The Total Vertical Stress Acting on a Horizontal Plane at Any Depth Equals

The total vertical stress acting on a horizontal plane at any depth equals:

where: total stress, effective stress and pore water pressure.

Total stress due to the weight of overlying rock and water usually is essentially constant so we can write:

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 () leads to a decrease in the volume () of a given mass of water.

The compressibility of water is defined as:

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:

where: is the density of water

COMPRESSIBILITY OF A POROUS MEDIUM

where: Es is the bulk modulus of compression for the aquifer skeleton,VT is the volume of the aquifer, and is the change in effective stress.

where: Vs is the volume of the solids and Vv is the volume of the voids.

If then 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:

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, , and for a unit decline in head we get:

SECOND MECHANISM

We can write:

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:

With VT =1, , and we get: