
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. 
Density Neutron
'SHALY SAND" Crossplot
CAUTION
DO NOT USE A SHALY SAND MODEL UNLESS
YOU ARE SURE IT IS SUITABLE FOR THE MINERAL MIXTURE. COMPLEX
LITHOLOGY MODELS ARE ALMOST ALWAYS BETTER THAN SHALY SAND MODELS
BECAUSE MOST SANDS ARE NOT PURE QUARTZ PLUS CLAY. MOST SANDS CONTAIN
OTHER MINERALS, SO THEY FIT THE COMPLEX LITHOLOGY MODEL BETTER. THE
FOLLOWING MATERIAL IS FOR REFERENCE ONLY, SINCE MANY SOFTWARE
PACKAGES CONTAIN "SHALY SAND" MODELS.
Most shaly sand cross plot models assume that the sand fraction
is pure quartz, so some of the models shown below will not be
appropriate for many situations. The
complex
lithology crossplot model is more robust and requires less
effort.
Density neutron crossplot methods involve simultaneous solution
of the response equations for the two logs. The response equation
for the density log in porosity units follows the classical form:
1:
PHID = PHIe * Sxo * PHIDw (water term)
+ PHIe * (1  Sxo) * PHIDh (hydrocarbon term)
+ Vsh * PHIDsh (shale term)
+ (1  Vsh  PHIe) * Sum (Vi * PHIDi) (matrix term)
WHERE:
PHIDh = log reading in 100% hydrocarbon
PHIDi = log reading in 100% of the ith component of matrix rock
PHID = log reading
PHIDsh = log reading in 100% shale
PHIDw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)
The response equation for the neutron porosity log also follows
the classical form:
2:
PHIN = PHIe * Sxo * PHINw (water term)
+ PHIe * (1  Sxo) * PHINh (hydrocarbon term)
+ Vsh * PHINsh (shale term)
+ (1  Vsh  PHIe) * Sum (Vi * PHINi) (matrix term)
WHERE:
PHINh = log reading in 100% hydrocarbon
PHINi = log reading in 100% of the ith component of matrix rock
PHIN = log reading
PHINsh = log reading in 100% shale
PHINw = log reading in 100% water
PHIe = effective porosity (fractional)
Sxo = water saturation in invaded zone (fractional)
Vi = volume of ith component of matrix rock
Vsh = volume of shale (fractional)
Various assumptions are made in order to solve these two equations
simultaneously for porosity. The second variable determined by
the pair is usually either of shale volume or matrix density,
which can determine rock type. If one of these is chosen, the
other must be assumed or previously calculated.
Very rapid methods for estimating porosity from density and neutron
log data are available, based on extensive use of assumptions.
They are easily memorized and should be used with mental arithmetic
or calculators. The approach simulates the results from chartbooks,
but more detailed formulae should be used in computers, as discussed
later in this chapter.
Read comments below BEFORE using this model.
In shaly sands, convert logs into sandstone units and enter the
uncorrected density and neutron log readings into the following
equations:
If no gas crossover exists, that is,
3: IF PHIN >= PHID
4: THEN PHIxdn = (PHID * PHINSH  PHIN * PHIDSH) / (PHINSH  PHIDSH)
If gas crossover is present, that is,
5: IF PHIN < PHID
6: THEN PHIxdn = ((PHIN ^ 2 + PHID ^ 2) / 2) ^ 0.5
WHERE:
PHID = density porosity log reading (fractional)
PHIdc = density log porosity corrected for shale (fractional)
PHIN = neutron porosity log reading (fractional)
PHInc = neutron log porosity corrected for shale (fractional)
PHIDSH = density log shale porosity (fractional)
PHINSH = neutron log shale porosity (fractional)
PHIxdn = porosity from density neutron crossplot (fractional)
COMMENTS:
Do not gas or shale correct the density or neutron data before
using this method.
This method calculates the shale correction based on the density
neutron crossplot shale volume (Vshx). If this is not the
shale volume method you want to use, DO NOT USE THIS POROSITY
METHOD.
This method is pessimistic if
heavy minerals exist in the sandstone. Since nearly every sandstone
contains minerals other than quartz to some degree (such as mica,
volcanic rock fragments, dolomite or calcite cement), the shaly
sand model is almost ALWAYS PESSIMISTIC. Methods listed below are also better.
However, this shaly sand model has been very widely used and it
can still be found in every computeraided log analysis package.
I strongly recommend NOT using this model.
Quick Density Neutron Porosity in Any Rock
An alternate quick look method for shaly sand or carbonates is:
7: PHIdc = PHID  Vsh * PHIDSH
8: PHInc = PHIN  Vsh * PHINSH
9: IF PHInc < 0
10: THEN PHInc = 0
11: PHIxdn = PHIdc + (PHInc  PHIdc) / 3
WHERE:
PHID = density porosity log reading (fractional)
PHIdc = density log porosity corrected for shale (fractional)
PHIN = neutron porosity log reading (fractional)
PHInc = neutron log porosity corrected for shale (fractional)
PHIDSH = density log shale porosity (fractional)
PHINSH = neutron log shale porosity (fractional)
PHIxdn = porosity from density neutron crossplot (fractional)
Vsh = shale volume (fractional)
COMMENTS:
This formula is effective even in gas zones and is based on the
premise that the neutron log sees deeper into the zone than the
density log, and thus has more gas effect.
Shale corrections could create
apparent gas crossover and this may be real or an artifact of
excessive correction. Check against known data from the well if
shale correction creates crossover.
RECOMMENDED
PARAMETERS: 


Range 
Default 
PHIDSH 
0.03
to +0.12 
0.00 
PHINSH 
0.10
to 0.40 
0.30 
NUMERICAL EXAMPLE:
1. Data for Shaly Sand "D"
PHID = 0.12
PHIN = 0.30
PHIDSH = 0.03
PHINSH = 0.30
PHIxdn2 = (0.12 * 0.30  0.28 * 0.03) / (0.30  0.03) = 0.100
The shale volume was 0.59 using this data. Since
this value is too high compared to the shale content from the GR, the porosity from this method will be too low.
2. If this porosity is unacceptably low, use the
complex
lithology formula
with Vsh of your choice, as shown below. Given Vsh = 0.33 from
GR, with other data as before, then:
PHInc = 0.30  0.33 * 0.30 = 0.20
PHIdc = 0.12  0.33 * 0.03 = 0.11
PHIxdn = (0.20 + 0.11) / 2 = 0.155
3. The alternate quick method gives, for Sand "D"
PHInc = 0.30  0.33 * 0.30 = 0.20
PHIdc = 0.12  0.33 * 0.03 = 0.11
PHIxdn3 = 0.11 + (0.20  0.11) / 3 = 0.140
Shaly Sand
Crossplot (Density Neutron) with Matrix Offset
This method is more longwinded than the quick method and accounts
for matrix offset to an arbitrary user defined matrix value.
Reconstitute density data from density porosity log.
12: DENS = (PHID * KD1 + (1  PHID)
* KD2
WHERE:
English Units
Metric Units Log Scale
KD1 1.00
1000
All
KD2 2.65
2650
Sandstone
KD2
2.71
2710
Limestone
KD2
2.87
2870
Dolomite
Calculate density porosity for desired matrix and fluid values:
13: PHIDm = (DENSMA  DENS) / (DENSMA  DENSW)
Calculate density offset for this matrix and fluid:
21: D = PHIDm  PHID
Calculate neutron offset for same matrix:
14: C = KN4 * D
WHERE:
KN4 = 1.00 for CNL 0.75 for SNP neutron log
Calculate neutron log reading for same matrix:
15: PHINm = PHIN  C
Adjust shale values for offsets:
16: PHIDSHm = PHIDSH + D
17: PHINSHm = PHINSH  C
Calculate porosity where there is no gas crossover, that is:
18: IF PHINm >= PHIDm
19: THEN PHIxdn = (PHIDm * PHINSHm  PHINm * PHIDSHm) / (PHINSHm
 PHIDSHm)
If gas crossover occurs:
20: IF PHINm < PHIDm
21: THEN PHIxdn4 = ((PHINm ^ 2 + PHIDm ^ 2) / 2) ^ 0.5
WHERE:
KD5 = 1.00 English units, 1000 Metric units
C = neutron log offset (fractional)
D = density log offset (fractional)
DENS = density log reading (Kg/m3 or gm/cc)
DENSMA = matrix density (Kg/m3 or gm/cc)
DENSW = fluid density (Kg/m3 or gm/cc)
PHID = density log reading in zone of interest (fractional)
PHIDm = density log reading corrected for matrix offset (fractional)
PHIDSH = density log reading in 100% shale (fractional)
PHIDSHm = density log reading in 100% shale corrected for matrix
offset (fractional)
PHIN = neutron log reading in zone of interest (fractional)
PHINm = neutron log reading corrected for matrix offset (fractional)
PHINSH = neutron log reading in 100% shale (fractional)
PHINSHm = neutron log reading in 100% shale corrected for matrix
offset (fractional)
PHIx = density neutron porosity with matrix and gas correction
applied (fractional)
PHIxdn = shaly sand crossplot porosity corrected for matrix offset
COMMENTS:
The graphical solution to these equations is shown below.
Chart for Estimating Shale Corrected Porosity From Density Neutron
Crossplot  Shaly Sand Model
The "Sand Point" can be shifted to account for Matrix Offset
Note
that this method, while called the shaly sand crossplot method,
will work for any lithology, providing it is constant over the
computed interval (shale may vary but not the mineral mixture).
Shale content is implicitly corrected by use of PHINSH and PHIDSH.
The method does not use the Vsh value calculated by any other
method.
The gas correction is usually applied automatically when crossover
of the density and neutron logs occurs. In shaly or dolomitic
sands, crossover may be inhibited by the nature of the
lithology, so the analyst may impose the corrections if gas is
known to be present. The analyst must select the matrix density
towards which the gas correction should be applied.
Because the shale volume is implied instead of an explicit input
parameter, I do not recommend this method.
RECOMMENDED
PARAMETERS: 


Range 
Default 
PHIDSH 
0.03
to +0.10 
0.00 
PHINSH 
0.20
to 0.40 
0.30 
DENSMA
(English) 
2.65
to 2.87 
2.71 
DENSMA
(Metric) 
2650
to 2870 
2710 
NUMERICAL EXAMPLE:
1. Assume data for Sand "C"
PHID = 0.33
PHIN = 0.24 (gas crossover occurs)
DENSMA = 2680 Kg/m3
DENSW = 1000 Kg/m3
D = 0.02
C = 0.02
PHIDm = 0.33 + 0.02 = 0.35
PHINm = 0.24  0.02 = 0.22
PHIxdn4 = ((0.22 ^ 2 + 0.35 ^ 2) / 2) ^ 0.5 = 0.30
2. Assume data from Sand "D" and impose gas correction
manually, since there is no gas crossover, due to shaliness.
PHID = 0.12
PHIN = 0.28
DENSMA = 2680 Kg/m3
DENSW = 1000 Kg/m3
D = 0.02
C = 0.02
PHIDm = 0.12 + 0.02 = 0.14
PHINm = 0.28  0.02 = 0.26
PHIDSHm = 0.03 + 0.02 = 0.05
PHINSHm = 0.30 = 0.02 = 0.28
PHIxdn4 = (0.12 * 0.28  0.26 * 0.05) / (0.28  0.05) = 0.11
