Abstract
In this work, we consider the transport of a surfactant in variably saturated porous media. The water flow is modelled by the Richards equations and it is fully coupled with the transport equation for the surfactant. Three linearization techniques are discussed: the Newton method, the modified Picard, and the L-scheme. Based on these, monolithic and splitting schemes are proposed and their convergence is analyzed. The performance of these schemes is illustrated on five numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are presented.
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References
Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3-4), 405–432 (2002)
Agosti, A., Formaggia, L., Scotti, A.: Analysis of a model for precipitation and dissolution coupled with a Darcy flux. J. Math. Anal. Appl. 431(2), 752–781 (2015)
Alt, W., Luckhaus, H.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)
Arbogast, T., Wheeler, M.F.: A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33(4), 1669–1687 (1996)
Barrett, J.W., Knabner, P.: Finite element approximation of the transport of reactive solutes in porous media. Part 1: error estimates for nonequilibrium adsorption processes. SIAM J. Numer. Anal. 34(1), 201–227 (1997)
Bause, M., Hoffmann, J., Knabner, P.: First-order convergence of multi-point flux approximation on triangular grids and comparison with mixed finite element methods. Numer. Math. 116(1), 1–29 (2010)
Berardi, M., Difonzo, F., Vurro, M., Lopez, L.: The 1D Richards’ equation in two layered soils: a Filippov approach to treat discontinuities. Adv. Water Resour. 115, 264–272 (2018)
Bergamaschi, L., Putti, M.: Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation. Int. J. Numer. Methods Eng. 45(8), 1025–1046 (1999)
Cances, C., Pop, I.S., Vohralik, M.: An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comput. 83, 153–188 (2014)
Celia, M., Bouloutas, E., Zarba, R.L.: A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation. Adv. Water Resour. 26(7), 1483–1496 (1990)
Christofi, N., Ivshina, I.B.: Microbial surfactants and their use in field studies of soil remediation. J. Appl. Microbiol. 93(6), 915–929 (2002)
Dawson, C.: Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35(5), 1709–1724 (1998)
Eymard, R., Gutnic, M., Hilhorst, D.: The finite volume method for Richards equation. Comput. Geosci. 3(3-4), 256–294 (1999)
Eymard, R., Hilhorst, D., Vohral, M.: A combined finite volume- nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105(1), 73–131 (2006)
Farthing, M.W., Ogden, F.L.: Numerical solution of Richards’ equation: A review of advances and challenges. Soil Sci. Soc. Am. J. 81, 1257–1269 (2017)
Gallo, C., Manzini, G., mixed finite, A: element/finite volume approach for solving biodegradation transport in groundwater. Internal Journal for Numerical Methods in Fluids 26(5), 533–556 (1998)
van Genuchten, M.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980)
Helmig, R.: Multiphase flow and transport processes in the subsurface: A contribution to the modeling of hydrosystems. Springer-Verlag, Berlin (1997)
Henry, E.J., Smith, J.E., Warrick, A.W.: Solubility effects on surfactant-induced unsaturated flow through porous media. J Hydrol 223(3-4), 164–174 (1999)
Husseini, D.: Effects of anions acids on surface tension of water, Undergraduate Research at JMU Scholarly Commons (2015)
Jones, J.E., Woodward, C.S.: Newton–Krylov-multigrid solvers for large-scale, highly heterogeneous, variably saturated flow problems. Adv. Water Resour. 24(7), 763–774 (2001)
Jenny, P., Tchelepi, H.A., Lee, S.H.: Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions. J. Comput. Phys. 228(20), 7497–7512 (2009)
Karagunduz, A., Young, M.H., Pennell, K.D.: Influence of surfactants on unsaturated water flow and solute transport. Water Resour. Res. 51(4), 1977–1988 (2015)
Klausen, R.A., Radu, F.A., Eigestad, G.T.: Convergence of MPFA on triangulations and for Richards’ equation. Int. J. Numer. Methods Fluids 58(12), 1327–1351 (2008)
Knabner, P., Bitterlich, S., Teran, R.I., Prechtel, A., Schneid, E.: Influence of surfactants on spreading of contaminants and soil remediation. Springer, Berlin (2003)
Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004)
Kumar, K., Pop, I.S., Radu, F.A.: Convergence analysis of mixed numerical schemes for reactive flow in a porous medium. SIAM J. Numer. Anal. 51(4), 2283–2308 (2013)
Lee, S.H., Efendiev, Y.: C1-Continuous relative permeability and hybrid upwind discretization of three phase flow in porous media. Adv. Water Resour. 96, 209–224 (2016)
Lie, K.-A.: An introduction to reservoir simulation using MATLAB: User guide for the Matlab reservoir simulation toolbox (MRST), SINTEF ICT (2016)
List, F., Radu, F.A.: A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016)
Mitra, K., Pop, I.S.: A modified L-Scheme to solve nonlinear diffusion problems. Comput. Math. Appl. 77, 1722–1738 (2019)
Nochetto, R., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25, 784–814 (1988)
Prechtel, A., Knabner, P.: Accurate and efficient simulation of coupled water flow and nonlinear reactive transport in the saturated and vadose zone - application to surfactant enhanced and intrinsic bioremediation. Int. J. Water Res. Dev. 47, 687–694 (2002)
Pop, I.S., Radu, F.A., Knabner, P.: Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168(1), 365–373 (2004)
Radu, F.A., Pop, I.S., Attinger, S.: Analysis of an Euler implicit, mixed finite element scheme for reactive solute transport in porous media. Num. Methods Part. Diff. Equ. 26 (2), 320–344 (2010)
Radu, F.A., Pop, I.S., Knabner, P.: Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation. SIAM J. Numer. Anal. 42(4), 1452–1478 (2004)
Radu, F.A., Pop, I.S., Knabner, P.: On the convergence of the Newton method for the mixed finite element discretization of a class of degenerate parabolic equation, Numerical Mathematics and Advanced Applications, pp. 1192–1200 (2006)
Radu, F.A., Suciu, N., Hoffmann, J., Vogel, A., Kolditz, O., Park, C.H., Attinger, S.: Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: a comparative study. Adv. Water Resour. 34(1), 47–61 (2011)
Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media, SIAM, pp. 35–106 (1983)
Slodicka, M.: A robust and efficient linearization scheme for doubly non-linear and degenerate parabolic problems arising in flow in porous media. SIAM J. Numer. Anal. 23(5), 1593–1614 (2002)
Smith, J.E., Gillham, R.W.: The effect of concentration-dependent surface tension on the flow of water and transport of dissolved organic compounds: A pressure head-based formulation and numerical model. Water Resour. Res. 31(3), 343–354 (1994)
Smith, J., Gillham, R.: Effects of solute concentration-dependent surface tension on unsaturated flow: Laboratory sand column experiments. Water Res. Res. 35(4), 973–982 (1999)
Suciu, N.: Diffusion in random velocity fields with applications to contaminant transport in groundwater. Water Res. Res. 69, 114–133 (2014)
Vohralik, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45(4), 1570–1599 (2007)
Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735 (2011)
Wang, X., Tchelepi, H.A.: Trust-region based solver for nonlinear transport in heterogeneous porous media. J. Comput. Phys. 253, 114–137 (2013)
Woodward, C.S., Dawson, C.N.: Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J. Numer. Anal. 37(3), 701–724 (2000)
Yong, W.A., Pop, I.S.: A numerical approach to porous medium equations, Preprint 95-50 (SFB 359), IWR University of Heidelberg (1996)
Younis, R., Tchelepi, H.A., Aziz, K.: Adaptively localized continuation-newton method–nonlinear solvers that converge all the time. SPE J. 15(02), 526–544 (2010)
Acknowledgments
Open Access funding provided by University of Bergen. We thank the members of the Sintef research group and in particular to Dr. Olav Moyner for the assistance with the implementation of the numerical examples in MRST, the toolbox based on MATLAB developed at Sintef itself.
Funding
The research of D. Illiano was funded by VISTA, a collaboration between the Norwegian Academy of Science and Letters and Equinor, project number 6367, project name: adaptive model and solver simulation of enhanced oil recovery. The research of I.S. Pop was supported by the Research Foundation-Flanders (FWO), Belgium, through the Odysseus programme (project G0G1316N) and Equinor through the Akademia grant.
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Illiano, D., Pop, I.S. & Radu, F.A. Iterative schemes for surfactant transport in porous media. Comput Geosci 25, 805–822 (2021). https://doi.org/10.1007/s10596-020-09949-2
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DOI: https://doi.org/10.1007/s10596-020-09949-2