Thermodynamic modeling of geological systems by convex programming under uncertainty

A new approach to formulating and solving thermodynamic-modeling problems b y the Gibbs-energy minimization under uncertainty of input information has been proposed. The proposed approach is as follows: 1. The effect of the uncertainty domain of input data on the solution is ma de by selecting a representative set of input-data combinations, which most completely describe possible variants of solutions. Here, the key operatio n is selection of the limited but sufficient number of points from a contin uous set of the possible values of prescribed parameters. To minimize the n umber of realizations in one problem, the method of point selection from a uniform lattice can be used for a unit hypercube of dimensionality equal to the number of all uncertainty elements at the Input. The method is based u pon the linear-code theory and provides point distribution in the hypercube with fulfillment of the following requirements: The nodes of the hypercube lattice are widely spaced; one point must be in the center of the hypercub e; for each point there exists a point symmetric about the center; all sele cted points have different coordinates. The latter condition provides a uni form distribution of the selected points along each axis. Transfer of the s elected points from the unit hypercube to the real region is performed prop ortionally to each uncertainty element. With minor additions, the method of regular point selection from a regular lattice in a unit hypercube can be applied to the problems with a priori prescribed distribution. 2. The input parameters with undetermined values, taken individually and in combination, are: isobaric-Isothermal potential, entropy, mote volume, coe fficient of activity and/or fugacity, mole quantity of independent componen ts, temperature, and pressure. 3. Correlation between the values of input quantities is taken into account . 4. Comparison of solution variants as well as statistic characteristics is made by a common scheme of the decision-making method under uncertainty. Th e payoff matrix is calculated, In which the efficiency of each variant is d efined by all selected combinations of input information. This provides a c orrect comparison between competing solutions. 5. The criteria function for the payoff matrix is not the minimum value of thermodynamic potential but the magnitude of difference in extreme values b etween direct and dual solutions for each variant in one problem. This crit erion, being one of the inequalities of the Khun-Tucker conditions for the chemical-equilibrium problem in convex-programming formulation, gives more pithy characteristics of solutions than the value of minimum in solving the direct problem. 6. A choice of preferable variants is made by characteristic estimation of variants from the payoff matrix by the Laplace, Wald, Savage, Hurwitz, and other criteria by the decision-making method under uncertainty. 7. A final problem - designing of a global software for personal computers - is posed. Models are built in the same way as for deterministic formulati on, and, if necessary, the system can be transferred to a chosen mode of un certainty just by cursor. All necessary additional instructions are given a utomatically by default. The proposed approach has been realized in the form of a special module "Un certainty" in the program complex Selektor-C. The operation of the module i s illustrated by a numerical example - calculation of equilibrium in the sy stem Fe-O-C-N at 500 degrees C and 1 bar.