Oscillations for an Equation Arising in Groundwater Flow With the Relaxation Time

Abstract

Groundwater flow problems are mostly formulated by means of massbalance equation combined with Darcy's law. In this way, the flow is governed by a parabolic equation. To prevent inaccuracies which may result from this formulation, the Cattaneo approach can be utilized. The paper presents groundwater flow equation adopting the Cattaneo approach. In both 2D and 3D cases, the equation is of hyperbolic type and contains a constant known as relaxation time. The article focuses further on energy solutions defined on unbounded time interval. It is shown that under certain conditions, such solutions are oscillatory. The conditions sufficient to ensure the oscillatory solutions are derived. An upper bound for the oscillatory time is proved to be independent of the particular solution.

Oscillatory Solutions of Some Autonomous Partial Differential Equations with a Parameter

“…The problem may be considered an abstract extension of the problem studied in [12] oscillations for an equation arising in groundwater flow with relaxation time.…”

A new mathematical model of asymmetric hydraulic fracture growth

“…The changes of the liquid-phase momentum can be taken into account by applying the Cattaneo principle and involving the relaxation time into equation (4); see the work of Cattaneo (1958), and for groundwater flow, see the work of Mls and Herrmann (2011). The changes of the liquid-phase momentum can be taken into account by applying the Cattaneo principle and involving the relaxation time into equation (4); see the work of Cattaneo (1958), and for groundwater flow, see the work of Mls and Herrmann (2011).…”

“…The studied process is non-stationary and particularly due to the growth of the fracture; the momentum of liquid in the fracture changes. The changes of the liquid-phase momentum can be taken into account by applying the Cattaneo principle and involving the relaxation time into equation (4); see the work of Cattaneo (1958), and for groundwater flow, see the work of Mls and Herrmann (2011). In this paper, we assume that the inertial forces are negligible and do not modify equation (4).…”

Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

“…There are experimental evidences that diffusive process takes place with finite velocity [5][6][7]. This issue can be ignored in some applications, but otherwise it is necessary to consider the wave nature of diffusive process [8]. Here we have considered a two dimensional hyperbolic transport equation in which both convection and diffusion are responsible for flow motion, which can be thought of as a more general telegrapher's equation.…”