed the best fit model for the gas breakthrough and the P10-90 envelopes in the forecast of the CO2 mole fraction in the produced gas. Our results contribute to the evaluation of the pilot test for a continuous CO2 injection.
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Development of Iterative Algorithms of Increased Convergence and Accuracy for Multiphase Flow Simulation
Authors D. Yu. Maksimov and M.A. FilatovMost of commercial simulators for multiphase flow in porous media use implicit or adaptively implicit discretization schemes which allow for rather large time steps. It is important that in the iterative process the equations being approximated may change, e.g. in cases of well target change, counterflow. To increase stability of nonlinear iterations convergence, we propose a method relying on the control of residual norm decreasing and taking into account the features of the problem under consideration. Final correction of recurrent solution increment is carried out after direct calculation of the residual with corresponding approximation before and after the point where equations change. Correction of the increment which tries to retain form of equations being approximated (e.g. well mode) makes convergence more stable and additionally allows avoiding convergence to nonphysical solution. In the paper a number of possibilities for safeguard retaining of approximation type from the previous iteration is pointed out for situations (e.g. for low filtration rate) where it has to be changed by algorithm in the strict sense, given constraints being controlled to obtain “physical” solution. Several simulation results are presented, demonstrating the robustness and effectiveness of the proposed method for challenging problems.
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Permeability Change Estimation from Microseismic Event Activity Variations
Authors S.B. Turuntaev, O.Y. Melchaeva, E.V. Zenchenko and E.I. Eremeevaactivated by the pore pressure change. It was found, that the probability distribution of these “potential fractures” can be approximated by a Weibull distribution. It was shown that it is possible to solve the inverse problem of defining local permeability from registered microseismic activity variation in a particular volume of porous medium.
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Modeling the Feasibility of Gas-Water or Gas-Oil Contact Control by Microgravity Monitoring during Enhanced Oil Recovery
By J. MrlinaMonitoring of fluids in reservoirs has become an essential tool for the control of active hydrocarbon fields, including EOR process. Repeated microgravity (time-lapse gravity, 4D gravity) can determine especially gas-water or gas-oil contact displacement in time. The technique can also be used in industrial and construction areas, contrary to 4D seismic and electromagnetics. The efficiency of the technique has been already proven by successful time-lapse gravity surveys, e.g. in Alaska, France, Italy, Oman and Qatar. Based on experience from water penetration to a sandstone formation in Egypt, gravity modelling was performed to simulate gas - water/oil contact movement in reservoirs related to pumping, water-flooding, etc. Various reservoir parameters were changed - depth, thickness, geometry, porosity and density. Gas or steam injecting/pumping was investigated, too. It was found that such processes can be observed by time-lapse microgravity, but the success depends on local geological conditions and reservoir parameters. New complex feasibility parameter cF was developed and established in graphic and tabular forms based on the time- and space-domain 3D and 4D gravity modelling. This parameter should provide fast pre-survey estimation of the effectiveness of microgravity monitoring. This procedure has not been developed before.
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Interactive Sketch-based Estimation of Stimulated Volume in Unconventional Reservoirs Using Microseismic Data
Authors Y. Hajizadeh, R. Amorim, N. Boroumand, E. Vital Brazil, D. Eaton and M. Costa SousaThe development of unconventional reservoirs has received tremendous attention from energy companies in recent years. Due to the low permeability nature of these resources, a hydraulic fracturing is often applied to stimulate the near-well region to enable economic production. The injection pressure, as it propagates, creates fractures that generate microseismic events. The monitoring of such events has become an important tool to better understand hydraulic fracture geometry, to estimate stimulated reservoir volume, to refine fracture treatment, and to optimize long-term field development. In the estimation of Stimulated Reservoir Volume (SRV) from microseismic data, recent literature highlights the importance of using time and uncertainty to achieve a more accurate estimation, as well as the influence of more complex geometries in understanding the microseismic event cloud. However, the current methods do not take any of these factors into consideration. In this work, we propose two different approaches to estimate the SRV that integrate spatial correlation together with time to obtain more accurate volume estimations. The first method is called alpha-shapes which is a generalization of the well-known shrink-wrap algorithm. The second approach is the density-based region reconstruction which considers the density of the microseismic samples in the space to reconstruct the SRV. The density-based approach uses radial basis function with Gaussian kernels to account for uncertainty in microseismic events. In addition to these two methods, we also developed a sketch-based tool to assist the users in filtering microseismic events that are visibly wrong. We molded these two approaches to allow for direct user changes to the final volume through sketch-based tools, and thus giving the expert the ability to guide the SRV estimation and to create "what-if" scenarios for a better understanding of the microseismic data. We also integrated the developed tools in this work with an interactive tabletop multitouch display to create a collaborative work environment for the experts.
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