Groundwater mound damping

M)2~7225/831040413+9$03.0010 0 1983 Pergamon Press Ltd.

hf. I. EIIRIIR SC;. Vol. 21. No 4. pp. 413-421. 1983 Printed in Great Bntain.

GROUNDWATERMOUNDDAMPING I. N. KOCHINA, N. N. MIKHAILOV and M. V. FILINOV I. M. Gubkin Institute of Petrochemical and Gas Technology, Moscow, U.S.S.R. (Communicated by 1. N. SNEDDON) Abstract-Damping of groundwater mounds in gas stratum has been considered taking into account possible partial retention of water in porousmedium previouslyoccupiedby water.This problem has been studied previously for the case of complete gas-water displacement. The later problem can be reduced to the Boussinesq nonlinear equation of parabolic type. INTRODUCTION

DAMPING OF groundwater mounds in a gas stratum has been considered taking into account

possible partial retention of water in porous medium previously occupied by water. This problem has been studied previously[l] for the case of complete gas-water displacement. The later problem can be reduced to the nonlinear Boussinesq equation of parabolic type. If the groundwater mound at the initial moment is concentrated in the infinitesimal vicinity of the symmetry axis the problem is self-similar and a solution of instantaneous source type can be obtained. This solution satisfies the conservation relation of the bulk water mass in the porous medium. The problem considered in the present paper is reduced to that of solving the Boussinesq equation with a coefficient which suffers a discontinuity at a point where ah/at = 0 (h is the mound height), whereas the condition of retention of the fluid mass assumes a nonintegrable form since a part of the fluid remains outside the mound. It has been shown that the Boussinesq equation with a discontinuous coefficient has an asymptotic self-similar solution of the second kind h = At-“f(r/BP) where the coefficients (Y and /? are not determined from dimensional considerations only, - they are found in the course of solution. The present paper presents also a numerical solution of the non-self-similar problem which turns out to converge asymptotically to the self-similar one at large times. I. DERIVATION

OF THE BASIC EQUATION

Suppose that initially, a groundwater mound exists in a gas stratum of infinite thickness with an impermeable horizontal bed. Due to the gravity force action it is damped over the stratum. Water is not displacing the gas completely but it occupies only a part of the pore volume equal to u. Water saturation in the original mound is supposed also equal to u. Some water remains in the porous volume that was previously occupied by water: the residual saturation being equal to coo< g (Fig. 1). We assume that initially the fluid occupied a certain volume symmetrical to the .z axis. We shall derive a differential equation for the fluid mound height (head) h(r,t), assuming the flow to be an axisymmetric one. We shall also assume that ah/Jr < 0 everywhere except at r = 0.

Stipulating the basic assumptions of hydraulic theory[5], i.e. neglecting the vertical velocity

Fig. I. Scheme of a groundwater mound damping. 413

414

I. N. KOCHINA

et N/.

component of filtration, and assuming that the vertical pressure obeys the hydrostatic can write an expression for the flow rate through the cylindrical surface 2nvh (Fig.2)

law, we

(1.1) where k is the permeability coefficient of the porous medium; p,‘the fluid viscosity; p, the fluid density; pressure p = pO+ pgh; p. is the pressure in the gas stratum. Equating the change in the water flow rate to the rate of change in mound volume due to reduction in saturation from v to the residual saturation a”, we obtain the Boussinesq equation

ah

1a

bg

at=2m&-&ar

ah2

( )

(1.2)

‘r

which is valid only for ah/at < 0, i.e. for r < r,. For r > r, (ah/at > 0) with an increasing water level the saturation and the differential equation takes the form:

increases

from zero to CT

(1.3) Hence the problem of damping of a groundwater mound where the residual saturation is taken into account is reduced to solving the Boussinesq equation with a discontinuous coefficient depending on the head time derivative

(1.4)

Where

(1.5) under the conditions

h(m, t) = 0

EL0 ar

h

(1.6) for

r = 0;

(rt

Fig. 2. Differential equation derivation for the mound height

Groundwater

mound

41.5

damping

(the last condition follows from the symmetry of the solution with respect to the z axis and the absence of the fluid inflow at the symmetry axis) and under the condition that the head h(r, t) and the derivative dh2/ar at the point where 8r/~?t = 0 are continuous. The continuity of dh’/ar follows from the constancy of the permeability coefficient k for the porous medium, the discontinuity of the coefficient c arises due to residual water saturation. The initial mound shape is also prescribed. Therefore the main difference of this problem from the one solved earlier[l, 41 is that the condition of conservation of the water mass in the mound is not satisfied because a part of water with a saturation nC,remains beyond the mound. This causes an additional nonlinearity due to the presence of a free boundary where the coefficient in the equation for the head suffers a discontinuity. 2.

NUMERICAL

SOLUTION OF THE DIFFERENTIAL EQUATION

For the Boussinesq equation with a constant coefficient c and the initial conditions instantaneous source type (h(r, 0) = 0 for rf0) a self-similar solution

of

(2.1) can be obtained from dimensional considerations. It would seem that for the Boussinesq equation with a discontinuous coefficient (1.4), under the conditions (1.6) and initial data of instantaneous source type, one might obtain a similar solution, since only one dimensionless constant parameter c2/cl is added in comparison with previous case. Therefore, the head h(r, t) should seemingly be represented as

h(rJ) =

($)I’? f(& 2)

(2.2)

where

Substituting the expression (2.2) in the first eqn (1.4) we obtain an ordinary can be integrated. Its solution is in the form

f(@) =-$+a,. Substituting

equation

which

(2.3)

(2.2) in the second eqn (1.4) we obtain

(2.4) where a, and az are the integration constants. Equating the r.h.s. of (2.3) and (2.4) and their derivatives at a point [ = & corresponding to ah/at = 0 we obtain a system of equations which for cl#cZ does not have a non-zero solution. Hence equation (1.4) under our condition has no solution of the type (2.2). TO solvethis paradox let us consider the numerical solution of a non-degenerate problem represented by equation (1.4), conditions (1.6), the condition of continuity of the functions h(r, t) and dh’/ar and the initial condition M

hko)=7hn

0

0Lrn

where M is the volume of water at the initial moment.

(2.5)

4lh

I. N. KOCHINA

In the calculations

the dimensionless

c/ t/i.

variables

H = h/h,,, R = r/r,,, T = 2cztlc, were used and the initial conditions H(R, 0) = I for R ~1, and H(R, 0)=0 for R>l. A two-layered scheme was used in computer-calculations. The distributions H(R, T) were calculated for different c2/c,. Figure 3 shows the plots for the case c2/c, = 0.9. The graphs of In H(0, T) and In R, as functions of In T (R, is a dimensionless coordinate of the waterfront) were plotted on the basis of these calculation results (Fig. 4); obviously, for large times values they are linear which proves that the time dependences of H(0, T) and R, are power functions. Most essential is that the graph of H(R, T)/H(O,T) as a function of R/R, for large T is presented by a single curve (Fig. 5).

Fig. 3. Space-time dktrihution

Fig. 4. Logarithm

of the mound height H(R. T) for c,/cI = 0.9. Code of curves-dimensionless time T.

of the mound height on the symmetry axis In H(0. T) and logarithm radius In R, versus logarithm of dimensionless time In T.

of damping front

417

Groundwater mound damping

e; cr-6 -II 05-

I 0

I

05 R R1

Fig. 5. Distribution of the mound height in self-similar variables H(R, T)/H(O, T), R( 7’)/R,( T),

3. ASYMPTOTIC

SELF-SIMILAR

SOLUTION

The numerical results show that there exists an asymptotic self-similar solution in a form different from (2.2). To find this solution we shall analyze the dimensions of the quantities in the non-degenerate problem. The head h depends on the dimensions of the characteristic parameters r, t, cl, c2, M, r,. Let L, T, [h] be the dimensions of length, time and head, respectively. For the sake of convenience, we introduce an independent dimensionality of the head. This is possible since for our problem it is not essential that length and head have the same dimensions. Hence [r] = L, [t] = T, [c,] = [cJ= [h]-‘L’T-‘,

[Ml = L*[h], [ro]= L

and three dimensionless combinations can be obtained c2

5=(c,Ait)114~ 77= (c,&“’ c, which determine the dimensionless head (3.1)

or h=

($)“*F(& :). 7,

(3.2)

The Cauchy problem given by eqn (1.4), conditions (1.6) and (2.9, and the condition of continuity of the functions h(r, t) and dh2/& has an asymptotic limiting solution in the form of a self-similar solution of the second kind to which the solution of the non-degenerate problem for large time values tend. Assuming that the solution (3.2) exists, let us consider its asymptotic expression as t+m;

I.N.KOCHINA etal.

418

hence n+O, and we may take r so large that [ remains finite. The same result is obtained by making r,, to tend to zero, and keeping r and t finite. However, the condition ro+O shows that the initial water distribution is of instantaneous source type and hence a solution of the type (2.2) should be obtained. It has been shown that there is no self-similar solution of the type (2.2) which is continuous, has a continuous derivative ah’lar and satisfies the conditions at infinity, symmetry and limiting initial condition. This means that consequently, the function (3.2) has no finite limit as n+O and [ is finite. We shall now seek a solution assuming incomplete self-similarity with respect to the arguments 5 and 7) [2], i.e. we assume that the solution is of the form

(3.3) or

Here y and E are certain real numbers which cannot, in principle be obtained from dimensional analysis; the new dimensionless variables h;lmy and 5~~’ remain finite as ro+O. Substituting the expression for F from (3.3) in (3.2) and the values of 5 and n, we obtain 0,

M~z-~)i4~; c2 t) = (c,t)(~+r)n f[ r;(Mc~*)“-‘“’ Cl

1

(3.4)

In order that h;lmy and 5~~’ may.remain finite as ro+O, M should be made to tend to zero or infinity (depending on the sign of yy and E) so that products M2-y/4ro and M’-“4ri remain finite. Let

Hence h = At-“f( l,

2)

The expression (3.6) is a self-similar solution of the second kind; it determines the asymptotics of the Cauchy problem for eqn (1.4) with conditions (1.6), (3.1) and continuity of the function and its r-derivative. The parameters (Yand fl are not known beforehand and should be estimated along with the coefficients A and B in the course of the solution. Substituting (3.6) into the first eqn of (1.4) we obtain (3.7)

Obviously,

for a self-similar

solution to exist, the following condition

must be satisfied

(Y+2p=1

(3.8)

A = B’/c,

(3.9)

but from (3.5) it follows that

419

Groundwatermound damping

and eqn (3.7) takes the form (3.10)

It is valid for J/~/C%< 0, or, if

df fff +P”di>o

(3.11)

From the second eqn of (1.4) we obtain (3.12)

which holds valid when

Moreover,

from (1.6) we get f’(0) = 0 (3.13)

fW = 0

and the function f(& cz/cl)as well as its derivative are everywhere continuous. Equations (3.10) and (3.12) for (3.13) can be solved only within to a constant, which cannot be determined from the integral condition of the mass conservation, since the mass conservation law is nonintegrable in our problem. Equations (3.10) and (3.12) were solved numerically under normalization condition f(1) = 0.

(3.14)

Hence, from (3.12) it follows that (3.15)

f’(1) = -c,PI2c*

The values of p were determined by the half-division method. The value of p corresponded to the condition of If’(O)-01~ E where E is the calculation accuracy. The graphs of f(& cz/cl) and p(c2/cI) are shown in Fig. 6 and 7, respectively. To determine the head with the help of (3.6) we need to know the constant A which can be found by matching the self-similar solution (3.6) with the non-self-similar one which asymptotics the solution (3.6) is. From (3.5) under the condition (3.14) we can determine the coordinate of the waterfront r, = 4. COMPARISON

@P

= BP = d(Ac,)P

OF NUMERICAL

AND SELF-SIMILAR

(3.16) SOLUTION

We shall now compare the results obtained for the non-self-similar and asymptotic selfsimilar solutions (4.6). The self-similar solution may be presented in a dimensionless form

H(RT)= AIT-"f(&:) where

,+A@$ 0

(4.1)

4X

1. N. KOCHINA

The front coordinate

el al.

is Rf= B,TP

(4.2)

where

Fig. 6. The graph of the function f([, c?/c,) for various c2/c, values.

Fig. 7. The dependence of the exponent fl on the

ratio c,/c,

Groundwater

In logarithmic

coordinates

mound damping

421

we obtain

InH(O,T)=-aInT+ln[A,f(O,~)]

lnR,=pInT+InB,

(4.3)

(4.4)

i.e. straight lines with slopes of - (Yand p. From the plot in Fig. 7 it follows that for cz/cl = 0.9 we obtain p = 0.243 and CY= I -2~ = 0.514. For the non-self-similar solution the slopes of the lines shown in Fig. 4 are -0.497 and 0.238, respectively, so that to about 3% accuracy they coincide with - (Yand /? values. Therefore, for large times the non-self-similar solution tends to a self-similar one and is given by (4.1) whereas the distributions presented in Fig. 4 satisfy eqns (4.3) and (4.4). The lengths intersected by the lines shown in Fig. 4 at the ordinate axis, In [A,f(O, c?/c,)] and In B,, can be used to determine A, and B, and verify whether the relationship A, = 2(cJc,)BT is satisfied. For c,/c, = 0.9, from the plot of Fig. 6 we obtain f(0; 0.9) = 0.0645 which corresponds to the ratio A,/BS within 6% accuracy. Acknowledgement-The authors thank Prof. G. I. Barenblatt for his valuable suggestions and Dr. V. F. Baklanovskaya her assistance in the computations.

for

REFERENCES I. G. I. BARENBLATT. V. M. ENTOV and V. M. RYZHIK. Theory of non-steady filtration of fluids and gases. Nedra, Moscow 1972). 2. G. I. BARENBLATT. Similarity, self-similarity, and intermediate asymptotics. Plenum Press, New York 1979. 3. G. I. BARENBLATT and G. I. SIVASHINSKY. English translation J. Appl. Math. and Mech. 33, (5) 861-870. (1969) 4 P. Ya. POLUBARINOVA-KOCHINA, Princeton University Press, New Jersey (1962). I. I. A. CHARNY. Subterranian fluid/gas dynamics. Gostoptechizdat. Moscow (1963). (Received 28 May 1981)