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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
41 - 60 of 136 results
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An Exploration of Porosity-permeability Relationships in 3D Fracture Network Using the Lattice-Boltzmann Method
By R.A. ArcherSummaryThis paper explores the relationship of permeability to fracture aperature, density and mineralisation in synthetic dual porosity media. Fluid flow is simulated using the Lattice-Boltzmann method which allows for detailed fracture geometries to considered in 3D. The geometry of the matrix part of the media is built using a Voronoi based approach similar to the work in 2D by Newman and Yin (2013) .
Stress sensitivity can be an important phenomena in dual porosity media. Reductions in reservoir pressure as production proceeds cause decreases in porosity and consequentially decreases in permeability. This effect is explored by considering the impact of fracture compressibility.
While it is not anticipated that full field modelling at this scale will be undertaken in the foreeable future, modelling such as this allows insight to defining appropriate REVs and into upscaling.
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Modelling of Irreversible Deformation near the Tip of a Crack in a Porous Domain Containing Oil and Gas
Authors A. Lukyanov, N. Chugunov, V.M. Sadovskii and O.V. SadovskayaSummaryThermomechanical processes observing in deformable solids under intensive dynamic or quasi-static loadings consist of coupled mechanical, thermal and fracturing stages. The fracturing processes involve formation, motion and interaction of defects in crystals, phase transitions, breaking of bonds between atoms, accumulation of micro-structural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear micro-fractures are caused by these processes. A dynamic fracturing is a complicated multistage process, which includes appearance, evolution and confluence of micro-defects and formation of embryonic micro-cracks, pores that can grow and lead to the breaking-up of bodies with formation of free surfaces. This results in a need to use more advanced mathematical and numerical techniques.
This talk presents modeling of irreversible deformation near the tip of a crack in a porous domain containing oil and gas during the hydraulic fracturing process. The governing equations for a porous domain containing oil and gas are based on constructing mathematical model of thermo-visco-elasto-plastic media with micro-defects (micro-pores) filled with another phase (e.g., oil or/and gas). The micropores can change their size during the process of dynamical irreversible deformation. The existing pores can expand or collapse. The model was created by using the fundamental thermodynamic principles and, therefore, it is a thermodynamically consistent model. All the processes (i.e., irreversible deformation, fracturing, micro-damaging, heat transfer) within a porous domain are strongly coupled. Therefore, explicit normalized-corrected meshless method is used to solve the resulting governing PDEs. The flexibility of the proposed technique allows running efficiently using a great number of micro- and macro-fractures. The results are presented, discussed and future studies are outlined.
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Reexamining CO2 Storage Capacity and Utilization of the Utsira Formation
Authors O.A. Andersen, H.M. Nilsen and K.A. LieSummaryIn this work we provide estimates of CO2 storage capacity of the Utsira Formation using the recently provided datasets from the Norwegian Petroleum Directorate, taking CO2 density variation into account. We also investigate strategies on how to realize as much as possible of this potential in large-scale injection scenarios. We base our study on the assumption that the limiting factor for CO2 storage at Utsira is the efficient use of available trapping mechanisms, with pressure buildup being of secondary concern. We consider the trapping mechanisms considered most important for the medium-to-long term (structural, residual and dissolution trapping), using a combination of modeling tools based on the Matlab Reservoir Simulation Toolbox (MRST).
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Reservoir Modeling through Fast Wavelet-based Stochastic Simulation
Authors H.M. Mustapha, S. Chatterjee and R. DimitrakopoulosSummaryStochastic simulation of complex geology is addressed through discrete wavelet transformation (DWT) that handles multiscale spatial characteristics in datasets and training images. The simulation of the proposed approach is performed on the frequency (wavelet) domain. A multiscale, multipoint simulation algorithm is proposed in this paper in which the scaling image and wavelet images across the scale are simulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in the space domain. The proposed algorithm reduces the computational time required for simulating large domain as compared to spatial domain multipoint simulation algorithm since the simulation is performed in the wavelet domain in which numbers of nodes to be simulated are significantly less as compared to spatial domain nodes. The algorithm is tested with an exhaustive dataset using unconditional simulation in two-dimensional data set. The realisations generated perform well and reproduce the statistics of the training image. The study conducted comparing the spatial domain filtersim multiplepoint simulation algorithm suggests that the proposed multiscale, multipoint algorithm generates equally good realisations at much lower computational cost.
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Shape Factor for Dual-permeability Reservoir Simulation - Effect of Non-uniform Flow in Fracture Network
Authors J. Gong and W.R. RossenSummaryThe flow properties of naturally fractured reservoirs are dominated by flow through the fractures. In a previous study we showed that even a well-connected fracture network behaves as one near the percolation threshold in some cases: i.e., most fractures can be eliminated but still form a percolating sub-network with virtually the same permeability as the original fracture network. In this study, we focus on the influence of eliminating unimportant fractures on the inferred characteristic matrix-block size. We model a two-dimensional fractured reservoir in which the fractures are well-connected. The fractures obey a power-law length distribution, as observed in natural fracture networks. For the aperture distribution, because information from the subsurface is limited, we test a number of cases: narrow and broad log-normal and power-law distributions and one where aperture correlates with fracture length. The matrix blocks in fractured reservoirs are of varying sizes and shapes; we calculate the characteristic matrix-block size from the ratio of matrix-block area to its perimeter. We test different criteria to determine the critical sub-network, such as aperture, flow simulation results, etc. We show how the characteristic matrix-block size increases from the original fracture network to the critical sub-network. An implication of this work is that the matrix-block size, or shape factor, used in dual-porosity or dual-permeability waterflood or EOR simulations should be based not on the entire fracture population but on the sub-network that carries almost of all the flow.
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Estimating Geo-modeling Control-parameters from Historical Data by Means of the EnKF
Authors R. Valestrand and D.R. RenardSummaryEstimating geo-modeling control-parameters from historical data by means of the EnKF.
Over the last decade the ensemble Kalman filter (EnKF) and other offspring ensemble based methods (here also referred to as EnKF), have attracted attention as promising methods for solving the reservoir history matching problem. The EnKF has successfully been used to estimate flow properties, such as permeability and porosity, of each grid cell in a history matching loop. For more complex problems, such as facies estimation problems, there are challenges. The direct use of EnKF to estimate e.g. porosity and permeability of a facies field could lead to a good history match, but, it would ruin the geological realism of facies fields, i.e., the boundaries between facies would be smeared out. Several papers have addressed this problem by estimating facies boundaries instead of, or in addition to, the petrophysical properties, in order to maintain the geological realism. The facies boundaries are typically reparameterized using variables that can be characterized by Gaussian to suit the assumption of EnKF.
In this work we attack the problem from a different angle; we perform ‘‘the Big-Loop’’ update, i.e., the geomodel control-parameters are updated using production data and updated facies models are generated with updated control-parameters, using the same geo-modeling work-flow, so that geological realism is naturally preserved. The implementation is referred to as the Big-Loop as we update the geological models not only reservoir models. To perform the investigation we have coupled a geostatistical simulator, a black oil flow simulator and the EnKF. The Big-Loop is tested on examples with facies models generated using the truncated PluriGaussian method. Based on the results found in this work the following can be stated:
* The update models satisfy geological constraints imposed in the geo-modeling work-flow. The match to historical data is improved by updating both local and global geo-parameters,
* It seems beneficial to include both global and local geo-parameters in the Big-Loop approach compared to only local or only global parameters.
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Multi-objective History Matching of a Deltaic Reservoir with Non-stationary Geostatistical Modelling
Authors M.H. da Silva Caetano Caeiro, V. Demyanov and A. SoaresSummaryHistory matching with a single objective function reflects only some global aggregated reservoir match quality and is not flexible enough to distinguish between local effects and provide a spread of diverse models for the forecast. On the contrary, multi-objective optimization (MOO) can distinguish between the contributions to the goodness to fit from some local parts of the model. History matching with MOO results in more diverse matched solutions and more robust prediction. Use of MOO in history matching also provides extra flexibility in matching local parts of the model especially in non-stationary cases. In this work we show how MOO improves the match quality of a non-stationary geostatistical model of a deltaic reservoir. Ensemble of multiple history match models achieved with MOO feature of better quality local matches. . Complex reservoirs descriptions with non-stationary characteristics are hard to match due to their high intrinsic heterogeneity and uncertainty associated with non-stationary description. Direct sequential simulation with local anisotropy correction (DSS-LA) provides a way to account for large scale trends, which are subject to uncertainty. The implementation of local anisotropy tackles the problem of non-stationary of the geostatistical model by imposing a trend in spatial correlation structure. The anisotropy change follows the trend in spatial correlation range and orientation across the reservoir model. While the uncertainty in the trend and spatial correlation is resolved through history matching using multi-objective optimisation.
The proposed adaptive stochastic sampling framework integrates DSS-LA with the multi-objective history matching. Multiple optima matched models of porosity and permeability were obtained allowing the uncertainty assessment of the anisotropy model parameters.
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Improved Realism of Channel Morphology in Object Modelling with Analogue Data Constraints
Authors A.K. Kuznetsova, J.A. Almeida and P. LegoinhaSummaryPreserving realism of geological structures is a challenge for the reservoir engineer when history matching. Object based models offer the chance of enforcing realism but are hard to constrain to observed data.
In this paper we present a novel approach for generating channels that is easier to match to observed data than the standard object-based approach. Our new technique uses Object Modelling as the first step, mimicking the geometry of fluvial sand channels. Object Modelling enables us to constrain channel geometry with analogue information of channel local orientation and dimension. However, the morphology of the output models is smoothed and the transition channel/no-channel zones are associated with uncertainty as they are not constrained by a variogram or multi-point statistics. The second step, Probability Field Simulation, adds realism in a final output model imposing a variogram to the transition zones. Probability Field Simulation preserves the channel geometry of the object models and shows a significant CPU advantage compared with currently available approaches.
The new method was applied to a synthetic problem in which 3 different channel scenarios were considered: low density, high density, and high density with thin channels. The results show that the proposed method can handle this range of channel densities, and we can therefore assume that the approach could be implemented for different channel density data. A further important finding was that there is no significant difference between the CPU time for each case.
In summary, our new technique is able to constrain object models to a variety of observed data types such as channels size, density and orientation, while showing significant CPU saving and preserving the geological realism of the matched model.
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Mathematical Modelling and Numerical Simulation of the Well-reservoir Coupling Flow
Authors C.R. Maliska, M.P. Tada, A.F.C. Silva and A.B. SopranoSummaryNormally, when reservoirs specialist solve the multiphase flow of oil, gas and water in petroleum reservoirs the existing wells are not treated with the required details. Well trajectories are poorly defined with respect to the reservoir grid and the well models for capturing the large gradients near the well are too simplified, among other simplifications. It is also true that when well specialists solve the flow in the well the reservoir is considered as a constant pressure body. However, the modern technologies for increasing oil production is focused on the design of the well completion, the so-called “smart wells”. These are complex devices installed along the column which are able to control and creating flow patterns from the reservoir to the well, increasing the production. The design of such completions required the knowledge of the flow characteristics near the well, what is only possible by solving the coupled flow between well and reservoir.
This paper presents a mathematical model for the oil, gas and water in the reservoir and in the well, following by the development of a numerical scheme for solving the coupled flow using a finite volume method. Full permeability tensor is considered, taking into account heterogeneities and anisotropies, with compressible fluids and rock. The unknowns are the mass fractions of the components, opposed to the saturation formulation, which has the numerical difficulty in dealing with the disappearance of the gas phase. For the well a drift flux method is employed solving a 1D flow along the horizontal well using a very stable approach for the pressure-velocity coupling in the well. A Newton-like method is used for both, reservoir and well variables, but using a segregated strategy for the well/reservoir coupling. This has proven to be suitable due to the large differences in the spatial and time scales of the problems. Several three-dimensional coupled multiphase flow were solved, including near well analysis with fractures, demonstrating the capabilities of the method.
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Use of Composite Elastic Modulus to Predict Inflow Performance
By T.A. JelmertSummaryThe proposed methodology is valid for any problem described by a diffusivity equation. As an application of the more general technique, we investigate fluid flow in a stress-sensitive reservoir. In case of production, the permeability, viscosity and fluid density may decrease in the near wellbore region. The reservoir thickness may also decrease. The objective of the present study is to quantify such changes and point out the effect on the inflow performance relationship.
The flow equation for stress-sensitive reservoirs may be highly non-linear. A non-linear variable shows up as a quadratic pressure gradient term in the diffusivity equation, Matthews and Russel (1967) . A logarithmic pressure transform may reduce the effect of the quadratic gradient term. The method involves the assumption of a constant elastic modulus. The equivalent assumption is that the variable may be approximated by an exponential function of pressure. Such variations have some experimental support. Many studies investigate the effect of a single pressure dependent variable. Then, a constant permeability modulus or compressibility is assumed. We propose a composite elastic modulus to investigate the simultaneous effect of an arbitrary number of pressure dependent variables in the transport term. The methodology depends on the assumption that every pressure-dependent variable may be approximated by an exponential function.
We investigate steady state flow, then the linearization of the diffusivity equation in terms of the transformed variable is complete, and analytical solutions are readily available. Solutions in terms of pressure are obtained by the inverse transform. Due to the non-linearities, the reservoir behaves different during production and injection. The pressure sensitivity depends on the value of the composite modulus and may be negligible. We provide equations to estimate the error incurred by neglecting the pressure sensitivity. The proposed methodology may be extended to time-dependent problems, but with reduced accuracy. Perturbations techniques are available to improve the accuracy.
Conclusions:
A composite elastic modulus has been proposed. The effect of non-linear terms may be estimated one by one and in combination.
A transformation based on the modulus will linearize the steady state diffusivity equation and the boundary conditions.
The non-linear pressure solutions may be obtained by the inverse transformation.
The degree of non-linearity may be characterized by the numerical value of the composite elastic modulus. Increasing values leads to decreased performance.
By use of L’Hospitals rule, we find that the conventional reservoir solution (without Stress-sensitivity) is included in the generalized equation as limiting behavior.
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Modeling of Multiphase Flow in Elastic Porous Media Based on Thermodynamically Compatible Systems Theory
Authors E.I. Romenskiy, Y.V. Perepechko and G.V. ReshetovaSummaryThe new computational model for multiphase flow in deforming elastic porous media is proposed. The derivation of the model is based on the thermodynamically compatible hyperbolic systems of conservation laws theory. The flow of the mixture of compressible fluids in the elastic medium is supposed to be a continuum, in which the multiphase character of flow is taken into account. This phenomenological approach of continuum mechanics modelling allows us to formulate the system of governing equations in a divergent form, which is advantageous for the mathematical study of the different problems and for the development of advanced numerical methods.
We present a thermodynamically compatible model for the flow of fluids mixture in elastic porous medium. The governing equations comprise balance laws for phase masses, total momentum and total energy supplemented by the equations for relative velocities in divergent form. The high accuracy Runge-Kutta-WENO numerical method for solving equations of the model is presened along the numerical test problem.
The proposed model and developed numerical framework can be used in the wide range of oil recovery problems. Examples are: tracking oil/water interfaces in oil reservoirs, modeling of flows in the well surrounding formation.
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Estimating Bottom Hole Damage Zone Parameters Based on Mathematical Model of Thermo-hydrodynamic Processes
Authors L.A. Gaidukov, D.V. Posvyanskii and R.R. TukhvatullinaSummaryIn order to find the optimal parameters of workovers for recovering well productivity it is important to know the properties of the bottom hole damage zone, such as its radius and permeability. The effect of the damage zone can be described by the skin factor, which can be indirectly estimated by well test analysis. Temperature and pressure are the most frequently observed physical parameters in well testing. During transient tests, both the pressure and temperature are measured at the well bottom hole. Typically, only pressure transient data is considered, and pressure transient data analysis is used to estimate the skin-factor. However, due to well bore effects, it is impossible to identify the structure of the damage zone. The reservoir pressure and temperature both recover after a well shuts. Temperature changes in the reservoir are induced by heat convection, heat conduction and the Joule-Thomson effect. The flow rate of these physical processes depends on the properties of both the reservoir and the damage zone. By considering the differences from the pressure buildup curve, temperature transient data can be used to estimate the radius of the damage zone, which can help give more accurate well test interpretation [1] . The mathematical model for well tests analysis is based on the solution of diffusivity equation for pressure and heat transfer equation for temperature. In [2] Green function technique was used to solve the pressure equation. In this study we present joint solution of the equations for the pressure and temperature dynamics. The calculations take into account the dependence of Joule-Thompson coefficient on reservoir pressure. Calculations were performed for the interpretation of well test data in a gas producing well. The results are qualitatively consistent with field examples.
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Modeling Of Flow in a Fracture Inside Porous Medium
Authors D.YU. Khanukaeva and A. N. FilippovSummaryThe modeling of oil and gas production processes such as hydraulic fracturing or oil displacement requires coupled description of free flow and flow in a porous medium.
In this work we used Stokes’ equation for free viscous flow coupled with Brinkman’s equation for liquid flow in a porous medium. The latter is differential equation of the same order as Stokes’ equation. Both equations together with continuity equation and proper boundary conditions present mathematically correct formulation of the coupled flow boundary value problem. For two-dimensional and axial symmetric geometry we showed analytically that the solution of Brinkman equation totally coincides with Darcy’s velocity in the whole region of a porous medium, except for a thin layer adjacent to impermeable wall. The thickness of the mentioned layer depends on specific permeability of porous material.
Stokes-Brinkman model was used for simulation of flows in finite dead-end channel with porous walls. This model is aimed at the simulation of flow through a hydraulic fracture or flow in a natural crack inside a porous medium. The results were compared with experiments fulfilled at the Institute of Geospheres Dynamics RAS and have demonstrated excellent agreement.
The work is partially supported by RFBR (grant #14-08-00893a) and CRDF-Global (grant #FSAX-14-60158-0).
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A New Model Reduction Technique Applied to Reservoir Simulation
Authors M.G. Ghasemi and E.G. GildinSummaryOne of the challenges in reservoir simulation is the study and analysis of large scale models with complex geology and multiphase fluid for considering real life applications. Even with recent increase in the computation power, the fast and reliable simulation of the fine scale models is still resource-intensive and hardly possible. Particularly, in optimization and field planning, it is necessary to simulate the system for varying input parameters. Here, model order reduction (MOR) can be used to significantly accelerate the repeated simulation. Although theory as well as numerical method for linear systems is quite well-established, for nonlinear systems, e.g. reservoir simulation, it is still a challenging problem.
We apply a recently introduced approach for nonlinear model order reduction to reservoir simulation. In order to overcome the issue of nonlinearity, we introduce the bilinear form of the reservoir model. The bilinear approximation is a simple form of the parent system and it is linear in the input and linear in the state but it not linear in both jointly. This technique is independent of input of the systems, and thus is applicable for wide range of input parameters without any training. Also, the formulation allows certain properties of the original models to be preserved in the reduced order models. The basic tools known from tensor theory are applied to allow for a more efficient computation of the reduced-order model as well as the possibility of constructing two-sided projection methods which are theoretically shown to yield more accurate reduced-order models.
Examples are presented to illustrate this recent approach for the case of two phase flow modeling, and comparisons are made with the case of linearized models and the full nonlinear models. We discuss the model reduction techniques to be applied to the two-phase flow system. We conclude the paper with some remarks and point out two ways to generalize the findings of this paper as a future work.
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Using Percolation Theory to Estimate Recovery from Poorly Connected Sands Using Pressure Depletion
Authors H. Wen, P.R. King, A.H. Muggeridge and E.S. VittoratosSummaryIn conventional waterflooding of low to intermediate net to gross reservoirs there is always some oil unswept even in the sands connected to both injection and production wells. This is oil trapped in “dangling ends”: flow units only poorly connected to the main flow path. In many cases the unswept volumes can be very large, depending on the properties of the reservoir and fluids and the well locations.
In this paper we show how percolation theory can be used to estimate the volumes of oil recovered and those left behind in these dangling ends following a conventional waterflood, without recourse to large scale simulation. Percolation theory is a general mathematical framework for connectivity and has been used previously to investigate the connectivity of flow units. The structure of these connected clusters in terms of backbones and dangling ends has not been previously studied. The results are also used to estimate the recovery of the unswept oil from dangling ends by a waterflood with a voidage replacement ratio <1.
We use a simple model of stochastically-distributed sandbodies to describe the reservoir. Many realizations for a range of net to gross ratio values and sandbody:system sizes were generated. In each realization the clusters connecting the injection and production wells were identified. These spanning clusters were subdivided into backbones and dangling ends. The volume fractions of the backbone and dangling end were then obtained. The statistical average and standard deviation of the volumes association with these clusters were obtained from the ensemble of realisations. These were used to determine the percolation scaling relationships in terms of simple algebraic formulae that cover the whole range of net to gross ratio and system sizes.
Our results show that the fraction of dangling ends can reach 20% of the clusters, and 80% among the spanning clusters, indicating a major proportion of the oil would be unswept by conventional waterflood. Recovery is improved for voidage replacement ratios less than 1 when the dangling ends are drained by oil expansion and solution gas drive.
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The Interplay between Viscous and Gravity Forces in Two-phase Stratified Reservoir Crossflow
More LessSummaryIn stratified reservoir models, crossflow describes how communicating layers of different petrophysical properties allow fluid to pass between them. Viscous crossflow is usually thought to be driven by pressure differentials due to permeability differences between layers, and gravitational crossflow is driven by density differences between phases. In this work, numerical simulations of basic two-dimensional models are presented to demonstrate the effect of gravity-driven crossflow.
Stratified reservoir models are very relevant today, particularly as the importance of offshore carbonate reservoirs grow, with their highly laterally homogenous and vertically cyclical beds with enormous ranges of permeability. Even in reservoirs which are not completely stratified, high-permeability streaks are common, and the crossflow between them and the adjacent low-permeability layers can be vital for maximising the recovery factor.
The results of this investigation serve to demonstrate that gravitational crossflow from high to low permeability layers may not necessarily improve a two-phase displacement sweep, when the low permeability layer is not bound by an impermeable layer beneath. This suggests that, in order to build up physically meaningful analytical models of crossflow during two-phase displacement in stratified reservoirs, a greater understanding of the relative importance of viscous and gravity forces is often required.
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Upgridding of Geological Models Based on Equations of Two-phase Flow
Authors S.P. Rodionov, O.N. Pichugin, L.N. Sokolyuk and Y. V. ShirshovSummaryThis paper presents an upgridding procedure that makes use of an analytical solution for a two-phase flow equation in the fine scale model. Both horizontal and vertical flow in fine scale cells are taken into account using appropriately defined boundary conditions. The method takes into account two-phase flow properties, cell sizes, diagonal tensor of absolute permeability, porosity, initial conditions and boundary conditions. We attempt to mimic the discretisation error using the reservoir heterogeneity concept. We discuss some of the difficulties encountered in applying the published methods to realistic reservoir models. We present several simple numerical cases to prove the advantages of the new method over other upgridding methods.
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Enhanced Decision Making for Chemical EOR Processes under Uncertainty - Applying the LSPC Method
Authors A. Alkhatib and P. KingSummaryThe Least Squares Monte Carlo method is a decision evaluation method that can capture the value of flexibility of a process. This method was shown to provide us with some insight into the effect of uncertainty on decision making and to help us capture the upside potential or mitigate the downside effects for a chemical EOR process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations.
The objective of this study was to enhance the performance of the Least Squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in producing the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. It approximates the random input distributions using polynomial chaos expansions and produces a proxy polynomial for the output parameter requiring a limited number of model responses that is conditional on the number of random inputs and the order of the approximation desired. The resulting proxy can then be used to generate the different statistical moments with negligible computational requirement. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus we introduce the least square probabilistic collocation method.
Both methods are then applied to chemical EOR problems using a number of stylized reservoir models. The technical uncertainties considered include the residual oil saturation to chemical flooding, surfactant and polymer adsorption and the viscosity multiplier of the polymer. The economic uncertainties considered were the oil price and the surfactant and polymer price. Both methods were applied using three reservoir case studies: a simple homogeneous model, the PUNQ-S3 model and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method.
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Hidden Information in Ill-posed Inverse Problems
Authors S.S. Kahrobaei, M. Mansoori, G.J.P. Joosten, P.M.J. Van den Hof and J.D. JansenSummaryIt is well known that parameter updating of large-scale numerical reservoir flow models (a.k.a. ‘computer assisted history matching’) is an ill-posed inverse problem. Typically the number of uncertain parameters in a reservoir flow model is very large whereas the available information for estimating these parameters is limited. The classic solution to this problem is to regularize the unknowns, e.g. by penalizing deviations from a prior model. Attempts to estimate all uncertain parameters from production data without regularization typically lead to unrealistically high parameter values and therefore to updated parameter fields that have little or no geological realism. However, it has been suggested that the application of unregularized reservoir parameter estimation may still add value, because it, sometimes, gives an indication of the location of significant missing features in the model. We investigated under which conditions this perceived added value might occur. We conducted several twin experiments and applied unregularized parameter estimation to update uncertain parameters in a simple two-dimensional reservoir model that contained a major deficiency in the form of a missing high or low permeability feature. We found that in case of low-permeability barriers or high-permeability streaks it is indeed sometimes possible to localize the position of the model deficiency. To further analyze this behavior we conducted one-dimensional experiments using a transfer function formalism to characterize the identifiability of the location and magnitude of model deficiencies (flow barriers).
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Efficient Inference of Reservoir Parameter Distribution Utilizing Higher Order SVD Reparameterization
More LessSummaryReservoir parameter inference is a challenging problem to many of the reservoir simulation workflows, especially when it comes to real reservoirs with high degree of complexity and non-linearity, and high dimensionality. In a history matching problem that adapts the reservoir properties grid blocks, the inverse problem leads to an ill-posed and very costly optimization schemes. In this case, it is very important to perform geologically consistent reservoir parameter adjustments as data is being assimilated in the history matching process. Therefore, ways to reduce the number of reservoir parameters need to be sought after.
In this paper, we introduce the advantages of a new parameterization method utilizing higher order singular value decomposition (HOSVD) which is not only computationally more efficient than other known dimensionality reduction methods such as, SVD and DCT, but also provides a consistent model in terms of reservoir geology. HOSVD power is due to its ability to supply a reliable low-dimensional reconstructed model while keeping higher order statistical information and geological characteristics of reservoir model. In HOSVD, we take the snapshots in a 2D or 3D approach, i.e., do not vectorize original replicates, and stack them up into a tensor form, i.e. a multi-way array in multilinear algebra which leads to implementing tensor decomposition. Technically, we performed HOSVD to find the best lower rank approximation of this tensor that is an optimization problem utilizing alternating least square method. This results in a more consistent reduced basis.
We applied this novel parameterization method to the SPE10 benchmark reservoir model to show its promising parameterization performance. We illustrate its advantages by comparing its performance to the regular SVD (PCA) in a history matching framework using EnKF, as well as characterization performance of the ensemble-based history matching approaches along with HOSVD. Overall, HOSVD outperforms SVD in terms of reconstruction and estimation performance.
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