On the relation between excess hydraulic head and flow in a conduit imbedded within a porous matrix

This article provides a partial answer to the question "What is the relation between excess hydraulic head and volume flux of water in a conduit within a porous matrix?", focusing on the case that the forcing is steady. The conduit is modelled as a horizontal circular cylinder, imbedded within a porous matrix of rectangular cross section, having constant head prescribed on the sidewalls and being confined top and bottom. Laminar flow in the matrix is assumed to obey Darcy's law, while turbulent flow in the conduit is quantified using the Darcy-Weisbach equation. Analysis of the latter equation shows that the length scale of variations in the direction of the conduit is large compared with the scale of lateral and vertical variations. This permits separation of the full three-dimensional non-linear problem into a two-dimensional linear problem for head within the matrix and a one-dimensional non-linear problem for head within the conduit. Analytic solutions are obtained for the distribution of head in the matrix and in a conduit of either infinite or finite length. In both cases, the volume flux of water is proportional to the excess head to the 2/3 power, the conduit radius to the 5/3 power, the matrix permeability to the 1/3 power and gravity to the 1/3 power. The scale of variation of head along the conduit is proportional to the excess head to the -1/3 power, the conduit radius to the 5/3 power, the matrix permeability to the -2/3 power and gravity to the 1/3 power.