Improved methods for water shutoff. Final technical progress report, October 1, 1997--September 30, 1998

In the United States, more than 20 billion barrels of salt water are produced each year during oilfield operations. A tremendous economic incentive exists to reduce water production if that can be accomplished without significantly sacrificing hydrocarbon production. This three-year research project had three objectives. The first objective was to identify chemical blocking agents that will (a) during placement, flow readily through fractures without penetrating significantly into porous rock and with screening out or developing excessive pressure gradients and (b) at a predictable and controllable time, become immobile and resistant breakdown upon exposure to moderate to high pressure gradients. The second objective was to identify schemes that optimize placement of the above blocking agents. The third objective was to explain why gels and other chemical blocking agents reduce permeability to one phase (e.g., water) more than that to another phase (e.g., oil or gas). The authors also wanted to identify conditions that maximize this phenomenon. This project consisted of three tasks, each of which addressed one of the above objectives. This report describes work performed during the third and final period of the project. During this three-year project, they: (1) Developed a procedure and software for sizing gelant treatments in hydraulically fractured production wells; (2) Developed a method (based on interwell tracer results) to determine the potential for applying gel treatments in naturally fractured reservoirs; (3) Characterized gel properties during extrusion through fractures; (4) Developed a method to predict gel placement in naturally fractured reservoirs; (5) Made progress in elucidating the mechanism for why some gels can reduce permeability to water more than that to oil; (6) Demonstrated the limitations of using water/oil ratio diagnostic plots to distinguish between channeling and coning; and (7) Proposed a philosophy for diagnosing and attacking water-production problems

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions

Commutativity Theorems Examples in Search of Algorithms

dedicated to the memory of John Hunte

Computer Simplification of Engineering Systems Formulas

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA (the 3 M’s). These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations. On the other hand, most of the computation involved in linear control theory is performed on matrices and these do not commute. A typical issue of IEEE TAC is full of A B C D type linear systems and computations with the A B C D’s or partitions of them into block matrices. The 3 M’s are weak in the area of non-commutative operations. They allow a user to declare an operation to be non-commutative, but provide very few commands for manipulating such operations and no powerful algorithmic tools. It is the purpose of this article to report on applications of a powerful tool: a non-commutative version of the Gröbner Basis Algorithm. The commutative version of this algorithm is implemented on each of the three M’s. It has many applications ranging from solving systems of equations to computations involving polynomial ideals. The non-commutative version is relatively new [Mora]. Our application to the simplification of expressions which occur in systems theory is unique. We will describe the Gröbner Basis for several elementary situations which arise in systems theory. These give (in a sense to be made precise) a “complete” set of simplifying rules for formulas which arise in these situations. We have found that this process elucidates the nature of simplifying rules and provides a practical means of simplifying some types of complex expressions. The research required the use of software suited for computing with non-commuting symbolic expressions. Most of the research was performed using a special purpose system developed for the project by J. Wavrik. This system uses a new approach to the development of mathematical software. It provides the flexibility needed for experimentation with algorithms, data representation, and data analysis. In another direction, Helton, Miller and Stankus have written packages for Mathematica called NCAlgebra which extend many of Mathematica’s commands to symbolic expressions in non-commutative algebras. We have incorporated in these packages some of the results on simplification described in this paper

Computer Simplification of Formulas in Linear Systems Theory

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations. On the other hand, most of the computation involved in linear control theory is performed on matrices, and these do not commute. A typical issue of IEEE TRANSACTIONS ON AUTOMATIC CONTROL is full of linear systems and computations with their coefficient matrices A B C D’s or partitions of them into block matrices. Mathematica, Maple, and MACSYMA are weak in the area of non-commutative operations. They allow a user to declare an operation to be non-commutative but provide very few commands for manipulating such operations and no powerful algorithmic tools. It is the purpose of this paper to report on applications of a powerful tool, a non-commutative version of the Gröbner Basis algorithm. The commutative version of this algorithm is implemented in most major computer algebra packages. The non-commutative version is relatively new

Rewrite Rules and Simplification of Matrix Expressions

This paper concerns the automated simplification of expressions which involve non-commuting variables. The technology has been applied to the simplification of matrix and operator theory expressions which arise in engineering applications. The non-commutative variant of the Gröbner Basis Algorithm is used to generate rewrite rules. We will also look at the phenomenon of infinite bases and implications for automated theorem proving.

Automated Simplification and Deduction for Engineering Formulas

The commutative version of the Gröbner Basis Algorithm has become one of the most powerful tools in computer algebra. The recent adaptation of the algorithm to noncommuting variables has potential applications to matrix and operator expressions. A Forth-based research system was used to implement and study applications of a noncommutative variant of the Gröbner Basis Algorithm. It has been used in automated simplification and deduction for formulas in engineering. The paper will discuss this work and the use of Forth in producing systems for mathematics research