Inertia in two simple problems of soil hydrodynamics

Role of inertia in falling head permeability test

“…The predicted distributions of height (h) and velocity (dh/dt) of flow in the column are reasonable from the physical point of view and were discussed in detail in [5,6].…”

“…Although the fluid velocity all along the test is variable and for high-permeability materials could be large, the model used in the standard interpretation of the test assumes the validity of the quasi-static linear approximation of the motion equation, e.g. [4][5][6]. Taking into account the well-known results of macroscopic description of fluid-saturated porous materials (see [7,8]), with disregarded deformations of solid skeleton, and results of modelling high-velocity flow in porous materials (e.g.…”

“…Since the first two forces depend on macroscopic acceleration of fluid they will be called macro-inertial forces as opposed to the third force called the micro-inertial force. The literature devoted to the analysis of role of different factors disregarded in the quasi-static linear approximation of the falling head test is limited to discussions of model and influence of inertia of the column of fluid above and in the sample on distribution of fluid velocity, [5,6]. The role of dynamic interaction force (also known as the added mass interaction force) and micro-inertial forces was not thoroughly discussed.…”

A comparison of dynamic and quasi-static results for the flow towards a well

“…In the case of the radial horizontal flow towards a well in a confined aquifer, the solution of = 2 H ¼ 0 delivering the piezometric head as a function of the radial coordinate r reads H(r) ¼˜H ln (r=r 1 )=ln (r 2 =r) 1 þ H(r 1 ), in which˜H ¼ H(r 2 ) À H(r 1 ) is the piezometric head difference between two arbitrary points 1 and 2 located at distances r 1 and r 2 from the well axis (Muskat, 1946). According to this theory, the flow rate into the well is given by Q ¼ 2ðDk˜H=ln (r 2 =r 1 ), where D is the aquifer thickness.It is known that the law v a ¼ Àk grad H corresponds to the differential equation motion with the acceleration term set equal to zero (Polubarinova-Kochina, 1962;Kolymbas, 1998;Nader, 2009a;2009b). Thus, the traditional analysis neglects the acceleration that occurs, even in a steady flow, as the water particles approach the well.…”

“…It is known that the law v a ¼ Àk grad H corresponds to the differential equation motion with the acceleration term set equal to zero (Polubarinova-Kochina, 1962;Kolymbas, 1998;Nader, 2009a;2009b). Thus, the traditional analysis neglects the acceleration that occurs, even in a steady flow, as the water particles approach the well.…”