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HYDROGEOLOGICAL
CALCULATION OF SPOUTING WELLS
Nver l. Melikyan
The
Underground waters department of the Research Institute of Water Problems and Hydraulic
Engineering,
125 Amaranotcain Str., Yerevan, 375049, Republic of
Armenia,
Tel. 589 462
In the paper is considered the problem of hydrogeological calculation of
spouting wells operating under the springy forces of artesian water-bearing
layer. A mathematical simulation procedure is given for operating spouting
wells on network models. The obtained formula (7) is proposed as a result of
generalised problem solution by this method for a single spouting well, which
is giving an opportunity to determine the discharge in the start of the well
operation as well as to forecast its further decreasing in a time.
In the
paper is considered also the redistribution of a piezometric pressure head in a
water-bearing layer at the acting of the spouting well. Also are obtained
formulas (15) and (16) as similar series for the determination of reduction of
a pressure accordingly in any point of water-bearing horizon and in the well at
any moment of time. The checking of the offered formulas by actual observations
and measurements of the spouting wells in the Ararat valley of Armenia, as well
as model investigations on the hydrointegrator permit to propose them for
using, in designing and operation process of spouting wells.
Introduction
The water intake from high-pressure artesian water-bearing layer is carry out with the help of
spouting wells under the action of springy compression forces of a
water-bearing layer.
The problem of exact designing and operation of spouting wells consists
in the determination of its initial discharge and forecasting of its further
reducing during in a time, as well as in the determination of a pressure
reduction in any point of a water-bearing
layer (including in a well), with the mandatory account of interwells
hydraulics.
The posed problem cannot be solved by the accounting formulas, which
nowadays are available in theory of underground hydraulics obtained for
vertical wells, acting with a constant discharge or with a constant pressure in
well [4, 8, 10]. Herewith the originating difficulties are connected to
nonlinearity of a boundary condition on a water intake well.
Let's
consider inflow of springy water to single spouting well placed in uniform, isolated
and unlimited water-bearing layer of a constant potency.
The unsteady planned motion of filtration water within the given layer
is described by the equation [4, 8, 9, 10]:
(1)
where K is the water-bearing layer filtration
factor; m - its power; H - piezometric head in the point with the co-ordinates
(x, y) on t moment of the time; m* - an elastic water yield factor of the water-bearing layer.
At inflow of water to spouting
well a boundary condition for the equation (1) will be:
At the
t=0, H(x,
y, 0) = He (2)
At the
t>0, 
(3)
On a contour of a well:
2pkmr0 
(4)
where Q (t) is the discharge of a well, which is an unknown function and
is the subject to the definition.
The discharge is expressed on formula (5) in internal hydraulics of
spouting well:
Q(t)
= jp
(5)
Or neglecting a resistance of an air we will have:
Q(t)
= p
(6)
where He - a pressure of underground waters in natural
conditions (at t=0); H0 - mark of a mouth of a well; H(r0,
t) - a pressure in filtered part of a well; r0 - a radius of a well,
- a height of a fountain above the mouth of a well, j - a factor of a velocity of water motion in its shaft; r = 
- a distance of a point of the water-bearing layer from a
well.
The analytical solution of the equation (1) under such boundary
conditions is absent because of nonlinearity of a condition (4).
In such cases usually resort to numerical methods of a solution.
Using the theory of mathematical simulation on network models and the
principle of hydraulic simulation, on the hydrointegrator of the Lukyanovs
system [1, 3], we have created a mathematical model of a united hydraulic
system: a water-bearing layer - a spouting well with making available the
appropriate condition of a water motion (laminar motion in water-bearing layer
and turbulent motion inside the well). Herewith is designed a special device
simulating a nonlinear hydraulic resistance, which is originating inside the
well. This device is connected with the hydrointegrator. Also the scale factors
of simulation have been obtained for the radius of a well coming from condition
of fountain and for nonlinear hydraulic resistances, coming from condition of
hydraulic losses of energy in the well [5, 6].
The generalised problem have been solved by the developed method for
single spouting wells.
We have investigated the modifications of a discharge of a well in the
time and pressure in water-bearing layer in a broad band of values of
parameters of a water-bearing layer (factor of a filtration, power, elastic
waterloss, initial positive pressure) and wells (depth, radius, imperfection,
nonlinear hydraulic resistance). The considered variants were divided on
separate groups, and in a limit of one group, the values only of one parameter varied
to straighten find out a character of its influence on a spout process.
The criteria equation, describing the spout process, is posed on the
Langaar method with the help of matrixes and analysis of dimensionality [5].
The dimensionless complexes are composed from the parameters
characterising a water-bearing layer and a spouting well, and are defined their
values by results of the solution of the generalised problem. The graphic
connection is composed between these complexes and by using a method of least
guadrates also is detected it analytical expression. As a result is obtained
the following calculated formula for the definition of the discharge of a
spouting well at any moment of time.
(7)
F0
= ln
+2xHC (8)
where (Hn=He
- H0) - an initial positive pressure above a mouth of a well;
a=
- factor of a piezoconductivity; xHC - a factor of an imperfection of a
well; h - a hydraulic resistance of a well.
Now let's
consider the redistribution of a piezometric pressure head in a water-bearing
layer.
M. Macket has received the formula (9) as the solution of the equation
(1) by using a method of sources and run-of and by accepting, that in
considered filtration medium acts constantly a point source, the discharge of
which varies under the law Q(t) since any moment t of the time:
S
= He-H = 
, (9)
where S is
a decrease of a pressure head in water-bearing layer on distance r from the water intake well.
This
solution is lead up to the calculated formula only for point sources with
constant discharge in time.
Below we
bring one solution at variable discharge, that is characteristic for spouting
well.
Let us to
consider, that in some interval of time the function Q(t) can be expanded by its Taylor series expansion following to E.Chorny
[6].
By marking:
=U; in it 
; 
; (10)
t = t - 
, dt=
, (11)
We'll
have:
(12)
where Q(n)(t)
n numbered derivative of a function Q(t).
The
substitution of Eq. (10)...(12) in (9) yields
(13)
Taking into account, that 
and substituting (12) in (13), we shall receive:
(14)
Integrating
the equation (13) piecemeal, we shall receive:
After some
transformations in a final view we shall receive decrease of a pressure head in
any point of the artesian water-bearing layer in any moment of the time at a
operation of a single spouting well:
(15)
where Ei(-b) is exponential function.
It is
enough to suppose, that in the Eq. (15) r=r0 for the determination
of a decrease of the pressure head inside of the spouting well itself. In this
case it is possible to accept with a big accuracy, that
; r0 << 0; e-b (r) 1; -Ei(-b) = ln 
= ln 
Then, the
Eq. (9) for the hole will be:
(16)
The checking of an accuracy of the formula (9) and (15) we have made on
the data of full-scale observations of spouting wells in the Ararat valley of
Armenia.
Conclusion
The
obtained equations (9) and (15) quite satisfactorily can be recommended at
various hydro-geological calculations of spouting wells at their designing and
exploration.
References
