Hidrológiai Közlöny 1968 (48. évfolyam)

8. szám - Varrók Endre: A nyomásvonal alakulása töltésezett folyók mellett

Varrók E.: A nyomásvonal alakulása Hidrológiai Közlöny 1968. 8. sz. 345 cover, first written up by Boussinesq (5) is The equation expresses the continuity of flow and states that the difference between the amounts of water seeping inwards and outwards through the verticai faces of a soil prism of unit base area under the ground water table (as shown in Fig 2.) is equal to the pore volume swept by the changing surface of the ground water. n is the pore volume, k the coefficient of permeability of the soil. Writing up the equation, Boussinesq assumed the absence of evapotranspiration and of recharge by precipitations through the ground water table. Impermeable boltom Fig. 2. Diagram to interpret the notations in the Boussinesq equation 2. ábra. Értelmező ábra a Boussinesq-egyenlethez The equation is based on the further assumptions that (1) inertial forces are negligible, (2) the components v

x and v

y of the rate of seepage do not change vertically: this is why the above expression is a function only of time and the two horizontal coordinates. The verticai component v

z of the seepage rate varies linearly with height. When treating unsteady ground water movement, the textbooks usually start from the Boussinesq equation (e. g. 3, 4). The results arrived at refer — in keeping with the initial assumptions — to ground water movement in uncovered aquifers. In the case to be treated here, the verticai movement of ground water is limited; under such conditions, the Boussinesq equation makes no sense. However, parts of the procedure applied in its approximate analytical solution are useful alsó in the case of the movement under pressure particularlv as regards the evaluation of measured data, and for this reason we shall briefly outline the elements of the said approximate solution. The originál form of the equation is unsolvable: it has to be linearized to make a solution possible. The operation by which linearization is achieved is the introduction of a constant in place of the quantity (H + h) depending on the coordinates x and y. This is justified if the change of h is negligible as compared to (H + h). Denoting this constant by m and considering ground water movement as a planar problem, we obtain the linearized, Fourier variety of the Boussinesq eqution: 9

h _ mk d

2h ~~dT~~n dx*' ( ' TTZrC where = a

2 is a constant, of cm

2/sec dimensin ' on, for a given soil prism. Eq. (2) is a partial differential equation of constant coefficients, belonging to the group called "parabolic". A "rigorous" solution of parabolic equations was given already by Fourier. We shall not go into the details of the solution here: we shall point out only a few steps of importance from the point of view of the problem in hand. The independent variables of the linearized eq. (2) are t and x. Let us denote the looked-for function by h(x: t). Fourier a solution is based on the separation of the variables, i. e. on finding functions f^x) and f

2(t), depending solely on x and t respectively, which satisfy the boundary conditions of the problem, and whose product is just the looked-for function h(x\ t): h(x-t)=f

1(x)-f

i(t) (3) As the pressure fluctuation is considered in a relatively short time-interval within an infinitely long chain of events, beginning at an arbitrary instant t

0, it may be assumed that the initial conditions do not affect the actual pressure distribution. Consequently, the problem in hand is one with no initial conditions. A damping of the pressure fluctuation will take place alsó in covered aquifer but in this case pressure will be stored in the form of an elastic deformation of water and of the solid rock rather than in the form of a water level rise. For this case Theis proposed the introduction into (2) instead of n of a likewise dimensionless quantity s by s=r°

m(í

+-iB where y

0 is the specific weight of water at atmospheric pressure, m is the thickness of the aquifer, e is the pore coefficient, and Es and Ek are the elastic moduli of the liquid and of the solid grains, respectively. From the point of view of the damping of a pressure fluctuation, the verticai displacement (uplifting) of the top layer by pressure plays almost the same role as elastic deformation, provided a decrease of pressure is accompanied by the subsidence of the deposits into their originál position. Thus, in effect, the authors treating unsteady ground water movement have used differential equations of the same structure to describe move-