T N Narasimhan
Articles written in Journal of Earth System Science
Volume 108 Issue 2 June 1999 pp 69-79
For nearly two centuries, the partial differential equation of heat conduction has constituted the foundation for analyzing many physical systems, including those involving the flow of water in geologic media. Even as the differential equation continues to be a powerful tool for mathematical analysis in the earth sciences, it is useful to look at the groundwater flow process from other independent perspectives. The physical basis of the partial differential equation is the postulate of mass conservation. Alternatively, it is possible to understand groundwater movement in terms of energy and work because mechanical work has to be done in moving water against the resistance to flow offered by the solid material and to store water by opening up pore spaces. To this end, the behaviors of steady-state and transient groundwater systems are sought to be understood in terms of postulates concerning the state of a groundwater system, its tendency to optimally organize itself in response to impelling forces and its ability to store and release energy. This description of groundwater occurrence and flow, it is shown, is equivalent to the variational statement of the Laplace equation for the steady-state case and is similar to Gurtin’s (1964) variational principle for the transient case. The approach followed here has logical similarities with Hamilton’s principle for dynamical systems. Though the variational statement of the transient groundwater flow process is appealing in that it provides a rationale for deriving the parabolic equation, intriguing questions arise when one attempts to understand the physical significance of the variational statement. This work is motivated in part by a desire to develop a better understanding of the groundwater flow process from an intuitive base pertaining to discrete systems. Also, as we show an increasing preference to numerically solve groundwater flow problems on the basis of integro-differential equations, it is likely that the work presented here may contribute to improving such integral solution techniques.
Volume 108 Issue 3 September 1999 pp 117-148
The equation describing the conduction of heat in solids has, over the past two centuries, proved to be a powerful tool for analyzing the dynamic motion of heat as well as for solving an enormous array of diffusion-type problems in physical sciences, biological sciences, earth sciences, and social sciences. This equation was formulated at the beginning of the nineteenth century by one of the most gifted scholars of modern science, Joseph Fourier of France. A study of the historical context in which Fourier made his remarkable contribution and the subsequent impact his work has had on the development of modern science is as fascinating as it is educational. This paper is an attempt to present a picture of how certain ideas initially led to Fourier’s development of the heat equation and how, subsequently, Fourier’s work directly influenced and inspired others to use the heat diffusion model to describe other dynamic physical systems. Conversely, others concerned with the study of random processes found that the equations governing such random processes reduced, in the limit, to Fourier’s equation of heat diffusion. In the process of developing the flow of ideas, the paper also presents, to the extent possible, an account of the history and personalities involved.
Volume 115 Issue 2 April 2006 pp 219-228
Hydrogeological systems are earth systems influenced by water. Their behaviors are governed by interacting processes, including flow of fluids, deformation of porous materials, chemical reactions, and transport of matter and energy. Here, coupling among three of these processes is considered: flow of water, heat, and deformation, each of which is represented by a diffusion-type of partial differential equation. One side of the diffusion-type equation relates to motion of matter or energy, while the other relates to temporal changes of state variables at a given location. The coupling arises from processes that govern motion as well as those that relate to change of state. In this work, attention is devoted to coupling arising from changes in state. Partial derivatives of equations of state constitute the capacitance terms of diffusion-type equations. Of the many partial derivatives that are mathematically possible in physical systems characterized by several variables, only a few are physically significant. Because the state variables are related to each other through an equation of state, the partial derivatives must collectively satisfy a closure criterion. This framework offers a systematic way of developing the coupled set of equations that govern hydrogeological systems involving the flow of water, heat, and deformation. Such systems are described in terms of four variables, and the associated partial derivatives. The physical import of these derivatives are discussed, followed by a description of partial derivatives that are of practical interest. These partial derivatives are then used as the basis to develop a set of coupled equations. A brief discussion is presented on coupled equations from a perspective of energy optimization
Volume 116 Issue 6 December 2007 pp 465-467
Volume 117 Issue 3 June 2008 pp 237-240
Some recent analyses of India ’s water budget are based on information attributed to the Ministry of Water Resources.An examination of the budget components indicates that they imply an evapotranspiration estimate that is signiﬁcantly lower than what one may expect based on information from other sources.If such is the case,India ’s water resources situation may be more dire than is otherwise perceived.For,higher evapotranspiration implies correspondingly reduced availability of water for human use.It should be worthwhile to investigate and reconcile the apparent discrepancy between water budget and evapotranspiration,considering the importance of water in the national context.