Analysis of Theoretical Concepts for Interpretation the Result of the Experimental Studies of Free Groundwater Level Oscillation in Wells

A number of publications present the results of an experimental study of free oscillations of groundwater piezometric level in wells with their eigenfrequencies. The oscillations were initiated by a pulsed impact on the aquifer through the well. Also in a number of publications a theoretical interpretation was proposed for the established phenomenon. However, the existing theoretical ideas about free oscillations of the groundwater level seem to be incorrect. In the present work, a critical analysis of these available theoretical concepts is performed. The analysis served as an impetus to the development of a consistent theory of relaxation filtration of groundwater.

It should be noted that in the process of such analysis it is not enough to declare the inconsistence of the available theoretical ideas being analyzed. The fact is that, according to the theory of evidence, such a statement is a negative statement. And it requires not only a detailed analysis of the available theoretical concepts, which allows answering the question why they are not correct, but also answering the next obvious question: which should be the correct theory? In other words, the negative statement made should be supported by the developed alternative theory that correctly represents the process in question, the phenomenon.

Formal Problems of Description of the Pressure Wave Propagation in Aquifer
By now, as it was already noted above, there are a significant number of publications that have studied wave processes in aquifers [2, 6, 20-22, 26, 32]. In these works, the data of studies of forced GW level oscillations in wells are mainly presented, although in [5,6,17,18,22,21] the results of field the presence of the oscillations having eigenfrequencies was assumed a priori, and the frequencies of forced oscillations were substantiated, which provide resonance with their own. In the majority of published works, mechanical analogs were used in the formal description of experiments (self-damped oscillations of the load attached to the free end of the spring). Such mechanical analogs are the basis for the formalization of forced level oscillations.
In some cases, the description of the propagation of waves with eigenfrequencies in water-bearing formations is carried out on the basis of the already accepted physical and mathematical concepts of GW motion (the main one being the linear filtration law and the filtration equation of parabolic type).
However, such a description is incorrect and can not always be used in the analysis of wave processes.
Let us explain what was said by example [7,13]. In [20] an attempt was made to find a solution to the problem of natural GW level oscillations in a disturbing well and in an aquifer near it. It was supposed that these oscillations are initiated by an instantaneous decrease or increase in pressure in the well ( Figure. 2). The process of wave propagation is described by the well-known parabolic equation of a one-dimensional axisymmetric nonstationary filtration, written in the piezometric level declining as: where   t r S , is the decrease (increase) in the level of GW at the point of the reservoir with coordinate r at the moment of time t from the beginning of the experiment; *    T -piezoconductivity of the reservoir (T-water conductivity, and μ*-elastic capacity of water-bearing sediments); (2) Here   t Q is the rate of the axial flow of water in the wellbore (inflow of GW into the well or outflow from it); 2 F F r F   -the area of the internal cross-section of the perturbing well; F r -radius of the internal cross section of the filter column; 0 r -well radius (outer radius of the filter column).
The solution of the problem with conditions (2) is presented in [20] as following: where   r f is a function of the r coordinate; -parameter; w  -eigenfrequency; -damping parameter of natural oscillations.
Substituting (3) into (1) leads to the ordinary differential equation: the solution of which [1, 28] is: From equation (6) it is obvious that the value of  should be positive. However, in this case it turns out from (3) that the function   t r S , increases indefinitely, and this is not possible, since additional energy does not enter into the well-aquifer system.
In equation (5) value , as it could be easily seen, is negative for any 1   . Therefore, when substituting (3) into (1) instead of equation (6), we get the following equation [7, 13]: The solution of the latter for natural oscillations, provided that the function   t r S , must remain finite It follows that with an instantaneous decrease (increase) of the level in the well (at 0  t ), the distribution of the decrease (increase) of the head in the reservoir will also instantly take the form: Figure 3). With time the change in the piezometric level at each point monotonously (without oscillation) decreases according to the law Obviously, the solution shown has no physical meaning, since it is difficult to explain the occurrence of standing waves (if they can be called waves at all) in a reservoir unlimited in the radial dimension. In other words, it does not satisfy the requirement of monotonicity of the characteristics of GW motion (in particular, the pressure distribution) within the filtration area for equations of parabolic type in the absence of a periodic component of forced oscillations.
If we specify  in the form of a complex number [7,13]: (here  is the oscillation damping parameter), similar to that specified in [20] from equation (4), and use the second condition from (2) to find the constant 1 C in (8), it is easy to show that this condition actually determines damped forced oscillations of the water level in a disturbing well. The detailed justification for such a problem for plane one-dimensional fluid motion is considered in [23]. In our case, the solution is represented as a cylindrical wave: So far, based on the assumption of presence of natural oscillations of the GW level, we have considered solutions corresponding to standing cylindrical waves. However, the latter can occur only in finite-volume media [23], and this contradicts the formulation of the problem adopted in [20] (the reservoir is assumed to be unlimited in the radial plane). For forced oscillations of the GW level in an unlimited aquifer, the solution of equation (7) should be sought in the form corresponding to a traveling diverging wave [23]. Therefore we get the following equation: It is not difficult to show that the latter equation in a simplified form in Laplace images (by modified second-order Bessel function of the second kind [1]. The remaining notation is the same. Solutions of a similar problem in a more general formulation (taking into account the imperfections of the perturbing well by the pattern and degree of opening of the aquifer, by the well and changes in the well filter radius), obtained using the method of integral transforms (Laplace and Laplace-Carson) are given in [16, 29] (in [29], moreover, a review of such solutions is given).
The results of the numerical transform from images of the form (9) to the original, presented in [16,29] in the form of reference curves make it possible to perform a qualitative analysis of the laws of the GW level restoration in a well after its instantaneous decrease or increase. Such an analysis clearly indicates, in particular (and this is the main thing), the absence of natural level oscillations. Thus, the description of such GW head oscillations in aquifers based on parabolic equations, as proposed in [20,21], is incorrect.
So, the formal problems of the pressure waves propagation in an aquifer were actually investigated above, but so far the consistency of the original physical and mathematical model of fluid motion has not been considered. Therefore, it remains unclear whether the noted discrepancy is a consequence of partial interpretation errors, or a consequence of the inadequacy of the physical and mathematical model used to describe the phenomenon observed in the experiments. We will try to analyze this physical and mathematical model here.

Description of the Piezometric Oscillations in a Perturbing Well in a Hydrodynamic Form
In accordance with the ideas of I. Krauss in the already mentioned papers, the process of damped oscillations of a load suspended from a spring fixed by one end acts as a mechanical analogue to the process of level oscillations in the well-aquifer system. This process is described by the following equation: where z is the coordinate of the displacement of the load relative to the equilibrium position (along the vertical axis z). The remaining notation is the same. . We will not dwell in detail on its presentation; we will consider only the physical assumptions used in the justification of equation (10) and well detailed in the same monograph [25]. In parallel, we will trace the derivation of the equation of motion obtained by I. Krauss on the basis of these assumptions.
In work [25] it is indicated that when the body moves in the medium, the latter provides resistance, tending to slow down the movement, which, strictly speaking, results in damped oscillations. The energy of a moving body in this case goes into heat or dissipates. The process of movement under these conditions is no longer purely mechanical; its consideration requires consideration of the movement of the medium itself and the internal thermal state, both of the medium and of the body. However, as noted in [25], there is a category of phenomena for which motion in a medium can be described using mechanical equations of motion by introducing some additional terms into them. These include, in particular, oscillations with frequencies that are small compared with those characteristic of internal dissipative processes in the medium. When this condition is fulfilled, it is permissible to assume that a friction force acts on the body, depending only on its speed.
If this speed is small enough, then the friction force can be expanded in its powers. The zero term of the expansion is equal to zero, since the friction force does not act on a fixed body, and the first non-vanishing term is proportional to the velocity [25]. Accordingly, we will assume that the fluid moving laminarly in the wellbore experiences a resistance proportional to the first degree of movement velocity. Considering the liquid being purely viscous, and neglecting the change in its mass in the well, we can present the force of resistance (friction) as: where  is a positive coefficient. The minus sign indicates that the force acts in the direction opposite to the direction of the velocity vector of the body (water column in the wellbore).
At the bottom of the water column in the wellbore there is a pressure force from the aquifer: where F r is, as before, the radius of the internal cross section of the filter column,   t r p F , is the current value of pressure in the fluid in the opened part of the aquifer (in the well itself), and 0 p is the pressure in the fluid in the undisturbed formation.
According to the Dalamber principle, the sum of the forces acting on the system of material points is equal to the inertia force [19]. The latter, in neglecting the change in the mass of fluid in the well, is: where m is the mass of fluid in the wellbore; It is necessary to pay attention to the fact that the term on the right-hand side of equation (11) is included in the algebraic sum on the left-hand side of equation (12)  When analyzing the obtained equations of free damped oscillations, it is necessary to pay attention to two circumstances that determine the physical features of the presented model, which were completely ignored in [20]. First, the model was based on a system with concentrated parameters. Accordingly, the mass of the moving body in (12) is assumed to be concentrated in a limited volume of space, so that, in accordance with the presentation of equations (10) and (12), the elastic force of the spring is applied to the whole body mass. Secondly, equation (12), as noted above, is true for an isolated dissipative system, and the system is limited only by the wellbore. In other words, in the model of damped level oscillations in the well, according to the condition, there is no provision for the exchange of energy and matter (water) between the well and the aquifer. The energy source of the level oscillations themselves can only be the energy of elastic compression of the fluid in the wellbore.
In order to overcome this difficulty, in [5,17,18,20,21], the effect on the reservoir in the model is proposed to take into account by applying the boundary condition of the form (2), written in [20] in the following form: Here   t r H , is the piezometric head at the point of the aquifer with the coordinate r at the moment of time t from the start of the disturbance; the remaining notation is the same.
We repeat, an isolated dissipative system cannot exchange mass with other systems, the exchange of energy for an isolated system is also prohibited. Thus, a condition of the form (14)  initially (when deriving the equation of motion), are not related. In this case, the damped oscillations are considered separately in an isolated oscillatory system-a well, and condition (14) -adiabatic values of the compression modulus;  is the density of the rod material (water in the well bore) as mentioned before. The remaining notation is the same.
In equation (15) there is no member that determines the resistance to the movement of fluid in the wellbore due to its wall friction . Accordingly, the oscillations are turn out to be continuous. Obviously, this is a definite abstraction, however, equation (15)  The solution of equation (15) where N is integer.
where  E is, as before, the water compression modulus (isothermal); Τ-absolute water temperature; p c is the specific heat of water at constant pressure;  is the relative change in the volume of water V when heated.
Taking into account the fact that the water temperature in the wellbore during the initiation of level From the above test, it follows that the characteristic minimum value of the eigenfrequency of the level in the well (we repeat, considered as an isolated system) significantly (approximately by 2-3 orders of magnitude) exceeds the frequencies obtained at the EFT of real aquifers and is equal in order of magnitudes   ·n·(10 -2 -10 -1 ) s -1 . This is primarily determined by the low value of water compressibility. Then it becomes unclear what, in fact, determines the water level oscillations in a well with eigenfrequencies of the order of   ·n·(10 -2 -10 -1 ) s -1 , recorded in experiments [20,21]?
So, the kinematic model of the oscillation of the load, suspended at the free end of the spring, used as an analogue for description of the free oscillations of the piezometric level in the well, does not allow to performe the test calculations to assess its applicability. It is simply not intended to calculate the eigenfrequencies using the "internal" parameters of the oscillating system-the mass of the suspended load, the geometric parameters (for example, the length) of the spring and spring rate; the model does not contain physical prerequisites for oscillations. By the way, for this reason, the entire physical interpretation of the natural oscillations of the level in the wells in the works by S.F. Grigorenko [5,17,18] is reduced to the arguments about some "transient processes" in the system of a well-aquifer.

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In addition, the absence of strict initial physical notions about the process allows to operate arbitrary and without any justification (including the introduction of new ones) with boundary conditions of the system. Thus, the water exchange condition is introduced into the kinematic model.
The need to ensure consistent interpretation of the oscillation process of the piezometric level in the well involuntarily causes a direct association with the oscillating load in the well, fixed on the free end of the spring (see Figure 4), prompted by the concepts of oscillatory processes used in [20,21]. Only in this case a source of energy necessary for initiation of oscillations appears, and the natural oscillations of the water column in the well will not be determined by the parameters of this column (its length 0 H and water compressibility), but by the characteristics of the spring and the inertia of the load suspended to it.
Attenuation of oscillations will be determined by the friction of the load on the water.
However, then the initial premise that both oscillations of the water column in the wellbore and pressure (or piezometric pressure) oscillations in the aquifer (outside the perturbating well itself) will become forced oscillations. And then it becomes indifferent what type of equation (parabolic or some other type) will describe the process of propagation of pressure waves in the aquifer.

Krauss
Thus, a system in which oscillations with eigenfrequencies can be observed should either be isolated but unified (i.e., oscillations should be considered as a single process at once in the entire system), or an open system, initially connected to another system, only in this case, the exchange of mass (water) and energy between the aquifer and the well is possible. In this case, the source of oscillations should be the aquifer, and the probe well should be considered as a reacting system, although the primary impact on the reservoir goes through the well.
In order to open the system, you must enter the flow component [27]. As such a streaming component, for opening a mechanical system, we can consider the equation of GW filtration. In this case, the flow component is considered as applied to the system (aquifer) with distributed parameters, therefore it should be of a gradient type.
Let us repeat the derivation of an equation that provides a description of free decaying oscillations of a piezometric level in a well with eigenfrequencies, taking into account the flow component.
So, the fluid in the well in accordance with the third law of Newton is acted upon by the fluid in the aquifer. Taking into account the relation of change in pressure in the aquifer in the process of its perturbation, and neglecting the change in the mass of fluid in the well, we will present this force as following: Here 0 t is, as before, the characteristic time; is the pressure at the point of the aquifer with the coordinate 0 r (on the borehole wall) at the moment of time t from the beginning of the disturbance;  is the Laplace operator. This is exactly the flow component that ensures the opening the well-aquifer system.
The moving laminarly in the wellbore, fluid experiences resistance proportional to the first degree of movement speed. Assuming the fluid is purely viscous, and again neglecting the change in its mass in the well, we can present the force of resistance (friction), as in (11), in the following form: As before, according to the Dalamber principle, we consider that the sum of the forces applied to the system of material points is equal to the force of inertia. The latter neglecting the change in the mass of fluid in the well can be written as: Hence, using the principle of continuity, in other words, assuming that is, as before, the piezometric head in the well, we obtain for Since all the forces in equation (18)

Result
A critical analysis of the available theoretical ideas about the process of pressure waves propagation in the aquifer is presented. These waves determine free oscillations of the GW piezometric level with eigenfrequencies in the disturbing well, through which the impulse impact on the aquifer was applied.
Firstly, the description of such a process based on a parabolic-type filtration equation that does not take into account the inertial component of the resistance to the GW motion, without the need to specify the boundary condition determining the oscillations of the piezometric GW level in the well due to some extraneous influence, in principle is possible. And then these oscillations are not free, but forced, and the frequencies of such oscillations are determined by the given frequencies of such an external source of oscillations.
Secondly, in the existing theoretical concepts, a perturbation well is actually considered as an isolated dissipative system. The estimation of the eigenfrequencies of the longitudinal free oscillations of the water column in the well significantly (by 2-3 orders of magnitude) exceeds the frequencies obtained by the EFT of real aquifer horizons. In other words, such a system cannot adequately represent the phenomenon of free oscillations of the piezometric level observed in the experiments in a probe well with eigenfrequencies.
Thirdly, the correct formulation of the problem leads to a completely different type of filtration equation for the GW, as compared to that used in the existing theoretical concepts, to a hyperbolic (wave) equation.
At the same time, the introduction of a flow component fundamentally changes the characteristics of a disturbing (experimental) well-aquifer system-it should become isolated, but uniform (ie, oscillations Published by SCHOLINK INC. should be considered as a single process within the whole system). Only in such a system the eigenfrequencies are determined by the geometrical and filtration parameters of the disturbed part of the aquifer. In this case, the source of oscillations should be the aquifer, and the probe well should be considered as a reacting system, although the primary impact on the reservoir has been made through this well.
Namely these provisions that were used as the basis for the theory of relaxation filtration, developed by the author, and will be presented in subsequent publications in English.

Discussion
I believe that in this article I was able to convincingly show the inconsistency of the currently used theoretical concepts of free oscillations in the GW level in disturbing wells to describe the interpretation of these oscillations. There is an obvious need to develop a consistent physical and mathematical theory that provides such a description and interpretation. I have proposed a version of such a theory, which will be presented in subsequent publications in English.