The Matrix-Density / U Plot

Part 12, Chapter 12: Lithology and Mineralogy

Learning objectives

Compute the apparent matrix grain density and matrix U cross section
Read a three-mineral mix as a point inside the quartz-calcite-dolomite triangle
Get the mineral fractions from the point's barycentric position
Recognize this as matrix identification solved as a mixture

The Litho-Density Triangle

The modern lithology crossplot trades the sonic for the photoelectric factor. Strip the fluid from the density and from the volumetric cross section $U = P_e,\rho_b$ , and you get two apparent matrix quantities, the grain density and the matrix cross section:

\rho_{maa} = \frac{\rho_b - \phi\,\rho_f}{1-\phi}, \qquad U_{maa} = \frac{P_e\rho_b - \phi\,U_f}{1-\phi}.

The three common minerals sit far apart here, quartz at (4.8, 2.65), calcite at (13.8, 2.71), dolomite at (9.0, 2.87), so they frame a wide triangle.

Position Is the Mixture

Any three-mineral mix is a point inside the triangle, a weighted average of the three vertices. Its position reads off directly as the three fractions, the barycentric coordinates: how close the point sits to each vertex is how much of that mineral the rock holds. Porosity moves it nowhere, because the strip removes the fluid exactly.

A Solved Mixture

This is matrix identification solved, for three minerals, by pure geometry. But a rock can hold more than three components, add clay, or anhydrite, or a second carbonate, and then a triangle is no longer enough. The next section generalizes the idea into a linear solver that takes any set of logs and any list of minerals and returns all the volume fractions at once, the algebra behind every one of these crossplots.

References

Schlumberger (2009). Log Interpretation Charts (chart CP-21). Schlumberger Educational Services.
Ellis, D. V. and Singer, J. M. (2007). Well Logging for Earth Scientists, 2nd ed. Springer.