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Houston, TX, United States

Chen C.,Kappa Engineering | Raghavan R.,Phillips 66
Journal of Petroleum Science and Engineering | Year: 2015

The work introduces a simple numerical scheme to evaluate the integral ∫x∞dλλμKν(λ) where Kν(λ) is the modified Bessel function of fractional order. We use this scheme to compute the response at a fractured well that describes transient diffusion in porous media considered to be fractal structures. © 2015 Elsevier B.V. Source


Chen C.,Kappa Engineering | Raghavan R.,Phillips 66
Journal of Petroleum Science and Engineering | Year: 2015

A one-dimensional, fractional-order, transient diffusion equation is constructed to model diffusion in complex geological media. Such a conceptual model permits for the incorporation of a wide range of velocities as fluid particles in high and low permeability paths perform complex motions. The transient diffusion equation is non-local in character with both spatial and temporal fractional derivatives. The pressure distribution is derived in terms of the Laplace transformation and the Mittag-Leffler function. Results are used to deduce expectations in the early-time response of a fractured well producing complex reservoirs such as unconventional shales. The flux law considered here allows for declines in rate that are faster or slower than models based on classical diffusion. A brief survey of the Mittag-Leffler function and its computation is provided. We apply the results derived to obtain solutions for the 'trilinear' model that is often used to evaluate horizontal well performance. © 2015 Elsevier B.V. Source


Raghavan R.,Phillips 66 | Chen C.,Kappa Engineering
Transport in Porous Media | Year: 2015

Flow to a horizontal well under subdiffusion is considered. The mathematical technique known as subordination is shown to provide a succinct method to consider transient flow and obtain pressure distributions. Subordination broadens the scope of existing solutions to consider a variety of problems of interest, particularly bounded systems. © 2015, Springer Science+Business Media Dordrecht. Source


Raghavan R.,Phillips 66 | Chen C.,Kappa Engineering
Journal of Petroleum Science and Engineering | Year: 2013

Assuming diffusion in the rocks to be anomalous, a flux law that is nonlocal in space and time is used to develop a mathematical model for fractured rocks drained by a horizontal well produced through multiple transverse hydraulic fractures. As a result the governing differential equation is fractional in character. The conductivity of the fractures is assumed to be finite and their properties (width, length, permeability, etc.) may be variable. Expressions for the well response that produces at a constant rate or at a constant pressure are derived in terms of the Laplace transformation. Approximate analytical solutions are derived and the analytical development provides perspectives on short and long-time well behaviors. In addition to outlining characteristic features of the model, the analytical solutions are useful in verifying numerical computations. The computational results obtained by the Stehfest algorithm establish the robustness and viability of the mathematical model. Comparisons with classical diffusion are noted. © 2013 Elsevier B.V. Source


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Kappa Engineering | Date: 2012-06-29

Computer software for use in the field of oil and gas prospecting and production and petrochemical mining and production operations, namely, computer software which provides 3D subsurface visualization data. Manuals provided with computer software for use in the field of oil and gas prospecting and production and petrochemical mining and production operations, namely, manuals in the field of software for use in the field of oil and gas exploration and production operations for real-time 3D distributed visualization of subsurface data using client server architecture. Computer software support services, namely, technical assistance, troubleshooting services in the field of computer software for use in the field of oil and gas industry, namely, software for use in the field of oil and gas exploration and production operations for real-time 3D distributed visualization of subsurface data using client server architecture.

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