The article presents a methodology for constructing a tectonic and a structural model with faults in the RN-GEOSIM. The basis of the approach is the interpolation of geometric objects, such as faults and horizons in an implicit form, which implies the interpolation of scalar quantities in a three-dimensional area of interest so that the resulting surfaces are their isosurfaces, or a set of points of equal function value. Each fault is described by its own function, and a family of disjoint geological horizons is described by one common function. Due to the fact that each fault is an isosurface of the zero value of the function specified in the entire modeling domain, it becomes convenient to describe the rules of fault junctions using a system of inequalities. However, this approach makes it necessary to interpolate additional auxiliary quantities to describe the fault area. Additional functions introduce a curved coordinate system on the fault surface, which makes it convenient to orient the fault in relation to the cardinal directions. Interpolation of scalar quantities responsible for faults and horizons is a solution to the optimization problem of minimizing the so-called smoothness functional with constraints set by the initial data. Variation of the optimization problem leads to a biharmonic differential equation with natural boundary conditions at the boundary of the modeling domain, which corresponds to the linear elasticity model.
The article describes ways to set constraints for an optimization problem in a linear form, according to the interpolated data. Due to the quadratic form of the functional and the linearity of the constraints, the optimization problem is reduced to a system of linear algebraic equations. The presented methodology makes it possible to model inclined faults, geological horizons with thrusts and overlaps.
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