TY - JOUR

T1 - A discrete, arbitrarily oriented 3D plane-source analytical solution to the diffusivity equation for modeling reservoir fluid flow

AU - Bao, Anqi

AU - Hazlett, Randy D.

AU - Babu, D. Krishna

N1 - Publisher Copyright:
Copyright © 2017 Society of Petroleum Engineers.

PY - 2017/10

Y1 - 2017/10

N2 - 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.

AB - 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.

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U2 - 10.2118/185180-pa

DO - 10.2118/185180-pa

M3 - Article

AN - SCOPUS:85032223336

SN - 1086-055X

VL - 22

SP - 1609

EP - 1623

JO - SPE Journal

JF - SPE Journal

IS - 5

ER -