References
- C. M. Oldenburg and K. Pruess, Plume Separation by Transient Thermohaline Convection in Porous Media, Geophys. Res. Lett., vol. 26, pp. 2997–3000, 1999.
- M. Kühn and A. Günther, Stratabound Rayleigh Convection Observed in a 4D Hydrothermal Reactive Transport Model Based on the Regional Geological Evolution of Allermöhe (Germany), Geofluids, vol. 7, pp. 301–312, 2007.
- E. Shalev, V. Lyakhovsky, and Y. Yechieli, Is Advective Heat Transport Significant at the Dead Sea Basin? Geofluids, vol. 7, pp. 292–300, 2007.
- F. Magri, U. Bayer, U. Maiwald, R. Otto, and C. Thomsen, Impact of Transition Zones, Variable Fluid Viscosity and Anthropogenic Activities on Coupled Fluid-Transport Processes in a Shallow Salt-Dome Environment, Geofluids, vol. 9, pp. 182–194, 2009.
- B. Elhajjar, A. Mojtabi, P. Costesèque, and M.-C. Charrier-Mojtabi, Separation in an Inclined Porous Thermogravitational Cell, Int. J. Heat Mass Transfer, vol. 53, pp. 4844–4851, 2010.
- J. Peterson, B. Murray, and G. Carey, Multi-Resolution Simulation of Double-Diffusive Convection in Porous Media, Int. J. Numer. Methods Heat Fluid Flow, vol. 20, pp. 37–65, 2010.
- M. Narayana, P. Sibanda, S. Motsa, and P. Siddheshwar, On Double-Diffusive Convection and Cross Diffusion Effects on a Horizontal Wavy Surface in a Porous Medium, Boundary Value Probl., vol. 2012, pp. 1–22, 2012.
- S. Bea, U. Mayer, and K. MacQuarrie, Reactive Transport and Thermo-Hydro-Mechanical Coupling in Deep Sedimentary Basins Affected by Glaciation Cycles: Model Development, Verification, and Illustrative Example, Geofluids, vol. 16, pp. 279–300, 2015.
- D. Nield and A. Bejan, Convection in Porous Media. Springer, New York, 1999.
- B. Straughan, Stability and Wave Motion in Porous Media, vol. 165. Springer Science & Business Media, New York, NY, 2008.
- B. Straughan, The Energy Method, Stability, and Nonlinear Convection, vol. 91. Springer Science & Business Media, New York, NY, 2013.
- C. Beckermann, C. Fan, and J. Mihailovic, Numerical Simulations of Double-Diffusive Convection in a Hele-Shaw Cell, Int. Video J. Eng. Res., vol. 1, pp. 71–82, 1991.
- A. M. Morega and T. Nishimura, Double Diffusive Convection by a Chebyshev Collocation Method, Technol. Rep. Yamaguchi Univ., vol. 5, pp. 259–276, 1996.
- R. Bennacer, A. Tobbal, H. Beji, and P. Vasseur, Double Diffusive Convection in a Vertical Enclosure Filled with Anisotropic Porous Media, Int. J. Therm. Sci., vol. 40, pp. 30–41, 2001.
- P. Ackerer, A. Younès, and M. Mancip, A New Coupling Algorithm for Density-Driven Flow in Porous Media, Geophys. Res. Lett., vol. 31, 2004.
- N. Zabaras and D. Samanta, A Stabilized Volume-Averaging Finite Element Method for Flow in Porous Media and Binary Alloy Solidification Processes, Int. J. Numer. Methods Eng., vol. 60, pp. 1103–1138, 2004.
- A. Mansour, A. Amahmid, M. Hasnaoui, and M. Bourich, Multiplicity of Solutions Induced by Thermosolutal Convection in a Square Porous Cavity Heated from Below and Submitted to Horizontal Concentration Gradient in the Presence of Soret Effect, Numer. Heat Transfer Part A Appl., vol. 49, pp. 69–94, 2006.
- J. Kramer, R. Jecl and L. Škerget, Boundary Domain Integral Method for the Study of Double Diffusive Natural Convection in Porous Media, Eng. Anal. Boundary Elem., vol. 31, pp. 897–905, 2007.
- X. Yu, Z. Guo, and B. Shi, Numerical Study of Cross Diffusion Effects on Double Diffusive Convection with Lattice Boltzmann Method, in Computational Science–ICCS 2007, Springer, 2007, pp. 810–817.
- I. Lunati and P. Jenny, Multiscale Finite-Volume Method for Density-Driven Flow in Porous Media, Comput. Geosci., vol. 12, pp. 337–350, 2008.
- J. C. Kalita and A. K. Dass, Higher Order Compact Simulation of Double-Diffusive Natural Convection in a Vertical Porous Annulus, Eng. Appl. Comput. Fluid Mech., vol. 5, pp. 357–371, 2011.
- E. M. Hirthe and T. Graf, Non-Iterative Adaptive Time-Stepping Scheme with Temporal Truncation Error Control for Simulating Variable-Density Flow, Adv. Water Resour., vol. 49, pp. 46–55, 2012.
- B. Kumar and S. Krishna Murthy, A Finite Element Study of Double Diffusive Mixed Convection in a Concentration Stratified Darcian Fluid Saturated Porous Enclosure under Injection/Suction Effect, J. Appl. Math., vol. 2012, 2012.
- A. W. Islam, M. A. Sharif, and E. S. Carlson, Numerical Investigation of Double Diffusive Natural Convection of CO2 in a Brine Saturated Geothermal Reservoir, Geothermics, vol. 48, pp. 101–111, 2013.
- T. Povich, C. Dawson, M. W. Farthing, and C. E. Kees, Finite Element Methods for Variable Density Flow and Solute Transport, Comput. Geosci., vol. 17, pp. 529–549, 2013.
- X. Albets-Chico and S. Kassinos, A Consistent Velocity Approximation for Variable-Density Flow and Transport in Porous Media, J. Hydrol., vol. 507, pp. 33–51, 2013.
- Z. Jamshidzadeh, F. T.-C. Tsai, S. A. Mirbagheri, and H. Ghasemzadeh, Fluid Dispersion Effects on Density-Driven Thermohaline Flow and Transport in Porous Media, Adv. Water Resour., vol. 61, pp. 12–28, 2013.
- Q. Shao, R. Fernández-González, A. Mikdam, L. Bouhala, A. Younes, P. Núñez, S. Belouettar, and A. Makradi, Influence of Heat Transfer and Fluid Flow on Crack Growth in Multilayered Porous/Dense Materials Using Xfem: Application to Solid Oxide Fuel Cell Like Material Design, Int. J. Solids Struct., vol. 51, pp. 3557–3569, 2014.
- Q. Shao, R. Fernández-González, J. Ruiz-Morales, L. Bouhala, D. Fiorelli, A. Younes, P. Núñez, S. Belouettar, and A. Makradi, An Advanced Numerical Model for Energy Conversion and Crack Growth Predictions in Solid Oxide Fuel Cell Units, Int. J. Hydrogen Energy, vol. 40, pp. 16509–16520, 2015.
- Q. Shao, L. Bouhala, D. Fiorelli, M. Fahs, A. Younes, P. Núñez, S. Belouettar, and A. Makradi, Influence of Fluid Flow and Heat Transfer on Crack Propagation in Sofc Multi-Layered Like Material with Anisotropic Porous Layers, Int. J. Solids Struct., vol. 78, pp. 189–198, 2016.
- M. Bellet, H. Combeau, Y. Fautrelle, D. Gobin, M. Rady, E. Arquis, O. Budenkova, B. Dussoubs, Y. Duterrail, A. Kumar, C. A. Gandin, B. Goyeau, S. Mosbah, and M. Založnik, Call for Contributions to a Numerical Benchmark Problem for 2D Columnar Solidification of Binary Alloys, Int. J. Therm. Sci., vol. 48, pp. 2013–2016, 2009.
- C. I. Voss, C. T. Simmons, and N. I. Robinson, Three-Dimensional Benchmark for Variable-Density Flow and Transport Simulation: Matching Semi-Analytic Stability Modes for Steady Unstable Convection in an Inclined Porous Box, Hydrogeol. J., vol. 18, pp. 5–23, 2010.
- F. Arpino, N. Massarotti, and A. Mauro, High Rayleigh Number Laminar-Free Convection in Cavities: New Benchmark Solutions, Numer. Heat Transfer Part B Fundam., vol. 58, pp. 73–97, 2010.
- J. Hao, M. H. Wu, A. Ludwig, and M. Grasser, Numerical Study of the Thermosolutal Convection Induced Macrosegregation During Columnar Solidification, in Advanced Materials Research, vol. 154, pp. 1415–1418, Trans Tech Publ, 2011.
- P. Weis, T. Driesner, D. Coumou, and S. Geiger, Hydrothermal, Multiphase Convection of H2O-NACl Fluids from Ambient to Magmatic Temperatures: A New Numerical Scheme and Benchmarks for Code Comparison, Geofluids, vol. 14, pp. 347–371, 2014.
- R. Bennacer, A. Mohamad, and M. El Ganaoui, Analytical and Numerical Investigation of Double Diffusion in Thermally Anisotropy Multilayer Porous Medium, Heat and Mass Transfer, vol. 41, pp. 298–305, 2005.
- B. Elhajjar, M.-C. Charrier-Mojtabi, and A. Mojtabi, Separation of a Binary Fluid Mixture in a Porous Horizontal Cavity, Phys. Rev. E, vol. 77, p. 026310, 2008.
- Y. Masuda, M. Yoneya, and S. Kimura, Multiple Solutions of Double-Diffusive Convection in Porous Media Due to Opposing Heat and Mass Fluxes on Vertical Walls, J. Therm. Sci. Technol., vol. 8, pp. 533–542, 2013.
- O. V. Trevisan and A. Bejan, Natural Convection with Combined Heat and Mass Transfer Buoyancy Effects in a Porous Medium, Int. J. Heat Mass Transfer, vol. 28, pp. 1597–1611, 1985.
- O. V. Trevisan and A. Bejan, Mass and Heat Transfer by Natural Convection in a Vertical Slot Filled with Porous Medium, Int. J. Heat Mass Transfer, vol. 29, pp. 403–415, 1986.
- A. Mchirgui, N. Hidouri, M. Magherbi, and A. B. Brahim, Entropy Generation in Double-Diffusive Convection in a Square Porous Cavity Using Darcy–Brinkman Formulation, Transp. Porous Media, vol. 93, pp. 223–240, 2012.
- G. Lauriat and V. Prasad, Natural Convection in a Vertical Porous Cavity: A Numerical Study for Brinkman-Extended Darcy Formulation, J. Heat Transfer, vol. 109, pp. 688–696, 1987.
- C. Beckermann, R. Viskanta, and S. Ramadhyani, A Numerical Study of Non-Darcian Natural Convection in a Vertical Enclosure Filled with a Porous Medium, Numer. Heat Transfer, vol. 10, pp. 557–570, 1986.
- P. Nithiarasu, K. Seetharamu, and T. Sundararajan, Double-Diffusive Natural Convection in an Enclosure Filled with Fluid-Saturated Porous Medium: A Generalized Non-Darcy Approach, Numer. Heat Transfer Part A Appl., vol. 30, pp. 413–426, 1996.
- D. Getachew, D. Poulikakos, and W. Minkowycz, Double Diffusion in a Porous Cavity Saturated with Non-Newtonian Fluid, J. Thermophy. Heat Transfer, vol. 12, pp. 437–446, 1998.
- Q. Shao, M. Fahs, A. Younes, and A. Makradi, A High-Accurate Solution for Darcy-Brinkman Double-Diffusive Convection in Saturated Porous Media, Numer. Heat Transfer Part B, vol. 69, pp. 26–47, 2016.
- M. Saghir and M. Islam, Double Diffusive Convection in Dual-Permeability, Dual-Porosity Porous Media, Int. J. Heat Mass Transfer, vol. 42, pp. 437–454, 1999.
- R. E. Showalter and D. Visarraga, Double-Diffusion Models from a Highly-Heterogeneous Medium, J. Math. Anal. Appl., vol. 295, pp. 191–210, 2004.
- A. Kuznetsov and D. Nield, The Effects of Combined Horizontal and Vertical Heterogeneity on the Onset of Convection in a Porous Medium: Double Diffusive Case, Transp. Porous Media, vol. 72, pp. 157–170, 2008.
- A. Kuznetsov and D. Nield, The Onset of Double-Diffusive Convection in a Vertical Cylinder Occupied by a Heterogeneous Porous Medium with Vertical Throughflow, Transp. Porous Media, vol. 95, pp. 327–336, 2012.
- K. Choukairy and R. Bennacer, Numerical and Analytical Analysis of the Thermosolutal Convection in an Heterogeneous Porous Cavity, Fluid Dyn. Mater. Process., vol. 8, pp. 155–172, 2012.
- D. Nield and A. Kuznetsov, The Effect of Heterogeneity on the Onset of Double-Diffusive Convection Induced by Internal Heating in a Porous Medium: A Layered Model, Transp. Porous Media, vol. 100, pp. 83–99, 2013.
- R. Bennacer, H. Beji, and A. Mohamad, Double Diffusive Convection in a Vertical Enclosure Inserted with Two Saturated Porous Layers Confining a Fluid Layer, Int. J. Thermal Sci., vol. 42, pp. 141–151, 2003.
- F. Bahadori and F. Rashidi, Analytical Solution of Thermo-Diffusion and Diffusion-Thermo Effects on Heat and Mass Transfer in a Composite Porous Media, Contemp. Eng. Sci., vol. 5, pp. 59–65, 2012.
- A. W. Islam, H. R. Lashgari, and K. Sephernoori, Double Diffusive Natural Convection of CO2 in a Brine Saturated Geothermal Reservoir: Study of Non-Modal Growth of Perturbations and Heterogeneity Effects, Geothermics, vol. 51, pp. 325–336, 2014.
- A. Islam, A. K. N. Korrani, K. Sepehrnoori, and T. Patzek, Effects of Geochemical Reaction on Double Diffusive Natural Convection of CO2 in Brine Saturated Geothermal Reservoir, Int. J. Heat Mass Transfer, vol. 77, pp. 519–528, 2014.
- E. Holzbecher, Numerical Studies on Thermal Convection in Cold Groundwater, Int. J. Heat Mass Transfer, vol. 40, pp. 605–612, 1997.
- C. T. Simmons, T. R. Fenstemaker, and J. M. Sharp, Variable-Density Groundwater Flow and Solute Transport in Heterogeneous Porous Media: Approaches, Resolutions and Future Challenges, J. Contam. Hydrol., vol. 52, pp. 245–275, 2001.
- A. Prasad and C. T. Simmons, Unstable Density-Driven Flow in Heterogeneous Porous Media: A Stochastic Study of the Elder [1967b] “Short Heater” Problem, Water Resour. Res., vol. 39, pp. SBH–4, 2003.
- D. Nield, General Heterogeneity Effects on the Onset of Convection in a Porous Medium, in Emerging Topics in Heat and Mass Transfer in Porous Media, pp. 63–84, Springer, 2008.
- P. Duhamel, A New Finite Integral Transform Pair for Hyperbolic Conduction Problems in Heterogeneous Media, Int. J. Heat Mass Transfer, vol. 44, pp. 3307–3320, 2001.
- S. Fagherazzi, D. J. Furbish, P. Rasetarinera, and M. Y. Hussaini, Application of the Discontinuous Spectral Galerkin Method to Groundwater Flow, Adv. Water Resour., vol. 27, pp. 129–140, 2004.
- J. Niessner, R. Helmig, H. Jakobs, and J. Roberts, Interface Condition and Linearization Schemes in the Newton Iterations for Two-Phase Flow in Heterogeneous Porous Media, Adv. Water Resour., vol. 28, pp. 671–687, 2005.
- L. Durlofsky, Y. Efendiev, and V. Ginting, An Adaptive Local–Global Multiscale Finite Volume Element Method for Two-Phase Flow Simulations, Adv. Water Resour., vol. 30, pp. 576–588, 2007.
- H. Hoteit and A. Firoozabadi, An Efficient Numerical Model for Incompressible Two-Phase Flow in Fractured Media, Adv. Water Resour., vol. 31, pp. 891–905, 2008.
- M. Fahs, A. Younes, and F. Delay, On the Use of Large Time Steps with Ellam for Transport with Kinetic Reactions Over Heterogeneous Domains, AIChE journal J., vol. 55, pp. 1121–1126, 2009.
- F. A. Radu, N. Suciu, J. Hoffmann, A. Vogel, O. Kolditz, C.-H. Park, and S. Attinger, Accuracy of Numerical Simulations of Contaminant Transport in Heterogeneous Aquifers: A Comparative Study, Adv. Water Resour., vol. 34, pp. 47–61, 2011.
- F. Boso, A. Bellin, and M. Dumbser, Numerical Simulations of Solute Transport in Highly Heterogeneous Formations: A Comparison of Alternative Numerical Schemes, Adv. Water Resour., vol. 52, pp. 178–189, 2013.
- N. Zhang, J. Yao, S. Xue, and Z. Huang, Multiscale Mixed Finite Element, Discrete Fracture–Vug Model For Fluid Flow in Fractured Vuggy Porous Media, Int. J. Heat Mass Transfer, vol. 96, pp. 396–405, 2016.
- Y.-X. Ding, A.-F. Shi, H.-S. Luo, and X.-H. Wang, Adaptive Mesh Refinement for Non-Isothermal Multiphase Flows in Heterogeneous Porous Media Comprising Different Rock Types with Tensor Permeability, Numer. Heat Transfer Part A Appl., vol. 69, pp. 31–50, 2016.
- X.-H. Wang, M. Quintard, and G. Darche, Adaptive Mesh Refinement for One-Dimensional Three-Phase Flow with Phase Change in Porous Media, Numer. Heat Transfer Part B, vol. 50, pp. 231–268, 2006.
- H.-S. Luo, X.-H. Wang, and M. Quintard, Adaptive Mesh Refinement for One-Dimensional Three-Phase Flows in Heterogeneous Fractured Porous Media, Numer. Heat Transfer Part B, vol. 54, pp. 476–498, 2008.
- D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, vol. 32. Springer Science & Business Media, New York, NY, 2013.
- D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers, Springer Science & Business Media, Netherlands, 2009.
- M. Y. Hussaini, C. L. Streett, and T. A. Zang, Spectral Methods for Partial Differential Equations, Transactions of the First Army Conference on Applied Mathematics and Computing, ARO Report, vol. 84, pp. 883–925, Washington, D.C., 1984.
- M. Y. Hussaini and T. A. Zang, Spectral Methods in Fluid Dynamics, Annu. Rev. Fluid Mech., vol. 19, pp. 339–367, 1987.
- M. Fahs and A. Younes, A High-Accurate Fourier-Galerkin Solution for Buoyancy-Driven Flow in a Square Cavity, Numer. Heat Transfer Part B, vol. 65, pp. 495–517, 2014.
- H. R. Henry, Effects of Dispersion on Salt Encroachment in Coastal Aquifers, in Sea Water in Coastal Aquifers, U.S. Geol. Surv. Supply, vol. 1613C, pp. 70–84, 1964.
- M. Simpson and T. Clement, Theoretical Analysis of the Worthiness of Henry and Elder Problems as Benchmarks of Density-Dependent Groundwater Flow Models, Adv. Water Resour., vol. 26, pp. 17–31, 2003.
- A. Zidane, A. Younes, P. Huggenberger, and E. Zechner, The Henry Semianalytical Solution for Saltwater Intrusion with Reduced Dispersion, Water Resour. Res., vol. 48, 2012.
- V. Numerics, IMSL Math/Library. 1994.
- A. Younes, M. Fahs, and S. Ahmed, Solving Density Driven Flow Problems with Efficient Spatial Discretizations and Higher-Order Time Integration Methods, Adv. Water Resour., vol. 32, pp. 340–352, 2009.
- S. Mahmud and I. Pop, Mixed Convection in a Square Vented Enclosure Filled with a Porous Medium, Int. J. Heat Mass Transfer, vol. 49, pp. 2190–2206, 2006.
- B. Goyeau, J.-P. Songbe, and D. Gobin, Numerical Study of Double-Diffusive Natural Convection in a Porous Cavity Using the Darcy-Brinkman Formulation, Int. J. Heat Mass Transfer, vol. 39, pp. 1363–1378, 1996.
- M. Saar and M. Manga, Depth Dependence of Permeability in the Oregon Cascades Inferred From Hydrogeologic, Thermal, Seismic, and Magmatic Modeling Constraints, J. Geophys. Res. Solid Earth, vol. 109, 2004.
- X.-W. Jiang, L. Wan, X.-S. Wang, S. Ge, and J. Liu, Effect of Exponential Decay in Hydraulic Conductivity with Depth on Regional Groundwater Flow, Geophys. Res. Lett., vol. 36, 2009.
- X.-W. Jiang, L. Wan, M. B. Cardenas, S. Ge, and X.-S. Wang, Simultaneous Rejuvenation and Aging of Groundwater in Basins Due to Depth-Decaying Hydraulic Conductivity and Porosity, Geophys. Res. Lett., vol. 37, 2010.
- X. Kuang and J. J. Jiao, An Integrated Permeability-Depth Model for Earth’s Crust, Geophys. Res. Lett., vol. 41, pp. 7539–7545, 2014.
- X.-S. Wang, X.-W. Jiang, L. Wan, S. Ge, and H. Li, A New Analytical Solution of Topography-Driven Flow in a Drainage Basin with Depth-Dependent Anisotropy of Permeability, Water Resources Research, vol. 47, 2011.
- M. J. Powell, An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives, Comput. J., vol. 7, pp. 155–162, 1964.
- M. Powell, On the Calculation of Orthogonal Vectors, Comput. J., vol. 11, pp. 302–304, 1968.
- W. H. Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, New York, NY, 2007.
- M. Fahs, A. Younes, and A. Makradi, A Reference Benchmark Solution for Free Convection in a Square Cavity Filled with a Heterogeneous Porous Medium, Numer. Heat Transfer Part B, vol. 67, pp. 437–462, 2015.
- M. W. Farthing, C. E. Kees, and C. T. Miller, Mixed Finite Element Methods and Higher-Order Temporal Approximations, Adv. Water Resour., vol. 25, pp. 85–101, 2002.
- L. J. Durlofsky, Accuracy of Mixed and Control Volume Finite Element Approximations to Darcy Velocity and Related Quantities, Water Resour. Res., vol. 30, pp. 965–973, 1994.
- A. Younes and P. Ackerer, Solving the Advection–Dispersion Equation with Discontinuous Galerkin and Multipoint Flux Approximation Methods on Unstructured Meshes, Int. J. Numer. Methods Fluids, vol. 58, pp. 687–708, 2008.