Publication History: This article was written especially for "Crain's Petrophysical Handbook" by E. R. Crain, P.Eng in 2007. This webpage version is the copyrighted intellectual property of the author.

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Cramer's Rule is a handy way to solve for any one of the variables in a set of linear simultaneous equations without having to solve the whole system of equations. Or it can be used to solve for all the unknowns. Such equation sets are often used to solve multi-mineral models for lithology and porosity.

Crossplot methods of the types discussed in other Chapters are actually solutions to three or four simultaneous equations. For example, the density neutron crossplot can be described by generalized forms of their response equations:

      a1 * X + a2 * Y +  a3 * Z = PHID
      b1 * X + b2 * Y +  b3 * Z = PHIN
     1.0 * X + 1.0 * Y + 1.0 * Z = 1.00

  a1, a2, a3 = density log porosity values for rock components X, Y, Z
  b1, b2, b3 = neutron log porosity values for rock components X, Y, Z
  X, Y, Z = rock volumes of the three components (fractional units)

The left-hand side of the equations with the variables is the coefficient matrix and the right-hand side is the answer matrix.

                  Coefficient Matrix |D|        Answer Matrix
                           | a1  a2  a3 |                           | PHID |
                           | b1  b2  b3 |                           | PHIN |
                           | 1.0 1.0 1.0 |                           |  1.0   |                  | a3   b3   1.0 | 

|Dx| is the determinant formed by replacing the X-column values with the answer-column values. Similarly, the |Dy| and |Dz} determinants are formed by replacing the Y-column and the Z-column, as shown below.



                X- Determinant |Dx|   Y- Determinant |Dy|   Z- Determinant |Dz|
                      | PHID   a2   a3 |              | a1   PHID   a3 |                | a1   a2   PHID |
                      | PHIN   b2   b3 |              | b1   PHIN   b3 |                | b1   b2   PHIN |
                      |   1.0    1.0  1.0 |              | 1.0    1.0   1.0 |                | 1.0  1.0     1.0  |

Cramer's Rule says that
      1: X = |Dx| / |D}
      2: Y = |Dy| / |D|
      3: Z = |Dz| / |D|.

The next step is to evaluate each determinant and calculate X, Y, and Z.

Solving for the value of a determinant is a matter of properly applying the arithmetic needed. Start with a sample, such as |Dx|. Extend the matrix by re-writing all the columns except the last one, as below. Then multiply the values in each "full" diagonal (coloured cells) and add these products together (honour the signs). This gives the sum of the "Down" diagonals, Dd.

                X- Determinant |Dx|                    X- Determinant EXTENDED
                      | PHID   a2   a3 |                             | PHID   a2   a3 | PHID   a2
                      | PHIN   b2   b3 |                             | PHIN   b2   b3 | PHIN   b2
                      |   1.0    1.0  1.0 |                             |   1.0    1.0  1.0  1.0    1.0   <==  a3 * PHIN * 1.0  
                                                                                                \         \== a2 * b3 * 1.0
                                                                                                 \== PHID * b2 *1.0  ADD Products together = Dd

Then do the same with the opposite diagonals. This gives the sum of the "Up" diagonals, Du.

                X- Determinant |Dx|                    X- Determinant EXTENDED
                      | PHID   a2   a3 |                             | PHID   a2   a3 | PHID   a2
                      | PHIN   b2   b3 |                             | PHIN   b2   b3 | PHIN   b2
                      |   1.0    1.0  1.0 |                             |   1.0    1.0  1.0 |   1.0    1.0

Obtain the products of the "Up" diagonals and ADD the products = Du.

THEN |Dx| = Dd - Du. Follow the same procedure for |Dy|, |Dz|, and |D|, then use Cramer's Rule to solve for X, Y, Z.
If you give the coefficients appropriate numerical values that correspond to calcite, dolomite, and water for example,, with particular values of PHID and PHIN, you will get the fraction of each component in the reservoir. Porosity will equal the volume of water.

To generalize the model, form the approptiate equations for each determinant using variable names instead of actual numerical values. The equations will look a little messy, but will work with any rational inputs. Negative answers for X, Y, or Z are illegal and suggest bad data or bad parameters. Small negative values can be trimmed to zero, but large negative answers will need more help.


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