This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

Accepted:

Published online:

Carmine Di Nucci ^{1}

@article{CRMECA_2018__346_5_366_0, author = {Carmine Di Nucci}, title = {Unsteady free surface flow in porous media: {One-dimensional} model equations including vertical effects and seepage face}, journal = {Comptes Rendus. M\'ecanique}, pages = {366--383}, publisher = {Elsevier}, volume = {346}, number = {5}, year = {2018}, doi = {10.1016/j.crme.2018.03.003}, language = {en}, }

TY - JOUR AU - Carmine Di Nucci TI - Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face JO - Comptes Rendus. Mécanique PY - 2018 SP - 366 EP - 383 VL - 346 IS - 5 PB - Elsevier DO - 10.1016/j.crme.2018.03.003 LA - en ID - CRMECA_2018__346_5_366_0 ER -

Carmine Di Nucci. Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face. Comptes Rendus. Mécanique, Volume 346 (2018) no. 5, pp. 366-383. doi : 10.1016/j.crme.2018.03.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.03.003/

[1] Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le debit des sources, J. Math. Pures Appl., Volume 10 (1904), pp. 5-78

[2] The Dynamics of Fluids in Porous Media, Dover, New York, 1988

[3] Unconfined Aquifer Flow Theory: From Dupuit to Present, Springer, New York, 2013

[4] Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristic response, Water Resour. Res., Volume 39 (2003), p. 1316

[5] Second order theory of shallow free surface flow in porous media, Q. J. Mech. Appl. Math., Volume 20 (1967), pp. 517-526

[6] Free-surface flow in porous media and periodic solution of the shallow-flow approximation, J. Hydrol., Volume 70 (1984), pp. 251-263

[7] Nonlinear diffusive surface waves in porous media, J. Fluid Mech., Volume 347 (1997), pp. 119-139

[8] Groundwater waves in aquifers of intermediate depths, Adv. Water Resour., Volume 20 (1997), pp. 37-43

[9] A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches, Adv. Water Resour., Volume 26 (2003), pp. 1239-1247

[10] Steepness expansion for free surface flows in coastal aquifers, J. Hydrol., Volume 309 (2005), pp. 85-92

[11] Extended Boussinesq equations for water-wave propagation in porous media, J. Eng. Mech., Volume 136 (2010), pp. 625-640

[12] Extended Boussinesq equations for waves in porous media: derivation of governing equations and generation of waves internally, Seoul, Korea (2014)

[13] A capillarity correction for free surface flow of groundwater, Water Resour. Res., Volume 23 (1987), pp. 805-808

[14] Groundwater waves in a coastal aquifer: a new governing equation including vertical effects and capillarity, Water Resour. Res., Volume 36 (2000), pp. 411-420

[15] An alternative Boussinesq equation considering the effect of hysteresis on coastal groundwater waves, Hydrol. Process., Volume 30 (2016), pp. 2657-2670

[16] Validation of a Boussinesq model of beach ground water behaviour, Mar. Geol., Volume 148 (1998), pp. 55-69

[17] Tidal water table fluctuations in a sandy ocean beach, Water Resour. Res., Volume 35 (1999), pp. 2313-2320

[18] Curved-streamline transitional flow from mild to steep slopes, J. Hydraulic Res., Volume 48 (2010), pp. 699-700

[19] Energy and momentum under critical flow condition, J. Hydraulic Res., Volume 49 (2011), pp. 127-128

[20] Unsteady friction and visco-elasticity in pipe fluid transients, J. Hydraulic Res., Volume 49 (2011), pp. 398-401

[21] Weakly undular hydraulic jump: effects of friction, J. Hydraulic Res., Volume 49 (2011), pp. 409-412

[22] Moment of momentum equation for curvilinear free-surface flow, J. Hydraulic Res., Volume 49 (2011), pp. 415-419

[23] Universal probability distributions of turbulence in open channel flows, J. Hydraulic Res., Volume 49 (2011), p. 702

[24] On the propagation of one-dimensional acoustic waves in liquids, Meccanica, Volume 48 (2013), pp. 15-21

[25] On transient liquid flow, Meccanica, Volume 51 (2016), pp. 2135-2143

[26] Theory of Ground Water Movement, Princeton University Press, Princeton, 1962

[27] Evaluation of the Polubarinova-Kochina formula for the dam problem, Water Resour. Res., Volume 21 (1985), pp. 395-398

[28] Is the Dupuit assumption suitable for predicting the groundwater seepage area in hillslopes?, Water Resources Research, Volume 50 (2014), pp. 2394-2406

[29] A free boundary problem: steady axisymmetric potential flow, Meccanica, Volume 48 (2013), pp. 1805-1810

[30] Erratum: a free boundary problem: steady axisymmetric potential flow, Meccanica, Volume 49 (2014), p. 253

[31] Steady free-surface flow in porous media: generalized Dupuit–Fawer equations, J. Hydraul. Res., Volume 49 (2011), pp. 821-823

[32] Seepage face height, water table position, and well efficiency at steady state, Ground Water, Volume 45 (2007), pp. 168-177

[33] Drainage of recharge to symmetrically located downstream boundaries with special reference to seepage faces, J. Hydrol., Volume 380 (2010), pp. 94-103

[34] On the solution of transient free-surface flow problems in porous media by finite-difference methods, J. Hydrol., Volume 12 (1971), pp. 177-210

[35] On the solution of transient free-surface flow problems in porous media by the finite element method, J. Hydrol., Volume 20 (1973), pp. 49-63

[36] Seepage face simulation using PLASM, Ground Water, Volume 25 (1987), pp. 722-732

[37] A Modular Three-Dimensional Finite-Difference Ground–Water Flow Model: U.S. Geological Survey Techniques of Water-Resources Investigations, 1988 (book 6, Chap. A1)

[38] SEEPAGE, a new MODFLOW DRAIN package, Ground Water, Volume 42 (2004), pp. 576-588

[39] Nonlinear programming approach for transient free boundary flow problem, Appl. Math. Comput., Volume 160 (2005), pp. 317-328

[40] Tidal effects on groundwater dynamics in unconfined aquifers, Hydrol. Process., Volume 15 (2001), pp. 655-669

[41] A study on unsteady seepage flow through DAM, J. Hydrodyn., Volume 21 (2009), pp. 499-504

[42] A fully coupled depth-integrated model for surface water and groundwater flows, J. Hydrol., Volume 542 (2016), pp. 172-184

[43] Theoretical derivation of the conservation equations for single phase flow in porous media: a continuum approach, Meccanica, Volume 49 (2014), pp. 2829-2838

[44] Natural Groundwater Flow, Lewis Publishers, Boca Raton, 1993

[45] Comment on: “Methods to derive the differential equation of the free surface boundary” by C. Chen, X. Kuang, J.J. Jiao, Ground Water, Volume 48 (2010), pp. 486-489

[46] Free boundary problems in the theory of fluid flow through porous media, Ann. Mat. Pura Appl., Volume 97 (1973), pp. 1-82

[47] Linear and Nonlinear Waves, Wiley, New York, 1974

[48] Steady periodic flow through a rectangular DAM, Water Resour. Res., Volume 12 (1981), pp. 1222-1224

[49] Analytic Functions, Elsevier, Amsterdam–London–New York, 1971 https://eudml.org/doc/219298

[50] Complex Variables, Dover, New York, 1999

[51] On cnoidal waves and bores, Proc. R. Soc. London A, Volume 224 (1954), pp. 448-460

[52] On the origin of the Korteweg–de Vries equation | arXiv

[53] Meccanica dei Fluidi – Principi e Applicazioni Idrauliche, UTET, Torino, 1996

[54] On the free overfall, J. Hydraul. Res., Volume 31 (1993), pp. 777-790

[55] On the non-linear unsteady water flow in open channels, Il Nuovo Cimento B, Volume 122 (2007), pp. 237-255

[56] Dissipative Boussinesq equations, C. R. Mecanique, Volume 335 (2007), pp. 559-583

[57] COMSOL Multiphysics 5.2a, COMSOL Inc., 2016

[58] Dynamic crack propagation in fiber reinforced composites, Proc. COMSOL Conference 2009 Milan, 2009

[59] A rigorous derivation of Dupuit's formula for unconfined seepage with seepage surface, Dokl. Akad. Nauk S.S.S.R., Volume 79 (1951), pp. 937-940 (in Russian)

*Cited by Sources: *

Comments - Politique