I was drawn to this DSG because about 20 years ago while I was working as an experimental research engineer in the oil and gas upstream industry. I was able to obtain an analytic solution to a system of PDEs that describe the measurement of capillary pressure (Pc) in cylindrical core samples (~2,5-3.0”L x 1.5”D) on a centrifuge using Mathematica. The core sample is mounted in a coreholder that can allow application of overburden pressure consistent with that in the subsurface. Typically, four core samples are used in each centrifuge run, The coreholders are hung on a trunyen head attached to the vertical centrifuge rotor. The coreholders are free to swing into a horizontal position once the centrifuge is started. Displaced fluid is collected in a calibrated transparent cup attached to the coreholder. The displacing fluid, also in the cup, is connected to the inlet face of core sample. The centrifuge can also operate at elevated temperatures to match that of the oil and gas reservoir of interest. The centrifuge is run at each angular velocity until the volume of fluid displaced from the core samples are constant. The centrifuge is then run at the next angular velocity. The simplest Pc measurement is to begin with the core sample pore volume filled completely with brine (to match the composition of that in the reservoir if possible). This initial condition is referred to as 100% water saturation; Sw = 1. This corresponds to the state of the reservoir before it was charged with hydrocarbons, a process known as primary drainage. The brine is displaced from the core sample with nitrogen. The cumulative volume produced at each centrifuge angular velocity is recorded, from which the average Sw in the core sample can be determined based on the measured porosity of the sample. One then assumes a constitutive equation that describes the capillary pressure as a function of Sw. For example, for primary drainage, Pc = a((Sw-Swir)/(1-Swir))^b + Pcth is often used, where a, b, Swir and Pcth are regression parameters. Swir, referred to as the irreducible water saturation, corresponds to infinite capillary pressure; i.e., Swir = Limit[Pc(Sw), Pc -> +Infinity]. Pcth is the threshold pressure, which corresponds to the minimum capillary pressure required before nitrogen can penetrate the inlet face of the core sample. In this primary drainage measurement, the inlet face of the core sample is closest to the centrifuge rotational axis. The boundary condition for the primary drainage Pc measurements is that Sw=1 at the outlet face of the core samples at all times; Sw(z=L) = 1, where L is the length of the core sample. Based on the geometric dimensions of the centrifuge and coreholders, Pc(z) can be easily calculated at each centrifuge angular velocity based on the density difference between the displacing and displaced fluids. The data analysis consists of minimizing the cumulative error between the measured Sw and that given by integrating Pc(z) from z=0 to z=L for each centrifuge speed, using regression to obtain the best fit of a, b, Swir, and Pcth. Until that time, this regression analysis was done numerically using cubic splines. The analytic solution contained a term containing the hypergeometric function, with which I was not familiar at the time. I feel I’m more familiar with it, and all the other special functions discussed in this DSG, now.
One thing I would like to see more are in depth examples of instances where these special functions provide solutions to the ODEs and PDEs that describe the physical phenomena of interest. I realize many phenomena were mentioned, but in areas in which I’m not particularly well versed. I suppose I can check out all the references provided in this DSG to start, but I’m afraid I’ll get stuck at some point and not know where to turn and end up spiraling down a rabbit hole. In any case, this DSG has piqued my interest in this rather esoteric branch of mathematics that are the special functions. Thanks for providing this valuable material!
