GROUNDWATER
HYDROLOGY II
WMA 302
Dr. A.O. Idowu, Dr. J.A. Awomeso and Dr O.Z. Ojekunle
Dept of Water Res. Magt. & Agromet
UNAAB. Abeokuta. Ogun State
Nigeria
oojekunle@yahoo.com
STEADY STATE FLOW AND
TRANSIENT FLOW


Steady-state flow occurs when at any point in a
flow field the magnitude and direction of the
flow velocity are constant with time.
Transient flow (Unsteady flow or non steady
flow) occurs when at any point in a flow field
the magnitude or direction of the flow velocity
changes with time.
STEADY STATE FLOW AND
TRANSIENT FLOW (Cont)

Fig. 1 below show a steady-state flow groundwater flow pattern
(dashed equipotentials, solid flowline) through a permeable
alluvial deposit beneath a concrete dam. Along the line AB, the
hydraulic head hAB = 1000m. It is equal to the elevation of the
surface of the reserviour above AB. Similar hAB = 900m (the
elevation of the tailrace pond above CD). The hydraulic head
drop h across the system is 100m. if the water level in the
reserviour above AB and the water level in the tailrace pond
above CD do not change with time. The hydraulic head at point
E, for example, will be hE = 950m and will remain constant.
Under such circumstances the velocity V = -Kdh/dl will also
remain constant through line. In a steady-state flow system, the
velocity may vary from point, but it will not vary with time at any
given point.
STEADY STATE FLOW AND
TRANSIENT FLOW (Cont)

Let us now consider the transient flow problem schematically
shown in fig. 2. At time t0 the flow net beneath a dam will be
identical to that of fig. 1 and the hE will be 950m. If the
reserviour level is allowed to drop over the period t0 to t1 until
the water level above and below the dam are identical at time t1,
the ultimate condition under a dam will be static with no flow of
water from the upstream to the down stream side. At point E the
hydraulic head hE will under a time-dependent decline from hE
= 950m at time t0 to its ultimate value of hE= 900m. There may
well be time lag in such a system so that hE will not necessarily
reach the value hE = 900m until sometime at t = t1.

DIFFERNCES BETWEEN
STEADY STATE FLOW AND
TRANSIENT
FLOW
One important difference between steady and transient lies in the relation
between their flowline and pathlines. Flowlines indicate the instantaneous
direction of flow throughout a system (at all times in a steady system, or at a
given instant in time in a transient system). They must be orthogonal to the
eqiuipotential lines throughout the region of flow at all times. Pathlines may
take the route that an individual particle of water follows through a region of
flow during a steady or transient event. In a steady flow system, a particle of
water enter the system at an inflow boundary will flow towards an outflow
boundary along a pathline that coincides with with a flowline such as that
shown in fig. 1. In Transient system, on the otherhand, pathline and flowline
do not coincide. Although a flow net can be constructed to describe the flow
conditions at any given instant in line in a transient system, the flowline
shown in such a snapshot represent only the configuration of the flowlines
changes with time, the flowlines cannot describes, in themselves, the
complete path of a particle of water as it transerves the system. The
delineation of transient pathline has obvious importance in a study of
groundwater contamination.
Note

The practical methodology that is presented later is
often based on theoretical equations, but it is not
usually necessary for practicing hydrogeologist to have
the mathematical equations at his/her fingertips. The
primary application of steady state techniques in
groundwater hydrology is the analysis of regional
groundwater flow. An understanding of transient flow
is required for the analysis of well hydraulic,
groundwater recharge, and many of the geochemical
and geotechnical application.
STEADY-STATE SATURATED
FLOW

Consider a unit volume of porous media such as
that shown in fig. 3 such an element is usually
called an elemental control volume. The law of
conservation of mass for a steady-state flow
through a saturated porous medium requires
that the rate of fluid mass flow into any
elemental control volume. The equation of
continuity that translate the law into
mathematical form can be written with reference
to fig. 3 as
STEADY-STATE SATURATED
FLOW (Cont)

A quick dimension analysis on the v terms will show
then to have the dimension of a mass rate of flow
across a unit cross-sectional area of the elemental
control volume. If the fluid is incompressible, (x,y,z) =
constant and the ’s can be removed from eqn 1. even if
the fluid is compressible and (x,y,z) is not equal
constant, it can be shown that the terms of the form
dvx/dx are much greater that terms of the form
vxd/dx, both of which arise when the chain rule is used
to expand eqn 1 in either case eqn 1 simplifies to
STEADY-STATE SATURATED
FLOW (Cont)

Substitution of darcy’s law for Vx, Vy and Vz in eqn. 2
yield the equation of flow for steady-state flow through
anisotropic saturated porous medium.

For an anisotropic medium, Kx=Ky=Kz, and if the
medium is also homogeneous, then k(x,y,z)=constant.
Eqn. 3 then reduces to the equation of flow for steadystate flow through a homogeneous isotropic medium.
STEADY-STATE SATURATED
FLOW (Cont)


Eqn. 4 is one of the most basic partial differential
equation to mathematician. It is called laplace’s
equation. The solution of the equation is a function of
h(x,y,z) that describes the value of hydraulic head h at
any point in a three dimensional flow field. A solution
of equation 4 allows us to produce a contoured
equipotential map of h1 and with the addition of
flowline, at a flow net.
For steady-state saturated flow in a two dimension flow
field, say in the xz plane, the central term of eqn. 4
would drop out and the solution would be a function of
h(x,z).
TRANSIENT SATURATED FLOW

The law of conservation of mass for transient
flow in a saturated porous medium requires that
the net rate of fluid mass flow into any
elemental control volume be equal to the time
rate of change of fluid mass storage with the
element with reference to fig 3, the equation of
continuity takes the form
TRANSIENT SATURATED FLOW
(Cont)

The first term on the right hand side of eqn. 6 is the mass rate of
water produced by the expansion of water under a change in its
density . The second term is the mass rate of water produced by
the compaction of the porous medium as reflected by the change
in its porosity n. The first term is controlled by the
compressibility of the fluid and the second term is controlled by
the compressibility of the aquifer . It is necessary to simplify the
second terms in the right of equation 6. We known that the
change in and the the change in n are both produced by the
change in hydraulic head h, and that the volume of water
produced by the 2 mechanisms for a unit decline in head in Ss,
where Ss is the specific storage given by Ss = g ( + n). The mass
rate of water produce (time rate of change of fluid mass storage)
is Ssdh/dt and the eqn. 6 becomes