Resistivity interpretations for conventional reservoirs depend upon an ability to estimate the resistivity of mud filtrate in flushed zones. An estimator, in use since 1953, was invented by Jan Jacob Arps. Arps worked out a method to use mud filtrate resistivity and its temperature as measured on the surface to estimate its resistivity at any other temperature, in particular the increased temperatures encountered at depth in wellbores. The method is known as Arps' approximation1. It is also indispensable for extrapolating formation water resistivity from one depth and temperature to another depth at a different temperature.
Arps' approximation is based upon resistivity-temperature-salinity data as recorded in the International Critical Tables of Numerical Data, Physics, Chemistry and Technology (known as the ICT). There are 81 such data points extracted by Arps from the ICT. It is Arps stated goal to fully document his method; still, it is quite difficult to grasp his technique. However, the resulting Arps approximation is so simple and useful that it has not been subjected to further analysis in the intervening 67 years.
A re-examination of the basic ICT data has led to the formulation of a "first principle" relating the conductivity of a sodium chloride solution to its temperature as a function of salinity. The first principle takes the form of a simple differential equation from which the Arps approximation can be derived. Whereas Arps uses four pages and 2970 words written on 579 lines, this is done on five lines using the first principle, straight-forward step-by-step algebra, and five (or six) additional, but simple, equations without reference2 to the ICT data.
A "reference temperature" arises in the derivation; this reference temperature is determined from the ICT data. In the Arps approximation the reference temperature is given as T0 = − 6.7707 °F. T0 depends upon two arbitrary choices in Arps' analysis. Using Arps' choices the least squares fit of these data result in T0 = − 6.7707 °F.
However, this value might have been (slightly) different had Arps made other choices.
These discoveries give impetus to a re-examination of the evaluation of this constant, not only to validate Arps' estimate, but also to re-examine the choice of which data to include. For example, the ICT data is weighted toward low salinity (21 samples at 60, 100, 300 ppm) data and includes 10 samples at 32 °F; would these low-salinity, low-temperature samples unduly bias estimates at temperatures and salinities more representative of oil-field brines? Answering these questions leads to a collection of possibly more appropriate values for T0.
A final question is: does any of this re-analysis of the T0 value have an impact on present and prior analysis using the Arps approximation? Fortunately, the answer to this question is "no"! Still, the new five-line derivation from a first principle and revaluation of T0 improves our understanding of this venerable and ubiquitous approximation.