The total vertical stress  acting on a horizontal plane at any depth
equals:
   p
where: 
pressure.
 total stress,   effective stress and p  pore water
Total stress due to the weight of overlying rock and water usually
is essentially constant so we can write:
d  dp
or change in effective stress is equal to the negative change in
pressure.
The negative sign indicates that a decrease in fluid pressure is
accompanied by an increase in intergranular pressure.
COMPRESSIBILITY OF WATER
An increase in pressure ( dp ) leads to a decrease in the volume
(Vw ) of a given mass of water.
The compressibility of water

is defined as:
dVw / Vw
1
 
Ew
dp
where: Ew is the bulk modulus of compression for water
Vw is the volume of water
dp is the change in water pressure
For a given mass of water we can say:
1
d  / 
 
Ew
dp
where:

is the density of water
COMPRESSIBILITY OF A POROUS MEDIUM 

dVT / VT
1

d
Es
where: Es is the bulk modulus of compression for the aquifer
skeleton,VT is the volume of the aquifer, and d is the change in
effective stress.
VT  VS  VV
where: Vs is the volume of the solids and Vv is the volume of the
voids.
If dVS  0 then dVT  dVV indicating that the change in
aquifer volume is represented by the change in pore space (voids).
A decrease in hydraulic head (h) infers a decrease in fluid pressure
and an increase in effective stress.
Water is produced from storage in a confined aquifer under the
conditions of decreasing head by two mechanisms:
1) Compaction of the aquifer caused by increasing  .
2) The expansion of water caused by decreasing p.
The first mechanism is controlled by aquifer compressibility  .
The second mechanism is controlled by the fluid compressibility
.
FIRST MECHANISM
We can write:
dVW  dVT  VT d
Amt of water produced = volumetric reduction of the aquifer=
compressibility of the aquifer X volume of the aquifer X change in
effective stress.
For a unit volume VT = 1,
in head dh  1 we get:
d    gdh , and for a unit decline
dVW   g
SECOND MECHANISM
We can write:
dVW   VW dp
Amt of water produced = compressibility of water X volume of
water X change in pressure.
The volume of water VW in the total unit volume VT is nVT where
n is the porosity.
So:
dVW    nVT dp
With VT =1,
dp   gdh ,
dVW   n g
and
dh  1 we get: