Hydraulic Theory
Exfiltration Trench Design


From the MIDUSS Version 2
Reference Manual  Chapter 8
(c) Copyright Alan A. Smith Inc. 
An exfiltration trench
is a facility that encourages the return of runoff to the ground
water. It may be a very simple "soakaway" and comprise only a trench
filled with clear stone (i.e. single sized gravel) into which runoff
is directed. A more complex facility might be incorporated inline
with a conventional storm sewer and include one or more perforated
pipes along the length of the trench to provide more uniform
distribution of the inflow over the length of the trench. It is this
latter type of facility which is described in the MIDUSS Trench
command.
Figure 8.16  A typical
exfiltration trench.
Figure 8.16
illustrates a typical arrangement of an exfiltration trench which
splits the inflow hydrograph into two components. One fraction is
transmitted downstream as an outflow hydrograph that is attenuated by
the storage within the voids of the clear stone fill. The balance of
the flow is transmitted to the ground water through the pervious walls
of the trench. The Trench form has an option to include or exclude
the base of the trench in estimating the area contributing to
exfiltration.
It is usual to provide
some form of outflow control device at the downstream end of the
trench to force the free surface in the trench to rise. This causes
(1) the volume of voids available for storage to be increased and (2)
the surface area along the walls of the trench is increased to allow
increased exfiltration. Figure 8.16 shows a typical outflow control
device with a small orifice at or near the downstream invert of the
trench to allow drainage of accumulated flow in the trench plus an
overflow weir to produce high water levels during the maximum inflow
rate. The trench may be thought of as a variation of the "superpipe"
facility with a permeable pipe wall.
Analysis of the
facility is based on a form of the continuity equation which takes
account of the outflow control, the rate of exfiltration and the rate
of change of storage within the trench. Thus
Inflow = Outflow + Exfiltration
+ Rate of change of Storage
or
where I
= Inflow rate
Q
= Outflow rate
X
= Exfiltration rate
V
= Volume stored
and the subscripts 1
and 2 define values at times
t
and
(t+Dt)
respectively.
Equation [8.71] can be expanded as:
or
For any specified outflow control device, the
water surface elevation in the trench is dependent on the outflow
Q. Both storage volume
V and exfiltration
X are therefore dependent on
Q and a solution for the
unknown outflow at time (t+Dt)
can be obtained from:
The method is similar to the graphical solution
described in Figure 8.7 Graphical illustration of equation
[8.48] in topic Theory of Reservoir Routing. One difference is
that it is convenient to construct curves (or tables) of both
f(V,Q,X) and
X as functions of the water
surface elevation. In order to do this we must first provide a method
of predicting the rate of exfiltration from the trench.
Trench Exfiltration Rate
Figure 8.17 
Exfiltration Trench Crosssection
Figure 8.17 shows the crosssection assumed in
MIDUSS. The shape is a trapezium of height
H and top width
T tapering symmetrically to a
bottom width B. The water table
is assumed to be horizontal and located at a depth
P=(ILG) below the downstream
invert level of the trench. If the depth of water in the trench voids
is y the wetted surface of the
trench wall has a length ay
where a is given by:
Flow through the porous soil is assumed to be
laminar and can be estimated using Darcy's Law
where K
= hydraulic conductivity of the soil
Sf
= friction gradient
Q/A
= volumetric flux.
Note that the volumetric flux is much smaller
than the actual velocity through the voids since only a fraction of
area A is available for flow.
The average driving head between the water in the
trench and the water table is P + y/2
and the path length is P so that
the available gradient is given by [8.77].
The exfiltration flow through a unit length of
trench can then be estimated as:
where
b
= 1 or 0 depending on whether the ‘Include
base width’ check box is checked or unchecked. Checked is the default
condition.
Figure 8.18  Idealized
Longitudinal section on an Exfiltration Trench.
If the trench invert has a finite slope it is
possible that for low flows which can be transmitted by the orifice in
the outflow control device, the horizontal free surface does not
extend over the full length of the trench. Figure 8.18 shows this
situation. Even if the downstream depth is greater than the invert
drop DZ
the available surface for exfiltration must be corrected to
allow for the reduced depth at the upstream end. This assumes that
the hydraulic gradient along the trench is negligible and that the
surface is essentially horizontal. The available wall surface through
which exfiltration can occur is therefore given by:[8.79] and [8.80].
for y³DZ
for y<DZ
Estimating the
Required Trench Volume
When the Trench
command is invoked MIDUSS tries to estimate the required trench volume
(i.e. voids plus stone) which is required to achieve the currently
defined target peak outflow. The process is similar to that described
in the topic Theory of Reservoir Routing; Estimating the Required Pond
Storage. However, an additional level of iteration is required as for
each estimate of storage volume the corresponding exfiltration must be
computed and the target outflow reduced by this amount.
As with the Pond
procedure, the iteration uses the secant method to solve a
relationship between
Q
and K
to yield the required value of
Q
and thus estimate the storage from the corresponding lag
K.
The algorithm is
summarized as follows.
1. Assume maximum exfiltration rate
Xmax = 0
2. Set desired
Qout = TargetQout 
Xmax
3. Initialize values of K and Q for
two points on the curve, i.e.
K1 = Hydrograph
Volume/(0.6*Imax)
K2 = 0.2 Inflow
hydrograph timebase
Q1 = 0.1 Imax
4. Route inflow through a linear
reservoir of lag K2 to get maximum outflow Q2
5. Interpolate between points
(K1,Q1) and (K2,Q2) to get K3 for required Qout
6. For next iteration
set K1 = K2
K2 = K3
Q1 = Q2
7. If change in Q2 >
e
go to step 4.
8. Solution found for Q2. Estimate
storage S = K2.Q2 and convert to trench volume.
9. From trench volume estimate
maximum water level Wlmax.
10. For WL calculate exfiltration
Xmax.
11. For 5 iterations go to step 2.
Because of the many
other quantities which can affect the routing operation the estimate
is only an approximate guide and trial and error is normally required.
