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Boundary layer analysis of unsaturated seepage into cylindrical cavities
Author(s) -
Ross Benjamin
Publication year - 2007
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2006wr005216
Subject(s) - discretization , curvature , bessel function , boundary layer , mathematical analysis , capillary action , mathematics , radius , boundary value problem , radius of curvature , mechanics , geometry , physics , mean curvature , mean curvature flow , thermodynamics , computer security , computer science
An analytical solution has previously been published for flat capillary barriers in uniform layered rock with a quasi‐linear relative permeability function. This solution can be extended to curved surfaces with a radius of curvature that is much greater than the characteristic length of the relative permeability curve α −1 because water flows around the cavity in a relatively thin boundary layer. A simple formula is obtained for the onset of seepage into cylindrical cavities. Published solutions to the quasi‐linear seepage problem, which take the form of infinite sums of Bessel functions, agree with this formula in appropriate limits. Studies of seepage into a proposed radioactive waste repository have used a finite difference model to solve the equations of unsaturated flow explicitly. Using the boundary layer solution, the discretization error in such numerical models is calculated. The numerical estimate of the diversion capacity of a capillary barrier is shifted away from the exact solution by a factor of αd cos ϕ/sinh( αd cos ϕ), where 2 d is the grid spacing and ϕ is the slope of the cavity wall.