Prediction of the Formation of Water-in-Oil Emulsions - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 66

Table 3.2 Model 1: values
used to correct oil property
input parameters

Parameter Form Correction value

Density Exponential 2.5

Viscosity Natural

logarithm

5.8

Saturates Standard % 45

Resins Standard % 10

Asphaltenes Standard % 4

Asphaltene/Resin ratio Standard 0.6

less the peak value. The values found for the transformations are listed in Table 3.2 .
It should be noted that the exponential of density was used and the natural log of the
viscosity. Previous modelling work had shown that these mathematical changes are
necessary to achieve higher correlations, cf. [ 4 ].
Having the transformed values, the newmodel proceeds by fitting a multiple linear
equation to the data. The choice of functions was achieved by correlating the stability
function directly with the data and taking the best of the functions (e.g. square, log,
etc.) into the regression process. The functionalities of square, logarithmic or expo-
nential curves are achieved by correlating the straight value of the input properties
plus their expanded values, taken here as the cube of the starting parameter as well as
the square of the exponential of the starting value; and their companded values, the
natural log (ln) and the logarithm (base 10) of the parameter divided by the square of
the value. Each parameter is correlated with the stability index in five sets of math-
ematical statements. This is similar to the standard Gaussian expansion regression
technique. In this method the regression is expanded to functionalities above and
below linear until the entire entity is optimized. For example a linear function would
be included, then a square and then a square root and so on until tests of the com-
plete regression show that there are no more gains in increased expansions. Using
this technique, six input parameters: exponential of density, ln of viscosity, saturate
content, resin content, asphaltene content and the asphaltene/resin ratio (A/R) were
found to be optimal. Thus with 4 transformation and the original values of these
input parameters, there are 6 times 5 or 30 input combinations.
Using Datafit, a multiple regression software, a maximum of 20 of these could
be taken at a time to test the goodness-of-fit. Values that yield Prob(t) factors of
greater than 0.9 were dropped until all remaining factors could be calculated at
once. The Prob(t) is the probability that input can be dropped without affecting the
regression or goodness-of-fit. Over twenty regressions were carried out until the
resulting model was optimal. The r2, the regression coefficient, was 0.75, which is
quite high considering the many potential sources of error, etc. The statistics on the
new model are shown in Table 3.3 , along with the parameters to create the model.
Table 3.3 shows that the 14 remaining parameters all contribute to the accuracy of
the final result and that none of them can be cut without affecting the outcome
of the model. The procedures for using model I are given below and summarized
in Table 3.4 .