
Publication History:
This article is based on
Chapter 7 of "The Log Analysis Handbook" by E. R. Crain, P.Eng., published by Pennwell Books 1986 Updated 2004.
This
webpage version is the copyrighted intellectual
property of the author.
Do not copy or distribute in any form without explicit
permission. 
Sonic DENSITY Crossplot
The sonic density crossplot method involves the simultaneous solution
of the sonic and density response equations for porosity. They
are similar in form to the density neutron pair, and will not
be repeated here.
The sonic density crossplot works best in shaly sands with no
gas. The resolution is poor in carbonates and gas will make the
result too high. The equations are shown graphically below the
math.
Calculate compaction correction.
1: KCP = max (1, CDTSH / (100 + 228 * (IF DEPTHUNIT$ = "METRIC")))
Calculate sonic shale porosity and total porosity.
2: PHISSH = (DELTSH  DELTMA) / (DELTW  DELTMA) / KCP
3: PHIS = (DELT  DELTMA) / (DELTW  DELTMA) / KCP
Calculate effective porosity:
4: PHIxsd = (PHID * PHISSH  PHIS * PHIDSH) / (PHISSH  PHIDSH)
Where:
CDTSH = sonic shale value for compaction correction (usec/ft or
usec/m)
KCP = compaction factor (fractional)
DELT = sonic log reading (usec/ft or usec/m)
DELTMA = travel time in rock matrix (usec/ft or usec/m)
DELTSH = sonic log reading in shale (usec/ft or usec/m)
DELTW = travel time in water (usec/ft or usec/m)
PHID = density log reading (fractional)
PHIDSH = density shale point (fractional)
PHIS = porosity from sonic log (fractional)
PHISSH = sonic porosity in shale (fractional)
PHIxsd = porosity from density sonic crossplot (fractional)
COMMENTS:
This method is pictured in below.
Chart for Sonic Density Porosity Model  shale corrected data
must be entered
This
method is analogous to the density neutron shaly sand method. It does not work well in mixed lithology, such
as limestone, dolomite and anhydrite mixtures. This method also
does not use the volume of shale (Vsh) determined by the analyst,
but uses the implicit shale correction determined by the sonic
shale point (PHISSH) and the density shale point (PHIDSH). The
sonic density crossplot should not be used in gas zones, since
both logs can read too high due to gas effect.
If a matrix offset is required for the density log, use the method
described in HERE.
RECOMMENDED PARAMETERS: 


Range 
Default 
PHIDSH 
0.03
to +0.20 
0.00 
DELTSH
(English)

75
to 140 
100 
DELTSH
(Metric)

225
to 460 
328 
Porosity From the Sonic Density Log (HuntRaymer
Method)
Calculate shale corrected density and sonic log readings and convert
to English units.
5: PHIdc = PHID  Vsh * PHIDSH
6: DELTc = (DELT  Vsh * (DELTSH  DELTMA))
/ KX2
7: DENSc = PHIdc * KD1 + (1  PHIdc) * KD2
Where:
KD1 = 1.00
KD2 = 2.65 for Sandstone Units
KD2 = 2.71 for Limestone Units
KX2 = 3.281 for Metric Units
KX2 = 1.00 for English Units
Calculate velocity data from sonic travel time data.
8: VELOGc = 10 ^ 6 / DELTc
9: VELMA = 10 ^ 6 / DELTMA
10: VELW = 10 ^ 6 / DELTW
Calculate sonic porosity.
11: C = 1  (VELOGc / (VELMA * ((DENSMA / DENSc) ^ 0.5))) ^ (1
/ 1.9)
12: D = DELTc ^ 2  DENSc * (DELTMA ^ 2) / DENSMA
13: E = DENSc * (DELTW ^ 2) / DENSW  DENSc * (DELTMA ^ 2) / DENSMA
14: IF C <= 0.37
15: THEN PHIxhr = C
16: OTHERWISE PHIxhr = ((0.47  E / D) / 0.1) * E / D + ((0.37
 C) / 0.1) * C
Where:
C = intermediate term
D = intermediate term
DELT = sonic log reading in zone of interest (usec/ft or usec/m)
DELTc = sonic log reading corrected for shale (usec/ft or usec/m)
DELTMA = sonic log reading in l00% matrix rock (usec/ft or usec/m)
DELTSH = sonic log reading in l00% shale (usec/ft or usec/m)
DELTW = sonic log reading in 100% water (usec/ft or usec/m)
DENSc = density log reading corrected for shale (gm/cc)
DENSMA = matrix density (gm/cc)
E = intermediate term PHID = density log reading in zone of interest
(fractional)
PHIDSH = density log reading in 100% shale (fractional)
PHIxhr = porosity from sonic log by HuntRaymer crossplot (fractional)
VELOGc = sonic velocity log reading corrected for shale (ft/sec
or m/sec)
VELMA = sonic velocity log reading in l00% matrix rock (ft/sec
or m/sec)
VELW = sonic velocity log reading in 100% water (ft/sec or m/sec)
Vsh = shale volume (fractional)
COMMENTS:
The HuntRaymer equations for density sonic crossplot porosity are an extension of
their work for the sonic log. The results are
too high in gas and can be corrected by a proper choice of DENSW
and DELTW. The method is not universally applicable and should
be tested in each area before use.
RECOMMENDED PARAMETERS:
Range Default
PHIDSH 0.03 to +0.20 0.00
DELTSH (English) 75 to 140 100
DELTSH (Metric) 225 to 460 328
NUMERICAL EXAMPLE:
1. Sonic Density Crossplot  data from Sand "D"
DELT = 300 usec/m
PHID = 0.12
DELTSH = 328 usec/m
PHIDSH = 0.03
Vsh = 0.33
DELTW = 616
DELTMA = 182
CP = 1.00
no matrix offset
PHISSH = (328  182) / (616  182) / 1.0 = 0.33
PHIS = (300  182) / (616  182) / 1.0 = 0.27
PHIxsd = (0.12 * 0.33  0.27 * 0.03) / (0.33  0.03) = 0.105
Vsh was 0.48 using this data, but Vsh from GR is
only 0.33. Therefore, porosity may be too low due to the shale
correction built into this method.
2. HuntRaymer Sonic Density Crossplot  data from Sand "D"
Vsh = 0.33
PHIdc = 0.12  0.33 * 0.03 = 0.11
DELTc = (300  0.33 * (328  182)) / 3.28 = 76.8 usec/ft
DENSc = 0.11 * 1.00 + (1  0.11) * 2.65 = 2.47
VELOGc = 10 ^ 6 / 76.8 = 13020 ft/sec
VELMA = 10 ^ 6 / 55.5 = 18020 ft/sec
VELW = 10 ^ 6 / 188.0 = 5320 ft/sec
C = 1  (13020 / (18020 * ((2.65 / 2.47) ^ 0.5))) ^ (1 / 1.9)
= 0.173
PHIxhr = 0.173
This is considerably higher than the standard sonic density crossplot
which over corrected for shale in this example.
MATRIX ROCK PROPERTIES
