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ECMOR XVI - 16th European Conference on the Mathematics of Oil Recovery
- Conference date: September 3-6, 2018
- Location: Barcelona, Spain
- Published: 03 September 2018
101 - 172 of 172 results
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Well Placement Optimization Under Uncertainty Using Opportunity Indexes Analysis And Probability Maps
Authors H.M. Mustapha and D.D. DiasSummaryImproving hydrocarbon recovery from green and mature fields through targeting potential drilling requires a computationally complex process of well placement optimization. Because these operational activities are expensive and particularly critical in the periods of low oil prices, a risk quantification analysis is often required for uncertainty considerations.
A new automated probabilistic workflow optimizes well placement using probability maps based on reservoir and simulation opportunity indexes. Well types include vertical wells and horizontal wells. The probability map concept aims at unifying the existing model realizations into a single probability map by establishing thresholds for key physical parameters and reservoir characteristics. The opportunity indexes method is a fast way to identify zones with high potential for production from both oil and gas reservoirs.
The workflow is generic and can be applied to oil and gas in both mature and green fields as a fast method of well placement optimization under uncertainty. Starting from a set of reservoir modelling scenarios, a pattern recognition algorithm is first applied to classify and rank the realizations. A representative subset of these realizations is then used in an ensemble-based method and, when needed, calibrated to existing observed data. Reservoir and simulation opportunity indexes are applied to all calibrated realizations. Finally, a single probability map is created to unify the opportunity index maps. Based on the pattern observed in the probability map, areas of interest (AOI) are outlined, and several realizations of well configurations are generated. The designed wells are screened based on engineering criteria and ultimately assessed using numerical simulations on each realization.
The workflow terminates by results analysis and well design selection. This considers not only the improvement in oil recovery, but also a measure of risk coming from the uncertainty assessment. In testing on several simulation models, the unswept reservoir regions were successfully identified and ranked for drilling targets. Compared to existing methods, the workflow showed superior results.
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Integrated Production Strategy Optimization Based On Iterative Discrete Latin Hypercube
Authors J.C. Hohendorff Filho and D.J. SchiozerSummaryMost problems of production strategy optimization in the oil industry are characterized by a large number of discrete random variables in discontinuous search spaces with non-necessarily monotonic objective functions, usually net present value or oil recovery, with many local maximums within a maximization problem. It demands a large number of simulations to adequately evaluate search space, what becomes more complex when integrating reservoir and production system. This paper evaluates a new iterative discrete Latin Hypercube (IDLHC) sampling based method to maximize the objective function in integrated production strategy optimization.
Inside decision making study, to evaluate the best placement of wells in the reservoir, we have used an optimization process evaluating some objective function. We compared the optimization between the IDLHC and the genetic algorithm method. We used both methodologies to maximize the net present value objective function for the same variable set and search space. We used the benchmark case UNISIM-II-D (carbonate field in Brazil) reservoir model as an application case. And we applied our explicit methodology to integrate reservoir and production system simulators during optimization process.
IDLHC adequately treated posterior frequency distributions of discrete random variables and maximizes nonnecessarily monotonic objective functions within great discontinuous search space and many local optimums set by the well placement problem.
Population based optimization using iterative discrete Latin Hypercube sampling best suited this problem, with consistent convergence to global optimum, few objective function evaluations and the simultaneous multiple numeric reservoir simulations runs.
The IDLHC method showed the advantage of being a simple methodology to maximize the objective function, reducing the search space gradually with each iteration, while addressing posterior frequency distributions of discrete variable levels.
The method successfully maximized the net present value in the well placement step of production strategy optimization, and more faster when compared with a well-established optimization methodology (genetic algorithm).
This easy to use, reliable methodology with lower computational time costs is an interesting option for optimization methods in problems of integrated production strategy design related to the oil industry.
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Application Of Adjoint-Based Optimal Control To Gas Reservoir With A Memory Effect
Authors A. Kadyrova and A. KhlyupinSummaryIn recent years researchers in oil-gas industry have established that the contribution of memory is significant for the modeling of fluid flow in unconventional reservoirs. According to modern works, the memory effect appears due to the contrast between high permeable fractures and nanoporous matrix leading to the gap of fluid velocities at the interface between these media. Also in the homogenization procedure from micro to macro scale the nonlocality in time (in other words, the memory) reflects the delay of fluid pressure and density between subdomains with different properties of pore space geometry.
Mathematically, a memory-based fluid flow model can be described by the system of integro-differential equations. Despite the fact that a large number of journal articles are devoted to numerical methods for the forward solution of such equations, the problems of optimization and optimal control of these systems are actual and insufficiently studied.
We consider the one-dimensional model of gas filtration and diffusion as a model with memory. The system includes a partial differential equation for filtration in fractures and weakly singular Volterra integral equation of the second kind, which describes the diffusion of gas from blocks with closed nanopores. Numerical simulation, obtained using a Navot-trapezoidal algorithm, shows that the effect of memory influences on the distribution and the time evolution of pressure and density in comparison with the classical double porosity model.
The pressure-constrained maximization of discounted cumulative gas production was chosen as a basic optimization problem. The appearance of memory in the model makes the standard adjoint-based approach not applicable since it was developed only for conventional systems of partial differential equations. The novel adjoint model for media with memory was obtained from the necessary conditions of optimality using the classical theory of calculus of variations and efficiently applied to production optimization problem.
In conclusion we compare optimal control scenarios for the model with memory and for the classical double porosity model. Analysis has shown the importance of memory accounting in reservoir optimization problems.
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Optimization Of Well Rates For Condensate Field Development
Authors A.I. Ermolaev, A. Nekrassov and I. TrubachevaSummaryWe solve a problem of optimal control of wells on a condensate field. The problem is defined as follows: maximize condensate recovery subject to (selected) well group rate, while individual well rates to are unknown. This problem could be solved using existing oilfield & gasfiled industry standard simulators. However, the use of these simulators and underlying optimization procedures require a lot of simulation runs. Even for the small number of wells (about 10) it may lead to sufficient simulation time consumption for solving this optimization problem.
We propose a method to significantly reduce the total simulation time, allowing to find almost (within a given tolerance) optimal solutions. This time reduction is achieved by swapping the criteria of condensate recovery maximization to the criteria of minimizing maximal well pressure draw-down. Furthermore, this method allows to take into account well-to-well influence by introducing additional equations. All that allow to convert the initial problem to a problem of solving a system of linear equations.
We show an example for a test case with five wells. With a simulator we get an optimal solution within 2.5 days; using our algorithm, we got a condensate recovery 0.5% less than optimal within 2 hours.
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An Efficient Method To Improve Oil Field Productivity Using Tracers And Dynamic Flow Control Valves
Authors H.M. Mustapha, D.D. Dias, K.M. Makromallis and T.M. ManaiSummaryImproved recovery methods such as waterflooding, gas injection or tertiary injection fluids are often used to extend production following primary recovery, either by maintaining reservoir pressure or changing the reservoir fluid properties for enhanced oil displacement. A common challenge exists in all these recovery methods: maximizing the sweep efficiency of the injected fluids and predicting flow patterns. This is increasingly difficult in heterogeneous and stacked reservoirs where complex connectivity between wells leads to poor operational choices, often resulting in early breakthrough and diminished ultimate recovery.
To achieve better control of the displacement, advanced completions aim to control flow around the wellbore and strategically allocate production and injection from different parts of the completion. There are currently various types of flow control devices that can be installed to improve overall oilfield productivity. These include static inflow control devices (ICDs) by which inflowing fluids are choked back with nozzles that remain fixed in aperture size during the entire production cycle; devices that can respond to a change in flow rate, density or viscosity of inflowing product that are known as autonomous inflow control devices (AICDs); and devices that can change their flowing area through independent surface control and are called flow control valves (FCVs). Despite recent advances in the optimization of secondary and tertiary recovery performance, optimization methods generally involve elevated computational costs mainly for field-scale development plans. Their strategy is often centered around several optimization variables, complex optimization algorithms, and a substantial number of iterations to succeed.
A new method to optimize the performance of fluid injection schemes uses reservoir simulation techniques. An optimization methodology involving the analysis of numerical tracers attached to wells and FCVs was used. A function optimization method was implemented by tracing the injected fluids through each device of the injectors towards the production well. The breakthrough of tracer, and consequently injected fluid, was measured in each production well. These data were implemented as the input for a feedback control on the device to reduce the injection of fluids accordingly. The operation was performed dynamically to account for changes in fluid distribution with time. The results presented reveal the significant importance of the method as a fast solution for field applications.
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Closed-Loop Reservoir Management Using Nonlinear Model Predictive Control: A Field Case Study
Authors R. Patel, J.L. Guevara and J. TrivediSummaryClosed-loop reservoir management (CLRM) consists of near-continuous data assimilation and real-time optimization to improve oil recovery and reservoir economics. In deep oil sands deposits using steam-assisted gravity drainage (SAGD) recovery process, CLRM involves real-time subcool (difference between actual and saturation temperature) control to develop the uniform steam chamber along the horizontal injector-producer well pair. Recently, model predictive control (MPC) has been implemented to maintain the optimal subcool; however, oversimplified models used in MPC are inadequate as reservoir dynamics in SAGD is highly complex, spatially distributed, and nonlinear. This provides an opportunity for the improved CLRM workflow which can incorporate the nonlinear physical/empirical models in MPC to represent the flow dynamics accurately over the reservoir lifecycle.
In this research, two novel workflows, comprising linearization and nonlinear optimization are proposed to implement nonlinear model predictive control (NMPC) in CLRM of SAGD reservoirs. Linearization basically reduces an NMPC problem to linear MPC by estimating an equivalent linear model of a nonlinear black box model for a given input signal in a mean-square-error sense. Due to linear approximation, cost function in the MPC can be minimized using quadratic programming (QP) over the specified time horizon. Another approach is to use nonlinear dynamic models directly for accurate prediction of the plant states and/or outputs. Resulting nonconvex, nonlinear cost optimization problem is solved using interior-point algorithm at each control interval. Proposed workflows are tested using the history-matched, field-scale model of a SAGD reservoir located in northern Alberta, Canada. The horizontal well pair with dual-tubing string completion is segmented and subcool in each section is considered as an output variable while steam injection rates in both tubings and liquid production rate are the input variables of the NMPC controller. Bi-directional communication link was established between the controller and thermal reservoir simulator, acting as a virtual process plant. Qualitative and quantitative analysis of the results reveals that nonlinear black-box models can successfully capture the nonlinearity of the SAGD process in CLRM. Also, both workflows can control the subcool above desired set-point while ensuring the stable well operations. Furthermore, net-present-value (NPV) is increased by 24% when proposed NMPC workflows are used in CLRM as compared to the base case with no closed-loop control. Overall, NMPC can be successfully employed in CLRM of SAGD reservoirs for improved real-time subcool control, energy efficiency, and greenhouse gas emissions while satisfying the constraints offered by the surface facilities.
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Well Optimisation With Goal-Based Sensitivity Maps Using Time Windows And Ensemble Perturbations
Authors C.E. Heaney, P. Salinas, C.C. Pain, F. Fang and I.M. NavonSummaryKnowledge of the sensitivity of a solution to small changes in the model parameters is exploited in many areas in computational physics and used to perform mesh adaptivity, or to correct errors based on discretisation and sub-grid-scale modelling errors, to perform the assimilation of data based on adjusting the most sensitive parameters to the model-observation misfit, and similarly to form optimised sub-grid-scale models. We present a goal-based approach for forming sensitivity (or importance) maps using ensembles. These maps are defined as regions in space and time of high relevance for a given goal, for example, the solution at an observation point within the domain. The presented approach relies solely on ensembles obtained from the forward model and thus can be used with complex models for which calculating an adjoint is not a practical option. This provides a simple approach for optimisation of sensor placement, goal based mesh adaptivity, assessment of goals and data assimilation. We investigate methods which reduce the number of ensembles used to construct the maps yet which retain reasonable fidelity of the maps.
The fidelity comes from an integrated method including a goal-based approach, in which the most up-to-date importance maps are fed back into the perturbations to focus the algorithm on the key variables and domain areas. Also within the method smoothing is applied to the perturbations to obtain a multi-scale, global picture of the sensitivities; the perturbations are orthogonalised in order to generate a well-posed system which can be inverted; and time windows are applied (for time dependent problems) where we work backwards in time to obtain greater accuracy of the sensitivity maps.
The approach is demonstrated on a multi-phase flow problem.
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Compositional Simulation With Capillary Pressure For Oil Production From Tight Formation
Authors D.R. Sandoval, W. Yan and E.H. StenbySummaryThe influence of porous media on phase behaviour is a topic of interest driven by the shale gas boom because many field observations suggest the saturation pressure in tight shale formation may change dramatically. There has acmally been such a concern for other low permeable tight formation, such as the Lower Cretaceous (LC) formation in the Danish North Sea. for decades. However, there is no consensus on the extent of the influence and also little analysis of the issue in the open literature.
The integration of the capillary pressure effect on phase equilibrium into a reservoir simulator is not entirely trivial. The modifications needed will depend on the implicitness level of the numerical model of the simulator, with an increasing complexity as the level increases. In general, the standard thermodynamic routines should be modified to handle the cases where the liquid pressure becomes negative as a result of the high capillary pressures. The flash and stability analysis routines involving capillary pressure need an efficient implementation to maintain the robustness and speed needed during simulation. For the linear solver, the derivatives of the selected pressure models must be obtained and implemented in a consistent way to avoid differences between the capillary pressure model used for phase equilibrium, and the capillary pressure used for the flow equations. A fully implicit compositional simulator was modified by adding the influence of the capillary pressure into the phase behavior. The customized tool served to investigate a natural depletion scenario of a shale reservoir and a tight reservoir from the LC formation in the Danish North Sea using different capillary pressure models.
In general, low to moderate deviations in the cumulative oil production, pressure profiles, and saturation profiles were observed for the cases with effective pore sizes less than 40 nm. For the producing gas oil ratio considerable deviations were found even for pore sizes close to 100 nm. Moreover, a pore size distribution was compared to the fixed pore size assumption in the capillary pressure model. A variable pore size capillary pressure model shows similar results to those obtained at fixed capillary radius. In the long term, the results are closer to effective pore size calculated at the bubble point given by the maximum value of the pore size distribution.
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Scaling Law For Slip Flow Of Gases In Nanoporous Media From Nanofluidics, Rocks, And Pore-Scale Simulations
More LessSummaryIn unconventional reservoirs, as the effective pore size becomes close to the mean free path of gas molecules, gas transport behavior begins to deviate from Darcy’s law. The objective of this study is to explore the similarities of gas flows in nanochannels and core samples as well as those simulated by direct simulation BGK (DSBGK). a particle-based method that solves the Bhatnagar-Gross-Krook (BGK) equation.
Due to fabrication difficulties, previous work on gas flow experiments in nanochannels is very limited. In this work, steady-state gas flow was measured in reactive-ion etched nanochannels on a silicon wafer, which have a controlled channel size. A core-flooding apparatus was used to perform steady-state gas flow measurements on carbonate and shale samples. Klinkenberg permeability was obtained under varying pore pressures but constant temperature and effective stress. Same gas was used in nanofluidic and rock experiments, making them directly comparable. Results from both experiments were then compared to gas flow simulations by DSBGK method carried out on several independently constructed geometry models. DSBGK uses hundreds of millions of simulated molecules to approximate gas flow inside the pore space. The intermolecular collisions were handled by directly integrating the BGK equation along each molecule’s trajectory, rather than through a sampling scheme like that in the direct simulation Monte-Carlo (DSMC) method. Consequently, the stochastic noise is significantly reduced, and simulation of nano-scale gas flows in complex geometries becomes computationally affordable.
The Klinkenberg factors obtained from these independent studies varied across three orders of magnitude, yet they all appear to collapse on a single scaling relation where the Klinkenberg factor in the slip flow regime is inversely proportional to the square root of intrinsic permeability over porosity. Our correlation could also fit the data in the literature, which were often obtained using nitrogen, after correcting for temperature and gas properties. This study contributes to rock characterization, well testing analysis as well as the understanding of rarefied gas transport in porous media.
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Systematic Hybrid Modelling Using Fracture Subset Upscaling
Authors D.L.Y. Wong, F. Doster, A. Kamp and S. GeigerSummaryMulti-scale fractured reservoirs can be modelled effectively using hybrid methods that partition fractures into two subsets: one where fractures are upscaled and another one where fractures are represented explicitly. Existing partitioning methods are qualitative or empirical.
In this paper, we present a novel and quantitative partitioning approach based on a single-porosity hybrid modelling workflow that uses numerical (Embedded Discrete Fracture Methods – EDFM) and semi-analytical (Effective Medium Theory – EMT) methods for fracture subset upscaling. We demonstrate this workflow using synthetic fracture data and realistic data sourced from outcrops of the Jandaira Carbonate Formation in the Potiguar Basin, Brazil.
Fracture subset upscaling with EDFM and EMT using three datasets (two real, one synthetic) shows that the smallest, most numerous fractures are poorly connected. The ability of fracture subset upscaling to identify these fractures is essential to the hybrid modelling workflow. EDFM and EMT methods give nearly identical results, but EMT enables us to greatly accelerate the calculations.
To validate our workflow, hybrid models were created with different partitioning sizes and compared against EDFM simulations where all fractures are represented explicitly. A single-phase pressure drawdown was used a test problem. The simulation results show that once the upscaled fractures begin to connect, deviations in flow response start to grow because single-porosity representations are inadequate to capture the separation of timescales between flow in a well-connected fracture subset and flow in the matrix. In some cases, the flow regime in the model were observed to change entirely.
Overall, the results justify the proposed workflow as a means for systematic and quantitative construction of hybrid models.
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An Efficient Hybrid Grid Cross-Flow Equilibrium Model For General Purpose Field-Scale Fractured Reservoir Simulation
Authors H.M. Mustapha, K.M. Makromallis and A.C. CominelliSummaryMultiphase flow simulation in fractured reservoirs at field scale is a significant challenge. Despite recent advances and a wide range of applications in both hydrology and hydrocarbon reservoir engineering, discussing efficient methods that cover computational complexity, accuracy and flexibility aspects is still of paramount importance for a better understanding of these complex media. In this work, we present a new method that handles both the topological and computational complexities of fractures, taking into consideration advantages of various existing approaches, which include hybrid grid and crossflow equilibrium models. The hybrid grid (HG) model consists of representing fractures as lower-dimensional objects that still are represented as control volumes in a computational grid. The HG model is equivalent to a single-porosity model with a practical solution for the small control volumes at the intersection between fractures; however, the overall simulation run time is still dominated by the remaining fracture small control volumes. To overcome single-porosity computational challenges, a crossflow equilibrium (CFE) concept between discrete fractures and a small neighborhood in the matrix blocks can be employed. The CFE model consists of combining fractures with a small fraction of the neighborhood matrix blocks on either side in larger elements to achieve a better computational efficiency than conventional single-porosity models. The implementation of a CFE model at field scale is not practical because of the fracture topological challenges associated with the construction of an accurate computational grid for the CFE elements.
In this work, we propose a method based on a combination of HG and CFE models to overcome the challenges associated with the HG fracture small control volumes and field-scale CFE computational grid construction. First, we assess the performance of the existing CFE model, and we propose an improved model. In addition, we suggest an input data handling method that is sufficient to account for fractional flow inside the CFE elements for flow in homogeneous fractured reservoirs without the need of any change in the simulator. Second, we describe the uniqueness of the proposed method, and we discuss different numerical examples to assess both the accuracy and computational efficiency. The results obtained are very accurate, and, computationally, one to two orders of magnitude speedup can be achieved. The improved CFE results are superior over the traditional CFE model. Combined with the HG model, the results are significantly improved while retaining a very good performance.
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Quantification Of Coarsening Effect On Response Uncertainty In Reservoir Simulation
Authors S. de Hoop, D.V. Voskov, F.C. Vossepoel and A. JungSummaryIn this study, an attempt is made to better understand the effect coarsening of the parameter space has on the uncertainty representation of the response. Firstly, an HF ensemble of channelized reservoir models is constructed using a Multi-Point Statistic (MPS) approach. Several levels of coarsening are generated using a flow-based xipscaling algorithm. A water injection strategy is simulated for each scale of the hierarchical ensemble. Dynamic analysis is performed on a reduced representation of the response uncertainty obtained via Multidimensional Scaling (MDS). We introduce an Uncertainty Trajectory (UT), which quantifies the coarsening effect in terms of deviation from the HF ensemble response uncertainty. The UT also includes the temporal beha\'ior of the response uncertainty of each ensemble scale. The mean integrated distance from the HF ensemble UT can be used as a measure of dissimilarity in the flow- behavior of consecutive coarser ensembles scales. Reducing the number of HF flow- simulations required for uncertainty quantification can be achieved via the proposed methodology and thereby greatly reducing the overall computational cost.
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Multifidelity Framework For Uncertainty Quantification With Multiple Quantities Of Interest
Authors F.F. Kostakis, B.T. Mallison and L.J. DurlofskySummaryA systematic framework, involving flow simulation and model selection at many fidelity (resolution) levels, is introduced to accurately quantify the impact of geological xincertainty on output quantities of interest (Qols). The methodology considers large numbers of realizations (0(1000) in the cases presented), though very few (0(10)) simulations are performed at the highest resolutions. We proceed from coarser to finer resolution levels, and at each stage simulation results are used to select a subset of realizations to simulate at the next (higher) fidelity level. Models are constructed at all resolution levels through upscaling of the underlying fine-scale realizations. A global transmissibility upscaling procedure is applied for this purpose. Approximate cumulative distribution functions (CDFs) are constructed for all Qols considered. The Qol values themselves are always computed at the finest scale, but corresponding percentile values are determined using results at a ‘rank-preserving’ fidelity level. Detailed results are presented for oil-water flow in a channelized system. Simulations at seven different fidelity levels are used, and eight Qols are evaluated. Results for the example considered demonstrate accurate reconstruction of fine-scale CDFs for all Qols. with a speedup factor of 12 relative to performing all simulations on the fine scale.
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Stochastic Oilfield Optimization For Hedging Against Uncertain Future Development Plans
Authors A. Jahandideh and B. JafarpourSummaryWe develop a new oilfield optimization framework that assumes future development plans are uncertain. To handle this uncertainty, we formulate a multi-stage stochastic optimization approach that considers several plausible scenarios for future development. These scenarios are used to predict the Net Present Value of the reservoir through its life-cycle. At each stage, the current decision variables (including well locations and controls) are identified by optimizing the predicted project NPV, which is computed based on stochastic descriptions of the number, location, and well control settings of future development stages. We compare the performance of the stochastic approach with optimization by assuming perfect information about the future development plans and by disregarding uncertainty future development activities, and draw important conclusion about the behavior of the proposed formulation.
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Monte Carlo Simulation For Uncertainty Quantification In Reservoir Simulation: A Convergence Study
Authors M.A. Cremon, M. Christie and M.G. GerritsenSummaryThe present work illustrates the convergence properties of a Monte Carlo Simulation (MCS) used to quantify the geological uncertainty in a 3D, 3-phase reservoir simulation test case. Our reservoir model along with fluid and numerical properties was obtained from a major oil and gas company. We generate 10,000 realizations of a geological model and run a Black-Oil flow simulation using a commercial reservoir simulator and a synchronous parallel implementation. The distributions of the moments and quantiles of the Net Present Value (NPV) are presented in the form of their Cumulative Density Functions (CDF). We also show the distributions of the break even time (BET) and the probability of breaking even in order to see the effect of considering quantities that are different in nature. We use log-plots to assess the convergence of the results, and verify that the convergence of the quantities of interest follows a squared-root law in the number of realizations used. We quantify the relative error made on various quantities and illustrate that the use of a small ensemble can yield errors of hundreds of percents, and that lowering the error to a given precision (e.g. below ten percent) can require thousands of realizations. For decision making and profitability assessments, using large sets of realizations is now feasible due to the availability of fast, distributed architectures and the parallel nature of MCS. Our results suggest that the improvement in the quality of the results is significant and well worth the extra effort. For optimization and sensitivity studies, running large ensembles is still intractable but yields sets of quantiles that can be used as a Reduced Order Model (ROM). Setting up a test case using this dataset is under consideration, and could provide an interesting integrated setup for comparisons of uncertainty quantification (UQ) methods.
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A Data-Space Approach For Well Control Optimization Under Uncertainty
Authors S. Jiang, W. Sun and L.J. DurlofskySummaryData-space inversion (DSI) methods provide posterior (history-matched) predictions for quantities of interest, along with uncertainty quantification, without constructing posterior models. Rather, predictions are generated directly from a large set of prior-model simulations and observed data. In this work we develop a data-space inversion with variable controls (DSIVC) procedure that enables forecasting with user-specified well controls in the prediction period. In DSIVC, flow simulations on all prior realizations, with randomly sampled well controls, are first performed. User-specified controls are treated as additional observations to be matched in posterior predictions. Posterior data samples are generated using a randomized maximum likelihood procedure, with some algorithmic treatments applied to improve performance. Results are presented for a channelized system. For any well control specification, posterior predictions can be generated in seconds or minutes. Posterior predictions from DSIVC are compared to reference DSI results. DSI requires prior models to be resimulated using the specified controls, while DSIVC requires only one set of prior simulations. Substantial uncertainty reduction is achieved through data-space inversion, and reasonable agreement between DSIVC and DSI results is consistently observed. DSIVC is applied for data assimilation combined with production optimization under uncertainty, and clear improvement in the objective function is attained.
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Performance Enhancement Of Gauss-Newton Trust Region Solver For Distributed Gauss-Newton Optimization Method
Authors G. Gao, H. Jiang, J.C. Vink, P.H. van Hagen and T.J. WellsSummaryDistributed Gauss-Newton (DGN) has been proved very efficient and robust for history matching (HM) and uncertainty quantification (UQ). In each iteration, DGN needs to solve an ensemble of hundreds to thousands of trust-region subproblems (TRS) concurrently, and it is extremely computational expensive, especially when applied to large-scale history matching problems. In this paper, different approaches are developed to reduce the computational cost, and their performances are compared with other well-known methods.
The original Gauss-Newton trust-region (GNTR) solver solves a nonlinear equation iteratively using the modified Newton-Rapson method, which involves solving a large-scale symmetric linear system twice. In this paper, we propose to estimate the nonlinear GNTR equation using either an inverse quadratic model or a cubic spline model, by fitting the nonlinear equation evaluated at different points in previous iterations. The computation cost can be cut by half, because it requires solving the symmetric linear system only one time in each iteration.
The proposed approach is validated on two sets of synthetic test problems, small-scale problems with 2000 to 5000 parameters and large-scale problems with 10000 to 100000 parameters. Each set contains 500 test problems with different number of parameters and observed data. The GNTR solver using an inverse quadratic model has performance that is comparable to the GNTR solver using a cubic spline model. Their performances are also compared with the well-known direct TRS solver using factorization and iterative TRS solver using conjugate-gradient approach of the GALAHAD optimization library. In terms of efficiency, robustness, and memory usage, the two newly proposed GNTR solvers outperform the two TRS solvers of the GALAHAD optimization library.
Finally, the proposed GNTR solver have been implemented in our in-house distributed HM and UQ system and has been validated on different real field HM examples. Our numerical experiments indicate that the DGN optimizer using the new GNTR solver performs quite stable and effective when applied to real field HM problems.
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Performance Assessment of Ensemble Kalman Filter and Markov Chain Monte Carlo under Forecast Model Uncertainty
Authors R. Patel, T. Jain, J. Trivedi and J. GuevaraSummaryEnsemble Kalman filter (EnKF) and Markov chain Monte Carlo (MCMC) are popular methods to obtain the posterior distribution of unknown parameters in the reservoir model. However, millions of simulation runs may be required in MCMC for accurate sampling of posterior as subsurface flow problems are highly nonlinear and non-Gaussian. Similarly, EnKF formulated on the basis of linear and Gaussian assumptions may also require a large number of realizations to correctly map the solution space of the unknown model parameters, ultimately resulting in the high computational cost. Data-driven meta/surrogate/proxy models provide an alternative solution to alleviate the issue of high computational cost. Since these models are not as accurate as numerical solutions of partial differential equations (PDE), their implementation may add an uncertainty in the forecast model. In literature, the effect of uncertainty in forecast model on data assimilation is not well studied, especially with field-scale reservoir models.
In this work, we propose the robust assisted history matching workflow using polynomial chaos expansion (PCE) based forecast model. Proposed forecast model relies on reducing parameter space using Karhunen–Loeve (KL) expansion which preserves the two-point statistics of the field. Random variables from KL expansion and orthogonal polynomials corresponding to the prior probability density function (pdf) form the set of input parameters in PCE. Further, non-intrusive probabilistic collocation method (PCM) is used to compute PCE coefficients. PCE forecast model is then used in EnKF and MCMC to calculate the likelihood of the samples in place of high fidelity full physics simulation runs.
A case study is performed using a 3D field scale model of a reservoir located near Fort McMurray in northern Alberta, Canada. Performance of EnKF and MCMC are assessed under forecast model uncertainty using rigorous qualitative and quantitative analysis and posterior distribution characterization. Results clearly depict that, although EnKF provided reliable mean and variance estimates of model parameters, MCMC outperformed the former even under the uncertainty associated with PCE metamodel. Inaccurate initial assumptions of model parameters were successfully handled by MCMC, although, with a longer burn-in period. Furthermore, characterization of posterior demonstrated reduced uncertainty in the estimation of model parameters using MCMC as compared to EnKF.
Practical implications of the proposed approach and performance assessment under forecast model uncertainty will be consequential in designing accurate and computationally efficient reservoir characterization and optimization workflows and hence, improved decision-making in reservoir management.
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A Multiscale Method For Data Assimilation
Authors R. Moraes, H. Hajibeygi and J.D. JansenSummaryIn data assimilation problems, various types of data are naturally linked to different spatial resolutions (e.g. seismic and electromagnetic data), and these scales are usually not coincident to the subsurface simulation model scale. Alternatives like down/upscaling of the data and/or the simulation model can be used, but with potential loss of important information. To address this issue, a novel Multiscale (MS) data assimilation method is introduced. The overall idea of the method is to keep uncertain parameters and observed data at their original representation scale, avoiding down/upscaling of any quantity. The method relies on a recently developed mathematical framework to compute adjoint gradients via a MS strategy. The fine-scale uncertain parameters are directly updated and the MS grid is constructed in a resolution that meets the observed data resolution. The advantages of the technique are demonstrated in the assimilation of data represented at a coarser scale than the simulation model. The misfit objective function is constructed to keep the MS nature of the problem. The regularization term is represented at the simulation model (fine) scale, whereas the data misfit term is represented at the observed data (coarse) scale. The performance of the method is demonstrated in synthetic models and compared to down/upscaling strategies. The experiments show that the MS strategy provides advantages 1) on the computational side – expensive operations are only performed at the coarse scale; 2) with respect to accuracy – the matched uncertain parameter distribution is closer to the “truth”; and 3) in the optimization performance – faster convergence behaviour due to faster gradient computation. In conclusion, the newly developed method is capable of providing superior results when compared to strategies that rely on the up/downscaling of the response/observed data, addressing the scale dissimilarity via a robust, consistent MS strategy.
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History Matching Of Real Production And Seismic Data In The Norne Field
Authors R. Lorentzen, T. Bhakta, D. Grana, X. Luo, R. Valestrand and G. NaevdalSummaryAutomatic history matching using production and seismic data is still challenging due to the size of seismic datasets. The most severe problem when applying ensemble based methods for assimilating large datasets, is that the uncertainty is usually underestimated due to the limited number of models in the ensemble compared to the dimension of the data, which inevitably leads to an ensemble collapse. Localization and data reduction methods are promising approaches mitigating this problem.
In this paper, we present a new robust and flexible workflow for assimilating seismic attributes and production data. The methodology is based on sparse representation of the seismic data, using methods developed for image denoising. The approach can be applied seismic data or inverted seismic attributes obtained from geophysical inverse methods. The seismic response in the forward model is computed using a petroelastic model, that depends on several petrophysical parameters, including lithology, porosity, and saturation.
We propose to assimilate production and seismic data sequentially, which makes scaling of different data types redundant, and allows for use of different localization techniques. We use traditional distance-based localization for production data, and a newly developed correlation-based localization technique for seismic data. The latter is necessary because the image denoising method utilize discrete wavelet transforms, which render the seismic data without spatial positions.
The workflow is successfully implemented for the Norne field, and an iterative ensemble smoother is used for the sequential assimilation of production data and acoustic impedance. We show that the methodology is robust and ensemble collapse is avoided. Furthermore, the proposed workflow is flexible, as it can be applied to seismic data or inverted seismic properties, and the methodology requires only moderate computer memory. The results show that through this method we can successfully reduce the data mismatch for both production data and seismic data.
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Geochemical Equilibrium Determination Using An Artificial Neural Network In Compositional Reservoir Flow Simulation
Authors D. Guérillot and J. BruyelleSummaryThe application of chemical method for hydrocarbons extraction has attracted increasing interest in the reservoir simulation community. To simulate such reactive transfer processes, compositional flows in porous media with a complex mineralogical must be coupled with the chemical equilibria in the aqueous phase and the precipitation / dissolution reactions of the minerals.
The most important time consumed during reactive transport simulation is the geochemical equilibrium (about 30% to 80%). Typically, chemical equilibria are computed for each cell at each time-step by solving an equations system with the iterative Newton-Raphson method. To reduce the computation time, the number of species in solution is often reduce. However, such assumption leads to a less of accuracy of results.
Instead of simplifying the geochemical model, an approach that mimic the resolution of geochemical equilibrium can be considered. The aim of the approach is to provide a substitute method to bypass the huge consuming time required to balance the chemical system. This paper focuses on the use of artificial neural networks (ANN) to replace the geochemical equilibrium package. It is widely admitted that ANN are the most efficient response surface model due to the no linear behavior of the output again the parameters.
This paper presents a complete workflow for compositional reservoir simulation using an artificial neural network to determine the chemical equilibrium instead of solving equations system. This approach substantially reduces the computation time while keeping an accurate equilibrium calculation.
To illustrate the proposed workflow, a case study of CO2 storage in geological formation is presented. The compositional system involves 11 aqueous species, 1 mineral component, 6 chemical equilibrium reactions and 1 mineral dissolution/precipitation reaction.
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Model-Order Reduction Applied To Coupled Flow and Geomechanics
More LessSummaryCoupled flow and geomechanics computations are highly complex and require solving large nonlinear systems. Such simulations are intense for both runtime and memory standpoint, which strongly hints at employing Model-Order Reduction (MOR) techniques to speed them up. Different types of Reduced-Order Models (ROM) have been proposed to alleviate this computational burden. MOR approaches rely on projection. We first execute a computationally expensive “offline” stage, during which we carefully study the full-order model (FOM). Upon creating a ROM basis, we then perform the cheap “online” stage. Our reduction strategy estimates a ROM using Proper Orthogonal Decomposition (POD). We determine a family of solutions of the problem, for a suitable sample of input conditions, where every single realization is so-called a “snapshot.” We then ensemble all snapshots to determine a compressed subspace that spans the solution.
Usually, POD employs a fixed reduced subspace of global basis vectors. The usage of global basis is not convenient to tackle problems characterized by different physical regimes, parameter changes, or high-frequency features. Having many snapshots to capture all these variations is unfeasible, which suggests seeking adaptive approaches based on the closest regional basis. We thus develop such strategy based on local POD basis to reduce one-way coupled flow and geomechanics computations. We focus on the mechanics and consider factors such as challenging variable boundary conditions and the role of the heterogeneity. We also assess how to tackle different degrees of freedom, such as the displacements (intercalated and coupled) and pressure, with MOR. It seems to be convenient to treat DOF separately since this reduces the rank of the snapshot matrix. Preliminary 3-D results show significant compression ratios up to 90% for the mechanics part. We formally compare FOM and ROM and provide time data to demonstrate the speedup of the procedure. Examples focus on linear and nonlinear poroelasticity. We employ continuous Galerkin finite elements for most of the discretizations.
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Data-Driven Modeling Of Fractured Shale Reservoirs
More LessSummaryPerforming robust modeling and forecasting is an overarching challenge for unconventional reservoirs. Due to the lack of efficient and dependable physical models for adequately describing fluid/rock interactions on fractured geometries, there has been an increasing interest in seeking alternative solutions via data-driven models. Despite a few encouraging outcomes reported in the literature, off-the-shelf data-driven models may not be able to generalize well in realistic reservoir scenarios.
In this work, we strive to emulate first-order flow dynamics with data-driven models that have recently emerged in model reduction and machine learning. We rely on the assumption that complex flows on fractured systems can be decomposed into a simple representation based on coherent spatiotemporal structures. When field and simulation data are both integrated with the proposed approach, it is possible to extract additional patterns that enhance our capabilities for understanding predictions on different unconventional reservoir systems. We implement a single-phase flow model on structured curvilinear grids to capture first-order physics associated with unconventional shale production dynamics.
Latin hypercube sampling is carried out to represent a different number of fractures (stages), fractures length and, geological uncertainty across distinct field scenarios. The data-driven model consists of the application of the recently proposed Dynamic Mode Decomposition (DMD) approach for modeling the evolution of pressure field and Long Short-Term Memory (LSTM) network, a powerful class of Recurrent Neural Network (RNN), to track gas production consistently and accurately.
Our experiments show that our approach is accurate for a relatively small number of samples and reflects the relevant dynamics determining the production. Our model may not be as practical as empirical models employed in decline curve analysis, but it offers the potential to be more reliable as it can be based on complex simulations and field data. Numerical results support the accuracy of our approach with the possibility to impact forecast, reserves estimation and economics studies of unconventional assets in much shorter turnarounds.
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Parametric Model Order Reduction For Adaptive Basis Selection Using Machine Learning Techniques During Well Location Opt
Authors H. Zalavadia and E. GildinSummaryModel Order Reduction (MOR) aims at accelerating the numerical simulations for solving large-scale dynamical systems that inherently are computationally expensive while preserving the accuracy of the underlying solution. The objective of this paper is to develop a Parametric Model Order Reduction technique for well location optimization problem that requires a large number of high fidelity simulation runs. Reduced order modeling methodologies for well control optimization have reached a good level of maturity, however, MOR for well placement optimization, required during the closed loop field development plan, is unexplored. MOR for well control optimization requires excitation of inputs to train the model, not drastically different from the test schedule, with fixed well configuration (locations). This reduced model is generally not robust to change in the well location, which we consider here as the system parameter.
This calls for Parametric Model Order Reduction (PMOR) where the goal is to generate reduced order models that characterize the system for different parameters i.e. well locations. We propose local parametric reduced order models for new parameters using a Machine Learning (ML) framework that prove to be more accurate than the global methods using snapshots/basis concatenation. Projection based PMOR using Proper Orthogonal Decomposition is implemented here. In this work, we use Artificial Neural Network and Random Forest to predict the surrogate model error at a new well location with previously computed reduced models that help us choose appropriate basis. We also propose qualitative prediction strategy that may be more useful than exact error prediction for well location optimization. To train the neural network, input features are efficiently selected to represent the change in well location which also includes basis dimensions to account for optimal basis dimension at new parameters. The information from snapshots concatenation is added while training the ML models considering the fact that it is a better choice for some parameters.
This methodology applied to a synthetic channelized reservoir demonstrate the workflow to be accurate to obtain a good reduced model for new well locations. Regularized Neural Network show better prediction performance than Random Forest for the cases shown. We show that the entire system can be represented by already existing ROMs from very few well locations. Machine Learning provide promising results to approximate the characteristics of the parametric reduced order models.
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Channelized Reservoir Estimation Using A Low Dimensional Parameterization Based On High Order Singular Value Decomposition
Authors B. Sebacher and R.G. HaneaSummaryPrior to any estimation process of channelized reservoirs, in the context of an Assisted History Matching method, the parameterization of facies fields is a necessary task. The parameterization of channelized reservoirs consists of defining a numerical field (parameter field) so that a projection function recovers the facies field from the parameter field. Mostly, the dimension of parameter field is equal to the dimension of reservoir domain. The issue of dimensionality is becoming relevant when the history matching method is applied, especially due to the tremendous number of parameters involved in the estimation process of the channelized reservoirs. In addition, one of the most important issue encountered is the loss of the multi-point geostatistical properties in the updates (channel continuity). In this study, we start from an initial parameterization of the channelized fields and infer from it a low-dimensional parameterization obtained after a high order singular value decomposition of a tensor built with the parameter fields. We show how the facies fields are fully characterized by a linear combination of a small number of coefficients with “basis functions”. The decomposition is followed by a truncation so that we keep the relevant information from the channel continuity perspective. This new parameterization is further introduced in the estimation process of facies fields, using the ensemble smoother with multiple data assimilation (ES-MDA), updating the coefficients of decomposition. For a fair assessment of the parameterization, we perform a comparison of the results with those obtained by applying the traditional singular value decomposition and the original parameterization. The comparison is done from the perspective of multipoint geostatistical characteristics of the updates and predictions (oil and water rates). We show that the new parameterization is able to better keep the multipoint geostatistical structure in the updates than the other two parameterizations, while the prediction capabilities are the same.
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Multi-Model Hybrid Compositional Simulator With Application To Segregated Flow
Authors O. Møyner, A. Andersen and H.M. NilsenSummaryIn cases with significant density difference between the injection mixture and the reservoir fluids, segregation can take place on short temporal scales relative to the typical time-step lengths of the simulation. A fully resolved 3D description is computationally demanding and is often difficult to achieve with buoyancy segregated flow. Vertical Equilibrium (VE) is one possible technique for effective upscaling in regions of gravity segregation and has been widely considered for CO2 storage applications. VE has connections to pseudo-relative permeability models for gas injection. In this work, we use a general pseudo-relative permeability model to couple conventional 3D and upscaled models, including VE formulations with support for compositional simulations. This gives a flexible framework where different choices of coarsening and pseudoization can be used locally throughout the domain, allowing for an optimal trade-off between runtime and accuracy. The new approach is demonstrated within the framework of a fully implicit compositional flow simulator with nonlinear equation-of-state, which provides a robust and stable base for inclusion of additional physical effects.
Possible applications for this methodology include gas injection for enhanced oil recovery, CO2 storage in aquifers, or other gas-storage scenarios. We demonstrate the approach for gas injection and migration on both conceptual unstructured grids as well as corner-point models taken from real fields and saline aquifers from the Norwegian Continental Shelf.
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Direct Pore-Scale Modelling Of Dissolution And Trapping Of Supercritical CO2 In Reservoir Brine
Authors J. Maes and C. SoulaineSummaryDissolution trapping is an important mechanism for the long term security of geological CO2 capture and storage in saline aquifers. When modelling this process, dissolution of CO2 in the surrounding brine is often assumed to be instantaneous with equilibrium phase partitioning. However, recent experiment in sandstone core samples have shown the importance of pore-scale concentration gradient. Therefore, investigating and upscaling CO2 dissolution at the pore-scale is critical to better constrain macro-scale models.
In this work, we present a novel compressible two-phase multicomponent pore-scale model based on Direct Numerical Simulation of the Navier-Stokes equations using the Volume-Of-Fluid method. Mass transfer across fluid interfaces is accounted for using the Continuous Species Transfer method and the resulting phase change is computed and injected within the phase distribution equation. The model is validated by comparison with analytical solutions of simple set-ups. Then, the approach is used to simulate and upscale CO2 gas dissolution and trapping into the surrounding reservoir brine in realistic 2D porous media.
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CO2 Convection In Oil Driven By Non-Monotonic Mixture Density
Authors S. Gasda and M.T. EleniusSummaryCarbon capture and storage with utilization in enhanced oil recovery (EOR), or CCUS, is perceived as the most cost-effective method of disposing captured carbon dioxide (CO2) emissions. CO2 injection for EOR has been performed for many decades with a focus on hydrocarbon recovery. However with CCUS, two new factors will bring additional challenges to a well-established industry. First, CO2 storage will be increasingly emphasized in order to meet greenhouse gas emissions targets, requiring operators to quantify the fate of CO2 in the reservoir. Second, CO2-EOR will be introduced into the offshore environment, where economic constraints may restrict the number and placement of wells thus impacting flow regimes in the reservoir. In both cases, the interaction between CO2 and hydrocarbons at the fine scale plays an important role, and therefore detailed understanding is required for effectively managing CO2 migration and storage efficiency in CO2-EOR reservoirs, or any storage reservoirs with existing hydrocarbons.
Fine-scale interactions between CO2 and hydrocarbons occur in zones where fluids mix in a fully miscible setting. The mixing process occurs at the sub-centimeter-scale and involves complex convective-diffusive processes. Most oils exhibit non-monotonic change in density when mixed with CO2, which leads to density instabilities in the mixing zone. These density differences can play a significant role in parts of the reservoir that are gravity dominated, either by design or constraint. One key factor is that density instabilities may occur that drive convective mixing and impact the flow behavior of mixed fluids in these regimes.
In this paper, we investigate gravity-driven mixing for different CO2-hydrocarbon mixtures using a highly accurate computational model. The simulation results are used to characterize the fine-scale behavior of convective mixing. We consider reservoir conditions similar to the current Snøhvit CCS project in offshore Norway (approximately 20 MPa and 80 C). In that storage project, CO2 is injected into the water leg of a gas-producing field, just below a 10-m oil layer. If CO2 migrates into the gas plume through the oil, there is a risk of contamination at the producing wells. Due to the resolution required to correctly capture convective cells, it is not possible to simulate the entire reservoir. Therefore, we consider a simplified scenario where CO2 mixes with oil from above or below given a completely static initial condition. The results from this study can help provide insight into the potential impact of non-monotonic mixture density for CO2 migration in hydrocarbon reservoirs.
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Optimization Of Co2 Injection Using Multi-Scale Reconstruction Of Compositional Transport
More LessSummaryThe current situation with green gas emission requires the development of low carbon energy solutions. However, a significant part of the modern energy industry still relies on fossil fuels. To combine these two contradictory targets, we investigate a strategy based on a combination of CO2 sequestration with Enhanced Oil Recovery (EOR) in the hydrocarbon reservoirs. In such technology, the development of miscibility is the most attractive strategy from both technological and economic aspects. Modeling of this process involves solving complex nonlinear problem describing compositional flow and transport in highly heterogeneous porous media. An accurate capture of the miscibility development usually requires an extensive number of components to be present in the compositional problem which makes simulation run-time prohibitive for optimization. Here, we apply a multi-scale reconstructing of compositional transport to the optimization of CO2 injection. In this approach, a prolongation operator, based on the parametrization of injection and production tie-lines, is constructed following the fractional flow theory. This operator is tabulated as a function of pressure and pseudo-composition which then is used in the Operator-Based Linearization (OBL) framework for simulation. As a result, a pseudo two-component solution of the multidimensional problem will match the position of trailing and leading shocks of the original problem which helps to accurately predict phase distribution. The reconstructed multicomponent solution can be used then as an effective proxy-model mimicking the behavior of the original multicomponent system. Next, we use this proxy-model in the optimization procedure which helps to improve the performance of the process in several folds. An additional benefit of the proposed methodology is based on the fact that important technological features of CO2 injection process can be captured with lower degrees of freedom which makes the optimization solution more feasible.
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Fast Flow Computation Methods On Unstructured Tetrahedral Meshes For Rapid Reservoir Modelling
Authors Z. Zhang, S. Geiger, M. Rood, C. Jacquemyn, M. Jackson, G. Hampson and F.M. De CalvalhoSummaryHydrocarbon reservoir models have a high degree of uncertainty regarding their reservoir geometry and structure. A range of conceptual models should therefore be generated to explore how first-order uncertainties impact fluids-in-place, reservoir dynamics, and development decisions. However, it is very time consuming to generate and explore a large number of conceptual models using conventional reservoir modelling and simulation workflows. Key reservoir concepts are therefore often locked in early and are difficult to change later.
To overcome this challenge, the Rapid Reservoir Modelling (RRM) software has been developed to prototype reservoir models across scales and test their dynamic behaviour. RRM complements existing workflows in that conceptual models can be prototyped, explored, compared, and ranked rapidly prior to detailed reservoir modelling. Reservoir geology is sketched in 2D with geological operators and translated in real-time into geologically correct 3D models.
Flow diagnostics provide quantitative information for these reservoir model prototypes about their static and dynamic behaviours.
Numerical well testing (NWT) is implemented to further interrogate the reservoir model.
The combination of surface-based reservoir modelling with geological operators, flow diagnostics and NWT on unstructured grids enable, for the first time, rapid prototyping of reservoir geologies with real-time feedback on fluid flow behaviour.
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Numerical Investigation Of Non-Homogenous Wettability Alteration In Naturally Fractured Rocks
Authors M.S. Sedaghat and S.A. AzizmohammadiSummaryWettability alteration is one of the most promising techniques to enhance oil recovery of oil-wet reservoirs. As contact angle decreases within the matrix, the flux of fracture-matrix counter-current imbibition increases significantly (Sedaghat et al., ECMOR 2016). However, wettability altering agent only influences the wettability of a portion of the matrix region, and not necessarily uniformly. The agent (water phase) makes the fractures and surrounding matrix water-wet, but this alteration is gradually reduced with the distance from the fracture-matrix interface. Beyond the fracture-matrix imbibition halo, wettability (contact angle) does not change anymore. So, this wettability transition zone needs to be considered when simulation of multiphase flow in naturally fractured rocks is performed.
Utilizing a Finite-Element-Centered-Finite-Volume (FECFV) numerical approach, we simulated a waterflooding scenario on a discrete fracture and matrix (DFM) model built based on an outcrop analogue. First, a wettability altering agent is flooded into the system until the change in the saturation gets negligible. The system is then equilibrated in a given time period. By weighting contact angle with wetting fluid saturation, wettability is updated for each element at the equilibrium state. Finally, waterflooding is performed with the updated wettability. Assuming that the wettability altering agent has 100% performance, it does not influence water properties and requires a considerable time to change the wettability. Therefore, reactive transport is left out of the computations.
Compare to the results of base case model in which wettability transition zone was neglected (Sedaghat et al., ECMOR 2016), partial wettability alteration associated with a non-homogenous distribution of contact angle significantly influences the oil recovery, fracture-matrix counter-current imbibition, and ensemble relative permeability. Moreover, velocity fluxes and flow behaviour is influenced dramatically as it leads to a heterogeneous distribution of capillary pressure over the matrix within the fracture-matrix counter-current imbibition halos.
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Analytical solution to steady state, viscosity-gravity driven horizontal flow
Authors A. Rabinovich, P. Bedrikovetsky and D.M. TartakovskySummaryReservoir injection of low density fluids often results in phase segregation at some distance from the well. Similarly, gravity effects can be observed in coreflooding experiments, e.g., drainage by N2 or CO2. It is important to model these processes as they have substantial impact, for example, on recovery factor and history matching results. We formulate equations for steady state immiscible two-phase flow where capillary forces are negligible while gravity and viscous effects are significant. Both phases are injected simultaneously at the inlet boundary with a given fractional flow.
A solution is derived using the method of characteristics allowing to predict pressure and saturation in two-dimensional space. It consists of linear boundaries separating three regions of constant saturation describing the transition between a mixed wetting/nonwetting zone and a segregated zone, in which the lighter phase is above the heavier. The solution is compared to numerical simulations and to an analytical formula presented by Stone [1982] and Jenkins [1984] . It is found that when segregation occurs relatively near the inlet (e.g., for low flow rates) the new solution is more accurate than the existing formula, while otherwise its accuracy is reduced.
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Quantitative Production Analysis and EUR Prediction From Unconventional Reservoirs Using a Data-Driven Drainage Volume Formulation
More LessSummary1. We developed a general analytic solution for bounded transient flow based upon an asymptotic pressure approximation which reduces to Fetkovich’s work on bounded radial flow and Wattenbarger’s work on bounded linear flow in those specific geometries. The methodology offers similar abilities for analysis without the restriction to specific flow regimes.
2. We illustrate how to use the asymptotic solution for EUR estimation. Our approach works for infinite acting transient flow and through the transition to boundary dominated flow. It is validated by synthetic models and conventional simulation.
3. We illustrated the improved methods for both drainage volume calculation as well as w(τ) inversion which shows more detailed features. The drainage volume calculation is no longer based on global curve fitting. The proposed approach improves numerical inversion for w(τ).
4. We improved quantitative interpretation from production data. The interpretation of matrix permeability is based on fracture cluster interference seen in the w(τ) high resolution diagnostic plot and is validated by buildup analysis. EUR is also predicted.
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A New Multiscale Computational Model For Coupled Flow And Geomechanics In Karst-Carbonates Containing Solution-Collapse
Authors T.V. Lopes, E.L.M. Garcia, M.A. Murad, P.A. Pereira, A.C. Rocha and C.L. CazarinSummaryWe developed a new three-scale (micro/meso/macro) computational model based on a reiterated homogenization procedure to describe coupled flow and geomechanics in carbonate rocks containing complex geological structures at multiple length scales, such as fractures and mainly solution-collapse breccias. Within this framework we construct a Hierarchical Karst-Fracture Model wherein larger geological objects are incorporated in the mesoscopic model explicitly, whereas the high density smaller structures, which prevail microcopically, are homogenized and replaced by equivalent continua with hydromechanical properties computed using a self-consistent homogenization procedure. Using the terminology adopted in the cave-collapse literature, in the upscaling method we subdivide the different clastic-arrangements in the breccia into crackle, mosaic, totally open and chaotic substructures, where equivalent properties, such as permeability and elastic constants, are assigned to each layer of the breccia.
After establishing the mesoscopic coefficients we upscale the problem to the macroscale, characterized by a length of order of hundreds of meters, associated with a representative cell of a coarse grid of a reservoir simulator. In this latter procedure we apply the Discrete Fracture
Model (DFM) combined with robust computationally schemes with ability to handle strong heterogeneity induced by the collapse-breccia.
In this setting equivalent properties are computed through a straightforward averaging process. Numerical results, hinging on outcrop-based flow simulations, are presented with the input small-scale features extracted from drone images. In particular our simulations aim at analyzing the influence of different scenarios on the magnitude of the macroscopic properties which are subsequently explored as pre-processors in reservoir simulators.
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Overview Of The Olympus Field Development Optimization Challenge
Authors R.M. Fonseca, E. Della Rossa, A.A. Emerick, R.G. Hanea and J.D. JansenSummarySince the early 2000’s there has been a significant focus from many groups around the world towards the development and application of innovative technologies in order to improve reservoir management strategies and optimize field development plans. Benchmark studies are a very valuable way of evaluating and demonstrating the status and potential of developing technology. Numerical optimization is seen as a valuable technology for decision support in various stages of the life cycle of hydrocarbon fields. Its potential has been demonstrated in previous benchmark studies such as the 2008 Brugge study on Closed-Loop Reservoir Management albeit for primarily well control problems. Additionally since the Brugge benchmark exercise also involved history matching it was difficult to separate and thus draw significant conclusions about the performance of the optimization methods. Thus the OLYMPUS optimization benchmark challenge was setup and aimed at field development (FD) optimization under uncertainty. In this talk we will provide an overview of the OLYMPUS case and the optimization problems defined. In addition we aim to provide an anonymized overview of validated results from the participants for the OLYMPUS workshop which takes place the day after ECMOR.
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Development Of Efficient Constraint-Handling Approaches For Well Placement Optimization
Authors M. Bellout and O. VolkovSummaryEfficient constraint-handling methodology is developed to solve for a set of concurrent geometric well placement constraints. This implementation enhances constraint-handling capability such that expert knowledge may more easily be incorporated into well placement problem formulations through realistic geology- and engineering-based nonlinear constraints. A well-defined collection of constraint definitions may, besides enforcing minimum feasibility, also make the optimization process more efficient by limiting the search to highly-relevant solution spaces. This is particularly important for well placement problems that commonly rely on time-consuming reservoir simulations for objective function evaluation. Constraints are imposed on parameters determining the configuration of multiple deviated wellbores in reservoir space, e.g., well length and inter-well distance.
A constraint-handling repair approach based on an alternating projections methodology that solves each restriction as an independent constraint-handling subproblem is implemented. The subproblems are solved in sequence, i.e., the solution from one subproblem is used as the initial point for solving the next feasibility problem. The entire sequence is performed in a loop until feasibility is achieved for all constraints. Results from two optimization procedures that implement algorithms with very distinct search characteristics are presented. Though the repair method is external to the sequential quadratic programming and differential evolution algorithms implemented, this work provides a practical framework for how to adapt and couple the constraint-handling methodology to these different types of algorithms in an efficient manner. Results show the standalone optimization procedures provide feasible solutions while performing effective searches of the solution space, both in terms of cost function evolution growth and progression of well configuration.
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Oscillation Mitigation In Subsurface And Surface Couplings Using PID Controllers
Authors K.G.R. Gurjao, E. Gildin, D. Schiozer and J.C. HohendorffSummarySimulation of coupled subsurface (reservoir) and surface (network) systems is a challenging problem and can become a daunting task if one considers computationally intensive multi-reservoir models and realistic surface network facilities. Accurate production forecast is especially important in long-term field development plans. Integration of production systems, including reservoir, wellbore and surface facilities can be done using separate simulators (explicit) or in a seamless fashion by creating a large scale model (fully implicit) that can take into account all of the individual components in a single software. Unlike the implicit formulation, the explicit method is very flexible, allowing the integration of commercial-of-the-shelf simulators. However, as a drawback, it can yield inaccurate and oscillatory solutions. In this work, a new framework for mitigating explicit coupling instabilities (oscillations) is developed by recasting the problem in a control setting. Results from this work allow fast turn arounds in large-scale simulation of coupled surface-subsurface models.
Explicit coupling can present error and consequently oscillation that can grow unmanageably throughout the simulation, because the IPR curve and operating point flow rate (q_OP) exchanged at the beginning of a time step between reservoir simulator and coupling program, may not be representative for the entire coupling interval. In order to mitigate the numerical oscillations, a feedback control system, namely a PID (i.e., proportional, integral and derivative) controller is applied. The PID controller, with parameters (K_C,τ_I,τ_D) tuned manually for a group of well settings, adjusts the IPR curves generated by the reservoir simulator so that the error between the bottom-hole pressure calculated by the reservoir simulator (BHP_RS) and the bottom-hole pressure obtained in the operating point (BHP_OP) is minimal. In this case, a corrected value of the operating flow rate (q_OP) is obtained.
The new methodology was tested in a synthetic numerical model (UNISIM-I-D) based on Namorado field (Campos Basin - Brazil), which is comprised by 36,739 active cells and 20 satellite wells (7 injectors and 13 producers). The results indicate that the PID control indeed reduce the rate and pressure oscillations as expected by a more theoretical control point of view, and outperforms the base scenario, which represents the network system of producer wells by proper pressure drop tables.
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Multiple Constraints In An Oilfield Reservoir Simulator Wellbore Model
Authors T.S. Stone and T.J. JonsthovelSummaryPetroleum recovery from oilfield assets increasingly involves wells that are very long in extent, have multiple branches and also multiple control points in order to prevent breakthrough of unwanted fluids and/or to optimize recovery. Instead of simply controlling rates at the wellhead, downhole devices are now available where, either autonomously or through surface intervention, apertures and other controlling parameters can be set. Various control points in a single wellbore includes numerous flow paths that requires technology to effectively control all local constraints at various measured depths. This paper examines the relation between primary control at the well head and local downhole flow control devices. In particular, unique solutions of all constraints require algorithms that may include feedback control, slack variables and other constructs. These methods are analysed and comparisons are presented. Algorithms and methods in this paper have been implemented and verified in a reservoir simulator but may be applicable to real time oilfield operations and also to other systems with similar multiple constraints.
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Application Of Diffuse Source Basis Functions To Multiscale Simulation
More LessSummary- We extend the diffuse source pressure transient upscaling approach to a multiscale framework where large scale changes in pressure are resolved on the coarse scale, and multiphase fluid transport simulation is performed on the fine scale using a subgrid velocity field generated from the coarse problem. This precludes the need to upscale saturations and relative permeability which are highly non-linear and strongly dependent upon flow history. This approach is similar to the multiscale mixed finite element literature where we have a basis function for each coarse face.
- The formulation enables us to capture the subgrid heterogeneity and local connectivity by distinguishing between the weakly connected and well connected volumes from each coarse face.
- The use of diffuse source basis functions also allows us to explore the localization effect and test if the local upscaling calculation is able to capture the global flow field without introducing any bias.
- We also extend previously proposed time of flight flow diagnostics from incompressible flow to slightly compressible flow.
- The approach is tested on the SPE10 synthetic reservoir model.
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Use Of Dynamically Adapted Basis Functions To Accelerate Multiscale Simulation Of Complex Geomodels
Authors Ø.S Klemetsdal, O Møyner and K.-A LieSummaryA number of different multiscale methods have been developed as a robust alternative to upscaling and as a means for accelerated reservoir simulation of high-resolution geomodels. In their basic setup, multiscale methods use a restriction operator to construct a reduced system of flow equations on a coarser grid, and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. The prolongation operator consists of basis functions computed numerically by solving localized flow problems. The resulting multiscale solver can both be used as a CPR-preconditioner in fully implicit simulators or as an efficient approximate iterative linear solver in a sequential setting; successful implementation of the latter approach in a commercial simulator was reported at ECMOR XV. Recently, it has been shown that significantly faster convergence is observed if one instead of using a single pair of prolongation-restriction operators, applies a sequence of such operators, where some of the operators are adapted to faults, fractures, facies, or other geobodies. Herein, we present how the convergence can be accelerated even further, if we also include additional basis functions that capture local changes in the pressure.
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Multiscale Finite Volume Method For Finite-Volume-Based Poromechanics Simulations
Authors I.V. Sokolova and H. HajibeygiSummaryAccurate reservoir simulations that quantify mutual effect of reservoir flow and mechanical deformation demand for large-scale heterogeneous computational models: while fluid flow occurs inside heterogeneous reservoirs, stress and deformation fields span the entire geological domain. A conservative nature of mass and momentum balance governing equations motivates a locally conservative representation of the unknowns in discrete space. Accurate simulation of these large-scale heterogeneous coupled phenomena with sufficient resolutions remains computationally challenging for state-of-the-art simulators.
To resolve this challenge, we develop the first multiscale finite volume (FV) method for elastic poromechanics model, where the displacement-pressure system is solved fully implicitly using finite volume method. This finite volume discrete fine-scale system is obtained based on the Biot’s theory. Independent coarse grids for flow and deformation are imposed on this fine-scale computational domain, allowing for targeting larger domains for mechanical deformation simulations. Fully implicit coarse-scale quantities are obtained via sets of local basis functions, for both flow and deformation unknowns. These basis functions are calculated once at the beginning of the simulation, and are re-employed to construct and solve the coarse-scale systems during the entire simulation. The coarse-scale solution is interpolated to the fine scale using the same basis functions. This method provides stable (fully implicit), efficient (multiscale), and locally conservative (FV for all unknowns) solution for the coupled flow-deformation system of equations. We study several test cases, including benchmarking ones, to illustrate consistency, order of accuracy, convergence, and applicability of our method. Importantly, we show that our multiscale method allows for quantification of the geomechanical behavior with using only a fraction of the fine-scale grid cells, even for highly heterogeneous time-dependent models.
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A Multi-Level Algebraic Multiscale Solver (ML-AMS) For Reservoir Simulation
More LessSummaryThe Algebraic Multiscale Solver (AMS) is very efficient in handling the highly heterogeneous pressure system that arises from incompressible flow in porous media ( Wang et al., 2014 ). The standard AMS method employs a single auxiliary coarse level on which the original problem can be projected and solved efficiently, given the huge reduction in its original ‘fine’ size. The coarse solution is then projected back to the fine level and its quality can be further enhanced by iteratively applying additional AMS passes until ultimately converging to the exact solution ( Zhou and Tchelepi, 2012 ; Wang et al., 2014 ). However, the efficiency of AMS drops significantly when the problem size becomes extremely large, as the size of the coarse-scale representation of the problem, and, in turn, its solution cost, becomes inevitably large. Thus, in this work, we propose an extension of the standard two-level AMS method: a Multi-Level AMS (ML-AMS) method, which continues to use additional coarser levels recursively as needed to solve the problem more efficiently. The performance of ML-AMS is demonstrated and compared with the standard two-level AMS using 2D and 3D heterogeneous problems derived from the SPE10 benchmark ( Christie et al., 2001 ). For all test cases, ML-AMS is shown to have comparable performance with the two-level AMS, while avoiding the expensive cost of handling the coarse-scale problem with a direct solution method. Since ML-AMS applies the original two-level AMS algorithm recursively, any existing multiscale implementation can be easily extended to use ML-AMS.
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Algebraic Dynamic Multilevel (ADM) Method For Simulations Of Multiphase Flow With An Adaptive Saturation Interpolator
Authors M. Cusini and H. HajibeygiSummaryAn Algebraic Dynamic Multilevel (ADM) method for simulations of multiphase flow in heterogeneous porous media with an adaptive enriched multiscale formulation for saturation unknowns is presented. ADM maps the fine-scale fully-implicit (FIM) discrete system of equations to a dynamic multilevel system, the resolution of which is defined based on the location of the fluid fronts. The map between the dynamic multilevel resolutions is performed algebraically by sequences of restriction and prolongation operators. While finite-volume restriction operators are necessary to ensure mass conservation at all levels, different interpolation strategies can be considered for each main unknown (e.g., pressure and saturation). For pressure, the multiscale basis functions are used to accurately capture the effect of fine-scale heterogeneities at all levels. In previous works, all other unknowns (e.g., saturation) were interpolated with piece-wise constant functions. Hence, the multiscale nature of saturation equation was not fully exploited. Here, an adaptive interpolation strategy, thus a multiscale transport formulation, is employed for the saturation unknowns that allows to preserve most details of the fine-scale saturation distribution even in regions where a coarser resolution is employed. In regions where the ratio between the coarse and the fine-scale saturation updates is detected to be constant throughout the time-dependent simulation, such ratio is stored and employed as interpolator for subsequent time-steps in which a coarser resolution is employed. Numerical results are presented to study the accuracy and efficiency of the method and the advantages of such interpolation strategy for test cases including challenging non-linear physics, i.e. gravitational and capillary effects.
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Numerical Assessment Of Water Alternating Gas Practices In The Presence Of Hysteresis Effects On Relative Permeability
Authors E. Ranaee, F. Inzoli, M. Riva, G. Maddinelli, A. Cominelli and A. GuadagniniSummaryWater Alternating Gas (WAG) injection is one of the most successful enhanced oil recovery approaches. Properly accounting for the hysteretic effects of relative permeabilities is a critical issue encountered in numerical simulations of WAG at the mesoscale. Ranaee et al. (2015) proposed a sigmoid-based model for three-phase oil relative permeability, incorporating key physical effects taking place at the pore scale. The model can then be jointly used with the Larsen and Skauge (1998) model, accounting for gas relative permeability hysteresis, to develop a formulation for three-phase relative permeability suitable for reservoir simulation.
In this study we illustrate the impact of this joint formulation on a field scale setting through a suite of numerical simulations of WAG injection targeting a reservoir model inspired to real life cases. The analysis is performed by embedding the illustrated relative permeability models in the black oil model implemented in the Matlab Reservoir Simulation Toolbox ( Lie et al., 2011 ). We assume non-hysteretic behavior for water relative permeability under water-wet conditions and characterize it upon relying on corresponding laboratory-scale data. As a baseline, the results are compared against a scenario in the absence of three-phase relative permeability hysteresis.
The computational domain is heterogeneous, the spatial distributions of porosity and absolute permeability varying across the ranges of [0.02 – 0.3] and [0.1 – 2600 mD], respectively. The model is set at equilibrium conditions, production being driven by three peripheral injectors and five up-dip producers. A given flow rate is assigned to each injector and a target value of liquid production rate is imposed at the producing wells. The numerically evaluated production rates constitute our target state variables. The schedule of the injectors is set to achieve a preliminary waterflooding phase followed by a WAG injection scheme. The latter is implemented by periodically switching the injected phase between water and gas for two injectors, the third injector continuously injecting water. The numerical simulations are performed through a fully implicit discretization of the equations governing the system dynamics. To minimize computational costs, we employ an algebraic multi-grid method and resort to a multi-processor high performance clustered computer system. Our results suggest that hysteretic effects are important across significant portions of the studied reservoir system. Field production responses are associated with a simultaneous increase of ultimate oil recovery and a corresponding decline of the gas-oil ratio when hysteretic effects are included in the simulations.
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Simulation Of Acid Fracturing Including Acid Propagation In Formation
Authors R.D. Kanevskaya and A.V. NovikovSummaryAcid fracturing is a quite cheap and easy-to-use stimulation technique in carbonates. It differs from proppant fracturing in part of setting up a high conductivity channel. In the case of acid fracturing it is influenced by two processes: acid transfer along the fracture and dissolution of carbonate minerals in porous medium. The first process is critical for effective length of the channel, the second one regulates the initial channel conductivity. Conductivity prediction is one of the main issue of acid fracturing simulation. Modern simulators often assume infinite reaction rate at fracture surfaces while evaluation of acid fracture conductivity is based on various empirical correlations.
In this study, we propose a model assumes strict evaluation of final porosity and permeability distributions in the affected area. The model includes calculation of two-phase multicomponent flow in porous media taking into account kinetics of acid-mineral interactions, porosity and absolute permeability changes, relative permeabilities modification. The water phase can contain acid, water, salt and carbon dioxide components. It is supposed that the last one is instantaneously dissolved in the water phase. Kozeny-Carman equation is used for an absolute permeability calculation but any other expression can be easily embedded. This model coupled with acid transfer model along the fracture that includes convection-diffusion problem, uses analytical expressions for velocity profile in the fracture and prescribed relation between fracture width and net pressure. One-dimensional and two-dimensional cases were considered.
Such a model allows rather fast calculation of the size, porosity and permeability of affected area in order to estimate its effective parameters and use them in full-field flow simulation. It also provides an opportunity for various optimization and design of acid fracturing based on solution of mass balance equations in porous medium and fracture. Calculations were performed for a number of cases.
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A Robust Mathematical Model For Heavy-Oil Well Stimulations Using Nanofluids: Modelling, Simulation And Validation At La
More LessSummaryRecent pilot studies in Colombian reservoirs have demonstrated the benefits of nanotechnology at improving heavy-oil mobility. Understanding the physical and chemical processes that occur in the reservoir during the injection/production of nanofluid is essential for the design of the stimulation operations. We developed a novel mathematical model for studying the behavior of nanoparticles injected into heavy-oil reservoirs. Adsorption of asphaltenes on the surface of the nanoparticle and their effect on the viscosity of crude oil is accounted for in the model formulation. In addition, nanoparticles are retained/mobilized as operating condition changes. The transport and retention models are coupled to a multiphase/multicomponent model. The finite volume method is used to solve the differential equations of the model. The resulting equations are solved using the Newton-Raphson method. Core-flooding tests were used to calibrate the model parameters. The model was used for simulating the deployment of nanofluids in oil wells.
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Coupling Vertex Centered Based Flow Elements With Poromechanical Finite Elements Using Unstructured Grids
More LessSummarySignificant work has been done for generating unstructured grids. Coupled geomechanics simulation and hydraulic fracture flow for gas shale simulation have given a new impulse for unstructured gridding.
The objective is coupling flow and geomechanics using unstructured grid models, demonstrate the ability to apply a more efficient pressure coupling using discretization at vertices on both sides (mechanical and flow equations).
Coupled equations are discretized and solved on an unstructured flow grid and a geomechanical finite element grid which are composed of various types of elements such as tetrahedrons and hexahedrons. On the flow side, a recent multi-point flux method, the Vertex Approximate Gradient (VAG) (Eymard 2012) is investigated for solving the reservoir equations on such unstructured grid. SPEJ paper 173309 (Samier, Masson Apr. 2017) presented the implementation of VAG scheme inside a next generation reservoir simulator designed for handling unstructured grids. This paper proposes an iterative coupling scheme with full pressure coupling at vertices. The geo-mechanics equations fully coupled to a single phase flow are solved using global pressure.
Then the resulting deformations are iteratively coupled to the multi-phase flow simulator.
Since most geo-mechanic simulators propose fully coupled single phase flow features, the main advantage of this method is the ability to use a full pressure coupling method with industrial simulators.
The convergence of this new scheme is discussed and results are presented for two cases described below. The first case is a validation case used by other SPE papers. The second case is a synthetic faulted and stress sensitive black oil reservoir simulation. Faults are modeled using specific cohesive elements. Material properties are nonlinear according to a Camclay elastoplastic model. Results are compared with standard loose iterative coupling method using TPFA cell centered elements.
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Two-Way Coupling Of Flow And Geomechanics Simulations With Local Grid Refinement
Authors A. Rodriguez, J. Monteagudo and N. NietoSummaryUnconventional reservoirs can only be economically exploited via massive stimulation by means of the creation of multiple hydraulic fractures. These systems are mechanically sensitive due to the presence of the hydraulic fractures (which tend to close during depletion), as well as natural and induced shear fractures which constitute the so-called stimulated reservoir volume (SRV). Another factor that contributes to the induced mechanical deformation is the ultra-low permeability of the reservoir rocks. This induces extremely steep pressure gradients in the vicinity of the hydraulic fractures which translate into large mechanical forces, able to locally modify the in-situ stresses as well as other effects such as modification to the formation permeability. Numerical simulation of hydrocarbon recovery in this kind of reservoirs must consider two-way coupling of flow and geomechanics. Another challenge in the simulation of stimulated unconventional reservoirs relates to the localization of pressure gradients around hydraulic fractures. The most efficient way to model and simulate flow in those regions relies on the use of local grid refinement (LGR). LGR, however, poses additional challenges in the implementation of the coupling of flow and geomechanics. In this paper, we propose that this coupling can be achieved by applying a higher integration order to the mechanical equations of LGR gridblocks. Flow equations are usually discretized following a two-point approximation with cell-centered properties. Mechanical equations are solved at the quadrature points which in a Q1 approximation correspond to the corners of the blocks while Q2, Q3 and so on, add additional nodes which correspond to the corners of the LGR cells. The order of the finite element and the LGR discretization can be chosen in a consistent manner. Next, an iterative coupling scheme is used until convergence is achieved. In this paper, we implemented this idea and show that the coupled LGR formulation satisfactorily replicates the reference solutions obtained using a fine grid to represent the volume. The method is validated by comparing the simulation of several examples with analytical and fine-grid solutions.
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A New Fixed-Stress Split Scheme In Poroplastic Media Incorporating General Plastic Porosity Constitutive Theories
Authors R.O.S. da Silva, M.A.M. Murad and J.A.L.O. ObregonSummaryOne of the most challenging issues in reservoir modeling relies on the development of proper numerical schemes for coupling flow and geomechanics with ability to handle highly heterogeneous coefficients with complex spatial distributions while preserving local conservation properties and computational efficiency. In this challenging context substantial progress has been recently accomplished within the framework of sequential methods, where the fully-coupled system is partitioned into sub-problems with coupling enforced in an iterative fashion mainly through source terms in the pressure and equilibrium equations. Among the class of proposed schemes where hydrodynamics is solved first, we may highlight the fixed strain and stress split, where the former behaves conditionally stable whereas the unconditional stable fixed stress split is more efficient, since the source term involving the time-derivative of the total mean stress admits a much slower characteristic time scale compared to the other poromechanical variables. The extension of the fixed-stress split algorithm to the poroplastic scenario has been accomplished in [1] where additional nonlinearity was incorporated in the pressure equation by replacing the bulk modulus by the nonlinear elastoplastic tangent modulus. Such an immediate extension is valid provided the same form of the effective stress principle seated on the Biot-Willis coefficient, originally proposed by Biot for elastic rock skeleton, is preserved in the regime of nonlinear plastic deformations and used as the proportionality constant between plastic porosity and plastic deformation [1]. The generalization of the fixed stress split scheme to the scenario where this assumption is relaxed is still an open issue. This work aims at filling this gap. We develop a new generalized fixed stress split scheme for single phase flow in reservoirs characterized by irreversible deformations with ability to incorporate the plastic porosity concept not necessarily ruled by same the Biot-Willis parameter. This approach gives rise to additional complexity in the iterative formulation which needs to be handled by appropriate algorithms which are proposed herein. Numerical results illustrate the effects of the additional source term in the pressure equation steaming from the transient component of the total mean stress upon oil production, reservoir compaction and surface subsidence. In particular we highlight the opposite roles of the source term during primary/secondary recovery in both elastic and plastic regimes.
[1] J. Kim, H. A. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits.
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Convergence And Error Analysis Of The Undrained-Split Iterative Coupling Scheme In Heterogeneous Poro-Elastic Media
More LessSummaryRecently, the accurate coupling between subsurface flow and reservoir geomechanics received more attention and importance in both academia and industry. This stems from the fact that incorporating a geomechanics model into upstream flow simulation is critical for obtaining accurate pore-predictions, predicting wellbore instabilities, and modeling hydraulic fracturing. One of the recently introduced iterative coupling algorithms to couple flow with geomechanics is the undrained split iterative coupling algorithm [1,2]. The convergence of this scheme is established in [1] for the single rate iterative coupling algorithm, and in [2] for the multirate iterative coupling algorithm, in which the flow takes multiple finer time steps within one coarse mechanics time step. All previously established results study the convergence of the scheme in homogeneous poro-elastic media. In this work, following the approach in [4], we will extend these results to the case of heterogeneous poro-elastic media, in which each grid cell is associated with its own set of flow and mechanics parameters for both the single rate and multirate schemes. Second, following the approach in [3], we will establish a priori error estimates for the single rate case of the scheme in homogeneous poro-elastic media. In subsequent work, we will supplement our mathematical analysis with numerical results, highlighting the efficiency of the multirate undrained split iterative scheme over the single rate scheme in heterogeneous poro-elastic media. To the best of our knowledge, this is the first rigorous and complete mathematical analysis of the undrained split iterative coupling scheme in heterogeneous poro-elastic media.
[1] Mikelic, A. and Wheeler, M. F. “Convergence of iterative coupling for coupled flow and geomechanics.” Computational Geosciences, 17:455–461, 2013.
[2] Kumar, K., Almani, T., Singh, G., and Wheeler, Mary F. “Multirate Undrained Splitting for Coupled Flow and Geomechanics in Porous Media,” Springer International Publishing, 43–440, 2016.
[3] Almani, T., Kumar, K., and Wheeler, M. F., “Convergence and error analysis of fully discrete iterative coupling schemes for coupling flow with geomechanics.” Computational Geosciences, 21: 1157–1172, 2017.
[4] Almani, T., Kumar, K., and Wheeler, M. F., “Convergence Analysis of Single Rate and Multirate Fixed Stress Split Iterative Coupling Schemes in Heterogeneous Poroelastic Media,” ICES REPORT 17–23, The University of Texas at Austin, Sep. 2017.
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A Computational Investigation Of Seismic Wave Focusing As A Novel Means To Fracture Shale Reservoirs
More LessSummaryOne of the big challenges of our time is the prudent development of unconventional hydrocarbon resources. Hydraulic fracturing currently serves as the main-staple enabling technology, but unfortunately, its practice leads to low recovery efficiency and is linked to serious environmental concerns, including those associated with the disposal of the fracture flow-back water. This work studies the potential for a novel water-free alternative to hydraulic fracturing.
Comminution of consolidated porous media is a well-studied phenomenon within several contexts including mining, tunneling, and projectile impact. In comminution, rock fragmentation occurs due to the release of the local kinetic energy of shear strain rate. Recently reported efforts explore applications of this phenomenon towards the stimulation of permeability in unconventional hydrocarbon reservoirs [1]. The engineering goals are to enable permeability enhancement without the need for water, while simultaneously impacting significant volumes of rock, and maintaining well bore integrity. A direct application would introduce seismic waves from detonations or electrohydraulic pulsed arc, and the rock surrounding the source would comminute. A more desirable application would exploit seismic wave interactions in order to focus the comminution zone onto target volumes that are at a distance from the source of energy thereby avoiding unintended consequences such as a breach of the structural integrity of the wellbore.
In order to investigate the feasibility, we develop a coupled transient poromechanics and flow model. The model implements several rock failure criteria (quasistatic as well as the macroscopic shear strain rate criteria) and is applied to a laboratory-scale setting. The dimensionless parameter space is explored by conducting a sequence of simulations. The simulations show that four regimes for damage can be observed. In one regime, the rock is predicted to comminute exclusively over the target volume within the sample. Using published hydromechanical properties for shales, the computational results indicate that a sequence of detonations can be timed to produce multiple seismic waves that interact, thereby localizing comminution over target zones away from the wave sources.
Several important questions regarding the process physics are framed and an experimental study is proposed. [1] Zdeněk P. Bažant, Ferhun C. Caner, “Shale fragmentation by kinetic shear energy”, Proceedings of the National Academy of Sciences, Nov 2013, 110 (48) 19291–19294
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A Coupled XFEM-EDFM Numerical Model For Hydraulic Fracture Propagation
Authors G. Ren and R.M. YounisSummaryFracture propagation (FP) occurs in various applications including hydraulic fracturing, underground disposal of liquid waste, in-situ stress estimation, and the failure of dams due to underwater cracks. This work develops a numerical model of the hydraulic fracturing process that is widely applied to the extraction of natural gas from shale. Tensile fractures are created due to the injection of the highly pressurized viscous fluid into the brittle and quasi-brittle rock materials.
One issue associated with the modeling of the FP using unstructured fitted meshes is the necessity to remesh each time fractures move forward. The developed XFEM-EDFM scheme allows fractures to propagate without any need for remeshing. The fully coupled and fully implicit schemes ensure the stability of the current method. The J integral is applied to extract the stress intensity factors (SIF) of the mode I and II in the 2D domain. The criterion of the FP is based on the comparison of the SIF to the fracture toughness.
We investigate the fracture propagation algorithm. The hydraulic fracture propagation is also presented to demonstrate the computational capability.
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An Unstructured Dual-Grid Model For Flow In Fractured And Heterogeneous Porous Media
Authors M. Karimi-Fard and L.J. DurlofskySummaryDiscrete representations of highly heterogeneous porous media require high-resolution models, which are computationally expensive to simulate. Model reduction through upscaling is an effective way to accelerate flow simulations. Although single-grid upscaling techniques can provide accurate results for the pressure field, they may fail to capture the details of the saturation distribution when highly coarsened models are used.
One approach to address this issue is to use the coarse grid only for the pressure solution, and the original fine grid for transport solutions. Such procedures, commonly referred to as multiscale methods, have been extensively investigated in the reservoir simulation community. In this work we present a new dual-grid model that shares many similarities with existing finite-volume-based multiscale methods. Our dual-grid approach is, however, formulated as an extension of our previously developed aggregation-based upscaling procedure.
First, a coarse model is constructed for the pressure solution. The main flow parameters for this model are the transmissibilities between adjacent coarse (aggregated) cells. These are obtained using a flow-based upscaling procedure that (typically) requires two or three global fine-grid pressure solutions. The pressure fields constructed for transmissibility upscaling are used not only to evaluate the coarse transmissibility, but also to extract a fine-grid flux profile for each coarse (aggregated) interface. In the second step, fine-grid fluxes are calculated for the transport equation.
This is done locally within each coarse aggregate by solving a pressure equation with flux boundary conditions. These fluxes are determined by scaling the profile for each interface to match the coarse rate provided by the pressure solution. The overall procedure is implemented for unstructured fine and coarse grids. Examples involving two-phase flow in heterogeneous and fractured two-dimensional models are presented. Numerical results demonstrate the capabilities and flexibility of the overall methodology.
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Optimization Of In-Depth Water Diversion Using A Fully Implicit Thermal-Compositional Approach
Authors M. Sangers, R. Trujillo, D. Voskov and O. LeeuwenburghSummaryWe investigate the potential for improved recovery of subsurface energy resources (hydrocarbons or heat) through in-depth diversion technology. A number of pilot studies in the North Sea have demonstrated in recent years that sodium silicate can be used to block preferential flow paths and divert water to previously unswept areas of a reservoir. Accompanying simulation studies based on an explicit weak coupling of a reservoir flow simulator and an external chemical module have attempted to replicate the observed behaviour. Since the development of silicate gels and the accompanying permeability reduction is essentially a coupled flow-chemical process, we first will present a fully implicit compositional-reactive flow and transport implementation and investigate the impact of the grid and time-stepping resolution on simulation performance in 2D subsurface reservoirs mimicking petroleum and geothermal applications. We proceed to investigate the sensitivity of the recovery to design parameters of the in-depth diversion strategy. Since adjoint gradients are not typically available for these parameters and uncertainties associated with an application of in-depth divergence are large, we use an ensemble-based methodology to perform an optimization study. This study aims to find optimal strategies for combined waterflooding and design of in-depth diversion under geological uncertainty. It is demonstrated that in-depth diversion can significantly extend the life-time of hydrocarbon or geothermal fields when the timing of injection and the size of the sodium silicate batch is optimized. Finally, we discuss methods that help to address an issue of computational cost associated with the high resolution required for accurate simulation of the coupled process.
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Robust chemical solver for fully-implicit simulations
Authors X. Raynaud, C. Mac Neece, M. Hesse and H. NilsenSummaryThe outcome of water-based Enhanced Oil Recovery (EOR) processes, such as polymer and surfactant injection, highly depends on the chemical composition of the displacing fluid in the reservoir. A robust and efficient solver for geochemistry is therefore crucial to design successful EOR strategies. In this talk, we present the implementation of a new geochemistry solver as a module of the open source MRST (Matlab Reservoir Simulation Toolbox). The solver includes non-isothermal aqueous speciation, solid and gas phase equilibrium, and full surface complexation chemistry. Automatic differentiation is used to assemble the linearized equations and enables to easily set up different systems and models. The residual equations are obtained after a log-log transformation. Such a transformation is likely to bring robustness to the solver by increasing the convexity of the governing equations. An other important ingredient is a-priori bounds on the concentrations which when incorporated in the solver, improve the convergence of the Newton iterations. The solver can be easily combined within MRST with a transport solver. Numerical simulation of low salinity water injection will be presented to illustrate the solver capabilities.
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Numerical Modelling Of Wormhole Formation In Rock Matrix Acidizing Under Two-Phase Conditions
Authors M. Babaei and M. SedighiSummarySecondary phase saturation (e.g., as a result of formation of a separate CO2 phase during acidizing) adds complexity to the prediction of wormhole development in application of acidic solutions in porous media. We develop an acid-rock interaction code (coupled flow-geochemistry) for three dimensional applications capable of accounting for varying mobility ratios between two phases, spatial distribution and relative permeability of the secondary phase. We use dimensionless set of governing equations that have been used commonly in literature to model wormholing processes, however, we add the following to the code: (i) CO2 solubility in solution, (ii) the density of aqueous solutions with the addition of CO2 , (iii) viscosity change due addition of CO2. We investigate how the immobile or less mobile (low relative permeability) secondary phase influences the development of wormholes for Péclet-Damköhler controlled dissolution.
While recently for a 2D numerical model we have shown that existence of a uniformly-distributed secondary phase will actually favour the development of wormholes across the porous systems (https://doi.org/10.1016/j.ces.2017.10.046), here we extend the code to three dimensional systems. We also implement (i) a realistic distribution of the secondary phase by a rigorously coupled flow-geochemistry where evolved CO2 is calculated from chemical reactions, and (ii) physical heterogeneity in simulated system in the form of spatially correlated heterogeneity (as opposed to commonly used random heterogeneity in modelling works in literature).
The findings of this work will have practical relevance to control and design the acidizing operations where the aim is to enhance hydrocarbon or heat recovery from hydrocarbon or geothermal reservoirs. An inevitable existence of secondary phases (such as evolved CO2) in porous media, mean that an acid operation can be carried out more efficiently by consuming less amount of acid. Moreover, ignoring the spatial correlation may result into over-consuming and jeopardising the integrity of reservoir barriers or caprock, consequently leading to contaminate underground water resources by acidic solutions. The formulation is applicable for CO2 sequestration when acidic solutions react with porous media under two phase conditions.
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Adaptive Pore Network Model With Localization Of Time-Step
Authors M. Regaieg and A. MoncorgéSummaryPore scale simulation is used to study phenomena that cannot be reproduced by conventional Darcy-based simulators. Dynamic Pore Network Modeling simulators (PNM) are still relatively slow and constrained by small time-steps to simulate Representative Elementary Volumes (REV) and small scale physics in a reasonable time. In a previous work ( Regaieg et al., 2017 ), an adaptive approach has been proposed to localize the pressure computations only in viscous dominated regions and use a fast quasi-static algorithm for the rest of the domain. We propose to extend this adaptive pore network model by introducing three levels of pressure computations. The domain is divided into sub-networks. We introduce a first level of pressure computation solving fine-scale dynamics effects on local sub-networks. A second intermediate level of pressure computation is used to solve pressure interactions between the sub-networks. The third level is used to solve the pressure on the entire pore network and to update the boundary conditions of the sub-networks. We define automatic criteria to decide when intermediate or global pressure solutions are needed.
We first describe our adaptive pore network model with the different levels of pressure solution. Subsequently, we present several test cases of this algorithm for different viscosity ratios, injection rates and wettability scenarios. Finally, we report speed-ups obtained for simulations on networks of different sizes.
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Fully Implicit WENO Schemes On Stratigraphic And Fully Unstructured Grids
Authors K.A. Lie, T.S. Mykkeltvedt and O. MøynerSummaryHigher-order spatial discretization has been used by many authors to reduce numerical diffusion and mitigate grid-orientation effects. EOR processes are particularly susceptible to numerical diffusion, since the active chemical substances are often transported by linear or weakly nonlinear waves. Most high-resolution methods reported in the literature are based on explicit temporal discretization, which imposes severe time-step restrictions. Herein, we investigate fully-implicit, second-order WENO schemes on unstructured grids. Accuracy and computational efficiency is compared to a standard first-order scheme for Voronoi and corner-point grids in 2D and 3D for black-oil type and compositional flow models.
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Estimation Of Reduced Pressure Build-Up Due To Brine Seepage Using A Convolution Technique
Authors I. Aavatsmark and S.E. GasdaSummaryCO2 sequestration involves injection of large volumes of CO2 into a storage reservoir bounded above and below by low-permeability seals. Although capillary effects are usually sufficient to prevent CO2 leakage through the caprock, the native brine will slowly migrate through the over- and underlying units when subject to overpressure in the reservoir. At large scales and over long time periods, diffuse fluid migration may have an important impact on large-scale pressure development.
Typically, simulation studies of CO2 injection often omit the possibility of brine migration through the top and bottom boundaries of the reservoir. One reason is that vertical fluid flow requires additional resolution outside of the storage reservoir that poses a large computational burden. Therefore, analytical methods are an attractive approach for capturing diffuse brine leakage. In low permeability layers, the flow is predominantly vertical, and the local system can be reduced to a 1D (vertical) equation. This can be solved on a semi-infinite (vertical) domain for a thick seal (> 10–20 m), with the reservoir overpressure applied as a boundary condition. Because the boundary condition is not constant in time, the resulting solution is a convolution integral that must be computed at regular time intervals.
In this paper, we couple the analytical solution for vertical brine leakage with a vertical equilibrium simulator (VESA). The VESA simulator is a reduced-dimension (2D) numerical model for two-phase flow in gravity-segregated systems, which is an appropriate assumption for large-scale CO2 storage. Coupling analytical solution for brine migration into a numerical simulator gives greater flexibility modeling injection into heterogeneous reservoirs. We find that the additional computational burden of the convolution integral is minimal compared to solving the full 3D system at the correct resolution. A solution for pressure development in the adjacent strata is also obtained analytically, leading to a fully 3D representation of pressure in the system. The coupled code is benchmarked with a fully analytical solution, and then applied to large-scale CO2 injection into the Utsira formation. We study the impact of diffuse brine leakage on development of large-scale overpressure in the storage reservoir for scenarios of high-volume CO2 injection.
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Fully Coupled Schemes Using Virtual Element And Finite Volume Discretisations For Biot Equations Modelling
Authors J. Coulet, I. Faille, V. Girault, N. Guy and F. NatafSummaryModelling the interactions between mechanical deformations and fluid flow in a porous media leads to the well known Biot system. This system involves two coupled equations obtained from the mechanical equilibrium and from the fluid mass conservation. The classical way to numerically solve this system is to use one discretisation method for each conservation equation, usually with a finite element method for the mechanical part and a finite volume method for the fluid part. However, the meshing of specific geometries encountered in underground medias related to heterogeneities, discontinuities or faults commonly lead to badly shaped cells not suited to finite element based modelling.
The recent development of the virtual element method, which can be seen as an extension of legacy finite element to more general meshes, makes it appear as a potential discretisation method for the mechanical part. More specifically, [ Andersen, Nilsen, Raynaud 2017 ] provided a first insight into virtual element method applied to the elastic problem in the context of geomechanical simulations. The originality of our work is to design and study a numerical scheme coupling the lowest order virtual element method applied to the mechanical conservation equation with a finite volume scheme method applied to the fluid conservation equation.
A mathematical analysis of this original coupled scheme is provided, including existence and uniqueness results and a priori estimates, for the case of a two points finite volume scheme modelling of fluid flow. The coupled scheme is illustrated by some computations on two or three dimensional grids coming from realistic cases. In the presentation, the coupling with more elaborate finite volume schemes such as multi point flux approximation schemes is also investigated.
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Hybrid Finite Volume discretization of two-phase Discrete Fracture Matrix models with nonlinear interface solver
Authors J. Aghili, K. Brenner, J. Hennicker, R. Masson and L. TrentySummaryA new Hybrid Finite Volume discretization is proposed in this work for two-phase Darcy flow in Discrete Fracture Matrix (DFM) models accounting for nonlinear transmission conditions at matrix fracture (mf) interfaces. This type of model is more accurate than alternative hybrid-dimensional two-phase Darcy flow models based either on continuous phase pressures at the mf interfaces assuming fractures acting as drains, or based on the elimination of the mf interface phase pressures by harmonic transmissibility. On the other hand, keeping the pressure and saturation unknowns and the nonlinear flux continuity equations at the mf interfaces increases the difficulty to solve the nonlinear and linear systems due to the highly contrasted permeabilities, capillary pressures, and scales between the fractures and the matrix. In order to solve efficiently the nonlinear systems arising at each time step from the fully implicit time integration, a Newton solver with linear elimination and nonlinear update of the mf interface unknowns is derived. Numerical experiments show the efficiency of our approach on several 2D test cases including an anisotropic matrix permeability and a large fracture network.
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Acute Boundary Aligned Unstructured Grid Generation And Consistent Flux Approximations
Authors S. Manzoor, M.G. Edwards and A. DogruSummaryGrids based on Voronoi diagrams, comprise the dual of Delaunay triangulations (DTs), and remain predominant in reservoir simulation. Voronoi-based grids are locally orthogonal, i.e., PErpendicular BIsectional(PEBI), and permit consistent two-point flux approximation if the permeability field is isotropic, or if the grid is generated to be K-orthogonal. In addition, Voronoi grids can be made to honor classical key constraints, minimizing discretization error, thereby aligning associated control volumes with solid walls, well-trajectories, and geological features: layers, shale barriers, fractures, faults, and pinch-outs.
Typically, in reservoirs formed by deposition, the directional trend in the horizontal plane is not very distinct, whereas across the layers, rock properties may jump by orders of magnitude. The PEBI property associated with DTs is of major significance, and can only be exploited provided the circumcenter is used as the approximation point. This requires that the grids generated be boundary aligned, and comprised of entirely acute simplexes. The control-volume centroid is commonly used as the approximation point, because a geological feature honored acute triangulation cannot be guaranteed in the general case; especially in the presence of complex geometries and geological constraints. Development of a geological feature-based acute DT technique is presented. A boundary-aligned grid generation method is augmented with a mesh reconstruction technique, which can ensure circumcenter containment of a DT. To honor the geological feature idea of protection-circle is used. In the mesh reconstruction technique, each mesh point is optimized iteratively, using a local-advancing front method incorporating the length of opposite edges of the set of simplexes sharing it. The methods presented generate boundary-aligned acute DT, where previously proposed methods fail to ensure the acute DT property. Details of the method will be presented, together with results for a number of test cases that verify consistency of the two-point flux on the resulting boundary-aligned acute Voronoi grids.
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Combining Face Based And Nodal Based Discretizations For Multiphase Darcy Flow Problems
More LessSummaryA new methodology is introduced in this work to combine face based (Hybrid Finite Volume, HFV or Two Point Flux Approximation, TPFA) and nodal based (Vertex Approximate Gradient, VAG) discretizations on hybrid meshes in order to adapt the numerical scheme to the different types of cells and medium properties in different parts of the mesh. The stability and convergence of the combined VAG-HFV schemes is studied in the gradient scheme framework and is shown to hold on arbitrary partitions of the cells for the unstabilised version and on arbitrary partitions of the faces for the stabilised version. The framework preserves at the interface the discrete conservation properties of the VAG and HFV schemes with fluxes based on local to each cell transmissibility matrices. This discrete conservative form allows to naturally extend the VAG and HFV discretizations of two-phase Darcy flow models to the combined VAG-HFV schemes. Numerical results on different types of meshes show the accuracy and efficiency of the combined schemes which are compared to the stand alone VAG and HFV (or TPFA) discretizations.
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Hybrid-mixed Mimetic Method For Reservoir Simulation With Full Tensor Permeability
Authors A.S. Abushaikha and K. TerekhovSummaryIn this work, we present a fully implicit hybrid mimetic finite difference formulation for general-purpose compositional reservoir simulation. The formulation is locally conservative, and the momentum and mass balance equations are solved simultaneously; including Lagrange multipliers on element interfaces. The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems and the mixed finite element (MFE) finite-element method assures the coupling of the mass and momentum balance equations. The method utilizes automatic differentiation for the Jacobian construction. This hybrid approach accommodates unstructured grids, and we apply compositional test cases with permeability tensors. We also discuss the accuracy for the new formulation. For all tests, we compare the performance and accuracy of the proposed approach with the trivial TPFA method.
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Fv-Mhmm: Local Adaptation Driven By An A Posteriori Error Estimator
Authors J. Franc, G. Debenest, L. Jeannin and R. MassonSummaryMultiscale methods for simulating groundwater flow and for predicting production of large scale reservoirs have known significant breakthrough over the last decades. These approaches successively solve coarse scale problems from fine-scale local solutions and map the coarse scale solution to the fine scale. They make it possible to include petrophysical information from the fine scale, while keeping acceptable computation time. However, information exchange between coarse scale and fine scale has to be improved to deal with highly heterogeneous reservoirs.
Recently, FV-MHMM method[1] has been derived as an adaptation to the finite volume formalism of the Mixed Hybrid Multiscale Method developed in [2].
The pressure field is obtained by solving a hybrid form of the parabolic system at a coarse scale. The mathematical formulation relies on Lagrange multipliers, viewed as coarse scale fluxes, to ensure pressure continuity between coarse blocks. This paper proposes different strategies for improving the performance of FV-MHMM on heterogeneous media.
On one hand, basis functions of the global problem can be adapted to account for heterogeneities. Two approaches are proposed: a transmissivity weighted (tw) scheme and a multiscale two point flux approximation (mstpfa) scheme. This last approach uses local simulation to build a weighting scheme based on estimates of the heterogeneous fluxes across coarse faces.
On the other hand, the number of degrees of freedom (that is to say of basis functions at the coarse scale) can be increased in order to improve the solution. Such an adaptive mechanism driven by an a posteriori error estimator has been developed. It will trigger locally the division of coarse faces and, hence, the addition of degrees of freedom.
These two approaches may also be combined to further improve the FV-MHMM.
Finally, different numerical tests are presented and different strategies to improve the multiscale FV-MHMM solution are discussed. For example, a compromise has to be found between gain in accuracy and the number of degrees of freedom added.
[1] Franc, J., Jeannin, L., Debenest, G., Masson, R. “FV-MHMM method for reservoir modeling.”, 21(5–6), 895–908, Comput. Geosc., 2017
[2] Harder, C., Parades, D., Valentin, F. “A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients”, Journ. of Comput. Phys., 2013
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History Matching Channelized Facies Models Using Ensemble Smoother With A Deep Learning Parameterization
Authors S.W.A Canchumuni, A.A Emerick and M.A.C PachecoSummaryEnsemble data assimilation methods have been successfully applied in several real-life history-matching problems. However, because these methods rely on Gaussian assumptions, their performance is severely degraded when the prior geology is described in terms of complex facies distributions. This work introduces a novel parameterization based on deep learning for history matching facies models with ensemble methods.
The proposed method consists on a parameterization of geological facies by means of a deep belief network (DBN) used as an autoencoder. The process begins with a large set of facies realizations which are used for training the DBN. The trained network has two parts: an encoder and a decoder function. The encoder is used to construct a continuous parameterization of the facies which is iteratively updated to account for observed production data using the method ensemble smoother with multiple data assimilation (ES-MDA). After each iteration of ES-MDA, the decoder is used to reconstruct the facies realizations.
The proposed method is tested in three synthetic history-matching problems with channelized facies constructed with multiple point geostatistics. We compare the results of the DBN parameterization against the standard ES-MDA (with no parameterization) and the recently proposed optimization-based principal component analysis (OPCA). Our results show that all procedures are able to match the observed production data. However, standard ES-MDA failed to generate channel facies with well-defined boundaries. OPCA and DBN parameterizations improved the facies description resulting in the expected bi-modal distributions of log-permeability. This paper reports our initial results on an ongoing investigation with deep learning. Nevertheless, the results presented here indicate a great potential on the use of deep learning technologies in the inverse modeling of petroleum reservoirs.
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Towards Automatic And Adaptive Localization For Ensemble-Based History Matching
More LessSummaryEnsemble-based history matching methods are among the state-of-the-art approaches to reservoir characterization. In practice, however, they often suffer from ensemble collapse, a phenomenon that deteriorates history matching performance. To prevent ensemble collapse, it is customary to equip an ensemble history matching algorithm with a certain localization scheme.
In a previous study (SPE Journal, SPE-185936-PA), we propose an adaptive localization scheme that exploits the correlations between model variables and simulated observations for localization. Correlation-based adaptive localization not only overcomes some longstanding issues arising in conventional distance-based localization, but also is more convenient to implement and use in real field case studies (SPE conference paper, SPE-191305-MS).
The aforementioned correlation-based localization is subject to two problems. One is that, it requires to run a relatively large ensemble in order to achieve decent performance in an automatic manner, which becomes computationally expensive in large-scale problems. As a result, certain empirical tuning factors are introduced in the previous work to reduce the computational costs. The other problem is that, the way used to compute the tapering coefficients in the previous work may induce dis-continuities, and neglect the information of certain still-influential observations for model updates.
The main objective of this work is to improve the efficiency and accuracy of correlation-based adaptive localization proposed in the previous work, making it run in an automatic manner but without incurring substantial extra computational costs. To this end, we introduce two enhancements to address the aforementioned problems. We apply the resulting automatic and adaptive correlation-based localization with the two enhancements to a 2D and a 3D case studies, and show that it leads to better history matching performance than that is achieved in the previous work.
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Gaussian Mixture Model Fitting Method For Uncertainty Quantification By Conditioning To Production Data
More LessSummaryFor most real-field history matching problems, the dynamic system of multi-phase flow in the reservoir induces strong nonlinear behavior in the data responses. Therefore, the posterior probability density function (PPDF) formulated within the Bayesian framework may have multiple local maxima. It is extremely challenging to properly quantify uncertainty of reservoir simulation forecast results for such real-field problems with this type of complex PPDF.
In this paper, our previously introduce Gaussian-Mixture-Model (GMM) method to approximate the PPDF is improved by adding an arbitrarily large number of additional Gaussian components to the superposition, where the relative heights and widths of these components are determined using a suitable fitting procedure. Simulation results of all reservoir models generated during the history matching process, e.g., using distributed Gauss-Newton (DGN) optimizer, are used as training data points for this GMM fitting. The distance between the GMM approximation and the actual posterior PDF is estimated by summing up the errors calculated at all training data points. The distance is an analytical function of unknown GMM parameters such as covariance matrix and weighting factor for each Gaussian component. These unknown GMM parameters are determined by minimizing the distance function. A GMM is accepted if the distance is reasonably small. Otherwise, new Gaussian components will be added iteratively to further reduce the distance until convergence. Finally, high quality conditional realizations are generated by sampling from each Gaussian component in the mixture, with the appropriate relative probability.
The proposed method is first validated using nonlinear toy problems and then applied to real-field cases. GMM generates better samples with a computational cost comparable to or less than other methods, including the well-known expectation-maximization (EM) algorithm for GMM, Randomized-Maximum-Likelihood (RML) method, and ensemble-based methods. More importantly, by adding more Gaussian components, the accuracy of the GMM approximation can always be further increased (at a higher computational cost). Hence, as is illustrated in our test cases, the samples yield production forecasts that match production data reasonably well in the history matching period and are consistent with production data in the blind test period.
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Introducing Stochastic Model Errors In Ensemble-Based History Matching
By G. EvensenSummaryIn reservoir history matching, we usually neglect model errors other than those associated with the parametrization of the model. Neglected model errors include the stochastic errors in the model forcing, such as errors in production-rate data that are used to force the simulation model, although we allow for the rate data to contain errors in the update step. Thus, we assume that the selected uncertain parameters of the model account for all the model errors. If significant errors are unaccounted for, there is a risk for an unphysical update, of some uncertain parameters, that compensates for other neglected errors.
When using EnKF or ES, it is relatively easy to include stochastic model errors as long as we update both the parameters and the state variables simultaneously. However, typically when we use ES and in particular the new iterative smoothers like IES (Chen and Oliver, 2012, 2013) and ESMDA (Emerick and Reynolds, 2013), it is standard to assume the model to be perfect. Thus, we update only the uncertain parameters and then rerun the ensemble simulation to obtain the final result. In this setting, it is not straightforward to consistently include stochastic model errors in the assimilation scheme.
This paper gives the theoretical foundation for introducing additive stochastic model errors in ensemble methods for history matching. We review recent works on including model errors in IEnKF by Sakov et al. (2017) and also an approach by Tarantola (1987) who account for additive model errors by combining them with the measurement errors. Based on these results we explain possible procedures for practically including model errors in the iterative smoothers (IES and ESMDA). Also, we demonstrate the impact of adding (or neglecting) stochastic model errors in applications with reservoir models.
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Seismic History Matching Uncertainty With Weighted Objective Functions
Authors Q. Zhang, R. Chassagne and C. MacBethSummaryHistory matching using 4D seismic time-lapse data provide a method to manage uncertainties of multi-models, but it remains a great challenge without reaching any clear statements or practical methodology or good practise to follow. What makes the SHM so difficult is mainly the nature of the seismic data. Indeed, acquisition, interpretation, processing, make this data embedded with uncertainties, due to the physics and measurement issues. One of the challenge is to be able to extract and quantify the uncertainties carried over by the seismic data and use it to guide decisions.
The way we insert seismic data and its inherent uncertainty into the workflow is the key to further enforce progress in 4D seismic history matching. A comprehensive workflow is implemented which allows shape to estimate uncertainties using weighted factor. In the proposed history matching workflow, 4D seismic attributes are converted to binary images which are representative of fluid saturations, then binary maps are compared using a pattern-matching objective function which can capture the main feature of the seismic data. Novelty is that weighted binary maps are generated on different estimation of the uncertainty within seismic attribute, to explore and screen performance on the seismic history matching procedure. Weighted maps are associated with error/uncertainty quantification of 4D seismic signature, which allow us to identify even specific governing region of seismic reflecting fluid properties, also it shows greatest general-usage potential. An adaptive derivative free optimisation has been applied for the history matching process. Global and local properties are parameterised in the history matching loop, which includes permeability, porosity, fault transmissibility and net-to-gross.
Numerical experiments with a UKCS field shows this methodology is quite flexible and efficient, which circumvent large uncertainty in seismic data and use of uncertain petro-elastic model. This study also implies that the seismic history matching achieves a reasonable production matching while constrains saturation changes derived from time-lapse data using weighted binary seismic history matching method.
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Seismic Data Assimilation With An Imperfect Model
Authors D.S. Oliver and M.A. AlfonzoSummaryData assimilation methods often assume perfect models and uncorrelated observation error. The assumption of a perfect model is probably always wrong for real problems, and since model error is known to generally induce correlated effective observation errors, then the assumption of uncorrelated observation errors is probably almost always wrong, too. Ignoring the correlation of observation errors, leads to suboptimal assimilation of observations. Common methods for dealing with correlated observation errors included thinning of data, creation of super-observations, and inflation of error variance. While those methods can reduce the tendency to underestimate uncertainty, they tend to exclude small-scale information in the data.
In this paper, we examine the consequences of model errors on assimilation of seismic data. To provide a controlled investigation, we investigate two sources of model error -- errors in seismic resolution and errors in the petroelastic model. Both errors result in correlated total observation errors, which must be accounted for in the data assimilation scheme. We show how to recognize the existence of correlated error through model diagnostics, how to estimate the correlation in the error, and how to use a model with correlated errors in a perturbed observation form of an iterative ensemble smoother to improve estimates of uncertainty after assimilation of seismic data. The methodology is applied to synthetic seismic data from the Norne Field model. Parameters of the seismic resolution and the observation noise are estimated from the actual inverted impedance. Using this approach, we are able to assimilate approximately 115,000 observations with correlated total observation error efficiently without neglecting correlations. The examples show that the iterative estimation of total observation error compensates for the model error and improves forecasts. The method requires the observation error to be non-diagonal, but we show that this is easily handled even for large problems.
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Reservoir Inverse Modeling by Ensemble Smoother with Multiple Data Assimilation for Seismic and Production Data
More LessSummaryTime-lapse seismic data has been widely used for the detailed reservoir characterization, and data assimilation algorithms are commonly used for petroleum reservoir history matching of production data. However, hardly any seismic data has been integrated into the reservoir inverse modeling workflow, due to the large data size, finer gridding, and especially the scarcity in time of the seismic data. Popular ensemble-based reservoir inverse modeling methods such as ensemble Kalman filter (EnKF) also face the problems of high computational cost due to the storage of the intermediate variables, restarting the reservoir simulating process and the inconsistency of the full-step and step-wise simulations. The intrinsic sequential data assimilations characteristics of the EnKF set obstacles to assimilate 4D seismic data. The alternative method can be the ensemble smoother (ES), which is an ensemble based method for data assimilation. The ES is based on a Bayesian updating scheme of the reservoir model to match the production history and improve the production forecast. To improve the algorithm convergence, a multiple data assimilation (MDA) method was proposed by Emerick and Reynolds (2012a) .
The algorithm we used is called ensemble smoother with multiple data assimilation (ESMDA), we modified the ESMDA to integrate the geophysical data, and created the new workflow to history match both the production and geophysical data. The available production data is well measurements, including oil production rate, well water cut and bottom-hole pressure, as well as time-lapse geophysical data, including P-wave impedance. In this paper, we first formulated the mathematical descriptions of ESMDA, we then down-scaled the seismic data, and modified the data assimilation algorithms, illustrated the history matching workflow for both production and geophysical data, finally, we showed how it works with a case study on a water flooding operation in a synthetic reservoir. In comparison, we also showed the history matching only on the production data, which yields an inferior results than the one matches both production and seismic data.
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