Hydrology and Earth System Sciences www.hydrol-earth-syst-sci.net 1027-5606 1607-7938 13 7 2009 10.5194/hess-13-1123-2009 http://www.hydrol-earth-syst-sci.net/13/1123/2009/ http://www.hydrol-earth-syst-sci.net/13/1123/2009/hess-13-1123-2009.html http://www.hydrol-earth-syst-sci.net/13/1123/2009/hess-13-1123-2009.pdf 1123 1132 2009-07-13 Averaging hydraulic head, pressure head, and gravitational head in subsurface hydrology, and implications for averaged fluxes, and hydraulic conductivity G. H. de Rooij ger.derooij@wur.nl Wageningen University, Centre for Water and Climate, Soil Physics, Ecohydrology and Groundwater Management, Wageningen, The Netherlands Current theories for water flow in porous media are valid for scales much smaller than those at which problem of public interest manifest themselves. This provides a drive for upscaled flow equations with their associated upscaled parameters. Upscaling is often achieved through volume averaging, but the solution to the resulting closure problem imposes severe restrictions to the flow conditions that limit the practical applicability. Here, the derivation of a closed expression of the effective hydraulic conductivity is forfeited to circumvent the closure problem. Thus, more limited but practical results can be derived. At the Representative Elementary Volume scale and larger scales, the gravitational potential and fluid pressure are treated as additive potentials. The necessary requirement that the superposition be maintained across scales is combined with conservation of energy during volume integration to establish consistent upscaling equations for the various heads. The power of these upscaling equations is demonstrated by the derivation of upscaled water content-matric head relationships and the resolution of an apparent paradox reported in the literature that is shown to have arisen from a violation of the superposition principle. Applying the upscaling procedure to Darcy's Law leads to the general definition of an upscaled hydraulic conductivity. By examining this definition in detail for porous media with different degrees of heterogeneity, a series of criteria is derived that must be satisfied for Darcy's Law to remain valid at a larger scale. Bear, J. and Bachmat, Y.: Introduction to modeling of transport phenomena in porous media, Kluwer, Dordrecht, The Netherlands, 1991. Bertin, H. and Quintard, M.: Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system, Transport in Porous Media, 5, 543–590, 1990. Bhattacharya, R. N. and Gupta, V. K.: Theoretical explanation of solute dispersion in saturated porous media at the Darcy scale, Water Resour. Res., 19, 938–944, 1983. Brooks, R. H. and Corey, A. T.: Hydraulic properties of porous media, Hydrol. paper no. 3, Colorado State Univ. Fort Collins, Colorado, USA, 1964. Brutsaert, W.: Hydrology. An introduction, Cambridge University Press, Cambridge, UK, 2005. Cressie, N. A. C.: Statistics for spatial data. Revised edition, John Wiley & Sons, Inc., New York, 1993. Crapiste, G. H., Rotstein, E., and Whitaker, S.: A general closure scheme for the method of volume averaging, Chem. Eng. Sci., 41, 227–233, 1986. Dane, J. H. and Hopmans, J. W.: Water retention and storage. Laboratory. General, in: Methods of soil analysis. Part 4. Physical methods, edited by: Dane, J. H. and Topp, G. C., SSSA, Madison, Wisconsin, USA, 675–680, 2002a. Dane, J. H. and Hopmans, J. W.: Hanging water column, in: Methods of soil analysis. Part 4. Physical methods, edited by: Dane, J. H. and Topp, G. C., SSSA, Madison, Wisconsin, USA, 680–683, 2002b. Dane, J. H. and Hopmans, J. W.: Pressure cell, in: Methods of soil analysis. Part 4. Physical methods, edited by: Dane, J. H. and Topp, G. C., SSSA, Madison, Wisconsin, USA, 684–688, 2002c. Dane, J. H. and Hopmans, J. W.: Pressure plate extractor, in: Methods of soil analysis. Part 4. Physical methods, edited by: Dane, J. H. and Topp, G. C., SSSA, Madison, Wisconsin, USA, 688–690, 2002d. de Rooij, G. H., Kasteel, R. T. A., Papritz, A., and Flühler, H.: Joint distributions of the unsaturated soil hydraulic parameters and their effect on other variates, Vadose Zone J., 3, 947–955, 2004. Gray, W. G.: A derivation of the equations for multi-phase transport, Chemical Engineering Science, 30, 229–233, 1975. Gray, W. G.: On the definition and derivatives of macroscale energy for the description of multiphase systems, Adv. Water Resour., 25, 1091–1104, 2002. Gray, W. G. and Miller, C. T.: Examination of Darcy's Law for flow in porous media with variable porosity, Environ. Sci. Technol., 38, 5895–5901, doi:10.1021/es049728w, 2004. Gray, W. G. and Miller, C. T.: Consistent thermodynamic formulations for multiscale hydrologic systems: Fluid pressures, Water Resour. Res., 43, W09408, doi:10.1029/2006WR005811, 2007. Groenevelt, P. H. and Bolt, G. H.: Non-equilibrium thermodynamics of the soil-water system, J. Hydrol., 7, 358–388, 1969. Harter, T. and Hopmans, J. W.: Role of vadose-zone flow processes in regional-scale hydrology: review, opportunities and challenges, in: Unsaturated-zone modeling. Progress, challenges, and applications, edited by: Feddes, R. A., de Rooij, G. H., and van Dam, J. C., Wageningen UR Frontis Series, Vol. 6, Kluwer Academic Publishers, Dordrecht, The Netherlands, 179–208, 2004. Hassanizadeh, S. M. and Gray, W. G.: Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Water Resour., 13, 169–186, 1990. Hillel, D.: Environmental Soil Physics, Academic Press, San Diego, California, USA, 1998. Howes, F. A. and Whitaker, S.: The spatial averaging theorem revisited, Chem. Eng. Sci., 40, 1387–1392, 1985. Jacquin, C. G. and Adler, P. M.: Fractal porous media II: Geometry of porous geological structures, Transport Porous Med., 2, 571–596, doi:10.1007/BF00192156, 1987. Jackson A. S., Miller, C. T., and Gray, W. G.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow, Adv. Water Res. 32, 779–795, doi:10.1016/j.advwatres.2008.11.010, 2009. Jury, W. A., Gardner, W. R., and Gardner, W. H.: Soil physics, 5th ed., John Wiley & Sons, Inc., New York, 1991. Kavvas, M. L.: General Conservation Equation for Solute Transport in Heterogeneous Porous Media, J. Hydrol. Eng., 6, 341–350, doi:10.1061/(ASCE)1084-0699(2001)6:4(341)), 2001. Kitanidis, P. K.: Groundwater flow in heterogeneous formations, in: Subsurface flow and transport: A stochastic approach, edited by: Dagan G. and Neuman, S. P., Unesco, International Associations of Hydrological Sciences, Cambridge University Press, 83–91, 1997. Lee, H., Zehe, E., and Sivapalan, M.: Predictions of rainfall-runoff response and soil moisture dynamics in a microscale catchment using the CREW model, Hydrol. Earth Syst. Sci. 11, 819–849, 2007. Liu, H. H. and Dane, J. H.: Improved computational procedure for retentions of immiscible fluids using pressure cells, Soil Sci. Soc. Am. J., 59, 1520–1524, 1995. Nordbotten, J. M., Celia, M. A., Dahle, H. K., and Hassanizadeh, S. M.: Interpretation of macroscale variables in Darcy's Law, Water Resour. Res., 43, W08430, doi:10.1029/2006WR005018, 2007. Quintard, M. and Whitaker, S.: Two-phase flow in heterogeneous porous media: The method of large-scale averaging, Transport in Porous Media, 3, 357–413, 1988. Quintard, M. and Whitaker, S.: Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients, Transport in Porous Media, 5, 341–379, 1990a. Quintard, M. and Whitaker, S.: Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system, Transport in Porous Media, 5, 429–472, 1990b. Raats, P. A. C. and Klute, A.: Transport in soils: The balance of mass, Soil Sci. Soc. Am. Proc., 32, 161–166, 1986a. Raats, P. A. C. and Klute, A.: Transport in soils: The balance of momentum, Soil Sci. Soc. Am. Proc., 32, 452–456, 1986b. Renard, P. and de Marsily, G.: Calculating equivalent permeability: a review, Adv. Water Res., 20, 253–278, 1997. Romano, N., Hopmans, J. W., and Dane, J. H.: Suction table, in: Methods of soil analysis. Part 4. Physical methods, edited by: Dane, J. H. and Topp, G. C., SSSA, Madison, Wisconsin, USA, 692–698, 2002. Roth, K.: Scaling of water flow through porous media and soils, Europ, J. Soil Sci., 59, 125–130, doi:10.1111/j.1365-2389.2007.00986.x, 2008. Vereecken, H., R. Kasteel, J. Vandenborght, and T. Harter: Upscaling hydraulic properties and soil water flow processes in heterogeneous soils: a review, Vadose Zone J., 6, 1-28, 2007. Veverka, V.: Theorem for the local volume average of a gradient revisited, Chem. Eng. Sci., 36, 833–838, 1981. van Genuchten, M. T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, 892–898, 1980. Wagenet, R. J.: Water and solute flux, in: Methods of soil analysis. Part 1. Physical and mineralogical methods, 2nd ed., edited by: Klute, A., ASA, SSSA, Madison, Wisconsin, USA, 1055–1088, 1986. Whitaker, S.: Diffusion and dispersion in porous media, AIChE Journal, 13, 420–427, 1967. Whitaker, S.: Flow in porous media I: A theoretical derivation of Darcy's Law, Transport in Porous Media 1, 3–25, 1986. Yeh, T.-C. J.: Scale issues of heterogeneity in vadose-zone hydrology, in: Scale dependence and scale invariance in hydrology, edited by: Sposito, G., Cambridge University Press, 224–265, 1998. Zehe, E., Lee, H., and Sivapalan, M.: Dynamical process upscaling for deriving catchment scale state variables and constitutive relations for meso-scale process models, Hydrol. Earth Syst. Sci., 10, 981–996, 2006.