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Dashtbesh, Narges, Guillaume Enchéry, and Benoît Noetinger. "A dynamic coarsening approach to immiscible multiphase flows in heterogeneous porous media." Journal of Petroleum Science and Engineering 201 (June 2021): 108396. http://dx.doi.org/10.1016/j.petrol.2021.108396.
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Cancès, Clément, Thomas O. Gallouët, and Léonard Monsaingeon. "Incompressible immiscible multiphase flows in porous media: a variational approach." Analysis & PDE 10, no. 8 (2017): 1845–76. http://dx.doi.org/10.2140/apde.2017.10.1845.
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Chaouche, M., N. Rakotomalala, D. Salin, and Y. C. Yortsos. "Capillary Effects in Immiscible Flows in Heterogeneous Porous Media." Europhysics Letters (EPL) 21, no. 1 (1993): 19–24. http://dx.doi.org/10.1209/0295-5075/21/1/004.
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Ghommem, Mehdi, Eduardo Gildin, and Mohammadreza Ghasemi. "Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media." SPE Journal 21, no. 01 (2016): 144–51. http://dx.doi.org/10.2118/167295-pa.
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Summary In this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather post-processes a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their proper-orthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobility-related term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization.
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Sandrakov, G. V. "HOMOGENIZED MODELS FOR MULTIPHASE DIFFUSION IN POROUS MEDIA." Journal of Numerical and Applied Mathematics, no. 3 (132) (2019): 43–59. http://dx.doi.org/10.17721/2706-9699.2019.3.05.
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Non-stationary processes of mutual diffusion for multiphase flows of immiscible liquids in porous media with a periodic structure are considered. The mathematical model for such processes is initial-boundary diffusion problem for media formed by a large number of «blocks» having low permeability and separated by a connected system of «cracks» with high permeability. Taking into account such a structure of porous media during modeling leads to the dependence of the equations of the problem on two small parameters of the porous medium microscale and the block permeability. Homogenized initial-boundary value problems will be obtained. Solutions of the problems are approximated for the solutions of the initial-boundary value problem under consideration.
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Èiegis, R., O. Iliev, V. Starikovièius, and K. Steiner. "NUMERICAL ALGORITHMS FOR SOLVING PROBLEMS OF MULTIPHASE FLOWS IN POROUS MEDIA." Mathematical Modelling and Analysis 11, no. 2 (2006): 133–48. http://dx.doi.org/10.3846/13926292.2006.9637308.
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In this paper we discuss numerical algorithms for solving the system of nonlinear PDEs, arising in modelling of two‐phase flows in porous media, as well as the proper object oriented implementation of these algorithms. Global pressure model for isothermal two‐phase immiscible flow in porous media is considered in this paper. Finite‐volume method is used for the space discretization of the system of PDEs. Different time stepping discretizations and linearization approaches are discussed. The main concepts of the PDE software tool MfsolverC++ are given. Numerical results for one realistic problem are presented.
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Parmigiani, A., C. Huber, O. Bachmann, and B. Chopard. "Pore-scale mass and reactant transport in multiphase porous media flows." Journal of Fluid Mechanics 686 (September 30, 2011): 40–76. http://dx.doi.org/10.1017/jfm.2011.268.
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AbstractReactive processes associated with multiphase flows play a significant role in mass transport in unsaturated porous media. For example, the effect of reactions on the solid matrix can affect the formation and stability of fingering instabilities associated with the invasion of a buoyant non-wetting fluid. In this study, we focus on the formation and stability of capillary channels of a buoyant non-wetting fluid (developed because of capillary instabilities) and their impact on the transport and distribution of a reactant in the porous medium. We use a combination of pore-scale numerical calculations based on a multiphase reactive lattice Boltzmann model (LBM) and scaling laws to quantify (i) the effect of dissolution on the preservation of capillary instabilities, (ii) the penetration depth of reaction beyond the dissolution/melting front, and (iii) the temporal and spatial distribution of dissolution/melting under different conditions (concentration of reactant in the non-wetting fluid, injection rate). Our results show that, even for tortuous non-wetting fluid channels, simple scaling laws assuming an axisymmetrical annular flow can explain (i) the exponential decay of reactant along capillary channels, (ii) the dependence of the penetration depth of reactant on a local Péclet number (using the non-wetting fluid velocity in the channel) and more qualitatively (iii) the importance of the melting/reaction efficiency on the stability of non-wetting fluid channels. Our numerical method allows us to study the feedbacks between the immiscible multiphase fluid flow and a dynamically evolving porous matrix (dissolution or melting) which is an essential component of reactive transport in porous media.
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Doorwar, Shashvat, and Kishore K. Mohanty. "Viscous-Fingering Function for Unstable Immiscible Flows." SPE Journal 22, no. 01 (2016): 019–31. http://dx.doi.org/10.2118/173290-pa.
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Summary Displacement of viscous oils often involves unstable immiscible flow. Viscous instability and its influence on relative permeability were studied in this work at different viscosity ratios, injection rates, and domain widths. Micromodels and pore-scale models were used to visually inspect the interplay of viscous and capillary forces in the viscous-dominated regime. A new dimensionless scaling parameter, NI=(vwμwσow)(μoμw)2(D2/K), was developed that is useful in predicting the recoveries of unstable displacements at various viscosity ratios and injection rates. The scaling parameter showed excellent fit with experimental data of 68 corefloods. A lumped finger model was developed to modify multiphase flow equations and to yield pseudorelative permeability functions that account for viscous fingering. The parameters of the lumped model can be estimated from the new dimensionless number, NI. This pseudorelative permeability function could be applied at each gridblock on the basis of the local NI to simulate large-scale unstable floods in water-wet porous media.
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Zakirov, T. R., O. S. Zhuchkova, and M. G. Khramchenkov. "Mathematical Model for Dynamic Adsorption with Immiscible Multiphase Flows in Three-dimensional Porous Media." Lobachevskii Journal of Mathematics 45, no. 2 (2024): 888–98. http://dx.doi.org/10.1134/s1995080224600134.
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Kozdon, J., B. Mallison, M. Gerritsen, and W. Chen. "Multidimensional Upwinding for Multiphase Transport in Porous Media." SPE Journal 16, no. 02 (2011): 263–72. http://dx.doi.org/10.2118/119190-pa.
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Summary Multidimensional transport for reservoir simulation is typically solved by applying 1D numerical methods in each spatial-coordinate direction. This approach is simple, but the disadvantage is that numerical errors become highly correlated with the underlying computational grid. In many real-field applications, this can result in strong sensitivity to grid design not only for the computed saturation/composition fields but also for critical integrated data such as breakthrough times. Therefore, to increase robustness of simulators, especially for adverse-mobility-ratio flows that arise in a variety of enhanced-oil-recovery (EOR) processes, it is of much interest to design truly multidimensional schemes for transport that remove, or at least strongly reduce, the sensitivity to grid design. We present a new upstream-biased truly multidimensional family of schemes for multiphase transport capable of handling countercurrent flow arising from gravity. The proposed family of schemes has four attractive properties: applicability within a variety of simulation formulations with varying levels of implicitness, extensibility to general grid topologies, compatibility with any finite-volume flow discretization, and provable stability (monotonicity) for multiphase transport. The family is sufficiently expressive to include several previously developed multidimensional schemes, such as the narrow scheme, in a manner appropriate for general-purpose reservoir simulation. A number of waterflooding problems in homogeneous and heterogeneous media demonstrate the robustness of the method as well as reduced transverse (cross-wind) diffusion and grid-orientation effects.
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Mozolevski, I., and L. Schuh. "Numerical simulation of two-phase immiscible incompressible flows in heterogeneous porous media with capillary barriers." Journal of Computational and Applied Mathematics 242 (April 2013): 12–27. http://dx.doi.org/10.1016/j.cam.2012.09.045.
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Monteagudo, Jorge E. P., and Abbas Firoozabadi. "Control-Volume Model for Simulation of Water Injection in Fractured Media: Incorporating Matrix Heterogeneity and Reservoir Wettability Effects." SPE Journal 12, no. 03 (2007): 355–66. http://dx.doi.org/10.2118/98108-pa.
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Summary The control-volume discrete-fracture (CVDF) model is extended to incorporate heterogeneity in rock and in rock-fluid properties. A novel algorithm is proposed to model strong water-wetting with zero capillary pressure in the fractures. The extended method is used to simulate:oil production in a layered faulted reservoir,laboratory displacement tests in a stack of matrix blocks with a large contrast in fracture and matrix capillary pressure functions, andwater injection in 2D and 3D fractured media with mixed-wettability state. Our results show that the algorithm is suitable for the simulation of water injection in heterogeneous porous media both in water-wet and mixed-wettability states. The novel approach with zero fracture capillary and nonzero matrix capillary pressure allows the proper prediction of sharp fronts in the fractures. Introduction This work is focused on the numerical treatment of two main physical aspects of multiphase flow in fractured porous media: heterogeneity in rock-fluid properties and reservoir wettability. In a previous work (Monteagudo and Firoozabadi 2004), a CVDF method was used to discretize the system of equations governing water injection in fractured media with strong-water-wettability state and homogeneous matrix and rock-fluid properties. The method was restricted to a finite contrast in matrix-fracture capillary pressure. In this work, we extend the CVDF model for simulation of water injection in fractured media comprised of heterogeneous rocks and wettability conditions from strong-water-wetting to mixed-wetting conditions. We also present a formulation for infinite contrast in capillary pressures of matrix and fractures (zero capillary pressure in the fracture and finite capillary pressure in the matrix). The control volume (CV) method, first proposed by Baliga and Patankar (1980), is a finite-volume formulation over dual cells (CVs) of a Delaunay mesh. It is locally conservative and suited for unstructured grids. It has been widely employed for the simulation of multiphase flow in porous media (Monteagudo and Firoozabadi 2004; Verma 1996; Helmig 1997; Helmig and Huber 1998; Bastian et al. 2000; Geiger et al. 2003) and the convergence of the method for two-phase immiscible flow in porous medium has already been proved (Michel 2003). Numerical treatment of heterogeneity in the framework of the CV method has been extensively studied in the past (Edwards 2002; Edwards and Rogers 1998; Prevost 2000; Aavatsmark et al. 1998a, b). Nevertheless, those works have focused on absolute permeability heterogeneity and anisotropy in single-phase flow. The main concern in those works is the use of full tensor permeability and the accurate generation of streamlines (required by the streamline numerical method). It is well known that the standard CV method produces inaccurate velocity fields around the interfaces of heterogeneous media as the contrast in permeability is increased (Durlofsky 1994). In the standard CV method, Delaunay triangles are locally homogeneous and the polygonal CV cell may be heterogeneous (see Fig. 1a). For accurate streamlines, several authors (Verma 1996; Edwards 2002; Edwards and Rogers 1998; Prevost 2000; Aavatsmark et al. 1998a) have proposed that the polygonal CV cell must be locally homogeneous, implying heterogeneous Delaunay triangles (see Fig. 1b). The latter configuration, however, generates additional problems in the simulation of multiphase flow in porous media. Basically, from mesh generation standpoint, it may not be possible to generate an unstructured mesh where the boundaries of the CV median-dual cell conform to heterogeneous interfaces in the domain. Conforming mesh is important for the discrete-fracture approach. Therefore, it would be necessary to first generate a standard CV cell mesh, and later a homogenization procedure would be required to obtain CV cells with constant permeability. The homogenization or upscaling of permeability is somehow possible, but the same is not true for rock-fluid properties; most challenging is capillary pressure with different endpoints. Therefore, the approach with the homogeneous CV cell may be suitable for single-phase simulation where rock-fluid interactions are not part of the problem. However, rock-fluid interactions have to be taken into account for simulation of multiphase flow in fractured porous medium. Frequently, capillary pressure is disregarded in two-phase flow simulations; however, capillary pressure is of importance for simulation of multiphase flow in fractured porous media (Monteagudo and Firoozabadi 2004; Karimi-Fard and Firoozabadi 2003). Predictions of flow pattern and oil recovery may be severely affected if capillary pressure effect is neglected.
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Filimonov, Sergey A., Maxim I. Pryazhnikov, Andrey I. Pryazhnikov, and Andrey V. Minakov. "Development and Testing of a Mathematical Model for Dynamic Network Simulation of the Oil Displacement Process." Fluids 7, no. 9 (2022): 311. http://dx.doi.org/10.3390/fluids7090311.
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Multiphase flows in porous media are widespread in nature and various technologies. One of the most common examples of this kind of task is the task of recovering oil from the rock. This article describes a mathematical model of the flow of a two-phase (immiscible) liquid based on a new approach of network hydrodynamics for a highly branched microchannel medium (simulating a porous space in the rock). The coupling of the flow and pressure fields in the network is performed using a well-proven SIMPLE algorithm in CFD problems; this approach allows us to use effective approaches to modeling 3D tasks. Phase transfer over the network is carried out by an explicit method with an adaptive time step. The article presents the results of verification of the model, with analytical calculations and in comparison with the results of experimental studies. As an experiment, the displacement of oil from a microchip (Dolomite: 3200284) simulating a porous medium was simulated. The good qualitative and quantitative compliance with the results calculated and the results of the experiment show the correct functioning of the model.
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Rasaei, M. Reza, and Muhammad Sahimi. "Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media." Computational Geosciences 13, no. 2 (2008): 187–214. http://dx.doi.org/10.1007/s10596-008-9111-0.
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Pinilla, Andrés, Miguel Asuaje, Camila Pantoja, Luis Ramirez, Jessica Gomez, and Nicolás Ratkovich. "CFD study of the water production in mature heavy oil fields with horizontal wells." PLOS ONE 16, no. 10 (2021): e0258870. http://dx.doi.org/10.1371/journal.pone.0258870.
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Excessive water production in mature heavy oil fields causes incremental costs, energy consumption, and inefficiency. Understanding multiphase flows near the wellbore is an alternative to improve production efficiency. Therefore, this study conducts a series of numerical experiments based on the full set of the Navier-Stokes equations in 3D to simulate multiphase flows in porous media for heavy oil production horizontal wells. The solution given by this advanced mathematical formulation led to the description of the movement of the fluids near the wellbore with unprecedented detail. A sensitivity analysis was conducted on different rock and fluid properties such as permeability and oil viscosity, assuming homogeneous porous media. The influence of these parameters on the prediction of the breakthrough time, aquifer movement, and the severity of water production was noticed. Finally, the numerical model was verified against field data using two approaches. The first one was conducting a history match assuming homogeneous rock properties. In contrast, the second one used heterogeneous rock properties measured from well logging, achieving a lower deviation than field data, about 20%. The homogeneous numerical experiments showed that the breakthrough occurs at the heel with a subsequent crestation along the horizontal well. Moreover, at adverse mobility ratios, excessive water production tends to happen in water connings at the heel with an inflow area less than 1% of the total inflow area of the completion liner. Different aquifer movement dynamics were found for the heterogeneous case, like the breakthrough through multiple locations along the horizontal well. Finally, critical hydraulic data in the well, such as the pressure and velocity profiles, were obtained, which could be used to improve production efficiency. The numerical model presented in this study is proposed as an alternative to conducting subsurface modeling and well designs.
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Ding, Yun-Xiao, An-Feng Shi, Hai-Shan Luo, and Xiao-Hong Wang. "Adaptive mesh refinement for non-isothermal multiphase flows in heterogeneous porous media comprising different rock types with tensor permeability." Numerical Heat Transfer, Part A: Applications 69, no. 1 (2015): 31–50. http://dx.doi.org/10.1080/10407782.2015.1023081.
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Wheeler, Mary F., Guangri Xue, and Ivan Yotov. "Accurate Cell-Centered Discretizations for Modeling Multiphase Flow in Porous Media on General Hexahedral and Simplicial Grids." SPE Journal 17, no. 03 (2012): 779–93. http://dx.doi.org/10.2118/141534-pa.
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Summary We introduce an accurate cell-centered method for modeling Darcy flow on general quadrilateral, hexahedral, and simplicial grids. We refer to these discretizations as the multipoint-flux mixed-finite-element (MFMFE) method. The MFMFE method is locally conservative with continuous fluxes and can be viewed within a variational framework as a mixed finite-element method with special approximating spaces and quadrature rules. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a nonsymmetric quadrature rule on rough grids. The framework allows for handling hexahedral grids with nonplanar faces defined by trilinear mappings from the reference cube. Moreover, the MFMFE method allows for local elimination of the velocity, which leads to a cell-centered pressure system. Theoretical and numerical results demonstrate first-order convergence on rough grids. Second-order superconvergence is observed on smooth grids. We also discuss a new splitting scheme for modeling multiphase flows that can treat higher-order transport discretizations for saturations. We apply the MFMFE method to obtain physically consistent approximations to the velocity and a reference pressure on quadrilateral or hexahedral grids, and a discontinuous Galerkin method for saturations. For higher-order saturations, we propose an efficient post-processing technique that gives accurate velocities in the interior of the gridblocks. Computational results are provided for flow in highly heterogeneous reservoirs, including different capillary pressures arising from different rock types.
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Sorbie, K. S., A. Y. Al Ghafri, A. Skauge, and E. J. Mackay. "On the Modelling of Immiscible Viscous Fingering in Two-Phase Flow in Porous Media." Transport in Porous Media 135, no. 2 (2020): 331–59. http://dx.doi.org/10.1007/s11242-020-01479-w.
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Abstract Viscous fingering in porous media is an instability which occurs when a low-viscosity injected fluid displaces a much more viscous resident fluid, under miscible or immiscible conditions. Immiscible viscous fingering is more complex and has been found to be difficult to simulate numerically and is the main focus of this paper. Many researchers have identified the source of the problem of simulating realistic immiscible fingering as being in the numerics of the process, and a large number of studies have appeared applying high-order numerical schemes to the problem with some limited success. We believe that this view is incorrect and that the solution to the problem of modelling immiscible viscous fingering lies in the physics and related mathematical formulation of the problem. At the heart of our approach is what we describe as the resolution of the “M-paradox”, where M is the mobility ratio, as explained below. In this paper, we present a new 4-stage approach to the modelling of realistic two-phase immiscible viscous fingering by (1) formulating the problem based on the experimentally observed fractional flows in the fingers, which we denote as $$ f_{\rm w}^{*} $$ f w ∗ , and which is the chosen simulation input; (2) from the infinite choice of relative permeability (RP) functions, $$ k_{\rm rw}^{*} $$ k rw ∗ and $$ k_{\rm ro}^{*} $$ k ro ∗ , which yield the same $$ f_{\rm w}^{*} $$ f w ∗ , we choose the set which maximises the total mobility function, $$ \lambda_{\text{T}}^{{}} $$ λ T (where $$ \lambda_{\text{T}}^{{}} = \lambda_{\text{o}}^{{}} + \lambda_{\text{w}}^{{}} $$ λ T = λ o + λ w ), i.e. minimises the pressure drop across the fingering system; (3) the permeability structure of the heterogeneous domain (the porous medium) is then chosen based on a random correlated field (RCF) in this case; and finally, (4) using a sufficiently fine numerical grid, but with simple transport numerics. Using our approach, realistic immiscible fingering can be simulated using elementary numerical methods (e.g. single-point upstreaming) for the solution of the two-phase fluid transport equations. The method is illustrated by simulating the type of immiscible viscous fingering observed in many experiments in 2D slabs of rock where water displaces very viscous oil where the oil/water viscosity ratio is $$ (\mu_{\text{o}} /\mu_{\text{w}} ) = 1600 $$ ( μ o / μ w ) = 1600 . Simulations are presented for two example cases, for different levels of water saturation in the main viscous finger (i.e. for 2 different underlying $$ f_{\rm w}^{*} $$ f w ∗ functions) produce very realistic fingering patterns which are qualitatively similar to observations in several respects, as discussed. Additional simulations of tertiary polymer flooding are also presented for which good experimental data are available for displacements in 2D rock slabs (Skauge et al., in: Presented at SPE Improved Oil Recovery Symposium, 14–18 April, Tulsa, Oklahoma, USA, SPE-154292-MS, 2012. 10.2118/154292-MS, EAGE 17th European Symposium on Improved Oil Recovery, St. Petersburg, Russia, 2013; Vik et al., in: Presented at SPE Europec featured at 80th EAGE Conference and Exhibition, Copenhagen, Denmark, SPE-190866-MS, 2018. 10.2118/190866-MS). The finger patterns for the polymer displacements and the magnitude and timing of the oil displacement response show excellent qualitative agreement with experiment, and indeed, they fully explain the observations in terms of an enhanced viscous crossflow mechanism (Sorbie and Skauge, in: Proceedings of the EAGE 20th Symposium on IOR, Pau, France, 2019). As a sensitivity, we also present some example results where the adjusted fractional flow ($$ f_{\rm w}^{*} $$ f w ∗ ) can give a chosen frontal shock saturation, $$ S_{\rm wf}^{*} $$ S wf ∗ , but at different frontal mobility ratios, $$ M(S_{\rm wf}^{*} ) $$ M ( S wf ∗ ) . Finally, two tests on the robustness of the method are presented on the effect of both rescaling the permeability field and on grid coarsening. It is demonstrated that our approach is very robust to both permeability field rescaling, i.e. where the (kmax/kmin) ratio in the RCF goes from 100 to 3, and also under numerical grid coarsening.
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Le Blay, Marine, Mokhtar Adda-Bedia, and Denis Bartolo. "Emergence of scale-free smectic rivers and critical depinning in emulsions driven through disorder." Proceedings of the National Academy of Sciences 117, no. 25 (2020): 13914–20. http://dx.doi.org/10.1073/pnas.2000681117.
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During the past 60 min, oil companies have extracted 6 trillion liters of oil from the ground, thereby giving a striking illustration of the impact of multiphase flows on the world economy. From a fundamental perspective, we largely understand the dynamics of interfaces separating immiscible fluids driven through heterogeneous environments. In stark contrast, the basic mechanisms ruling the transport of fragmented fluids, such as foams and emulsions, remain elusive with studies mostly limited to isolated droplets and bubbles. Here, we demonstrate that the mobilization of emulsion driven through model disordered media is a critical plastic depinning transition. To elucidate this collective dynamics, we track the trajectories of hundreds of thousands of microfluidic droplets advected through random lattices of pinning sites. Their dynamics reveals that macroscopic mobilization only requires the coordinated motion of small groups of particles and does not involve any large-scale avalanches. Criticality arises from the interplay between contact and hydrodynamic interaction, which channel seemingly erratic depinning events along smectic river networks correlated over system spanning scales. Beyond the specifics of emulsion transport, we close our article discussing the similarities and profound differences with the plastic depinning transitions of driven flux lines in high-Tcsuperconductors, charged colloids, and grain transport in eroded sand beds.
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Beaude, L., R. Masson, S. Lopez, and P. Samier. "Combined face based and nodal based discretizations on hybrid meshes for non-isothermal two-phase Darcy flow problems." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 4 (2019): 1125–56. http://dx.doi.org/10.1051/m2an/2019014.
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In the last 20 years many discretization schemes have been developed to approximate the Darcy fluxes on polyhedral cells in heterogeneous anisotropic porous media. Among them, we can distinguished cell based approaches like the Two Point Flux Approximation (TPFA) or the Multi Point Flux Approximation (MPFA) schemes, face based approaches like the Hybrid Finite Volume (HFV) scheme belonging to the family of Hybrid Mimetic Mixed methods and nodal based discretizations like the Vertex Approximate Gradient (VAG) scheme. They all have their own drawbacks and advantages which typically depend on the type of cells and on the anisotropy of the medium. In this work, we propose a new methodology to combine the VAG and HFV discretizations on arbitrary subsets of cells or faces in order to choose the best suited scheme in different parts of the mesh. In our approach the TPFA discretization is considered as an HFV discretization for which the face unknowns can be eliminated. The coupling strategy is based on a node to face interpolation operator at the interfaces which must be chosen to ensure the consistency, the coercivity and the limit conformity properties of the combined discretization. The convergence analysis is performed in the gradient discretization framework and convergence is proved for arbitrary cell or face partitions of the mesh. For face partitions, an additional stabilisation local to the cell is required to ensure the coercivity while for cell partitions no additional stabilisation is needed. 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. The efficiency of our approach is tested for single phase and immiscible two-phase Darcy flows on 3D meshes using a combination of the HFV and VAG discretizations as well as for non-isothermal compositional liquid gas Darcy flows on a vertical 2D cross-section of the Bouillante geothermal reservoir (Guadeloupe) using a combination of the TPFA and VAG discretizations.
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Zhou, Hui, and Hamdi A. Tchelepi. "Operator-Based Multiscale Method for Compressible Flow." SPE Journal 13, no. 02 (2008): 267–73. http://dx.doi.org/10.2118/106254-pa.
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Summary Multiscale methods have been developed for accurate and efficient numerical solution of flow problems in large-scale heterogeneous reservoirs. A scalable and extendible Operator-Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework. It is natural and convenient to incorporate more physics in OBMM for multiscale computation. In OBMM, two operators are constructed: prolongation and restriction. The prolongation operator is constructed by assembling the multiscale basis functions. The specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finitevolume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators to the fine-scale flow equations. Solving the coarse-scale equation results in a high-quality coarse-scale pressure. The finescale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport (saturation) equation. We describe the OBMM approach for multiscale modeling of compressible multiphase flow. We show that extension from incompressible to compressible flows is straightforward. No special treatment for compressibility is required. The efficiency of multiscale formulations over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrated using several numerical examples including a challenging depletion problem in a strongly heterogeneous permeability field (SPE 10). Introduction The accuracy of simulating subsurface flow relies strongly on the detailed geologic description of the porous formation. Formation properties such as porosity and permeability typically vary over many scales. As a result, it is not unusual for a detailed geologic description to require 107-108 grid cells. However, this level of resolution is far beyond the computational capability of state-of-the-art reservoir simulators (106 grid cells). Moreover, in many applications, large numbers of reservoir simulations are performed (e.g., history matching, sensitivity analysis and stochastic simulation). Thus, it is necessary to have an efficient and accurate computational method to study these highly detailed models. Multiscale formulations are very promising due to their ability to resolve fine-scale information accurately without direct solution of the global fine-scale equations. Recently, there has been increasing interest in multiscale methods. Hou and Wu (1997) proposed a multiscale finite-element method (MsFEM) that captures the fine-scale information by constructing special basis functions within each element. However, the reconstructed fine-scale velocity is not conservative. Later, Chen and Hou (2003) proposed a conservative mixed finite-element multiscale method. Another multiscale mixed finite element method was presented by Arbogast (2002) and Arbogast and Bryant (2002). Numerical Green functions were used to resolve the fine-scale information, which are then coupled with coarse-scale operators to obtain the global solution. Aarnes (2004) proposed a modified mixed finite-element method, which constructs special basis functions sensitive to the nature of the elliptic problem. Chen et al. (2003) developed a local-global upscaling method by extracting local boundary conditions from a global solution, and then constructing coarse-scale system from local solutions. All these methods considered incompressible flow in heterogeneous porous media where the pressure equation is elliptic. A multiscale finite-volume method (MsFVM) was proposed by Jenny et al. (2003, 2004, 2006) for heterogeneous elliptic problems. They employed two sets of basis functions--dual and primal. The dual basis functions are identical to those of Hou and Wu (1997), while the primal basis functions are obtained by solving local elliptic problems with Neumann boundary conditions calculated from the dual basis functions. Existing multiscale methods (Aarnes 2004; Arbogast 2002; Chen and Hou 2003; Hou and Wu 1997; Jenny et al. 2003) deal with the incompressible flow problem only. However, compressibility will be significant if a gas phase is present. Gas has a large compressibility, which is a strong function of pressure. Therefore, there can be significant spatial compressibility variations in the reservoir, and this is a challenge for multiscale modeling. Very recently, Lunati and Jenny (2006) considered compressible multiphase flow in the framework of MsFVM. They proposed three models to account for the effects of compressibility. Using those models, compressibility effects were represented in the coarse-scale equations and the reconstructed fine-scale fluxes according to the magnitude of compressibility. Motivated to construct a flexible algebraic multiscale framework that can deal with compressible multiphase flow in highly detailed heterogeneous models, we developed an operator-based multiscale method (OBMM). The OBMM algorithm is composed of four steps:constructing the prolongation and restriction operators,assembling and solving the coarse-scale pressure equations,reconstructing the fine-scale pressure and velocity fields, andsolving the fine-scale transport equations. OBMM is a general algebraic multiscale framework for compressible multiphase flow. This algebraic framework can also be extended naturally from structured to unstructured grid. Moreover, the OBMM approach may be used to employ multiscale solution strategies in existing simulators with a relatively small investment.
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Carpenter, Chris. "Artificial Neural Network Models and Predicts Reservoir Parameters." Journal of Petroleum Technology 73, no. 01 (2021): 44–45. http://dx.doi.org/10.2118/0121-0044-jpt.
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This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper IPTC 19854, “Modeling and Prediction of Resistivity, Capillary Pressure, and Relative Permeability Using Artificial Neural Network,” by Mustafa Ba alawi, SPE, King Fahd University of Petroleum and Minerals; Salem Gharbi, SPE, Saudi Aramco; and Mohamed Mahmoud, King Fahd University of Petroleum and Minerals, prepared for the 2020 International Petroleum Technology Conference, Dhahran, Saudi Arabia, 13–15 January. The paper has not been peer reviewed. Copyright 2020 International Petroleum Technology Conference. Reproduced by permission. Capillary pressure and relative permeability are essential measurements that affect multiphase fluid flow in porous media directly. The processes of measuring these parameters, however, are both time-consuming and expensive. Artificial-intelligence methods have achieved promising results in modeling extremely complicated phenomena in the industry. In the complete paper, the authors generate a model by using an artificial-neural-network (ANN) technique to predict both capillary pressure and relative permeability from resistivity. Capillary Pressure and Resistivity Capillary pressure and resistivity are two of the most significant parameters governing fluid flow in oil and gas reservoirs. Capillary pressure, the pressure difference over the interface of two different immiscible fluids, traditionally is measured in the laboratory. The difficulty of its calculation is related to the challenges of maintaining reservoir conditions and the complexity of dealing with low-permeability and strong heterogeneous samples. Moreover, unless the core materials are both available and representative, a restricted number of core plugs will lead to inadequate reservoir description. On the other hand, resistivity can be obtained by either lab-oratory analysis or through typical and routine well-logging tools in real time. Both parameters have similar attributes, given their dependence on wetting-phase saturation. Despite many studies in the literature that are reviewed in the complete paper, improvement of capillary pressure and resistivity modeling remains an open research area. Artificial Intelligence in Petroleum Engineering In addition to labor and expense concerns, conventional methods to measure resistivity, capillary pressure, and relative permeability depend primarily, with the exception of resistivity from well logs, on the availability of core samples of the desired reservoir. The literature provides several attempts to model these parameters in order to avoid many of the requirements of measurement. However, the performance of many of these models is restricted by assumptions and constraints that require further processing. For example, the accuracy of prediction of capillary pressure from resistivity is highly dependent on the tested core permeability, which requires measuring it as well to achieve the full potentiality of the model.
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Pedretti, Daniele, and Marco Bianchi. "Preliminary results from the use of entrograms to describe transport in fractured media." Acque Sotterranee - Italian Journal of Groundwater, December 18, 2019. http://dx.doi.org/10.7343/as-2019-421.
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Fractured media are heterogeneous systems in which water flows primarily across rock fractures. Flow dynamics and transport of dissolved substances are controlled by the topological distribution and hydraulic properties of the fracture network (including aperture , hydraulic conductivity K and porosity). These topological and hydrodynamic properties are usually insufficiently characterized in field applications, generating uncertainty in the predictions of flow and solute transport. This paper explores a possible application of the concept of geological entropy, in particular the entrogram, as an approach to describe and potentially predict flow and transport in fractured media. In porous media, the entrogram was proven to be an effective approach to represent the spatial persistence and connectivity of high K patterns, enabling predictions for solute transport when proper correlations are established. Given the similarities between high K patterns in porous media and water-bearing fractures in fractured media, preliminary tests were realized to evaluate an idealized two-dimensional fractured system with regular distribution of two sets of fracture networks, one with a more persistent spatial distribution of fractures than the other. A multiphase flow model based on discrete fracture network is used to simulate a tracer test during which a conservative species displaces an immiscible one injected through a well. The analyses of the breakthrough curves (BTCs) of the relative saturation of each phase at another well allows evaluating the relationship between entrogram metrics and the shape of the BTCs. The initial results are promising and push for a more rigorous evaluation of the link among the metrics. This would require primarily the reproduction of more realistic fracture network including multidimensional systems.
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Mathiesen, Joachim, Gaute Linga, Marek Misztal, François Renard, and Tanguy Le Borgne. "Dynamic Fluid Connectivity Controls Solute Dispersion in Multiphase Porous Media Flow." Geophysical Research Letters 50, no. 16 (2023). http://dx.doi.org/10.1029/2023gl105233.
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AbstractSolute transport in multiphase flow through porous media plays a central role in many natural systems and geoengineering applications. The interplay between fluid flow and capillary forces leads to transient flow dynamics and phase distributions. However, it is not known how such dynamic flow affects the dispersion of transported species. Here, we use highly resolved numerical simulations of immiscible two‐phase flow to investigate dispersion in multiphase flows. We show that repeated activation and deactivation of different flow pathways under the effect of capillary forces accelerates the spreading of solutes compared to single phase flow. We establish the transport laws under dynamic multiphase flows by linking the dispersion coefficient to the Bond number, the ratio of the force driving the flow and the surface tension. Our results determine the controlling factors for solute dispersion in porous media, opening a range of applications for understanding and controlling transport in porous geological systems.
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"Optimal displacement of immiscible two-phase fluids in a pore doublet." Physics of Fluids 35, no. 5 (2023). http://dx.doi.org/10.1063/5.0149182.
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Multiphase fluid flow in a pore doublet is a fundamental problem and is important for understanding the transport mechanisms of multiphase flows in porous media. During the displacement of immiscible two-phase fluids in a pore doublet, the transport process is influenced not only by the capillary and viscous forces, but also the channel geometry. In this paper, a mathematical model is presented of the two-phase fluid displacement in a pore doublet considering the effects of capillary force, viscous force, and the geometric structure. This leads to a new and more general analytical solution for the pore doublet system, and it is found that the displacement process is dominated by the capillary number, viscosity, and radius ratios. Also, the optimal displacement is defined, which refers to the wetting fluids in the two daughter channels breaking through the branches simultaneously (i.e., both having the same breakthrough time). Also, the critical capillary number corresponding to the optimal displacement is obtained, which is related to the radius ratio of the two daughter channels and the viscosity ratio of the two immiscible fluids. Finally, analytical results are presented for the displacement in the pore doublet, which can be used to explain and understand the preferential flows in porous media, such as for improving oil recovery from porous media; these are usually observed in oil recovery, groundwater pollution, and the geological sequestration of carbon dioxide.
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Rajabi, Farzaneh, and Hamdi A. Tchelepi. "Probabilistic Forecast of Multiphase Transport Under Viscous and Buoyancy Forces in Heterogeneous Porous Media." Water Resources Research 60, no. 3 (2024). http://dx.doi.org/10.1029/2023wr034449.
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AbstractWe develop a probabilistic approach to map parametric uncertainty to output state uncertainty in first‐order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two‐phase transport in heterogeneous porous media in the presence of a stochastic velocity field. The uncertainty in the velocity field can arise from incomplete descriptions of either porosity field, injection flux, or both. This uncertainty leads to spatiotemporal uncertainty in the saturation field. Given information about spatial/temporal statistics of spatially correlated heterogeneity, we leverage the method of distributions to derive deterministic equations that govern the evolution of pointwise cumulative distribution functions (CDFs) of saturation for a vertical reservoir, while handling the manipulation of multiple shocks arising due to buoyancy forces. Unlike the Buckley‐Leverett equation, the equation describing the fine‐grained CDF is linear in space and time. Ensemble averaging of the fine‐grained CDF results in the CDF of saturation. Thus, we give routes to circumventing the computational cost of Monte Carlo simulations (MCS), while obtaining a pointwise description of the saturation field. We conduct a set of numerical experiments for one‐dimensional transport, and compare the obtained saturation CDFs, against those obtained using MCS as our reference solution, and the statistical moment equation method. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, that is, standard deviation and correlation length of the underlying random fields, whereas the corresponding low‐order statistical moment equations significantly deviate from the MCS results, except for very small values of standard deviation and correlation length.
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Rabbani, Harris Sajjad, and Saideep Pavuluri. "Semi-analytical Model to Predict Dynamic Capillary Pressure–Saturation Relationship for Flows in Heterogeneous Porous Media." Transport in Porous Media, February 20, 2024. http://dx.doi.org/10.1007/s11242-024-02058-z.
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AbstractThe capillary pressure defines pressure difference between non-wetting and wetting fluids. The capillary pressure is part of the flow governing equations, and its definition can have a profound impact on the nature of fluids displacement in a multiphase flow environment. Conventionally, capillary pressure–saturation relationships are determined under equilibrium conditions which signify that all the fluid–fluid interfaces that exist at the pore scale maintain a static configuration at a certain instant in time. However, there exist experimental and numerical evidences that state that the dynamic nature of fluid flows indeed plays a prominent role in defining the trends of the capillary pressure–saturation relationships. In this paper, we develop a first of a kind semi-analytical model to predict the capillary pressure–saturation curves during drainage displacement by integrating the dynamics of fluid flow based on fundamental laws of fluid mechanics. The proposed semi-analytical model can potentially be incorporated into existing multiphase flow simulators to rapidly compute the capillary pressure at various saturations of the flow medium under dynamic flow conditions. The presented semi-analytical model has been validated against experimental and numerical data sets available in the literature at various flow conditions and considering different sets of fluid properties. We noticed a satisfactory match of the results predicted by the proposed semi-analytical model against the literature data. After performing a holistic sensitivity analysis, we notice that the properties of the porous medium, and the fluid–solid interactions play a significant role in defining the trends of the capillary pressure–saturation curves.
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Ahmed, E., Ø. Klemetsdal, X. Raynaud, O. Møyner, and H. M. Nilsen. "Adaptive Timestepping, Linearization, and A Posteriori Error Control for Multiphase Flow of Immiscible Fluids in Porous Media with Wells." SPE Journal, October 1, 2022, 1–21. http://dx.doi.org/10.2118/203974-pa.
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Summary This work focuses on the development of adaptive timesteps, stopping criteria, and error control strategies for reservoir simulations with fully implicit (FIM) solvers. Using a rigorous error control framework, an adaptive time selector combined with nonlinear stopping criteria is used to control nonlinear iterations as well as to balance accuracy and robustness for challenging nonlinear simulations. In reservoir simulation, efficiently solving a system of nonlinear equations arising from the FIM method can be computationally burdensome for complex recovery processes. Theoretically, an FIM reservoir simulator has no stability limit on the timestep size. In practice, standard Newton’s method often fails to converge for large timestep sizes and must therefore cut the timestep multiple times to achieve convergence, resulting in a large number of unnecessary iterations. Another cause of nonlinear convergence issues is the presence of wells, which are often presented as singular point/line sources strongly coupled to the reservoir model, posing additional restrictions on the timestep choice. Here, we use a posteriori error estimators to avoid unnecessary nonlinear iterations and timestep cuts when solving immiscible multiphase flow. First, we estimate error components (e.g., spatial, temporal, and nonlinear) and then apply these to balancing criteria, providing us with dynamic and adaptive strategies to control timestep and nonlinear iterations. The error estimators are fully and locally computable, inexpensive to use, and target the various error components, including well singularities. The method provides an adaptive criterion for stopping the nonlinear iteration process whenever the linearization error does not significantly affect the overall error. Simultaneously, timesteps are adapted to maintain a constant size of the temporal discretization error with respect to the total error. Altogether, this avoids using unnecessary linearization iterations, wasteful timestep cuts, and too small timesteps. To demonstrate the effectiveness of these adaptive features, we present results for a suite of cases, covering both standard benchmarks and conceptual problems incorporating highly heterogeneous media with multiple wells. The proposed timestep selector cooperates with the new stopping criteria to improve nonlinear solver performance and increases robustness for cases with high nonlinearity. Perhaps most important, the adaptive features ensure balanced temporal and spatial errors while maintaining sufficiently small nonlinear errors, which ensures solution accuracy by accurately reproducing saturation fronts, production plateau, and breakthrough times.