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ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery
- Conference date: September 8-11, 2014
- Location: Catania, Sicily, Italy
- Published: 08 September 2014
81 - 100 of 136 results
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The Hyperbolic Nature of a System of Equations Describing Three-phase Flows in Wellbores
Authors K. Sinkov, P. Spesivtsev and A.A. OsiptsovSummaryThe problem of the formulation of compressible, isothermal, multi-phase (3-phase) flow in wellbores is considered. One such approach is that provided by the drift-flux model. According to this model, in the three-phase case (typically, oil, water, and gas) the governing system of equations consists of three continuity equations, one for each phase, and a single equation for the conservation of momentum for the mixture. The system is closed by equations-of-state and algebraic relations for determining the individual phase velocities. The detailed characteristic analysis of both two- and three-phase problems is carried out to determine the domains where the system of equations is hyperbolic and, therefore, whether they are suitably well-posed, stable and robust for application in the numerical solution of this type of hyperbolic problem. The transient, steady-state, and non-inertial forms of the momentum conservation equation encountered in the literature are considered. The influence of mass exchange terms, responsible for the solubility of the gas-phase in liquid on the hyperbolicity is also studied. The analysis demonstrated that the system is found to be hyperbolic in the two-phase case and conditionally hyperbolic in the three-phase case, with eigenvalues being functions of the problem variables. It is also shown that the mixture momentum equation can be transformed to the so-called “advection” equation for pressure which possesses a real eigenvalue. The analysis presented suggests recommendations on the domains to which hyperbolicity is valid to a system of equations and on the specification of boundary conditions for the drift-flux model.
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Thermal Simulation of CO2 Storage with Generalized Cubic Equation of State
Authors I. Aavatsmark, B.K. Kometa and S.E. GasdaSummaryRelationships between pressure, temperature and density are generally described by an equation of state. For CO2, the Span-Wagner equation is generally assumed to give the best fit to measured data. The high accuracy of this equation does not come without a cost. Therefore, this equation of state is mostly used as a reference for comparisons with other formulations.
In reservoir simulations, cubic equations of state such as the Peng-Robinson and the Soave-Redlich-Kwong equations are widely used. They are fairly accurate, and computation of the solution is fast. In this paper, a generalized cubic equation of state is introduced. This equation is computationally precisely as efficient as the traditional equations of state. With the generalized equation of state, improved approximations of the density of CO2 in predefined temperature-pressure domains may be obtained. The parameters of the generalized cubic equation of state are determined through comparison with the Span-Wagner equation.
We show applications of the generalized cubic equation of state for different temperature-pressure domains. When compared with the Peng-Robinson equation, the root mean square density deviation is reduced by a factor 2 for domains containing the critical point, and a facto 7 for supercritical domains. Similarly, thermal simulations with the generalized cubic equation of state show large improvements in density and improvements in saturation close to the CO2 front.
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Negative-saturation Approach for Non-isothermal Compositional Three-phase Flow Simulations in Porous Media
Authors H. Salimi and J. BruiningSummaryThis article deals with developing a three-phase solution approach, called the three-phase non-isothermal negative-saturation (NegSat3) solution approach. The NegSat3 solution approach solves efficiently any non-isothermal compositional-flow problem in porous media that involves phase transitions between different phase states when the maximum number of phases is less than four. The advantage of the solution approach is that it circumvents using different equations and primary variables for single-phase, two-phase, and three-phase regions in porous media. Consequently, the NegSat3 solution approach avoids switches and the ensuing-unstable procedure. The NegSat3 solution approach can be implemented efficiently in numerical simulators to deal with modeling issues for thermal recovery processes, CO2-sequestration process, and for multicontact miscible gas injection in oil reservoirs if the number of phases is less than four. We illustrate the NegSat3 solution approach by way of example to steam injection in a 1D heavy-oil reservoir. The solution is compared with a standard numerical solution that is analytically verified by the method of characteristics, and shows excellent agreement. The results show that the oil recovery depends critically on whether the boiling temperature of the volatile oil is around the water boiling temperature, or much below or above it. These boiling-temperature ranges give rise to three different types of wave structures. When the boiling temperature of the volatile oil is near the boiling temperature of water, the striking result is that the speed of the evaporation front is equal or somewhat larger than the speed of the steam-condensation front. Thus, the volatile oil condenses at the location where the steam condenses too, yielding virtually complete oil recovery. Conversely, if the boiling temperature is too high or too low, there is incomplete recovery.
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Modelling In-situ Upgrading of Heavy Oil Using Operator Splitting Methods
Authors J. Maes, A.H. Muggeridge, M.D. Jackson, M. Quintard and A. LapeneSummaryThe In-Situ Upgrading (ISU) of bitumen and oil shale is a very challenging process to model numerically because a large number of components need to be modelled using a system of equations that are both highly non-linear and strongly coupled. Operator splitting methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as splitting error. Obviously the best splitting method for modelling a given process is the one that minimises the splitting error whilst improving computational performance over that obtained from using a fully implicit approach.
Although operator splitting has been widely used for the modelling of reactive-transport problems, it has not yet been applied to the modelling of ISU. One reason is that it is not clear which operator splitting technique to use. Numerous such techniques are described in the literature and each leads to a different splitting error. While this error has been extensively analysed for linear operators for a wide range of methods, the results observed cannot be extended to general non-linear systems. It is therefore not clear which of these techniques is most appropriate for the modelling of ISU.
In this paper we investigate the application of various operator splitting techniques to the modelling of the ISU of bitumen and oil shale. The techniques were tested on a simplified model of the physical system in which a solid or heavy liquid component is decomposed by pyrolysis into lighter liquid and gas components. The operator splitting techniques examined include the Sequential Split Operator (SSO), the Strang-Marchuk split operator (SMSO) and the Iterative Split Operator (ISO). They were evaluated on various test cases by considering the evolution of the discretization error as a function of the size of the time-step compared with the results obtained from a fully implicit simulation. We observed that the error was minimum for a splitting scheme where the thermal conduction was performed first, followed by the chemical reaction step and finally the heat and mass convection operator (SSO-CKA). This method was then applied to a more realistic model of the ISU of bitumen with multiple components.
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Vertically Integrated Models with Coupled Thermal Processes
Authors S.E. Gasda, W.G.G. Gray and H.K.D. DahleSummaryCO2 storage in geological formations involves coupled processes that affect the migration and ultimate fate of injected CO2 over multiple length and time scales. For example, coupling of thermal and mechanical process has implications for storage security, including thermally induced fracturing and loss of caprock integrity in the near wellbore environment. This may occur when CO2 is injected at a different temperature than reservoir conditions, e.g. Snøhvit injection, potentially leading to large temperature, density and volume changes within the plume over space and time. In addition, thermally induced density changes also impacts plume buoyancy that may affect large-scale migration patterns in gravity-driven systems such as Utsira storage site. This interaction becomes particularly important at temperatures and pressures near the critical point. Therefore, coupling thermal processes with fluid flow should be considered in order to correctly capture plume migration and trapping within the reservoir.
A practical modeling approach for CO2 storage at the field scale is the vertical-equilibrium (VE) model, which solves partially integrated conservation equations for flow in two lateral dimensions. This class of models is well suited for strongly segregated flows, as can be the case for CO2 injection. In this paper, we extend the classical VE model to non-isothermal systems by vertically integrating the coupled heat transport equations, focusing on the thermal processes that most impact the CO2 plume. The model allows for heat exchange between the CO2 plume and the surrounding environment assuming thermal equilibrium across the plume thickness for relatively thin plumes. We investigate the validity of simplifying assumptions required to reconstruct the fine-scale thermal structure from the coarse-scale model solution. The model concept is verified for relatively simple systems. The results of this work demonstrate the potential for reduced models to advance our understanding of the impact of thermal processes in realistic storage systems.
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Numerical Aspects of Polymer Flood Modeling
Authors T.S. Mykkeltvedt, I. Aavatsmark and K.A. LieSummaryWe discuss the application of modern high-resolution schemes to a hyperbolic system that models polymer flooding. This system consists of a pair of non-strictly hyperbolic conservation laws. In general, high-resolution schemes are often used for model problems where high accuracy is required in the presence of shocks or discontinuities. Polymer flooding is a difficult process to model, especially since the dynamics of the flow lead to concentration fronts that are not self-sharpening. Because the water viscosity is strongly affected by the polymer concentration, it is crucial to capture polymer fronts accurately to resolve the nonlinear displacement mechanism correctly and its efficiency for enhanced recover.
The main objective of this work is to compare different first- and higher-order methods in terms of how the discontinuities are treated. The discussion will focus on the validity, convergence and robustness of the schemes. Especially, different initial conditions and the inclusion of adsorption and permeability reduction can change not only the solution, but also the behavior of the different numerical methods. We show that these effects also can influence the applicability of a solver and we investigate of how suitable different numerical methods are for different polymer flooding situations.
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Non-equilibrium Based Compositional Simulation of Carbonated Water Injection EOR Technique
Authors J. Foroozesh, M. Jamiolahmady, M. Sohrabi and S. IrelandSummaryCarbonated water injection (CWI) is an augmented water injection strategy, which has great potentials for EOR and CO2 storage purposes. When carbonated water, CO2 enriched water, is injected into oil reservoirs, due to higher CO2 solubility in oil compared to that in water, CO2 migrates from carbonated water into oil. This improves oil mobility, due to swelling and viscosity reduction, which consequently increases oil production.
Our core flood experiments show that during CWI, CO2 is transferred and distributed between water and oil gradually and thermodynamic equilibrium is not reached. Available compositional reservoir simulators, which are based on the instantaneous equilibrium assumption, do not capture the actual physics of CWI. In this work, a non-equilibrium based compositional simulator was developed to simulate the performed core flood CWI experiments more realistically. The developed two-phase flow simulator is currently suitable for one dimensional core experiments. It includes a mass transfer term to capture the kinetics of CO2 transfer between phases. Governing equations were derived based on the water, oil and CO2 components balance, and solved using the fully implicit finite difference technique. Black-oil (without mass transfer) and compositional (with mass transfer) modes of the simulator can be used for simulation of conventional water injection (WI) and CWI, respectively. A genetic algorithm based optimization software was also developed that can be linked to the simulator to history match the available production data and obtain the unknown parameters of the model.
The simulator was used to model WI and CWI coreflood experiments conducted on a water-wet sandstone core fully saturated by Decane (with well-defined fluid properties). First, the WI experiment was simulated when an oil-water relative permeability (kro-w) was obtained by history matching of WI production data. The WI test was re-simulated by ECLIPSE100 (E100) commercial simulator using optimized kro-w. E100’s predictions of production data reasonably matched model’s results, which verify its integrity. Next, the obtained kro-w was used for simulation of CWI. Mass transfer coefficient, as the only remaining unknown parameter, was tuned to match the available CWI production data leading to an acceptable match. The simulator shows promising potentials for simulation and better understanding of CWI for practical field applications. Moreover, the structure of this simulator offers a solid foundation for other EOR methods where kinetics of mass transfer is important.
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Numerical Simulation of Mutually Soluble Solvent-aided Spontaneous Imbibition in Fractured Reservoirs
Authors M. Chahardowli, R. Farajzadeh and H. BruiningSummaryMutually-soluble solvents can enhance oil recovery both in completely and partially water wet fractured reservoirs. When a strongly or partially water-wet matrix is surrounded by an immiscible wetting phase in the fracture, spontaneous imbibition is the most important production mechanism. Initially, the solvent moves with the imbibing brine into the core. However, upon contact with oil, diffusion occurs and the solvent is transported in the oleic phase. Through the migration of the mutually soluble component from the aqueous phase into the oleic phase, oil properties and/or rock-fluid interactions are modified. The hypothesis in this work is that a mutually-soluble solvent improves the ultimate recovery and the imbibition rate in partially and completely water-wet cores. The main recovery mechanisms are the wettability change of the partially water-wet cores, oil swelling and oil viscosity reduction in both partially and completely water-wet cores.
This paper considers the numerical modelling of experiments that use Amott imbibition cells to study solvent enhanced spontaneous imbibition in completely water-wet and partially water-wet cores. In the first stage of the experiment, the completely water-wet core was exposed to brine. In a second stage, the core was put in another Amott cell that was filled with a solvent/ brine mixture. The extra recovery by a solvent/brine mixture strongly depends on the residual oil saturation after brine imbibition and it is relatively insensitive to the permeability of the core or the oil viscosity. We implemented the swelling mechanism, the oil viscosity reduction mechanism, and the IFT reduction mechanism in the numerical model. Our studies show that the most important production mechanism in the completely water wet system is the oil swelling and the second most important mechansism is the oil viscosity reduction. The effect of the IFT reduction in the oil production is not significant. The numerical results show an improvement of 10%.
For the partially water-wet samples, we also started with exposing the core to pure brine without solvent. Contrary to the completely water-wet samples, the numerical results show a significant increase in recovery rate when the sample is transferred to another Amott cell where it is exposed to a mixture of solvent and brine. The main modification in simulating the recovery process is that now a wettability change mechanism is taken into account. Our numerical studies show that the contribution of the wettability change is the main contributor to the oil recovery, which is enhanced by 35 %.
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Microbial Enhanced Oil Recovery - A Mathematical Study of the Potential of Spore-forming Bacteria
Authors S.M. Nielsen, I. Nesterov and A.A. ShapiroSummaryMicrobial enhanced oil recovery (MEOR) utilizes microbes for enhancing the recovery by two main mechanisms: 1) reduction of oil-water interfacial tension (IFT) by produced biosurfactant, and 2) selective plugging by microbes and metabolic products. A mathematical model for MEOR has been built. Our model organism is a surfactant-producing spore-forming bacterium that shows a great potential for MEOR. Application of spore-forming bacteria is an advantageous novelty of the present approach. A spore is a dormant, very resistant version of the cell. Sporulation (turning bacteria into spores) occurs when bacteria are exposed to stresses, such as starvation. The remains of a sporulated cell serve as a substrate for the rest bacteria. An inverse process, reactivation, occurs when spores sense favorable conditions.
The model accounts for growth, substrate consumption, surfactant production, attachment/filtering out, sporulation and reactivation. The mathematical setup is a set of 1D transport equations involving reactions and attachment. Characteristic sigmoidal curves are used to describe sporulation and reactivation in response to substrate concentrations. Surfactant decreases IFT, modifying the relative permeabilities and thus decreasing the fraction of water in the flow. Attachment of bacteria reduces the pore space available for flow, i.e. the effective porosity and permeability. Clogging of specific areas may occur.
In order to obtain sufficient local concentrations of surfactant, substantial amounts of substrate should be supplied; however, massive growth of bacteria increases the risk for clogging at the well inlet areas, causing injectivity loss. In such areas starvation may cause sporulation, reducing the risk of clogging. Substrate released during sporulation can be utilized by attached vegetative bacteria and they will continue growing and producing surfactant, which prolongs the effect of the injected substrate. The simulation scenarios show that application of the spore-forming bacteria gives a higher total production of surfactant and the reduced risk of clogging, leading to an increased period of production and a higher oil recovery.
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Unstructured Adaptive Mesh Refinement for Flow in Heterogeneous Porous Media
Authors M. Karimi-Fard and L.J. DurlofskySummaryThe spatial resolution required in a simulation model depends on the type and degree of geological heterogeneity, along with the flow physics and well locations. The use of adaptive mesh refinement (AMR) enables the grid to adapt dynamically during the course of the simulation, which facilitates the efficient use of computational resources. In this work we present a finite-volume-based AMR procedure that is suitable for complex reservoir models such as those characterized by fractures or channels. The approach uses a fully unstructured hierarchical grid to accurately represent complex geological features. An important aspect of the method is the use of flow-based global transmissibility upscaling, which provides accurate cell-to-cell transmissibilities at any level of coarsening. The performance of the method is illustrated for problems involving two-dimensional channelized and fractured systems. High levels of solution accuracy relative to reference fine-scale results are observed for both cases, and the AMR models are shown to consistently use less than 30% of the cells in the underlying fine-grid representation. For fractured systems, the method essentially provides a dual-continuum coarse-scale representation, and can thus be viewed as a dual-continuum-AMR technique.
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Indirect Unstructured Hex-dominant Mesh Generation Using Tetrahedra Recombination
Authors A.B. Botella, B.L. Lévy and G.C. CaumonSummaryCorner-point gridding of reservoir and basin models is widely used but generally yields approximations in the geological interfaces representation in flow simulation. This paper introduces an indirect method to generate a hexdominant mesh conformal to 3D geological surfaces suitable for Finite-Element and Control-Volume Finite-Element simulations. By indirect, we mean that the method first generates an unstructured tetrahedral mesh whose tetrahedra are then merged into primitives (hexahedra, prisms and pyramids). More specifically, we focus on determining the optimal set of primitives that can be recombined from a given tetrahedral mesh. First, we detect in the tetrahedral mesh all the feasible volumetric primitives using a pattern-matching algorithm that we re-visit and extend with configurations that account for degenerated tetrahedra (slivers). Then, we observe that selecting the optimal set of primitives among the feasible ones can be formalized as a maximum weighted independent set problem, known to be N P-Complete. We propose five heuristic optimizations to find a reasonable set of primitives in a practical time. All the tetrahedra of each selected primitive are then merged to build the final unstructured hex-dominant mesh. This method is demonstrated on complex 3D geological models including a discrete fracture network.
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Boundary Aligned Grid Generation and CVD-MPFA - Cell-centred Versus Cell-vertex on Unstructured Grids
Authors S. Manzoor, M.G. Edwards, A. Dogru and T.M. Al-ShaalanSummaryDevelopment of boundary aligned grid generation is presented together with comparative performance of cell-vertex versus cell-centred CVD-MPFA finite-volume formulations using equivalent degrees of freedom. When generating structured or unstructured grids for reservoir simulation, classical key constraints involve boundary aligned grid generation with control-volume boundaries aligned with solid walls and geological features such as layers, shale barriers, fractures, faults, pinchouts and multilateral wells.
The schemes used are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). Consequently a natural choice is for primal grid cells to act as control-volumes, then grid generation can be performed with primal grid cell boundaries being aligned with key interior constraint boundaries. This naturally leads to cell-centred approximation, where flow variables and rock properties are associated with grid cell centres.
The alternative is to employ cell-vertex approximation which uses far fewer approximation points on a given unstructured grid. In this case control-volumes are constructed around primal grid vertices. The grid generation process is less straight forward since control-volumes must be constrained to satisfy interior boundary alignment. A novel grid generation procedure is proposed that automates control-volume boundary alignment and yields a Voronoi mesh. The actual grid is then generated such that dual boundaries are aligned with key internal constraint boundaries and cell-vertex approximation becomes the natural choice. In this case flow variables and rock properties are associated with grid cell vertices and their dual control-volumes.
The relative benefits of both types of approximation is made clear in terms of flow resolution and degrees of freedom required.
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A Volume-conserving Representation of Cell Faces in Corner Point Grids
More LessSummaryCorner point grids are currently the standard representation for reservoir simulation grids. Cell faces in corner point grids are traditionally represented as bilinear surfaces where the edges between the corner points all are straight lines. This representation has the disadvantage that at pillar-gridded faults with varying dip the cell faces on either side will not precisely match, giving overlapping cells or gaps between cells. We propose an alternative representation for the cell faces. The four vertical cell faces are still represented as bilinear surfaces, but instead of having linear edges between the cell corners along the top and bottom faces, we propose a representation where all horizontal intersections through the grid give straight lines between the grid pillars, giving column faces whose shape is independent of the corner point locations of the individual grid cells. This ensures that the grid columns match up and that there are no gaps or overlapping volumes between grid cells. This representation gives a local parameterization for the whole grid column, and the top and bottom grid cell surfaces are modelled as bilinear within this domain. Using this representation we get a parameterization of the grid cell which we use to calculate the cell volume.
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Ensemble Level Upscaling for Compositional Flow Simulation, with Application to Uncertainty Quantification
Authors H. Li and L.J. DurlofskySummaryUncertainty quantification is typically accomplished by simulating multiple geological realizations, which can be very expensive computationally if the flow process is complicated and the models are highly resolved. Upscaling procedures can be applied to reduce computational demands, though it is essential that the resulting coarse-model predictions correspond closely to reference fine-scale solutions. In this work, we develop an ensemble level upscaling (EnLU) procedure for compositional systems, which enables the efficient generation of multiple coarse models for use in uncertainty quantification. This requires us to first develop an accurate global compositional upscaling method for individual realizations. This global upscaling entails transmissibility and relative permeability upscaling, along with the computation of α-factors to capture component fluxes. A procedure for iterating on coarse-scale α-factors is introduced, and this is shown to improve accuracy. In EnLU, global upscaling is applied for only a few selected realizations. For 90% or more of the realizations, upscaled functions are assigned statistically based on quickly-computed flow and permeability attributes. A sequential Gaussian co-simulation procedure is incorporated to provide coarse models that honor the spatial correlation structure of the upscaled properties. The resulting EnLU procedure is applied for multiple realizations of 2D models, for both Gaussian and channelized permeability fields. Results demonstrate that EnLU provides P10, P50 and P90 results for phase and component production rates that are in close agreement with reference fine-scale results. Less accuracy is observed in realization-by-realization comparisons, though the models are still much more accurate than those generated using standard coarsening procedures.
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Analytical Expressions for Upscaled Relative Permeabilities in Three-phase Flow
Authors E. Bianchi Janetti, M. Riva and A. GuadagniniSummaryWe present analytical solutions for the relative permeabilities governing a Darcy scale description of three-phase flow of immiscible fluids. We consider flow taking place within a capillary tube with circular cross-section for two settings corresponding to (a) a water wet and (b) an oil wet configuration. Momentum transfer between the moving phases, which leads to the phenomenon of viscous coupling, is explicitly accounted by imposing continuity of velocity and shear stress at the fluid-fluid interfaces. The macroscopic model describing the system at the Darcy scale includes three-phase effective relative permeabilities, Kij,r, accounting for the flow rate of the ith-phase due to the presence of the jth-phase. These effective coefficients are function of phases saturation, fluids viscosity and wettability of the medium. Our results show that (i) the relative permeability Kii,r of the wetting phase is only a function of its own saturation while Kii,r of the non-wetting phases depend on the saturations of all the fluids; (ii) viscous coupling effects (elucidated by Kij,r with i ≠ j) can be relevant in water wet and oil wet systems.
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An Upscaling Methodology for EOR
Authors A.H. Muggeridge and P. HongtongSummaryThe outcome of immiscible enhanced oil recovery (EOR) processes such as low salinity water injection and polymer flooding can be very sensitive to geological heterogeneity. In some cases the resulting reduction in macroscopic sweep (versus a waterflood) can be more than the improvement in microscopic displacement efficiency. It is therefore important to be able to model the impact of this heterogeneity during simulation studies. This may require an upscaling step if the geological heterogeneity is smaller than the simulation grid block size or simply to compensate for numerical diffusion on the coarse grid.
This paper proposes an upscaling methodology that can be applied to different EOR processes and demonstrates its application to low salinity water injection examples. The methodology involves a hierarchy of upscaling steps. First the absolute permeability is upscaled with the objective of predicting the correct changes in pressure. This may also involve near well bore upscaling. Next pseudo relative permeability curves are generated to capture the shock front behaviour. In this study we compare results obtained from using traditional pore volume weighted pseudos with those obtained using pseudos determined analytically using Buckley-Leverett theory. Finally the simulator models relevant to the EOR process of interest are upscaled. In low salinity waterflooding this is the choice of low and high salinity thresholds.
The upscaled models are in better agreement with the fine grid models in terms of pressure, water saturation production and production of either salinity or polymer than are the outputs of coarse grid models without upscaling. The results using the analytical pseudo relative permeabilities are comparable to those obtained from simulations using the pore volume weighted pseudo in many of the cases tested. These models are obviously less time-consuming and complex to generate as they do not need the engineer to run a full fine grid simulation of the EOR process to calculate the pseudo relative permeabilities.
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A Fast and Accurate Upscaling of Transmissivities for Field Scale Reservoir Simulation
Authors D. Guérillot and J. BruyelleSummaryDescription:
For representing reservoir heterogeneities, most of geological models need several tens of million of cells.
Even though reservoir simulators are quicker and quicker, it is most of the time requested to average the properties representing these heterogeneities on larger cells for simulating the multi-phase flow.
Methods to average reservoir simulation have been studied for several years by many authors. For averaging absolute permeabilities, two main approaches are met: 1) averaging first the permeabilities and using the averaged value when discretizing the equations, leading to average transmissivities, or 2) calculating directly the averaged transmissivities from the high resolution permeabilities.
This paper is in line with the second approach. Compared to previous work, we combined the finite volume principles with algebraic methods providing upper and lower bounds of the upscaled transmissivities considering adjacent volumes on both sides of the boundaries between cells in a manner that the contrast of heterogeneities is better conserved.
Comparisons with current upscaling approaches will confirm that this approach is more accurate than calculating transmissivities from upscaled permeabilities.
The competitive advantages of the algebraic methods are that the upscaling step will be fast and the use of bounds provided by algebraic method allows controlling the quality of the upscaling and gridding.
The main innovation of the presented method is that the calculations of transmissivities are performed on smaller local domain than those proposed in previous work. This approach increases contrast of transmissivity and thus gains in accuracy for field scale flow simulation in heterogeneous media.
Applications:
This approach had been successfully implemented in an industrial E&P software platform and tested on actual fields.
Results and conclusions:
This method calculating upscaled transmissivities is the natural end point of any workflow devoted to the geological modeling to determine the petrophysical parameters of the field scale simulation grid.
Being based on algebraic method, this approach is fast and a control of its quality is possible.
Technical contributions:
- An innovative method calculating on smaller domains the upscaled transmissivities is proposed to increase the accuracy of reservoir simulations.
- This fast approach avoid calculation of local fluid flow simulation everywhere
- This approach allows controlling the quality of upscaling and gridding.
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Reservoir Management Optimization Using Calibrated Transmissibility Upscaling
Authors S. Krogstad, H.M. Nilsen and X. RaynaudSummaryOptimizing the objectives of long-term reservoir management typically requires a high number of forward reservoir simulations. Two important remedies to reduce the runtime (and make the optimization problem manageable) are model reduction/upscaling and efficient computation of gradients. Adjoint methods are generally considered to be the most efficient means for obtaining gradients.
Furthermore, there has been much interest in reduced-order modelling (e.g., based on POD) for reservoir management optimization. Very promising results have been reported for models with somewhat limited complexity. Application to industry-standard cases is inhibited in part by the invasive nature of the approach with respect to simulator code, and the fact that the number of required basis functions is correlated with the degree of nonlinear dynamics.
In this work, we utilize ideas from POD to compute upscaled transmissibilities for a coarse model, rather than using the method directly to build a basis for the fine-scale state space. The proposed coarsening strategy takes as input any number of fine-scale states (pressure fields from e.g., a previous simulation), and produces a coarse-scale model calibrated to the specific flow scenario(s) dictated by the input. We argue that compared to traditional general-purpose upscaling approaches, much more aggressive coarsening can be applied in this type of scenario-specific upscaling. Utilizing a fully-implicit, three-phase, black-oil simulator with adjoint capabilities, we investigate the performance of our methodology by optimizing the net-present-value (NPV) for a real-field model. We consider multiple coarsening strategies and coarsening factors, and conclude that for the model considered, at least two orders of magnitude speed-up can be achieved whilst retaining sufficient accuracy.
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Data-driven Model Inference and its Application to Optimal Control under Reservoir Uncertainty
More LessSummaryMany real-world oil reservoir simulations rely on either legacy code that is difficult to modify or commercial software without available source code. Such black-box simulators solve a set of nonlinear partial differential equations and generally involve the use of many unknown variables, functions or types of discretization methods. For the purpose of carrying out history matching or optimal control in reservoir management studies, black-box simulators are usually employed as function evaluators within the context of derivative-free optimization methods. However, these methods can be computationally unaffordable when the number of parameters or controls becomes high dimensional. The present work describes an efficient optimization methodology for black-box simulations consisting of inferring and calibrating a “twin model” representation of the black-box simulator. The twin model is a non-intrusive model that mirrors the behavior of the black-box simulation using data assimilation techniques. Once the inferred twin model is available, adjoint operators based on gradient-driven techniques can be easily computed to perform efficient optimization. Computational experiments illustrate the significant reduction of simulations to estimate the optimal water injection strategy under various geological uncertainty scenarios when compared with traditional approaches using gradient-free methods.
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Evaluation of Non-intrusive Generalized Polynomial Chaos Expansion in the Context of Reservoir Simulation
Authors O. Pajonk, F.C. Yanou Ngongang and R. Schulze-RiegertSummaryUncertainty quantification workflows which map input uncertainties to output uncertainties are an important part of many reservoir simulation applications. Examples are estimation of prediction uncertainties including history data, reserves estimation, or optimization under uncertainty. Over the last few years, proxy modeling techniques have turned out to be essential for improving the efficiency of the employed methods, with one example being stochastic sampling techniques like Markov Chain Monte Carlo. An established proxy model family is the generalized Polynomial Chaos Expansion (gPCE). Some applications to reservoir simulation exist - also in the context of history matching.
In this work we focus on a non-intrusive, efficient way to construct such gPCE representations for multi-dimensional input uncertainties: sparse-grid spectral projection – also known as Smolyak grid. In standard approaches, which derive their multi-dimensional cubature rules from full tensor-products of one-dimensional rules, the number of required points grows exponentially with the input dimension. This causes a significant computational effort since each point represents a forward simulation. On the other hand, sparse-grid techniques used in this work show polynomial scaling behavior by “thinning out” the full tensor-product. However, in order to avoid possible numerical instabilities of this technique a design of simulation cases needs to be applied. Practical consequences and guidelines for real applications will be described in detail.
The numerical framework for constructing gPCEs is applied to uncertainty quantification workflows in reservoir simulation. We investigate performance criteria for representing key performance indicators using gPCEs, e.g., cumulative properties like oil production total or transient properties such as bottom hole pressure. Practical application designs are derived and described.
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