### Drain-spacing formula for transient

```Drain-spacing formula for transient-state
flow with ellipse as an initial condition
JACEK UZIAK1 and SIETAN CHIENG2
xDepartment of Machine Theory and Automatics, Academy of Agriculture, Lublin, Poland; and 2Bio-Resource Engineering
Department, University ofBritish Columbia, Vancouver, BCf Canada V6T1W5. Received1 June 1988, accepted 12 December1988.
Uziak, J. and Chieng, S. 1989. Drain-spacing formula for transient
state flow with ellipse as an initial condition. Can. Agric. Eng. 31:
101-105. A new solution of the differential equation describing the
drainage problem was presented. An initial condition in the form of
an ellipse, approximated by a negativeexponentialfunction, was used.
From this solution, a new formula was proposed for drain-spacing cal
culations. It was found that the proposed formula is more general for
drain-spacing calculations. Two well-known drain-spacing formulae,
Glover's equation (Van Schilfgaarde 1974), and Glover-Dumm's
(Dumm 1960, 1964) equation, are two special cases of the proposed
Y= 0
(for x = 0 and X = L)
(2)
where L = drain spacing.
The initial condition is much more complicated and three types
of initial water table shapes, as shown in Fig. 2 (Dass and MorelSeytoux 1974), are usually assumed:
(a) A constant water table height, Y0, exists everywhere
between the two adjacent drains except above the drains where
the water table abruptly drops to zero (to drain level).
formula.
(for 0 < X < L where t = 0)
(3)
INTRODUCTION
Within the pastthreedecades, a considerable amountof research
on drainage problems has been done. So far, drainageproblems
have been divided into steady-stateand transient-state flow con
ditions. A steadystate exists when the boundaries and flow rates
of a system do not changewithtime. Otherwise, a transient state
exists. As steady state seldom exists under actual field condi
tions, solutions using the transient-state condition should be
tions basedon elliptical initialconditions have been widely used
in subsurface drainage design. Currently, there is no transientstate drain-spacing equation basedon elliptical initial conditions.
tions cannot be meaningfully compared. This study addresses
thisneedby using the ellipse-initial conditionin transient-state
flow and developing a new drain-spacing equation.
(b) The water table shape is a fourth degree parabola with
the following form:
Y= -^ (L3X - 3L2 X2 + 4LX3 -2X4) (for t = 0) (4)
L4
(c) The water table is a parabola with the inflexionpoint given
by the expression:
v
16*o X1 (L-X)2
(for t = 0)
(5)
The initial water table shape with a fourth degree parabola
expressed by Eq. 4 is most widely used. Using this initial con
ditionand the boundarycondition in the form of Eq. 2, the solu
tion of Eq. 1 in the following form is obtained (Luthin 1978):
Y =
192r0
v
nV-8
n2ir2at
exp
n=l,3,5,
TRANSIENT FLOW
rnrX
Transient or nonsteady-state conditionsoccur when the ground
(6)
sin
water table fluctuates with time; therefore, the hydraulic head
is changing constantly. The linearized differential equation for
transient flow, as derived on the basis of Dupuit-Forchheimer
assumptions, can be expressed as (Luthin 1978):
Atmid-spacing, X=L/2, the water table height canbe expressed
as:
192
£
Ym =
dy
(1)
(-D
az2-8/tt2
« = 1,3,5,...
dx2
dt
ir2n2at
where:
a = K.D/f
K = soil-saturated hydraulic conductivity,
D
d
Y0
/
=
=
=
=
average depth of flow region = d + Y0/2,
depth to impermeable layer below drain,
watertable heightat mid-spacing as shown in Fig. 1,
drainable porosity or specific yield,
t
= time, and
Y = water table height above the datum.
The solution of Eq. 1 depends on the initial and boundary con
ditions. The boundary conditions are relatively simple and are
generally set up as follows:
(7)
exp
ELLIPSE AS AN INITIAL CONDITION
In the steady-state condition it is assumed thatthe recharge and
drainage rates are equal, and the hydraulic head does not vary
with time. This situation is described by Hooghoudt (Luthin
1978) as an ellipse equation with a semi-major axis of
d2 +
8L2
AK
and a semi-minor axis of fd2 +
RL
AK
101
nonsteady-state condition when the rechargeand drainage rates
JL
GROUND
are not balanced. Hooghoudt's equation and other steady-state
SURFACE
equations have been used widely in subsurface drainage design.
It therefore seems logical to assume the ellipse as an initial con
WATER TABLE
y
when t » 0
y
when t > 0
DRAIN
IMPERMEABLE
dition for transient-state conditions. This enables the compar
ison of drain-spacings calculated from two different approaches
(i.e., ellipse). The equation of this ellipse-condition can be
expressed as:
(X-L/2)2
+
L2(Y0+d)2
4[(Y0+d)2-d2]
LAYER
(X+d)2
(Y0 + d)2
=
(8)
1
Figure 1. Definition sketch for drainage problem.
Unfortunately, because of the mathematical complications, it
is extremely difficult to obtain the analytical solution of Eq. 1
in the normal manner using an ellipse as the initial condition.
For this reason the authors approximate the ellipse by the wellestablished negative-exponential function as:
Y
k
e>o
T
I
I
I
-^X
DRAIN
I
-
(for 0 < X < LI2)
Y = A{\ - exp (-B(L-X))
I
DRAIN
Y = A{\ - exp (-BX))
1
(for LI2 < X < L)
(9a)
(9b)
From Eqs. 8, 9a and 9b the following expressions are obtained:
L •
(a)
Y0
(Y0 + d)2
1 — exp
I
DRAIN
^b
1
B =
(10)
2(Y0 + d)2
(11)
d2-L
Figure 3 compares the shapes of ellipses (Eq. 8) andnegativeexponential functions (Eqs. 10 and 11). The agreement between
the curves depends on the value of d/Y0. Comparisons were
done for d/Y0 = 0 to d/Y0 = oo. The best agreement is found
for d/Y0 = 2. Although the ratio of d/Y0 could vary from 0 to
infinity in theory, a range of 0-5 is more frequently encoun
tered in practice. As d/Y0 = 2 is about half-way between 0 and
5, it is felt that the proposed negative-functions can be used to
Substituting equations 10and 11 into Eqs. 9a and9b, yields:
T
Y0
Y =
Y0 + d
1 — exp
d
DRAIN
1 —exp ( —
2 (Y0 + d)2X
Ld2
(12a)
(for 0 < X < LI2)
Figure 2. Three initial water table conditions studied by Dass and
Morel-Seytous.
Y0 + d
1 — exp
2 (Y0 + d)A (L-Xj
where R is the drainage coefficient or recharge rate. These are
shown in Fig. 1. The steady-state condition becomes a
102
1 —exp ( —
Ld2
(for LI2 < X < L)
(12b)
UZIAK AND CHIENG
SOLUTION OF TRANSIENT STATE EQUATION
A solution of the transient state equation (Eq. 1) which satis
fies the boundary conditions (Eq. 2) and proposed ellipse ini
i+2o
tial condition (Eqs. 12a and 12b), is obtained as follows:
4F0
y =
exp
-
1+^0\2
d
exp
air2n2t
-
(14)
=1,3,5,
l-exp
where Ym is the water table height above drains at mid-spacing,
-1 +
at time t.
Figure 4 shows the relationship between the dimensionless
parameters Y/Y0 and KDt/fL2 for different ratios of d/Y0. As
rnr - (-1)
nic
a comparison, the same relationship for the U.S. Bureau of
Reclamation's equation (Eq. 6) is also included. It can be seen
from Fig. 4 that when d/Y0 increases the time required for
„2_2
4 1 + ^0\4
4 + nV
2
1 +
Y0
exp
-
i+
water table height to drop from Y0 to Y decreases. When the
drains are placed on the impermeable layer (d=0), the value
of d/Y0 is zero and Eq. 13 becomes:
2oV
d
av2n2t
ft
Equations 12a and 12b which have been used as initial con
exp
-
(13)
sin
ditions for Eq. 13 are an approximation of an ellipse given by
Eq. 8. It can be seen in Fig. 3 that the agreement between the
original ellipse's function and the proposed approximation
appears to be very good. Therefore, Eq. 13 can be treated as
a solution of Eq. 1 with an ellipse as the initial water table shape.
Since the main interest is in the height of the water table at the
midpoint between the drains, we can obtain the following
expression for y, at x = L/2:
n-\
00
41o
Ym =
E
■♦?)'
1—exp
(-D
=1,3,5,,
nir - (-1)
nir
4
1 +
+ nV
Y =
onr2n2t
47n
E
n = l,3,5,.
Texp(n
nirX
(15)
and the water table height at mid-spacing can be obtained as:
n-\
Ym =
4Y0
°°
1
D (-1) 2 7exp(-
air2n2t
(16)
n=l,3,5,...
Equations 15 and 16 are the solutions of Eq. 1 with an initial
condition in the form of Eq. 3 discussed above (Carslaw and
Jaeger 1959). In fact, when d=0 is substituted into Eq. 12, we
can obtain Y= Y0 for all values of X, except for X=0 and X—L.
This result is exactly the same condition as described by Eq.
3. It is interesting to find that the difference between the pro
posed solution (Eq. 13) and the USBR's equation is less than
2%.
0.8-
0.6-
0.4-
0.2-
Figure 3. Shapes of ellipse and of negative-exponential functions for different d/Y0 values.
103
NEF WHEN d/yQ - 0
GLOVER-DUMM (d/j^= 0.9166)
NEF WHEN d/yQ - 0
0.8-
NEF WHEN d/yQ • 5
NEF WHEN d/y0 • °°
0.6-
i
0.4-
0.2-
0.0001
KDVflf
Figure 4. Relationship between Y/Y0 and KDt/fL for different d/Y0 values.
NEW DRAIN-SPACING FORMULA
It has been found that the sum of the second and remaining terms
in the series, in Eq. 14, is very small and can be neglected.
This gives the approximate solution of Eq. 14, by taking only
the first term of the series, which yields:
1974).
It was interesting to find that when the ratio of
d/Y0 = 0.9100, is used in Eq. 18, the following equation is
obtained:
1
4Fn
ym =
It should be mentioned that Eq. 19 is exactly the same as the
well-known Glover's drain-spacing equation (Van Schilfgaarde
l-exp[-(l + y0/J)2] l*
2
4(l+ Yo/d)4 +r2
(XT t
In (1.15
x-2 (Y+Yo/d)2 exp (~(\-YQld)2
exp (-air2t/L2)
(17)
From Eq. 17, a new formula for drain-spacing calculation is
developed as follows:
air t
•2 _
(18)
1%.
CONCLUSION
xym(4co2 + 7r2)[l-exp(-o3)]
where a = (1 + Y0/d)2.
In the case of d = 0 (i.e., co approaches infinity) Eq. 18
becomes:
L2 =
In (4Y0/irYJ
In (1.27Y0/Ym)
(19)
When d approaches infinity, the value of co approached unity
and Eq. 18 becomes:
r2
8r0[2 + xexp (-1)]
In
7rrm(4 + 7r2)(l-exp(-l))
air t
In (0.91 IVI'm)
104
A new solution of the differential equation (Eq. 13) describing
the transient-state drainage problem was presented. The ellipse
initial condition was used and was approximated by a negative
exponential function in the form of Eq. 12. Good agreement
between the negative exponential function and the true ellipse
was found. Owing to this good agreement, the solution obtained
with this negative exponential function as an initial condition
could be considered as the solution with a true ellipse as the
initial condition.
cnr2t
_
-)
Equation 21 is almost the same as the Glover-Dumm formula
reported by Dumm (1960). The only difference between Eq.
21 and Glover-Dumm's equation is the "constant" term in the
equation (i.e., 1.15 in Eq. 21 and 1.16 in Glover-Dumm). The
difference between these two equations is found to be less than
8K0w[2o3 + ir exp ( —«)]
In
(21)
L2 =
(20)
It was found that when the ellipse was used as the initial con
dition, its shape was influenced by the ratio of the semi-major
and semi-minor axes. For this reason, the shape of the initial
water table will depend not only on water table height at midspacing but also on the distance to the impermeable layer below
drains. These features are also represented by the negative
exponential function.
UZIAK AND CHIENG
The new drain-spacing formula (Eq. 18) proposed in this
paper can beconsidered as ageneral formula for drain-spacing
calculations. It was found that two well-known drain-spacing
formulae, Glover's equation (Van Schilfgaarde 1974) and
Glover-Dumm's (Dumm 1960, 1964) equation, are covered by
this new formula. They can be considered as the solution of
two special cases of the proposed formula as given in Eqs. 19
REFERENCES
CARSLAW, H. S. and J. C. JAEGER. 1959. Conduction of
heatin solids. 2nd ed. Oxford University Press, London, U.K.
510 pp.
DASS, P. and H. J. MOREL-SEYTOUX, 1974. Sub
surface drainage solutions by Galerkin's Method. J. Irr. Drain.
and 21, respectively.
Div., ASCE 100 (IR1):1-15.
Currently, a nonsteady-state drain-spacing equation based
on elliptical initial conditions is not available. Drain spacings
be meaningfully compared. This need was successfully
addressed in this study by using the ellipse-initial condition
in nonsteady-state flow and developing a new drain-spacing
DUMM, L. D. 1960. Validity and use of the transient flow
equation.