
Publication History:
This article is based on
"Crain's Seismic Petrophysics" by E. R. (Ross) Crain, P.Eng., first
published in 2003, and updated annually until 2016.
This
webpage version is the copyrighted intellectual
property of the author.
Do not copy or distribute in any form without explicit
permission. 
Modeling Sonic and Density Logs From Resistivity
Resistivity is sometimes transformed into an apparent velocity
log with a number of different equations:
Faust Method
This method is very old, but is useful in shallow rock
sequences, especially clastics. You may need to determine new
parameters for each major geologic horizon.
_____1: Vc = KR1 * RESS ^ (1/KR2) * DEPTH ^ (1/KR3)
2: DTCsyn = 10^6 / Vc
Where:
__Vc = compressional
velocity (ft/sec)
DTCsyn = synthetic (modelled) sonic travel time
from Faust equation (usec/ft)
__KR1 = Faust constant (2000
to 3400 for depths in feet)
__RESS = resistivity from
shallow investigation log (ohmm}
__DEPTH = depth of layer (ft
or m)
__KR2 and KR3 = 6.0 or as
determined by regression analysis
NOTE: Constants given are for depths in FEET.
If depths are in meters, convert depth to feet. by multiplying
depth in meters by 3.281.
To obtain DTCsyn in usec/m from DTC in usec/ft, divide by 3.281.
The
Faust transform can be used when the sonic log is missing, and
can be calibrated with offset well data, check shots, or
vertical seismic profiles. The method does not account for gas
effect.
Smith Method
This method uses a simple correlation between resistivity and
sonic travel time:
_____ 3: DTCsyn = KR4 * (RESS ^ KR5)
Where:
DTCsyn = synthetic (modelled) sonic travel time
from Smith equation (usec/ft)
__KR4 = Smith constant (90
to 100 for depths in feet)
__RESS = resistivity from
shallow investigation log (ohmm}
__KR5 = 0.15 or as
determined by regression analysis
NOTE: Constants given are for depths in FEET.
If depths are in meters, convert depth to feet. by multiplying
depth in meters by 3.281.
To obtain DTCsyn in usec/m from DTC in usec/ft, divide by 3.281.
The
method does not account for gas effect. You may need to
determine new parameters for each major geologic horizon.
Fischer  Good Method
This method assumes a fairly sophisticated log analysis can be
run on the well in question or on a nearby well. This is needed
to obtain a list of water resistivity (RWA) versus depth. Since
most sonic log problems are in shales due to bad hole or rock
alteration, this calculation is usually possible and should be
done continuously or at least zone by zone.
Similarly, the apparent RW in shale (RWSH) is needed, based on
an estimate of the shale total porosity (BVWSH). This can be
computed continuously or zone by zone from one of the following:
If
neutron and density logs are both available and correct:
_____4: BVWSH = (PHIDSH + PHINSH) / 2
_____5: PHIt = (PHID + PHIN
) / 2
If
density log is missing or bad:
_____6: BVWSH = 0.95 *
PHINSH
_____7: PHIt = PHIN
Where the sonic log is behaving properly or from an offset well
that is OK:
_____8. BVWSH = (DTCSH 
DTCMA) / (DTCW  DTCMA)
_____9. PHIt = (DTC 
DTCMA) / (DTCW  DTCMA)
Then, for each shale zone:
_____10: RWSH = (BVWSH ^ M) *
RSH / A
And,
for each clean zone:
_____11: RWA = (PHIt ^ M) *
RESD / A
For
all digitized intervals or computation layers:
_____12: Vshg = (GR  GR0) /
(GR100  GR0)
_____13: Vshs = (SP  SP0) /
(SP100  SP0)
_____14: Vsh = Min (Vshg,
Vshs)
_____15: RMIX = 1 / (Vsh /
RWSH + (1  Vsh) / RWA)
_____16: DTCsyn = DTCMA + (DTCW
 DTCMA) * (A * RMIX / RESD) ^ (1/M)
_____17: DTCyn = Min (DTCsyn,
DTC)
_____18: DENSsyn = DENSMA + (DENSW
 DENSMA) * (A * RMIX / RESD) ^ (1/M)
_____19: DENSsyn = Min
(DENSsyn, DENS)
When
the zone is 100% shale, this equation should return a reasonable
travel time. If it doesn't match the log where it is believed to
be good, then adjust RWSH or Vsh. In clean zones, adjust DELTMA
or RWA if needed. When zones are hydrocarbon bearing, RWA and
RESD will both be too high, and the result will be close to
correct, but may give a DELTmod that is too low (too high a
velocity) or a DENSmod that is too high.
To
overcome some of this effect, you could substitute the shallow
resistivity RESS for RESD and RMF@FT for RWA. You may still need
to calibrate the RMF@FT with its own RMFA equation:
_____20: RMFA = (PHIt ^ M) *
RESS / A
_____21: RMIX = 1 / (Vsh /
RWSH + (1  Vsh) / RMFA)
_____22: DTCsyn = DTCMA + (DTCW
 DTCMA) * (A * RMIX / RESS) ^ (1/M)
_____23: DTCsyn = Min (DTCsyn,
DTC)
_____24: DENSsyn = DENSMA + (DENSW
 DENSMA) * (A * RMIX / RESS) ^ (1/M)
_____25: DENSsyn = Min
(DENSsyn, DENS)
Where:
DTCsyn = synthetic (modelled) sonic travel time
from Fischer and Good equation (usec/ft or usec/m)
DENSsyn = synthetic (modelled) density from Fischer and Good
equation (g/cc or kg/m3)
DTC, DENS = original sonic or density log readings if available
DTCSH, DTCMA, DYCW = compressional sonic travel time of shale,
matrix, water values
DENSSH, DENSMA, DENSW = density of shale, matrix, water values
PHID, PHIN = density and neutron porosity
A, M, N = Archie water saturation exponents
Vsh, PHIt = shale volume and total porosity
RWSH, RNFA, RNIX = resistivity of water in shale, nudfiltrate, and
invaded zone
Neither method accounts for the effect of gas, which must be
handled separately.
