Lithology from PE-Density-Neutron Logs
When the density neutron log is recorded with a photo electric curve, it is often called a litho-density log. Most density neutron logs run today are of this type. The photo electric effect (PE) log curve is sometimes used directly for determining lithology, because it has very definitive matrix values, and linear interpolation between two end points works quite well. However there is a small porosity effect, so a quantity called the photoelectric absorption coefficient (U) is used instead:
       1: U = PE * DENS

The response equation for the absorption coefficient (U) is:
      2: U = UMA * (1 - PHIe - Vsh)
             + UW * PHIe * Sxo
             + UH * PHIe * (1 - Sxo)
             + USH * Vsh

A common assumption is that UH and UW are very small, and that the shale term can be lumped into the matrix term. Thu equation 1 is sometimes used directly.  Equation 6 below is more accurate in shaly zones.:

WHERE:
  PHIe = effective porosity (fractional)
  U = photoelectric absorption cross section (barns/cm3)
  UH = photoelectric absorption of hydrocarbon (barns/cm3)
  UMA = photoelectric absorption of matrix rock (barns/cm3)
  USH = photoelectric absorption of shale (barns/cm3)
  UW = photoelectric absorption of water (barns/cm3)
  Vsh = volume of shale (fractional)

Note - 1 barn = 1*10^- 24 square centimeters - pretty small cows!

      3: DENSSH = PHIDSH * KD1 + (1 - PHIDSH) * KD2
      4: USH = PESH * DENSSH
      5: DENS = PHID * KD1 + (1 - PHID) * KD2
      6: Uma = (PE * DENS - Vsh * USH) / (1 - PHIe - Vsh)

WHERE:
  DENS = density log reading (gm/cc)
  DENSSH = density log reading in 100% shale (gm/cc)
  PE = photoelectric cross section (barns/cm3)
  PESH = photoelectric cross section  in 100% shale (barns/cm3)
  PHIe = effective porosity (fractional)
  U = photoelectric absorption cross section (barns/cm3)
  UH = photoelectric absorption of hydrocarbon (barns/cm3)
  Uma = computed matrix photoelectric absorption cross section (barns/cm3)
  USH = photoelectric absorption of shale (barns/cm3)
  UW = photoelectric absorption of water (barns/cm3)
  Vsh = volume of shale (fractional)

COMMENTS
The Uma values can be used in crossplots with matrix density (DENSma), to determine lithology fractions in a two or three mineral model.

 
 Matrix Density vs Matrix Cross Section Crossplot for Lithology

Rock Volume from PE Density Neutron Models
The PE values can be linearly interpolated between any two assumed end points of a two mineral model in the usual way.
      7: Min1 = (PE - PE2 - Vsh * PESH) / (PE1 - PE2)
      8: Min2 = 1.00 - Min1

This is the ONLY lithology model that works in gas zones, since PE is not affected by gas. All methods that use density, neutron or sonic are invalid in gas zones.

The Uma values can also be linearly interpolated between any two assumed end points of a two mineral model in the usual way.
      9: Min1 = (Uma - UMA2 - Vsh * USH) / (UMA1 - UMA2)
      10: Min2 = 1.00 - Min1

DENSma and Uma values can be linearly triangulated between any three assumed end points of a three mineral model in the usual way.
      11: D = (Uma * (DENS2 - DENS1) + DENSma * (UMA1 - UMA2)
             + UMA2 * DENS1 - UMA1 * DENS2) / (UMA1 * (DENS3 - DENS2)
             + UMA2 * (DENS1 - DENS3) + UMA3 * (DENS2-DENS1))
      12: E = (D * (DENS3 - DENS1) - DENSma + DENS1) / (DENS1 - DENS2)
      13: Min1 = MAX(0, 1 - D - E) / (MAX(0, 1 - D - E) + MAX(0, D) + MAX(0, E))
      14: Min2 = MAX(0, E) / (MAX(0, 1 - D - E) + MAX(0, D) + MAX(0, E))
      15: Min3 = (1 - Min1 - Min2)

WHERE:
  PHIe = effective porosity from any method (fractional)
  PE = measured PE log value of rock mixture
  PE1 = PE of first mineral (fractional)
  PE2 = PE of second mineral (fractional)
  Min1 = relative volume of first mineral (fractional)
  Min2 = relative volume of second mineral (fractional)
  Min3 = relative volume of third mineral (fractional)
  Vsh = volume of shale (fractional)
  Uma = computed UMA value of rock mixture
  UMA1 = UMA of first mineral (fractional)
  UMA2 = UMA of second mineral (fractional)
  UMA3 = UMA of third mineral (fractional)
  DENSma = computed matrix density value of rock mixture
  DENS1 = matrix density of first mineral (fractional)
  DENS2 = matrix density of second mineral (fractional)
  DENS3 = matrix density of third mineral (fractional)
 

COMMENTS:
The relative Vmin values must be multiplied by Vrock to get absolute values of V1, V2, V3. Vrock = 1- PHIe -Vsh..

NUMERICAL EXAMPLE:
1. Assume data as follows:
PE = 1.68 barns/cm3
DENS = 2.20 gm/cc
PHIN = 0.27
U = 1.68 * 2.20 = 3.69
Uma = 1.68 * 2.20 / (1 - 0.27) = 5.20

Both PE and Uma are close to the  quartz values. If it is dolomitic sandstone, assume:
Vsh = 0.10
PHIe = 0.24
Uqrtz = 4.79
Udolo = 9.00

Min1 = (5.20 - 9.00) / (4.79 - 9.00) = 0.90
Mun2 = 1.00 - 0.90 = 0.10

Vrock = 1 - 0.10 - 0.24 = 0.66.
Vmin1 = 0.90 * 0.66 = 0.60
Vmin2 = 0.10 * 0.66 = 0.06

The rock matrix is 90% quartz, 10% dolomite, but 34% of this is made up of porosity and shale, so the actual volumes of matrix rock are reduced by this amount.

 

MATRIX ROCK PARAMETERS


 

 

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