Vijay K. Gupta

On the formulation of an analytical approach to hydrologic response and similarity at the basin scale

Abstract The hydrologic response at the basin scale represents the temporally and spatially averaged behavior of water transport in a channel network. Some theoretical studies have recently appeared in the literature which attempt to formulate and solve the problem of how the averaging of the network geometry and the dynamics determines the hydrologic response. A review of these recent papers shows that the incorporation of the network geometry in terms of the Strahler ordered channels is not appropriate. An alternate analytical approach is formulated which requires the use of the path number classification for this purpose. This approach also makes it apparent that the analytical geomorphology (via Shreves postulates) provides a natural avenue for undertaking analytical investigations on the structure of the hydrologic response. These connections are explored in the vein of formulating and solving the problem of hydrologic similarity.

Runoff generation and hydrologic response via channel network geomorphology — Recent progress and open problems

Abstract Recent theoretical emphasis on quantifying the processes of runoff generation in space-time in river basins has given a new focus to the fundamental importance of channel network geomorphology in river basin hydrology. Of particular significance in this context is the need for a comprehensive quantitative theory of channel networks in three dimensions reflecting the constraints of space filling and available potential energy as well as climatic, hydrologic and geologic controls which are in dynamical equilibrium with channel network forms. Recent progress related to these issues is discussed in a nontechnical way and illustrated with examples, and important open problems are identified.

On Scales, Gravity and Network Structure in Basin Runoff

Runoff generation and its transmission to the outlet from an ungaged river basin having an identifiable channel network are considered at the basin scale. This scale is much larger than the hydrodynamic scale, where the equations governing the transport of water overland and in saturated and unsaturated soils are best understood. Gravity, via altitude, plays the fundamental role in both the transport of water as well as in network formation via erosion and sediment transport. So, here altitude is identified as the natural parameter for physically rigorous descriptions of network structures in the context of hydrologic investigations at the basin scale. In this connection an empirical postulate is made on the link heights as being independent but possibly non-homogeneous random variables having an exponential distribution. Data from six river basins ranging in sizes from 1 sq. km to 100 sq. km and from different climatic regions are used to test the suitability of this postulate. The drainage scaling parameter D N is introduced as the number of links per unit area density in an infinitesimal increment of the altitude at the basin scale. Data from five of the six basins is analyzed to show qualitatively that these basins are homogeneous with respect to D N . This homogeneity along with that in the exponential nature of the link heights are used to illustrate that the total runoff generated by the sub-basin associated with any link of a basin, has a gamma distribution with parameters λ/\( \overline \mu \) and 2 (link magnitude) - 1., The parameters denoting the link magnitude, the mean link height λ-1, and the long time average volume of runoff per unit elevation of a link, \( \overline \mu \), are meaningful only at the basin scale.

Foundation theories of solute transport in porous media: a critical review

Abstract The theories that have been employed to derive the macroscopic differential equations that describe solute transport through porous media are reviewed critically. These foundational theories may be grouped into three classes: (1) those based in fluid mechanics, (2) those based in kinematic approaches employing the mathematics of the theory of Markov processes, and (3) those based in a formal analogy between statistical thermodynamics and hydrodynamic dispersion. It is shown that the theories of class 1 have had to employ highly artificial models of a porous medium in order to produce a well-defined velocity field in the pore space that can be analysed rigorously or have had to assume that well-defined solutions of the equations of fluid mechanics exist in the pore space of a natural porous medium and then adopt an ad hoc definition of the solute difusivity tensor. The theories of class 2 do not require the validity of fluid mechanics but they suffer from the absence of a firm dynamical basis, at the molecular level, for the stochastic properties they attribute to the velocity of a solute molecule, or they ignore dynamics altogether and make kinematic assumptions directly on the position process of a solute molecule. The theories of class 3 have been purely formal in nature, with an unclear physical content, or have been no different in content from empirically based theories that make use of the analogy between heat and matter flow at the macroscopic level. It is concluded that none of the existing foundational theories has yet achieved the objectives of: (1) deriving, in a physically meaningful and mathematically rigorous fashion, the macroscopic differential equations of solute transport theory, and (2) elucidating the structure of the empirical coefficients appearing in these equations.

A molecular approach to the foundations of the theory of solute transport in porous media: I. Conservative solutes in homogeneous isotropic saturated media

Abstract A molecular model is developed for the transport of a conservative solute at low concentrations in a homogeneous isotropic water-saturated porous medium. In this model, a solute molecule undergoes contact collisions with the solid grains in the medium at successive random times; in between these collisions, the velocity of the molecule is governed by the Langevin equation; the effect of the collisions with the solid grains is to scatter a molecule in a random direction. This molecular model is employed to derive rigorously a parabolic differential equation for the solute concentration at the macroscopic level. In the absence of solute convection, the coefficient of molecular diffusion in a porous medium is proved to be less than the coefficient of molecular diffusion in bulk solution, a finding which is in agreement with experimental observations. For non-zero convection, the assumption of isotropicity of the medium is employed to prove that the solute dispersion tensor is diagonal. For small magnitudes of the liquid velocity, the isotropicity assumption also implies that the coefficients of longitudinal and transverse dispersion are approximately parabolic functions of the liquid velocity. The expressions derived for the dispersion coefficients, when compared with their experimentally observed values, suggest that their dependence on the liquid velocity comes primarily through solute—liquid molecular collisions instead of through collisions of the solute molecules with the grains of the solid phase. The solute convective velocity is shown to be less than or equal to the liquid velocity. Precisely when the difference between the two velocities will be significant remains to be established.