Improving of the Method of Calculating of Linearly-Extended Constructions of the Perfect Watertight Type Constructed by the Method “Wall in the Ground” With Initial Filtration Gradient

There is calculation method of basic parameters of linear-extended anti-filtration structures taking into account the initial filtration gradient of the filling material of “wall” in two cases. The first one is that the “wall” is defective type and it overlaps water-yielding heavily filtrated stratum with embedment into the second weakly filtrated stratum. The second one is the particular case of the decision of the common task and is the most effective. The second “wall” is a “wall” of a higher kind with embedment into an aquitard. The offered calculation method bases on the model which takes into account the non-linear way of water filtration through the “wall”. It lets to avoid the inaccuracies in calculations. The water filtration through the “wall” will take place when the excess gradient exceeds the initial gradient. The calculation method, described in regulatory documents, doesn’t take into account the initial gradient value. It influences on the choise of “wall” construction parameters. Taking into account the initial filtration coefficient lets to increase the calculations accuracy of anti-filtration structures. If the initial filtration gradient value is neglected, the calculation result from the obtained formulas for particular case will coincide with values, obtained by formulas, described in regulatory document.


Introduction
The calculation methodology proposed below is based on a model different from what is recommended by the existing standards [1,2] in that it considers the initial threshold level of filtration in writing diaphragm's material. Filtration of water through the diaphragm will only occur after the said threshold level is overcome and the surplus gradient exceeds the initial filtration gradient of the diaphragm -once it is overcome, filtration of water begins. Documents [1 and 2] use only the filtration ratio to describe the material, which is quite obviously insufficient. Some European nations, such as Germany, have regulated that the level on the diaphragm's outer side may be lower by up to 10 cm, or within 20 cm in some particular cases, whereas inflow of water into the pit over the diaphragm is restricted. Respective Russian regulations overlook such parameters, while commonly practiced methods of diaphragm parameter calculation tend to recommend oversized values. The issue of the non-linear nature of water filtration in low-permeability soils, considering the initial filtration gradient, was studied by: Karman, B.V. Deryagin, S.A. Roza, S.V. Nerpin, B.P. Gorbunov, M.Yu. Abelev and others. The writer adopted the same approach to the engineering problem in question.
It has to be said that issues related to impervious structures are covered in [6][7][8][9][10][11][12][13][14][15][16][17][18][19], where authors discussed the issues of technology improvement, researched parameters of ground waters moving around the impervious screen, factors bearing on production and filtration performance of aggregate materials and their composition, examined construction methods and problems related to organization of underground space.

Research Objective
The objective is to estimate dependencies underlying design of linear-type impervious structures, considering the initial gradient of filtration. Another objective is to find the principal parameters: flow velocity, head distribution before the diaphragm and in the material thereof, width and characteristics of the aggregate, water inflow to the enclosed outline. If necessary, the diaphragm can be designed with zero water inflow. In an earlier study of the method, the author proposed a way of finding key parameters of an imperfect-design impervious structure with plane-radial one-dimensional filtration [3].
An imperfect design diameter is now examined in the environment of plane-parallel onedimensional filtration. For the purposes of solution, we assume that the established filtration flow passing through the diaphragm is composed of three fragments: first is gravity filtration in the top layer, the other is under pressure around the bottom layer; the third fragment is pressure filtration constrained by the diaphragm base and the confining layer.

Results of the Study
Theoretical research resulted in estimated dependencies used to find key parameters needed to design impervious structures: flow velocity, water inflow, head distribution before the diaphragm and in the material thereof, width and characteristics of the aggregate. The study was assisted with a computer software pack; it calculated and analyzed the results for a wide range of soil conditions, with various engineering solutions of impervious structures, embedded in the low-impermeability layer based on the amount of water inflow to the enclosed outline.
Design diagram for the solution is represented in Fig. 1 The following conventions were used in the problem solution: Н0 -static head (read from the reference plane of the confining layer); H (b) -head in section before the diaphragm; H(y) -head distribution in the diaphragm material; H (y) -same but in the ground mass before the diaphragm; h0 -head in the ground mass enclosed by the diaphragm around the pit as water level is ultimately lowered; h1 -thickness of second layer (II); hi -height of i-th (1, 2, 3) fragment; V- height from the second layer's roof to lowered water level in the pit; d -distance from ground surface to the static level; Z -diaphragm perimeter around the pit; L -distance from diaphragm's inner edge to the point where H (y) = H0; b -diaphragm width; K, K1, K2 -filtration ratios of diaphragm's aggregate material (К) and soil (top K1 and bottom K2 layer); i0 -initial filtration gradient of the diaphragm's aggregate material; Vx(i),Vx -filtration rates in i-th fragment in the ground mass and in diaphragm material, respectively; q1, q2, q3 -filtration rate per length unit of the diaphragm outline, filtered through fragments 1, 2 and 3, respectively; Q -total water filtrate into the diaphragm-enclosed ground mass; r -distance from diaphragm to the pit center.
(1) Amount of filtrate through the i-th fragment is found as: qi = hi * Vx (0) .
(2) Filtration rate through the i-th fragment is known from the following equations: general velocity equation, for flat one-dimensional filtration: and flow continuity equation: In the light of (3), equation (4) is going to appear as: By integrating equation (5) by х, we can find head distribution in diaphragm material: where: С1 -integration constant. A singular water inflow filtered through the diaphragm body in the first fragment, according to (2), (3) and (6) shall be found to be: Integration constant С1 must consider (6); having divided our variables we get: In the first fragment, head Н is read from the second layer's roof Н(0) = h0 -h1. For this equation, the integral will be within the given range: Thus the equation for С1 will appear as: Assuming in equation (9) that x = b, we get the ratio of head H(b) in the section before the diaphragm, and C1: To find H(b) we need to know distribution of head in the ground mass before the diaphragm; seeing that i0 = 0, equation (5) appears to be: Distribution of head in the ground mass before the diaphragm is determined by integrating equation (11) with the premises that   To express ' C 1 through С1, we use the balance of flows q = q' in section x = b: whence: Now with substitute ' C 1 from (14) in (12), and assume that x = b, to find the head at the entry point of the diaphragm body: Reading the head from the confining layer, the entry point coordinate is: Now we find a singular inflow and distribution of heads in the second fragment (assuming the flow is independent by the fragments). Thus, velocity is known from: where head Н is found separately for each fragment.
Head distribution in the diaphragm material and in the groundmass of the second fragment is found as: where: The singular water inflow in the second fragment is known from the equation: (20) And finally, the singular water inflow in the third fragment should be found in the same way; for this, we use equation (5), which, assuming i0 = 0 and h = const, appears to be: We integrate (21), to get: Н=C1X+C2 . (23) Finding head distribution in the third fragment: Finding the singular water inflow in the third fragment: Now let us take the practical approach and examine a case most frequently used in the construction sector.
Perfect design: A diaphragm embedded in the confining layer, considering initial gradient of the aggregate material, the head falling in the ground mass in the section before the diaphragm.
We find b from the transcendent equation (17), assuming that h1=0: where: Our singular water inflow will be: q = K * C1 (27) The head in the groundmass before the diaphragm is found from equation (16), assuming that h1=0: International Considering the above, we propose a method to find the key parameters of diaphragm design and distribution of heads before the diaphragm. This method can prevent future adverse consequences and structural deformation in buildings or structures adjacent to the construction site. Some European nations, e.g. Germany, have regulated that the level on the diaphragm's outer side may be lower by up to 10 cm, or within 20 cm in some particular cases, whereas inflow of water into the pit over the diaphragm is restricted. Respective Russian regulations overlook such parameters, while commonly practiced methods of diaphragm parameter calculation tend to recommend oversized values. Error is caused by the fact that aggregate material of the diaphragm is described only by the filtration ration, while error actually depends on the level of initial filtration gradient and the diaphragm width, and tends to grow dramatically as they increase.
We have received certain dependencies (16,17,18,20,24,25) based on the initial filtration gradient of the aggregate material in order to calculate an imperfect-design diaphragm that penetrates the heterogeneous water-saturated layer (strong filtration (I)), embedded (in the second layer h1, weak filtration (II)). Efficiency of this diaphragm type, as supported by computer-assisted digital experiments, is ensured by varying diaphragm thickness b, and embedding depth in the second lowfiltration layer h1 (II). The second layer's filtration ratio must be lower than that of the strongly filtered layer (I) by the magnitude of five (10‾ 5 cm/sec).
A series of partial solutions (26, 27, 28) has been produced for a perfect-design diaphragm embedded in the confining layer, as the most efficient option. As a particular case of general problem solution, should we neglect the initial filtration gradient (i0 =0), the result of calculation to find the water inflow will match the results of the formulas recommended by the regulators [1, 2].
We need to emphasize that one may not neglect the initial gradient (i0 = 0), as also stated by the above researchers. If water inflow calculation overlooks the initial filtration gradient of the aggregate, the error of its estimate for the pit will increase. In other words, as diaphragm thickness grows and the initial filtration gradient of the aggregate increases along with the flow's elevation in the ground mass relative to the confining layer, and as the difference of the heads (static and reduced) falls, all this generates an error in estimation of the amount of water inflow to the pit enclosed by the diaphragm. Quantitative evaluation of such errors and its determining factors is covered in depth in [4] on a calculation example of a perfect-design diaphragm embedded in the confining layer (with plane-radial filtration). As an illustration, we are going to quote one of the studies.

Conclusions:
1. Dependencies (16, 17, 18, 20, 24, 25) have been identified that consider the initial filtration gradient of aggregate material to make calculations for impervious structures of an imperfect-design diaphragm that penetrates the heterogeneous water-saturated layer (I -strong filtration), embedded in the second layer (II -weak filtration) h1.
2. Efficient use of imperfect design diaphragms, as demonstrated with computer-assisted numeric tests, is ensured by varying diaphragm thickness b, and embedding depth in the second low-filtration layer h1 (II), the second layer's filtration ratio must be lower than that of the strongly filtered layer (I) by the magnitude of five (10‾ 5 cm/sec).
3. A series of partial solutions (26, 27, 28) has been produced for a perfect-design diaphragm embedded in the confining layer. The results matched those of the officially recommended formulas, if the initial gradient is neglected (i0 =0) [1, 2]. 4. Engineering calculations may not neglect the values of the initial filtration gradient of the aggregate material, as this causes overestimation of the diaphragm's design parameters, inaccurate head before the diameter, adoption of over-optimistic characteristics of the aggregate measurement, and too high estimated water inflow to the enclosed outline (pit).