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CRAMER'S RULE
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
Where:
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.
RESOLVING DETERMINANTS
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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