Part III, page 4
Effect of Porosity:
The effect of porosity is to reduce the
velocity of stress waves. Wyllie et al. (1956, 1958) relates bulk velocity Vb
to matrix velocity Vma and fluid velocity Vf to formation porosity (F) as follows:
1/Vb
= F/Vf + (1-F)/Vma (3-1)
Where:
F = Porosity
Vma
= velocity of sound in matrix
Vf = velocity of sound in fluid
Vb = Bulk Velocity
Equation 3-1 is often referred to as
the time-average formula or Wyllie’s equation.
Note also that when laboratory data was
obtained (Gregory, 1977) comparing the time-average formula, 90 percent of the
data was within 5 percent of calculated, 50 percent of the data was within 2
percent of calculated and 30 percent of the data was within 1 percent of
calculated porosity for a quartz-water system. The time-average line represents
the best fit for the laboratory data.
Rearranging equation 3.1 and using Dt for calculation of
porosity (F);
Dt - Dt(ma)
F = -------------- (3-2)
Dt(f) - Dt(ma)
Where:
F = Porosity (%)
Dt = travel time per ft from acoustic log
Dt
(ma) = matrix travel time
Dt (f) = fluid travel time
Dt
(ma) shaly sandstone varies from 60 to 70 usec per ft.
Dt
(ma) sandstone varies from 50 to 58 usec per ft.
Dt
(ma) limestone varies from 43 to 50 usec per ft.
Dt
(ma) Dolomite varies from 40 to 43 usec per ft.
This equation is often referred to as
the “time-average” formula. It is the
formula for the velocity obtained from the average time for a wave to propagate
through a formation of thickness = 1, of which fraction p is fluid and (1-p) is
matrix.
Equations 3-1, 3-2 are the most common equations used in
acoustic logging. Most of the applications using acoustic logging have the
purpose of obtaining porosity.
Restrictions applied to equation 3-2
are as follows. This equation was
developed using unconsolidated sedimentary rock. It is applicable only to fully
saturated pore space up to 35 percent porosity. Pore fluid must be liquid – not
gas-filled.
Equation 3-2 does not apply to
secondary porosity in the form of fractures and vugs as referenced below.
Fractures on a scale of 5 cm per meter
of depth represent 5 percent porosity. Acoustic tool response may show
porosities on a scale of 50 percent porosity.
Correction for excessive shale:
Shale affects acoustic velocity.
Laminar or structural shale is denser and therefore has a matrix velocity
different than the rest of the matrix.
Shale volume (Vsh) is obtained from
gamma-ray or SP logs obtained in shale formations adjacent to the formation of
interest.
Using
Dt(sh),
the apparent change in porosity (dfsh)
may be determined as follows:
dfsh
= vsh * (Dt(sh) - Dt(ma))/ (Dt(f) - Dt(ma)) (3-3)
Equation 3-2 may be corrected for shale
when Dt in shale, Dt(sh) exceeds 100
usec/ft. using:
100
---- (3-4)
Dt(sh)
Effect of gas and shale Chart
Uncertainties in acoustic porosity:
Studies have shown matrix velocities in a given Limestone
or Sandstone matrix may vary 4.3 usec. per ft. (Gregory, 1977). Considering the
velocity uncertainty gives a porosity uncertainty of 2.15 percent.
Other acoustic porosity models include
the Gassmann, Biot, and Brandt.
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