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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
41 - 60 of 114 results
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Adjoint-Based Optimization of a Foam EOR Process
Authors J.F.B.M. Kraaijevanger, M. Namdar Zanganeh, H.W. Buurman, J.D. Jansen and W.R. RossenWe apply adjoint-based optimization to a Surfactant-Alternating-Gas foam process using a linear foam model introducing gradual changes in gas mobility and a nonlinear foam model giving abrupt changes in gas mobility as function of oil and water saturations and surfactant concentration. For the linear foam model, the objective function is a relatively smooth function of the switching time. For the nonlinear foam model, the objective function exhibits many small-scale fluctuations. As a result, a gradient-based optimization routine could have difficulty finding the optimal switching time. For the nonlinear foam model, extremely small time steps were required in the forward integration to converge to an accurate solution to the semi-discrete (discretized in space, continuous in time) problem. The semi-discrete solution still had strong oscillations in gridblock properties associated with the steep front moving through the reservoir. In addition, an extraordinarily tight tolerance was required in the backward integration to obtain accurate adjoints. We believe the small-scale oscillations in the objective function result from the large oscillations in gridblock properties associated with the front moving through the reservoir. Other EOR processes, including surfactant EOR and near-miscible flooding, have similar sharp changes, and may present similar challenges to gradient-based optimization.
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High Order Adjoint Derivatives using ESDIRK Methods for Oil Reservoir Production Optimization
Authors A. Capolei, E.H. Stenby and J.B. JørgensenIn production optimization, computation of the gradients is the computationally expensive step. We improve the computational efficiency of such algorithms by improving the gradient computation using high-order ESDIRK (Explicit Singly Diagonally Implicit Runge-Kutta) temporal integration methods and continuous adjoints . The high order integration scheme allows larger time steps and therefore faster solution times. We compare gradient computation by the continuous adjoint method to the discrete adjoint method and the finite-difference method. The methods are implemented for a two phase flow reservoir simulator. Computational experiments demonstrate that the accuracy of the sensitivities obtained by the adjoint methods are comparable to the accuracy obtained by the finite difference method. The continuous adjoint method is able to use a different time grid than the forward integration. Therefore, it can compute these sensitivities much faster than the discrete adjoint method and the finite-difference method. On the other hand, the discrete adjoint method produces the gradients of the numerical schemes, which is beneficial for the numerical optimization algorithm. Computational experiments show that when the time steps are controlled in a certain range, the continuous adjoint method produces gradients sufficiently accurate for the optimization algorithm and somewhat faster than the discrete adjoint method.
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Simultaneous Optimization of Well Placement and Control Using a Hybrid Global-local Strategy
Authors T. D. Humphries, R.D. Haynes and L.A. JamesOptimal placement and control of wells is essential to ensuring maximal net present value (NPV) or total oil recovery when developing an oil field. The majority of academic literature treats optimal placement and control as two separate problems; however, treating the problems simultaneously may allow us to achieve better results. The objective function (i.e. NPV) in this joint problem tends to vary nonsmoothly as positional parameters are varied, but smoothly in the control parameters. This suggests an approach that utilizes both global and local optimization techniques. In this paper we address the placement and control optimization problem simultaneously with two approaches combining a global search strategy (particle swarm optimization, or PSO), which operates over all variables, along with a local generalized pattern search (GPS) strategy, which operates primarily on the control parameters. The first approach is a hybrid PSO/GPS algorithm which optimizes over all positional and control variables simultaneously, while the second approach decouples the problem into separate placement and control problems, and attempts to solve them sequentially. Simulation experiments show that both approaches tend to outperform PSO in simple problems, while the decoupled approach may be the most suitable for more complicated cases.
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Ensemble Based Multi-Objective Production Optimization of Smart Wells
Authors R.M. Fonseca, O. Leeuwenburgh and J.D. JansenIn a recent study two hierarchical multi-objective methods were suggested to include short-term targets in life-cycle production optimization. However this previous study has two limitations: 1) the adjoint formulation is used to obtain gradient information, requiring simulator source code access and an extensive implementation effort, and 2) one of the two proposed methods relies on the Hessian matrix which is obtained by a computationally expensive method. In order to overcome the first of these limitations, we used ensemble-based optimization (EnOpt). EnOpt does not require source code access and is relatively easy to implement. To address the second limitation, we used the BFGS algorithm to obtain an approximation of the Hessian matrix. We performed experiments in which a water flood was optimized in a geologically realistic multi-layer sector model. The controls were inflow control valve settings at pre-defined time intervals. Undiscounted Net Present Value (NPV) and highly discounted NPV were the long-term and short-term objective functions used. We obtained an increase of approximately 14% in the secondary objective for a decrease of only 0.2-0.5% in the primary objective. The study demonstrates that ensemble-based multi-objective optimization can achieve results of practical value in a computationally efficient manner.
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Mathematical Modeling of Microbial Processes for Oil Recovery
Authors J. Monteagudo and C. HuangMicrobial recovery processes involve the usage of microorganisms, either indigenous or injected into the reservoir, to produce metabolic reactions that trigger a variety of mechanisms conducting to the production of hydrocarbons and/or enhanced oil recovery. In this work we have developed a mathematical model that accounts for several mechanisms involved both in the Microbial Gas Generation (MGG) and Microbial Enhanced Oil Recovery (MEOR) processes. This involves a kinetics model that predicts the cell growth and the metabolite production of gas, bio-surfactants and bio-polymers. Additionally, the model considers the reduction of the residual oil saturation due to the bio-surfactant and the change of water viscosity by the bio-polymer. An adsorption model depicts the retention of solutes in the aqueous phase thus altering the porosity and permeability. The model was implemented in a full-field 3-D compositional and black-oil reservoir simulator. We performed validations against experimental data available in the literature and then used the model to simulate MGG and MEOR processes with synthetic field cases. Sensitivity studies were conducted to assess the influence of the microbial kinetic model parameters in the predictions.
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A 2D Model for the Effect of Gas Diffusion on Mobility of Foam for EOR
Authors L.E. Nonnekes, S.J. Cox and W.R. RossenTransport of gas across liquid films between bubbles is cited as one reason why CO2 foams for enhanced oil recovery (EOR) are usually weaker than N2 foams and why steam foams are weaker than foams of steam mixed with N2. We examine here the effect of inter-bubble gas diffusion on flowing bubbles in a simplified model of a porous medium (a periodically constricted tube in 2D) and in particular its effect on the bubble-size distribution and capillary resistance to flow. Bubbles somewhat smaller than a pore disappear by diffusion as the bubbles move. For bubbles larger than a pore, as expected in EOR, diffusion does not affect bubble size. Instead, diffusion actually increases capillary resistance to flow (i.e. makes foam stronger): lamellae spend more time in positions where lamella curvature resists movement. When fit to pressures and diffusion and convection rates representative of field application of foams, diffusion is not expected to alter the bubble-size distribution in a foam, but instead modestly increases the resistance to flow. The reason for the apparent weakness of CO2 foam therefore evidently lies in factors other than CO2's large diffusion rate through foam.
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Using Dimensionless Numbers to Assess EOR in Heterogeneous Reservoirs
Authors B. Rashid, O. Fagbowore and A.H. MuggeridgeDimensionless numbers such as mobility ratio and the viscous to gravity ratio provide a convenient way of assessing the flow regime and thus ranking performance when designing secondary and tertiary oil recovery processes. Until recently, however, their application has been limited to homogeneous reservoirs due to a) the lack of a robust heterogeneity index and b) the fact that the viscous to gravity ratio depends upon reservoir permeability and thus heterogeneity. In this paper we present 3D phase diagrams showing how recovery and breakthrough time depend upon mobility ratio, viscous-to-gravity ratio and heterogeneity. We review the literature on the application of dimensionless numbers to identify flow regime in oil recovery processes and select a recently developed heterogeneity index based upon vorticity to characterize heterogeneity. The index has been previously verified using heterogeneous reservoir descriptions taken from SPE10 model 2. We use the phase-diagrams to identify dominant flow regimes and provide criteria based on the dimensionless numbers for identifying those flow regimes when assessing alternative EOR processes.
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Molecular Dynamics as a Tool to Deal with Thermogravitation
Authors G. Galliero and F.M. MontelAbstract: A precise description of the initial state of a petroleum reservoir is crucial to optimize its development plan. This relies on an accurate modeling of the spatial distribution of the fluid components within the reservoir which is mainly influenced by gravitational segregation and thermo-diffusion phenomena (thermogravitation). An alternative to the classical thermodynamic modelling to provide further information on thermogravitation without the need of any EoS or any correlation to describe transport properties is to use Non-Equilibrium Molecular Dynamics (NEMD) simulations on systems representing an idealized 1D reservoir fluid column. We will show how such a molecular based approach can shed light on some the underlying physical mechanism (evolution/stability) of the thermo-gravitational process in idealized situations. In particular, it will be shown, on a n-alkane mixture and a acid gas mixture, that the thermodiffusion effect can affects the vertical distribution of the different compounds as much as segregation with the same characteristic time and can even lead to an unstable (i.e. convective) situation in a CO2 rich reservoir.
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Modeling Compositional Compressible Two-phase Flow in Porous Media by the Concept of the Global Pressure
Authors B. Amaziane, M. Jurak and A. Zgaljic-KekoThe modeling of multiphase flow in porous formations is important for both the management of petroleum reservoirs and environmental remediation. More recently, modeling multiphase flow received an increasing attention in connection with the disposal of radioactive waste and sequestration of CO2. In this talk, we will discuss a new formulation for modeling compositional compressible two-phase flow in porous media such as immiscible gas injection in oil reservoirs or gas migration through engineered and geological barriers in a deep repository for radioactive waste . The focus is on the problems arising due to Newton-Raphson's flash calculations and the phase appearance and disappearance . Compositional compressible two-phase flows in porous media are usually modeled by the mass balance law written for each component, Darcy-Muscat's law, and the thermodynamic equilibrium between the phases . The obtained equations represent a set of highly coupled nonlinear partial differential equations. In order to model both saturated and unsaturated zones, one has to change the main unknowns of the system. In the saturated zones, the pressure and the saturation of one of the phases are commonly chosen as the main unknowns, whereas in the unsaturated zones the saturation may be replaced by the mass density of one of the component in its phase. To avoid changing the main unknowns, and to make the system coupling weaker, we derive a new formulation of the compositional compressible liquid and gas flow. The formulation considers gravity, capillary effects and diffusivity of each component. The main feature of this formulation is the introduction of a new variable called the global pressure. The derived system is written in terms of the global pressure and the total gas mass density that partially decouples the equations and is able to model the flows both in the saturated and unsaturated zones with no changes of the primary unknowns. The mathematical structure is well defined: the system consists of two nonlinear degenerate parabolic equations. The derived formulation is fully equivalent to the original equations and is more suitable for mathematical and numerical analysis. The accuracy and effectiveness of the new formulation is demonstrated through numerical results.
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Thermal Adaptive Implicit Method: Temperature Stability Criteria
Authors J. Maes and A. MoncorgéWe present new linear-stability criteria for the Thermal Adaptive Implicit Method (TAIM) for thermal multiphasic compositional displacement. The analysis is applied to the mass and energy equations. Moncorgé and Tchelepi’s work (2009) is based on the assumption of divergence-free total velocity, and accounts for compressibility effects. Our analysis shows that the criteria proposed do not guarantee oscillation-free numerical solutions in case of displacement that involves steep temperature and saturation fronts. We derive new criteria that result from the analysis of a simplified coupled pressure-temperature linearized system, obtained by decoupling from saturations and compositions unknowns. The new criteria explains instabilities that were undetected by the previous analysis. Moreover, we demonstrate through scaling analysis and numerical examples that for most problems of practical interest, a simple temperature stability criterion obtained by assuming incompressible multiphase flow is quite robust. The relationship between the full and simplified stability criteria is analyzed in detail. The methodology is demonstrated using several thermal-compositional examples, including Steam Assisted Gravity Drainage.
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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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