Kappa Engineering

Kappa, United States

Kappa Engineering

Kappa, United States

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Raghavan R.,R. Raghavan Inc | Chen C.,Kappa Engineering
Transport in Porous Media | Year: 2017

We address the influences of heterogeneity and complex geology of the matrix of fractured rocks on transient flow. Fractional constitutive flux laws reflect the stochastic framework that we consider. We model transient diffusion in both the matrix and fracture systems in terms of a continuous time random walk. Our procedure is particularly suited to address a complex geology that may exist on a number of scales and which may include dead ends and discontinuities. The performance of a horizontal well produced through arbitrarily located, multiple hydraulic fractures with distinct properties (length, width, permeability) forms the basis for our thesis. The pressure distribution in a rectangular drainage region where the well may be placed arbitrarily is expressed in terms of the Laplace transformation. The required solutions are obtained numerically. The focus of our work is on long-term behaviors for production at a constant pressure. In addition to numerical solutions, asymptotic results that provide information on the structure of the solutions are presented. Agreement between the asymptotic and numerical solutions is excellent. We show that long-term responses are governed by two distinct, two-parameter Mittag–Leffler functions and are an outcome of the complexities we desire to model in both the matrix and fracture systems. As a consequence, power-law behaviors that reflect the heterogeneity inherent in the system define long-time expectations. We show that the new solutions we derive do reduce to those of classical diffusion; that is, results corresponding to classical diffusion are a subset of the new results obtained here. Our results are particularly suited to model transient flow in shale reservoirs. © 2017 Springer Science+Business Media Dordrecht


Raghavan R.,R. Raghavan Inc. | Chen C.,Kappa Engineering
Society of Petroleum Engineers - SPE Low Perm Symposium | Year: 2016

A two-dimensional, mathematical model that considers transient flow under subdiffusion towards a horizontal well that produces a rectangular drainage region through multiple fractures is considered. Subdiffusive behavior expressed through a fractional constitutive flux law is intended to reflect a complex geology that may exist on a number of scales in the form of obstacles and channels as well as attendant changes induced by hydraulic fracturing, and considers transient diffusion as a continuous time random walk (CTRW). The hydraulic fractures intersect the horizontal well at arbitrary locations with distinct properties (length, width, permeability). Pressure distributions expressed in terms of the Laplace transformation are inverted numerically. Computational issues are addressed and numerical solutions are compared with asymptotic results. The focus of the work is on the boundary-dominated period for production at a constant pressure. Long-time responses described in terms of a two-parameter, Mittag-Leffler function lead to a decline in rate that exhibits power-law behavior - an outcome of a complex geological structure with significant variations in properties; that is, a complex geology where cracks, fissures and discontinuities abound. Conventional expectations of an exponential decline that until now has been the norm for the flow of a slightly compressible liquid in bounded regions are a subset of the results presented here. Copyright 2016, Society of Petroleum Engineers.


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.


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.


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.


Chen C.,Kappa Engineering | Raghavan R.,Phillips 66
SPE Reservoir Evaluation and Engineering | Year: 2013

The model developed by Raghavan et al. (1997) is used to exam ine the characteristics of horizontal wells produced through multi ple, transverse vertical fractures. The features of models of horizontal wells that drain a reservoir through multiple fractures are highlighted from the viewpoint of the use of commonly used approximate models. Rigorous and approximate predictions are compared. It is shown that scaling is needed to ensure that vol umes drained by approximate models are indeed correct. To that end, information on volumes drained by individual fractures is noted. Copyright © 2013 Society of Petroleum Engineers.


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.


Trademark
Kappa Engineering | Date: 2012-06-29

Computer software for use in oil and gas prospecting and production in the field of drilling operations and in the production of petrochemicals for the analysis of dynamic data, namely, a software including reservoir and production engineering modules for pressure transient analysis interpretation and design, production log interpretation and design, production data analysis and forecasting, full-field reservoir numerical simulation, wellbore flow modeling, and formation test analysis and design. Manuals provided with computer software for use in oil and gas prospecting and production, drilling operations and in the production of petrochemicals, namely, manuals in the field of computer software for use in oil and gas prospecting and production in the field of drilling operations and in the production of petrochemicals for the analysis of dynamic data, namely, a software including reservoir and production engineering modules for pressure transient analysis interpretation and design, production log interpretation and design, production data analysis and forecasting, full-field reservoir numerical simulation, wellbore flow modelling, and formation test analysis and design. Computer services, namely, technical assistance, information and advice relating to computer software; testing services for computer software; computer services, namely, updating computer software; computer services, namely, providing on-line information relating to computer software.


Trademark
Kappa Engineering | Date: 2012-06-29

Computer software for use in oil and gas prospecting and production, namely, computer software used in the field of drilling operations and in the production of petrochemicals for accessing and preparing permanent downhole gauge data for reservoir engineering studies. Manuals provided with computer software for use in oil and gas prospecting and production, drilling operations and in the production of petrochemicals, namely, manuals in the field of computer software for use in oil and gas prospecting and production, namely, computer software used in the field of drilling operations and in the production of petrochemicals for accessing and preparing permanent downhole gauge data for reservoir engineering studies. Computer services, namely, technical assistance, information and advice relating to computer software; testing services for computer software; computer services, namely, updating computer software; computer services, namely, providing on-line information on computer software.


Trademark
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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