Quarks in a Polar Model

Current-quark masses are compared to the rest masses allowed by the Helmholtz equation in a polar model. Within the uncertainty of the current u quark mass determination, the current quark mass coincides with the rest mass allowed by the Helmholtz equation in the polar model in accordance with the second root of the zero Neumann function. Current d quark mass coincides with the rest mass calculated in accordance with the third root of the Bessel zero function. On the basis of a comparison of these results with the results obtained earlier for ordinary real particles u and d quarks stability is discussed.

A Discrete, Arbitrarily Oriented 3D Plane-Source Analytical Solution to the Diffusivity Equation for Modeling Reservoir Fluid Flow

Summary A highly accurate and efficiently computable analytical solution to the diffusivity equation is presented for modeling fluid flow into a 3D, arbitrarily oriented plane sink within a box-shaped, anisotropic medium with Neumann boundary conditions. The plane sink represents a gathering system for a well stimulated by means of hydraulic fracturing. Our plane-source Neumann function arises from analytic double integration of the point-source solution to the diffusivity equation along two vectors, forming a parallelogram. A Neumann boundary condition is achieved by means of the method of images, resulting in triple infinite summations that are reduced with mathematical identities to a combination of closed-form expressions and infinite sums with exponential damping. Our solution forecasts time-dependent behavior of fractured wells, useful in interpreting field experiments for the characterization of fracturing efficacy, reservoir size, and matrix fluid-transport properties. We demonstrate our model with two applications. One is pressure-transient analysis with identified flow regimes from a pressure vs. time plot. The other is pseudosteady-state (PSS) pressure mapping, simulating inflow from multiple fractures along the trajectory of a single horizontal well, which is achieved with superposition theory and adjustment of flux strength of each plane source to achieve a common pressure at each well/fracture intersection.