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Simulation of Wave Propagation Along Fluid-Filled Cracks Using High-Order Summation-by-Parts Operators and Implicit-Explicit Time Stepping
Department of Geophysics, Stanford University, Stanford, USA.
Institute for Computational and Mathematical Engineering, Stanford University, Stanford, USA.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
2017 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, no 4, p. B675-B702Article in journal (Refereed) Published
Abstract [en]

We present an ecient, implicit-explicit numerical method for wave propagation insolids containing uid-lled cracks, motivated by applications in geophysical imaging of fracturedoil/gas reservoirs and aquifers, volcanology, and mechanical engineering. We couple the elastic waveequation in the solid to an approximation of the linearized, compressible Navier{Stokes equationsin curved and possibly branching cracks. The approximate uid model, similar to the widely usedlubrication model but accounting for uid inertia and compressibility, exploits the narrowness of thecrack relative to wavelengths of interest. The governing equations are spatially discretized usinghigh-order summation-by-parts nite dierence operators and the uid-solid coupling conditions areweakly enforced, leading to a provably stable scheme. Stiness of the semidiscrete equations can arisefrom the enforcement of coupling conditions, uid compressibility, and diusion operators requiredto capture viscous boundary layers near the crack walls. An implicit-explicit Runge{Kutta scheme isused for time stepping, and the entire system of equations can be advanced in time with high-orderaccuracy using the maximum stable time step determined solely by the standard CFL restriction forwave propagation, irrespective of the crack geometry and uid viscosity. The uid approximationleads to a sparse block structure for the implicit system, such that the additional computationalcost of the uid is small relative to the explicit elastic update. Convergence tests verify highorderaccuracy; additional simulations demonstrate applicability of the method to studies of wavepropagation in and around branching hydraulic fractures.

Place, publisher, year, edition, pages
SIAM PUBLICATIONS , 2017. Vol. 39, no 4, p. B675-B702
Keywords [en]
uid-lled crack, wave propagation, summation-by-parts, high-order accuracy, implicit-explicit
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-140022DOI: 10.1137/16M1097511ISI: 000408922600005OAI: oai:DiVA.org:liu-140022DiVA, id: diva2:1136316
Note

Funding agencies: Baker Hughes; Stanford Natural Gas Initiative; Chevron fellowship in the Department of Geophysics at Stanford University

Available from: 2017-08-27 Created: 2017-08-27 Last updated: 2017-09-19

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