THERMODYNAMIC MODELING OF GEOLOGICAL SYSTEMS BY CONVEX PROGRAMMING UNDER UNCERTAINTY

Abstract

A new approach to formulating and solving thermodynamic-modeling problems by 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 made by selecting a representative set of input-data combinations, which most completely describe possible variants of solutions. Here, the key operation is selection of the limited but sufficient number of points from a continuous set of the possible values of prescribed parameters. To minimize the number 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 upon 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 hypercube; for each point there exists a point symmetric about the center; all selected points have different coordinates. The latter condition provides a uniform distribution of the selected points along each axis. Transfer of the selected points from the unit hypercube to the real region is performed proportionally 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, mole volume, coefficient of activity and/or fugacity, mole quantity of independent components, 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. The payoff matrix is calculated, in which the efficiency of each variant is defined by all selected combinations of input information. This provides a correct 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 between direct and dual solutions for each variant in one problem. This criterion, 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 formulation, and, if necessary, the system can be transferred to a chosen mode of uncertainty just by cursor. All necessary additional instructions are given automatically by default. The proposed approach has been realized in the form of a special module "Uncertainty" in the program complex Selektor-C. The operation of the module is illustrated by a numerical example - calculation of equilibrium in the system Fe-O-C-N at 500°C and 1 bar.