Shale Compaction Kinetics

The grain-to-grain stress vertically in sediments is given by the overburden less the pore fluid pressure, σ, divided by the fraction of the horizontal area which is the supporting matrix , (1 − φ), where φ is the porosity. It is proposed that the fractional reduction of this ratio, Λ, with time is given by the product of φ 4m/3 , (1 − φ) 4n/3 , and one or more Arrhenius functions A exp(−E/RT ) with m and n close to 1. This proposal is tested for shale sections in six wells from around the world for which porosity-depth data are available. Good agreement is obtained above 30-40 C and porosities less than 0.5. Single activation energies for each well are obtained in the range 15-33 kJ/mole, close to pressure solution of quartz, 24 kJ/mol. Values of m and n are in the range 1 to 0.8, indicating nearly fractal pore-matrix spaces and water-wet interfaces. Results are independent of over- or under-pressure of pore water. This model explains shale compaction quantitatively. Given porosity-depth data and accurate activation energy, E, one can infer paleo-geothermal-gradient and from that organic maturity, thus avoiding unnecessary drilling.

Linear k-power Preservers on Tensor Products of Matrices

Invariants and the study of the map preserving a certain invariant play vital roles in the study of the theoretical mathematics. The preserver problems are the researches on linear operators that preserve certain invariants between matrix sets. Based on the result of linear $k$-power preservers on general matrix spaces, in terms of the advantages of matrix tensor products which is not limited by the size of matrices as well as the immense actual background, the study of the structure of the linear $k$-power preservers on tensor products of matrices is essential, which is coped with in this paper. That is to characterize a linear map $f:M_{m_{1}\cdots m_{l}}\rightarrow M_{m_{1}\cdots m_{l}}$ satisfying $f(X_{1}\otimes \cdots \otimes X_{l})^{k}=f\left( (X_{1}\otimes \cdots \otimes X_{l})^{k}\right) $ for all $X_{1}\otimes \cdots \otimes X_{l}\in M_{m_{1}\cdots m_{l}}$.

Uniqueness of minimal projections in smooth expanded matrix spaces

Let us consider the space M(n, m) of all real or complex matrices on n rows and m columns. In 2000 Lesław Skrzypek proved the uniqueness of minimal projection of this space onto its subspace $$M(n,1)+M(1,m)$$ M ( n , 1 ) + M ( 1 , m ) which consists of all sums of matrices with constant rows and matrices with constant columns. We generalize this result using some new methods proved by Lewicki and Skrzypek (J Approx Theory 148:71–91, 2007). Let S be a space of all functions from $$X\times Y \times Z$$ X × Y × Z into $${\mathbb {R}}$$ R or $${\mathbb {C}}$$ C , where X, Y, Z are finite sets. It could be interpreted as a space of three-dimensional matrices M(n, m, r). Let T be a subspace of S consisting of all sums of functions which depend on one variable. Let S be equipped with a smooth norm $$\Vert .\Vert $$ ‖ . ‖ . We show that there exists the unique minimal projection of S onto its subspace T.

Shale Compaction Kinetics

1 Abstract The grain-to-grain stress vertically in sediments is given by the overburden less the pore fluid pressure, σ, divided by the fraction of the horizontal area which is the supporting matrix , (1 − φ), φ being the porosity. It is proposed that the fractional reduction of this ratio, Λ, with time is given by the product of φ 4m/3) , (1 − φ) 4n/3 , and one or more Arrhenius functions A exp(−E/RT ) with m and n close to 1. This proposal is tested for shale sections in six wells from around the world for which porosity-depth data are available. Good agreement is obtained above 30-40 C. A single activation energy of 23+-5 kJ/mole, indicating pressure solution of quartz, 24 kJ/mol, was obtained. The average value of m is 1, indicating fractal pore-matrix spaces and water-wet interfaces. Grain-to -grain interfaces may be fractal with m close to 1, but can have lower values suggesting smooth surfaces and even grain-to-grain welding. Results are independent of over- or under-pressure of pore water. This model explains shale compaction quantitatively.

Detecting Similarities in Posts Using Vector Space and Matrix

This study discusses the application of two linear algebraic materials, namely vector and matrix spaces. The application of the two materials is related to an article, the writing can be in the form of an article, book, and so on. The writings examined in this study use example sentences made by the author. Two materials of linear algebra, namely the vector space and the matrix are used to analyze whether there is a similarity between the writing made with other writing. As a result, vector space and matrix can be used to detect similarities in a text.

Detecting Similarities in Posts Using Vector Space and Matrix

This study discusses the application of two linear algebraic materials, namely vector and matrix spaces. The application of the two materials is related to an article, the writing can be in the form of an article, book, and so on. The writings examined in this study use example sentences made by the author. Two materials of linear algebra, namely the vector space and the matrix are used to analyze whether there is a similarity between the writing made with other writing. As a result, vector space and matrix can be used to detect similarities in a text.