On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer

We show that the coupled continuum pipe flow model (CCPF) for flows in karst aquifers is ill-posed in the sense that no reasonable solution exists. We also demonstrate that Hua's modified CCPF model is ill-posed in 3D although it is well-posed in two spatial dimensions. A new modification of the original CCPF model that is consistent with basic physics is proposed and its well-posedness is proved here. We believe that this is the first physically relevant well-posed CCPF type model in 3D.

[1]  M. Gunzburger,et al.  Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition , 2010 .

[2]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[3]  A. Quarteroni,et al.  Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations , 2003 .

[4]  E. Greene,et al.  Quantitative Approaches in Characterizing Karst Aquifers , 2001 .

[5]  G. Teutsch,et al.  A combined continuum and discrete network reactive transport model for the simulation of karst development , 1996 .

[6]  Sebastian Bauer,et al.  Modelling of karst development considering conduit-matrix exchange flow , 2000 .

[7]  Fei Hua,et al.  Modeling, analysis and simulation of the Stokes -Darcy system with Beavers -Joseph interface condition , 2009 .

[8]  Ivan Yotov,et al.  Coupling Fluid Flow with Porous Media Flow , 2002, SIAM J. Numer. Anal..

[9]  D. Joseph,et al.  Boundary conditions at a naturally permeable wall , 1967, Journal of Fluid Mechanics.

[10]  Georg Teutsch,et al.  Hydraulic boundary conditions as a controlling factor in karst genesis: A numerical modeling study on artesian conduit development in gypsum , 2003 .

[11]  Yanzhao Cao,et al.  Analysis and finite element approximation of a coupled, continuum pipe‐flow/Darcy model for flow in porous media with embedded conduits , 2011 .

[12]  D. Loper,et al.  Contaminant sequestration in karstic aquifers: Experiments and quantification , 2008 .

[13]  Elemer Bobok,et al.  Fluid mechanics for petroleum engineers , 1993 .

[14]  J. Bear,et al.  Modeling groundwater flow and pollution , 1987 .

[15]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[16]  G. I. Barenblatt,et al.  Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] , 1960 .

[17]  Bill X. Hu,et al.  Laboratory analog and numerical study of groundwater flow and solute transport in a karst aquifer with conduit and matrix domains. , 2009, Journal of contaminant hydrology.

[18]  Georg Teutsch,et al.  Simulation of the development of karst aquifers using a coupled continuum pipe flow model , 2003 .

[19]  A. W. Harbaugh.  MODFLOW-2005 : the U.S. Geological Survey modular ground-water model--the ground-water flow process , 2005 .

[20]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[21]  Sebastian Bauer,et al.  Documentation of a Conduit Flow Process (CFP) for MODFLOW-2005 , 2007 .

[22]  Sebastian Bauer,et al.  Modeling of karst aquifer genesis: Influence of exchange flow , 2003 .

[23]  J. Kevorkian,et al.  Partial Differential Equations: Analytical Solution Techniques , 1990 .

[24]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[25]  P. Saffman.  On the Boundary Condition at the Surface of a Porous Medium , 1971 .

[26]  A. Quarteroni,et al.  Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations , 2004 .

[27]  Paul Williams,et al.  Karst Geomorphology and Hydrology , 1989 .

[28]  Thomas R. Gatliffe,et al.  Calibration and Reliability in Groundwater Modelling: Coping with Uncertainty , 2002, Technometrics.