Computational frameworks for hydraulically fractured well production analysis




Hu, Jing, Ph. D.

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Modeling for predicting hydraulic fracture growth and forecasting unconventional resources extraction is of great interest to the petroleum industry. Numerous celebrated accomplishments have been achieved over the decades, however there is still a lack of rigorous numerical models to simulate the well production process in fractured subsurface reservoirs and relatively little work touched upon accurately simulating intersecting and merging cracks due to the complex geometrical considerations that are involved. To address these challenges, we propose three computational frameworks in this dissertation.

In the first part of this dissertation, a computational framework for modeling steady state well production by coupling steady state Darcy flow with channel flow through the law of mass conservation, is presented. The governing equation of steady state Darcy flow is formulated as a weakly singular integral equation with symmetric Galerkin boundary element method (SGBEM) and that of channel flow is cast in a weak form by finite element method (FEM). An asymptotic analysis is conducted for the steady state flux field around the tip front in a porous matrix and a special crack tip element is developed correspondingly. A numerical integration scheme is elaborated for the singular integrals taking account of the special tip element shape functions. To regularize the coupled system in the bounded layer domain as the production zone, a datum condition is introduced. The numerical implementation is comprehensively validated through decoupled Darcy flow equation, decoupled channel flow equation and coupled equations, respectively.

Secondly, a computational framework for modeling transient state well production by coupling transient state Darcy flow with channel flow is proposed. In this framework, the governing equation of transient state Darcy flow is formulated as a convolutional integral equation with SGBEM and the channel flow equation is cast into the same form as in the steady state analysis. The coupling of SGBEM-FEM renders a time marching scheme which involves a slowly converging series kernel at large but finite times for the bounded layer domain. A fast algorithm for evaluating the bounded layer kernel is proposed based on Ewald summation. An asymptotic analysis is conducted showing that the transient state flux field is of the same order as the steady state flux field and the same special tip element is employed in the transient analysis. The numerical implementation is validated with the solutions to both the decoupled transient Darcy flow equation and the coupled equations respectively.

Lastly, we propose a computational framework to model the growth of intersecting and merging hydraulic fractures. We revisit the existed efficient SGBEM-FEM framework to simulate the growth of isolated height-contained hydraulic fractures. We formulate the efficient model in a variational form and augment it with Lagrange multipliers to prescribe physical constraints on intersections of hydraulic fractures. Pertinent numerical implementation techniques, including space and temporal discretizations, special tip element, crack propagation criteria and remeshing strategy, are presented. We validate the proposed framework with the analytical results of both star shaped cracks and symmetric branched cracks. The validation shows the necessity to properly enforce the physical constraints on the intersections of fractures.


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