In this section, the full equation of the transient w atertable on an inclined impervious base with and without recharge will be discussed and solutions given in detail.
The simplified equation of the groundwater will not be discussed here since these issues have been extensively studied by many investigators.
Consider the general case of unsteady flow in an aquifer over an impervious base as shown in Fig. 4.2.1.
Land surface
Impervious layer
x = L
Fig. 4.2.1. The cross-section of an unsteady groundwater aquifer on an inclined impervious base
For the unconfined groundwater aquifer on an inclined impervious base, the basic equation governing the flow is Boussinesq's equation (Werner, 1957; Bear, et al., 1968; Marino and Luthin 1982).
Let the aquifer be referred to as a coordinate system with positive values in the direction of flow and aligned with the im pervious layer of slope X. The aquifer is assumed to flow in a two-dimensional and homogeneous medium disregarding the effects of capillarity, air com pression due to recharged water, etc. The fundam ental law governing the m ovem ent of unsteady flow in such an aquifer, first derived by Boussinesq (1877), and also shown by W erner (1957), Bear, et al., (1968), Marino and Luthin (1982), can be written as:
3h k f - = - \ hk[~ 9 2h 0 L 9x2 and and Q b- - 0x. The initial condition is
h = h(X(0) at t = 0 The boundary conditions are h = Ho at x = 0
h = H[ at x = L
The symbols used in these equations are h is the height of watertable;
t is time;
x is the horizontal distance from the origin of coordinates; k is the hydraulic conductivity of the unconfined aquifer;
(4.2.1)
(4.2.2)
(4.2.3)
(4.2.4)
0 is the specific yield or porosity of the media;
X is slope gradient of the impervious layer, and X = tana; ä is the angle between the impervious layer and horizontal base; Re(t) is the recharge rate; and
Qb is the groundwater discharge from the aquifer.
Equation (4.2.1) can be solved in several ways, depending on the problems concerned. One of the methods is detailed below which is relevant to the groundwater flow on an inclined impervious base subject to spatially uniform time-varying recharge.
M ost o f the analytical solutions reported in the literature are based on the assumptions of constant rate of accretion and horizontal impervious base. We present complete solutions to the Boussinesq's equation which involve a time-varying recharge rate to groundwater on an inclined impervious base.
4 .3 DEVELOPMENT OF ANALYTICAL SOLUTIONS TO
Transient Groundwater Levels on an in c l in e d Impervious Base with t im e-varying Re c h a r g e
In this section, solutions will be presented to the basic equation, i.e. Eq. (4.2.1). Equation (4.2.1) can be simplified using a transform of the form (Werner, 1957)
z = h2 (4.3.1)
where z is a transformed variable. From Eq. (4.3.1), we have 3h 1 3z
(4.3.2)
and
3 h __ 1 3z 3x ~ 2Vz 3x
then differentiating Eq. (4.3.3) with respect to x once again yields
32h __ 1 32z 1 r3z-|2 3x2 ~~ 2 Vz 3x2 4zVz
Inserting Eqs. (4.3.2), (4.3.3) and (4.3.4) into Eq. (4.2.1) yields
(4.3.3) (4.3.4) 3z 3t = kVz 32z kxr3zi 0 3x2 0 ^3x-* 0 (4.3.5)
If we make an assumption that h deviates only a small amount from a weighted depth, D, of the groundwater aquifer, that is, we write D = h, then, Eq. (4.3.5) can be further simplified into the following form
3z Tr 32z 3z ?n Re(t) 3t 0 3x2 3x 0 with Tr = kVz = kD (4.3.6) (4.3.7)
being the transmissivity; and
s kX
0 (4.3.8)
Equation (4.3.6) is the equation presented by Werner (1957) who obtained several analytical solutions to this equation for several special cases such as for a sudden change in tail-water levels. For more general cases of time-varying recharge to groundwater on an inclined impervious base, no report has been found in the literature in relation to an analytical solution to Eq. (4.3.6).
The initial condition specified for the problem is 2
Z(x,0)= h(X.o) at t = 0 (4.3.9)
and the boundary conditions are
Z(o,t) = Hq at x = 0 (4.3.10)
Z(L,t) = H1 at x = L (4.3.11)
We introduce the following transform
2D 1
z = U + ^ f R e (t-)dt'+ q (4.3.12)
lg
where
(I is a small perturbation; t' is recharge time instant; and tg is the recharge lag time.
Equation (4.3.12), then, can reduce Eq. (4.3.6) into
3U _ Tf32U _ 3U
dt 0 d x 2 " (4.3.13)
The initial condition is
U = U(x,o) at t = 0 (4.3.14)
and boundary conditions
U = U(o,t) at x = 0 (4.3.15)
and
U = U(L,t) a tx = L (4.3.16)
We do not solve Eq. (4.3.12) directly. By means of a transform proposed by Fürth (1931) as reported by Jost (1960),
U = U * e x p [ |^ ( x - x 0) -
4Tr0 (4.3.17)
Here we set x0 = 0, and substitution of Eq. (4.3.17) into Eq. (4.3.12) yields a partial differential equation for U*
au* _ Tr a2u*
The initial condition for Eq. (4.3.18) is
u * = u ; , 01 att = 0 (4.3.19)
and the boundary conditions are
U* = u ; , 0) at x = 0 (4.3.20)
and
u* = u ; , 0) at x = L (4.3.21)
Equation (4.3.18) is the ordinary heat conduction equation. Hence, there are no difficulties in dealing with a problem of this type.
There are several standard methods which can be used to solve Eq. (4.3.17). One of these methods is the method o f separation of variables. This is a standard method assuming the two variables are separable, that is, Eq. (4.2.17) can be written
U* = XT (4.3.22)
where
X is a function of x and T a function of t only. Substituting Eq. (4.3.22) into Eq. (4.3.18) yields
3T Tr 32X 3t 0 5x2 1 3T Tr 1 32X T dt - 0 X d x 2 (4.3.23) (4.3.24)
Both sides of Eq. (4.3.24) must be equal to the same constant which, for the sake
T
of convenience o f subsequent algebra, is taken as - £ 2-£. Therefore, we have two
ordinary differential equations
I Ü _ e 2 Xe
T 3t _ ’ 0
and
1 d2X - 2
X 3 ? = - « of which solutions are
(4.3.25)
(4.3.26)
T = exp[- £ 2 ^ t ] (4.3.27)
and
X = A sin ( £x) + B cos ( £x) (4.3.28)
which lead to a solution to Eq. (4.3.22)
U* = [a sin ( ^x) + B cos ( £ x)]exp [- ^ 2-^-1] (4.3.29)
where
Since Eq. (4.3.29) is a linear equation, the most general solution is obtained by summing up solutions of the form of Eq. (4.3.29)
0 0 r p
U*= ^ [An sin (£ nx) + Bncos ( £nx)] exp[- 5 „ ^ t ] (4.3.30)
n=l
Then from Eqs. (4.3.30), (4.3.17), (4.3.12) and (4.3.1), we have
h2 = X [A"sin( £n x) + B„cos( £n x)]exp [ | ^ x -( + | ^ ) t ] n=l
2D 1
0
S
Re(t')dt' (4.3.31)lg
for accretion of an unsteady watertable on an inclined impervious base with recharge and
h2= X [A„sin(^nx) + B„cos(<5nx ) ] e x p [ |^ x - ( £ 2 + | ^ ) t ] (4.3.32)
for recession of such a watertable without recharge.
Equations (4.3.31) and (4.3.32) present the height of the watertable in a form of a summation. These equations can be simplified for practical use. For example, van Schilfgaarde (1974) has presented a solution to the unsteady groundwater flow on a horizontal base that is referred to as the Glover equation which simplifies an equation similar to Eqs. (4.3.31) and (4.3.32) in a way that for all but the smallest time periods, all terms but the first may be neglected, resulting in a simple expression. That is n = 1 in Eqs. (4.3.31) and (4.3.32).
Tapp and Moody, as reviewed by Dumm (1964), observed that the initial watertable shape encountered in the field was more parabolic than flat. They modified the initial condition accordingly and found a solution that differed from the simplified expression only in that the numerical constant 4 is replaced by approximately 3.7. These results indicate that the simplification made in this way is accurate enough to be accepted.
We use the same approximation to simplify Eqs. (4.3.31) and (4.3.32). That is for n = 1, Eqs. (4.3.31) and (4.3.32) can be respectively simplified as
h2= [ A,sin( § x)+BiCOs( £ x)]exp { ^ x - [ | 2^ + j ^ J t } 2D 1
0 f Re(t')dt' (4.3.33)
lg
(4.3.34) h2= [ A 1sin(<Jx) + B 1c o s ( ^ x ) ] e x p { | ^ x - \ %2j + | j r ^ ] t }
There are several methods to determine constants Ai and B i. A simple way to determine Ai and Bi is directly based on the initial and boundary conditions (Crank,
1956) rather than an initial height of the watertable (Derrick and Grossmand, 1987). From initial and boundary conditions, Bi = H$ and Ai = 0 for t = 0 and x = 0, then Eqs. (4.3.33) and (4.3.34) can be respectively written
h2= H § c o s ( f x ) e x p { ||; x - [ | * + | ^ ] t } + ® f R e(,)dt' (4.3.35)
r r lg
and
h2= Hocos( £ x)exp { x - (4.3.36)
The boundary conditions demand h2= H? for x = L at t = 0. Then Eq. (4.3.35) or Eq. (4.3.36) gives
Hl = H ? c o s ( £ L ) e x p [ ^ L ] (4.3.37)
Rearranging Eq. (4.3.37) yields
^ = ^ - a r c c o s [ ( ^ ) 2e x p ( - ||- L ) ] (4.3.38)
Inserting Eq. (4.3.38) into Eqs. (4.3.35) and (4.3.36) respectively yields
h2= cos { ^ arccos[ ( ^ ) 2e x p ( -| j r L )] } exp { ^ x -
[ i a r c c o s [ ( ^ - ) 2exp(- ^ L ) ] ] ^ t - ^t} + ^ J R e(f)dt' (4.3.39)
r f lg
for accretion of an unsteady watertable subject to recharge on an inclined impervious base and
h 2= H$ cos { ^ arccos[ ( ^ ) 2e x p ( - L ) ] } exp { x -
for recession of an unsteady watertable on an inclined impervious base without recharge. Obviously, Eqs. (4.3.39) and (4.3.40) yield
h 2= H$ for x = 0 at t = 0 (4.3.41)
Eqs. (4.3.39) and (4.3.40) yield
h 2 = H? for x = L at t = 0 (4.3.42)
and
h?x.o) = H? cos { j- a r c c o s [ ( ^ ) 2exp(- L )] } e x p [|^ - x] (4.3.43) for t = 0.
4 . 4 DEVELOPMENT OF ANALYTICAL SOLUTIONS