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PfEFFER ConceptsGeneral compositional solutions |
The mineral compositions of zones can be seen on log plots such as the density-neutron porosity crossplot when zones are referenced to specific mineral points and lithology lines. Unfortunately, the depth information of the log overlay is lost in the crossplot, but this loss is offset by the ability to make semiquantitative estimates of mineral and porosity proportions. On a crossplot the log data are plotted in a Cartesian space, where the measurement logs are orthogonal reference axes. If reference minerals are located in this log space, then the plotted zones can be remapped into a "composition space". Their coordinates in composition space are estimates of the proportions of the reference minerals.
Crossplots are methods of geometry, but the compositional proportions can be calculated from simple algebra. The composition of any zone in terms of volume of pore fluid (F), and proportions of two minerals, J and K, given two log readings signified by L1 and L2 can be solved from the equations:
Notice that the three unknown proportions are solved uniquely from only two logs, because the composition space is a closed system. The third ("given") unity equation states that the sum of all the proportions must total one. The closed system also explains why a coordinate location in two-log space can be remapped into a three component composition space (see Figure 57).
The set of equations could be solved by simple algebra. However, a more
efficient approach is to use matrix algebra (especially when we move from
this toy domain to some realistic applications). In matrix algebra terms the
three equations above could be represented as :CV = L
where C is a matrix of the log responses of the components, V is a vector
of the component proportions, and L is a vector of the log readings for the
zone of interest . The unknowns of the vector V, are solved by:![]()
where -1 is the inverse of the C-matrix.
When coded in a computer program, this little algorithm solves the problem neatly and efficiently. In most lithologic units, the set of descriptive minerals does not change as quickly as the log data. So, once the minerals are defined, the matrix, C can be constructed from their logging responses, using information such as supplied in the table in the appendix. The matrix is then inverted to compute the matrix, C-1. Volumetric proportions of mineral components can then be generated on a production line principle, by input of each successive zone reading, and multiplication by the inverse matrix to generate a string of volumetric proportions ordered with depth.
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| Figure 25: Different ways of plotting two log responses, L1 and L2: as a log overlay, “log space” crossplot, and “composition space” with three components (two minerals J and K, and a pore fluid component, F. |
How does this procedure work out in practice? Some ideas can be gained by making the concept more concrete. Let us identify the two logs as density and neutron porosity measurements, and the mineral components J and K to be calcite and dolomite. The fluid component is the pore space that is filled with mud filtrate. The component of water should not be considered differently from other mineral components. As far as the logging tools are concerned it is the mineral of hydrogen oxide. Links between the three components of calcite, dolomite, and water now define the boundaries of all limestones, dolomite, and dolomitic limestones that are physically possible and impossible. The log responses of any zone that plot within this composition triangle can be transformed into compositional proportions.
What are the possible explanations for any zones that lie outside the composition triangle? Zone A is close to the triangle, and could be a pure limestone whose logging responses have small errors. Alternatively, it could reflect the presence of quartz or some other lighter mineral that has not been use in the solution. How does the algebraic solution resolve this anomaly? The algorithm would predict a small negative proportion of dolomite for this zone, and the proportions of the other two components would be inflated slightly to maintain the proportional sum of unity. Clearly, this is impossible, but then the model decrees that all possible solutions must lie within the composition triangle. However, the solution has some useful and important consequences. The algorithm does not break down when confronted with “impossible” logging data, and the erroneous solution may contain clues to a correct composition.
Zone B is clearly a major problem and would be caused either by gross errors in logging tool readings or the presence of significant residual gas in the pore space. The solution would be a large negative dolomite proportion and an impossibly high calcite proportion (greater than a equivalent 100%). Zone C could be a dolomite with perfect log readings, but has a negative proportion of calcite, because the dolomite line is modeled by the algebra as a straight line rather than the curve of the real response. Zone D will generate a higher proportion of negative calcite and could be caused by a shale or some dense mineral. Finally, Zone E will not be a problem zone in the sense of generating an impossible solution. However, it could be flat wrong. For example, these log responses would be equally consistent for a zone of cherty dolomite.
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| Figure 26: A generalized neutron-density crossplot marked with some hypothetical “problem zones” (A-E) |
All these examples lead to conclusions that match common sense. The user defines a system of components to be solved by the algorithm in a computer program. The computer dutifully provides a solution in terms of that system. The overlay and crossplot methods described in the last section are therefore an important first step to the identification of mineral components to be used in the model. If strange solutions are generated by the algorithm, then crossplots of the log data will be the natural medium to detect the source of the problem.
As a log example of this same procedure, we can analyze the composition of a Viola Limestone section, using a matrix algebra representation. The Viola is a compositional mixture of dolomite, quartz (chert) and calcite. In cases where a lithodensity-neutron log suite was run through the Viola, it is possible to link three logging measurements with four unknowns as a set of simultaneous equations. The four unknown components are : dolomite, chert, calcite and porosity. The porosity component is the fluid in the pore space of the flushed zone, which is primarily mud filtrate. It can be considered to be a mineral called "water".
Although there are four unknowns, only three logging measurements are required,
because the unknowns are proportions which collectively form a closed system.
The three measurements are neutron porosity, density, and volumetric photoelectric
absorption. Note that neutron porosity can be entered in either percentage
or proportional units ; density can be expressed in units of grams/cc (as
used here) or in porosity units. However, the units must be CONSISTENT throughout.
So, if the density properties of the components are entered as grams/cc, then
the density log readings used in the calculations must ALSO be in grams/cc,
NOT porosity units. The photoelectric index, Pe, is converted to the volumetric
photoelectric measurement, U, in barns/cc. from the multiplication of the
photoelectric index by the bulk density :![]()
because the density closely approximates the electron density, which has units
of electrons/cc. The equations that link the unknown component proportions
multiplied by their physical properties with the log responses measured at
any depth zone are:
| Neutron: | |
| Density: | |
| UPhoto: | |
| Unity: |
Rewritten as matrices:

The inverse matrix, C-1, of the coefficient matrix is found and the solution for the vector of unknown proportions is then:
The proportional composition of any zone in the sequence can be found immediately by pre-multiplying a column vector of the zone log readings and a unity value by the inverse of the coefficient matrix.
This simple compositional algorithm is coded as an option in PfEFFER. In all cases, n components will require (n-1) logs for an explicit solution. Typically, these logs will be drawn from either/and/or the neutron, density, lithodensity and sonic logs, coupled with the gamma ray. When resolving reservoir lithologies, pore fluid is a distinct component that is identified primarily as mud filtrate because the investigation depth of most of the tools used is limited to the flushed zone. As far as the logging tool is concerned, water is a mineral in its own right and its liquid nature is immaterial. Logging properties of common minerals can be found in logging service company manuals and log analysis texts.. The quantitative evaluation of shale is a different problem, because its properties are a function both of its composition and degree of compaction. When used as a component, the coefficients of shale should be drawn from the analytical section itself, through the use of crossplots and log trace segments which are indicated to be typical shales by the gamma ray log.
In reviewing the results of a compositional solution, recall that the computed proportions of components are constrained to sum to unity, but there may be cases where individual proportions are negative. Although these may be physically impossible, they are not only mathematically possible, but even give valuable feedback information! The proportions will always sum to unity and, when all are positive, will lie within the space defined by the component coefficients. When one or more are negative or exceed unity, then the point lies outside the composition space. Small negative quantities signify minor errors and can be ignored. Larger values are diagnostic of systematic factors in addition to the components used for the solution. In the first cut, the log readings should be verified to see that they are both correct and have units which are consistent with the component log coefficients which were entered. The coefficients should also be checked to see if they are reasonable. This last point should be examined carefully when "shale" is selected as a component. Washouts usually produce outrageous values, but can be recognized either by their excessive log readings or on the caliper log.
If these initial tests are passed, then the indications are that an additional component is present. The identity of the "negative component" is usually a useful hint as to the nature of the extra component, as can be understood in this example by reference to the RHOmaa - Umaa crossplot. The quartz - calcite - dolomite triangle can be thought of as the base of a tetrahedron in log space, where the missing vertex is the fluid point. The presence of gas results in a decrease in neutron porosity and increase in density porosity which shifts the zone outside the triangle into "negative dolomite space". The clay mineral content of shales will draw zones in the opposite direction into "negative calcite space". In Permian sequences, zones plotting in "negative quartz space" will often signify the occurrence of anhydrite.
In the graphic compositional profile generated by PfEFFER, all negative components are made zero, and the proportions then recalculated to sum to one.
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| Figure 27: Location of “impossible solution” fields for
the quartz-calcite- dolomite system on a RHOmaa-Umaa plot. Negative fields are indicated for calcite (C), quartz (Q), and dolomite (D). |