Denis M. Filatov

Fuzzy expert system for solving lost circulation problem

Lost circulation is the most common problem encountered when drilling. This paper describes a distributed hybrid intelligent system, called SmartDrill, using fuzzy logic, expert system framework and Web services for helping petroleum engineers to diagnose and solve lost circulation problems. The fuzzy algebra of strict monotonic operations is used as an underlying model for expert system development. Its realization in inference procedures of expert systems is simpler than for expert systems based on lexicographic operations. Overall, the system architecture is discussed and implementation details are provided. The system is aimed to help in making decisions at the operational level and is at field testing phase in PEMEX, Mexican Oil Company.

Splitting-based schemes for numerical solution of nonlinear diffusion equations on a sphere

We provide an advanced study of our recently developed method for the numerical solution of nonlinear diffusion equations on a sphere. In particular, we analyse the method in detail when applied to solving diverse diffusion phenomena, with specific conditions on the smoothness of the solution, the degree of nonlinearity, and the initial data and sources. The main idea of the method consists in splitting the original differential operator by coordinates and subsequent constructing finite difference schemes for the split one-dimensional problems using different coordinate maps for the sphere at the two split time intervals. The essential advantage of this technique is that each split 1D equation can be equipped with a periodic boundary condition, despite the sphere being not a doubly periodic domain. Therefore, unlike the existing methods, this one does not require applying special numerical procedures for careful computing the solution near the poles, which is always a challenge. Each split 1D equation is approximated by a second- or a fourth-order finite difference scheme that keeps all the substantial properties of the differential problem: it is balanced and dissipative, since the spatial finite difference operator is negative definite. The developed algorithm is cheap-to-implement from the computational point of view. The theoretical results are confirmed numerically by simulating various nonlinear diffusion processes. A numerical example, in particular, shows that the competition of the three basic mechanisms-the nonlinear interaction, forcing and dissipation-can generate wave solutions, whose spatial structures on the sphere are subjected to the alternating influence of the processes of self-organisation and self-destruction. The accuracy of the method is evaluated by comparing the numerical solutions versus the analytical ones obtained with specially chosen forcings. The convergence of the numerical solution to the analytics is verified by refining spatial grids.

An uncertainty model for a diagnostic expert system based on fuzzy algebras of strict monotonic operations

Expert knowledge in most of application domains is uncertain, incomplete and perception-based. For processing such expert knowledge an expert system should be able to represent and manipulate perception-based evaluations of uncertainties of facts and rules, to support multiple-valuedness of variables, and to make conclusions with unknown values of variables. This paper describes an uncertainty model based on two algebras of conjunctive and disjunctive multi-sets used by the inference engine for processing perception-based evaluations of uncertainties. The discussion is illustrated by examples of the expert system, called SMART-Agua, which is aimed to diagnose and give solution to water production problems in petroleum wells.

Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere

A new method for the numerical modelling of physical phenomena described by nonlinear diffusion equations on a sphere is developed. The key point of the method is the splitting of the differential equation by coordinates that reduces the original 2D problem to a pair of 1D problems. Due to the splitting, while solving the 1D problems separately one from another we involve the procedure of map swap — the same sphere is covered by either one or another of two different coordinate grids, which allows employing periodic boundary conditions for both 1D problems, despite the sphere is, actually, not a doubly periodic domain. Hence, we avoid the necessity of dealing with cumbersome mathematical procedures, such as the construction of artificial boundary conditions at the poles, etc. As a result, second-order finite difference schemes for the one-dimentional problems implemented as systems of linear algebraic equations with tridiagonal matrices are constructed. It is essential that each split one-dimentional finite difference scheme keeps all the substantial properties of the corresponding differential problem: the spatial finite difference operator is negative definite, whereas the scheme itself is balanced and dissipative. The results of several numerical simulations are presented and thoroughly analysed. Increase of the accuracy of the finite difference schemes to the fourth approximation order in space is discussed.

Perception Based Hybrid Intelligent Systems in Petroleum Applications

We describe the methods of processing of perception based information in hybrid intelligent systems. Several innovative techniques like a multi-set based algebra of qualitative perception-based uncertainties and perception-based data mining form the technological framework of the approach. In the paper, we discuss the algebra of strict monotonic operations and inference procedures based on perception-based evaluations of uncertainty of facts and rules. They are characterized by multi-set-based representation of evaluations of uncertainty and by multi-valued inference of conclusions in expert system rules. The proposed method is implemented in the CAPNET expert system shell. We also discuss the method of evaluation of perception-based patterns in time series data bases. The approach is illustrated by examples of diagnostics of excessive water production in petroleum wells combining both methods

Simulation of soliton‐like waves generated by topography with conservative fully discrete shallow‐water arbitrary‐order schemes

Purpose – The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.Design/methodology/approach – The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.Findings – Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the st...

Phenomena of Nonlinear Diffusion in Complex 3D Media

Abstract Unbounded solutions (critical blow-up regimes) simulated by the 3D nonlinear diffusion equation in a spherical shell are studied. The coordinate splitting of the differential operator coupled with two spherical coordinate maps makes it possible to use periodic boundary conditions in the latitudinal and longitudinal directions and employ the computationally efficient Sherman-Morrison formula and Thomas algorithm. The resulting finite difference method is direct, with implicit and unconditionally stable schemes of second-order approximation in all the variables. Numerical tests demonstrate that it allows simulating different blow-up regimes in complex computational domains.

Modelling of Combustion and Diverse Blow-Up Regimes in a Spherical Shell

Physical phenomena with critical blow-up regimes simulated by the 3D nonlinear diffusion equation in a spherical shell are studied. For solving the model numerically, the original differential operator is split along the radial coordinate, as well as an original technique of using two coordinate maps for solving the 2D subproblem on the sphere is involved. This results in 1D finite difference subproblems with simple periodic boundary conditions in the latitudinal and longitudinal directions that lead to unconditionally stable implicit second-order finite difference schemes. A band structure of the resulting matrices allows applying fast direct (non-iterative) linear solvers using the Sherman-Morrison formula and Thomas algorithm. The developed method is tested in several numerical experiments. Our tests demonstrate that the model allows simulating different regimes of blow-up in a 3D complex domain. In particular, heat localisation is shown to lead to the breakup of the medium into individual fragments followed by the formation and development of self-organising patterns, which may have promising applications in thermonuclear fusion, nonlinear inelastic deformation and fracture of loaded solids and media and other areas.

Numerical Simulation of Coastal Flows in Open Multiply-Connected Irregular Domains

We develop a numerical method for the simulation of coastal flows in multiply-connected domains with irregular boundaries that may contain both closed and open segments. The governing equations are the shallow-water model. Our method involves splitting of the original nonlinear operator by physical processes and by coordinates. Specially constructed finite-difference approximations provide second-order unconditionally stable schemes that conserve the mass and the total energy of the discrete inviscid unforced shallow-water system, while the potential enstrophy results to be bounded, oscillating in time within a narrow range. This allows numerical simulation of coastal flows adequate both from the mathematical and physical standpoints. Several numerical experiments, including those with complex boundaries, demonstrate the skilfulness of the method.

Distributions of functional and content words differ radically

We consider statistical properties of prepositions—the most numerous and important functional words in European languages. Usually, they syntactically link verbs and nouns to nouns. It is shown that their rank distributions in Russian differ radically from those of content words, being much more compact. The Zipf law distribution commonly used for content words fails for them, and thus approximations flatter at first ranks and steeper at higher ranks are applicable. For these purposes, the Mandelbrot family and an expo-logarithmic family of distributions are tested, and an insignificant difference between the two least-square approximations is revealed. It is proved that the first dozen of ranks cover more than 80% of all preposition occurrences in the DB of Russian collocations of Verb-Preposition-Noun and Noun-Preposition-Noun types, thus hardly leaving room for the rest two hundreds of available Russian prepositions.