Artificial boundary conditions for simulations of seismic air-gun bubbles

Marine seismic exploration is a method employed by the hydrocarbon industry to find geological structures in the sub-surface with the potential to contain trapped hydrocarbons. A source of seismic energy is towed behind a ship. The energy produced by the source propagates as a sound wave through the sea into the sub-surface. Within the sub-surface the energy is reflected, refracted and diffracted. The ship also tows an array of hydrophones behind the seismic source, and these are used to measure the wavefield. If the source signal is known, then the received signal at each hydrophone can be deconvolved for the source signal to obtain the impulse response of the earth between the source and the hydrophone. These impulse responses can highlight some of the structures in the subsurface. Maps of the subsurface built up from these impulse responses are then interpreted to estimate the locations of trapped hydrocarbons. The most commonly used seismic source is the seismic air gun, which is a canister containing highly compressed air. The air is released into the sea, forming an oscillating bubble. There are two methods used by industry to determine the signal produced by an air gun or air gun array: (1) modelling, and (2) extrapolation from near-field measurements. Traditionally, industry uses the first method. With broader bandwidth data that are being recovered in data processing by removing the sea-surface reflection at the source and receiver (source and receiver ghosts), it has been found that modelling is inferior to extrapolation from near field measurements, although industry has been slow to adopt the second method. Despite this change, modelling remains a valuable tool in the design of air gun arrays, where designs can be optimised by adjusting

[1]  Ronald Fedkiw,et al.  A Boundary Condition Capturing Method for Multiphase Incompressible Flow , 2000, J. Sci. Comput..

[2]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[3]  S. J. Shaw,et al.  Shock emission from collapsing gas bubbles , 2010, Journal of Fluid Mechanics.

[4]  M. Liou,et al.  A New Flux Splitting Scheme , 1993 .

[5]  Andrew B. Wardlaw,et al.  Spherical solutions of an underwater explosion bubble , 1998 .

[6]  Mhamed Souli,et al.  Numerical simulation of underwater explosions using an ALE method. The pulsating bubble phenomena , 2012 .

[7]  Thomas Hagstrom,et al.  Accurate Radiation Boundary Conditions for the Linearized Euler Equations in Cartesian Domains , 2002, SIAM J. Sci. Comput..

[8]  Ronald Fedkiw,et al.  An Isobaric Fix for the Overheating Problem in Multimaterial Compressible Flows , 1999 .

[9]  Universitext.  An Introduction to Ordinary Differential Equations , 2006 .

[10]  M. S. Plesset,et al.  A Nonsteady Heat Diffusion Problem with Spherical Symmetry , 1952 .

[11]  Nikolaus A. Adams,et al.  On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow , 2009, J. Comput. Phys..

[12]  M. Y. Hussaini,et al.  TOWARDS PERFECTLY ABSORBING BOUNDARY CONDITIONS FOR EULER EQUATIONS , 1997 .

[13]  A. Zhang,et al.  Simulation of bubble motion in a compressible liquid based on the three dimensional wave equation , 2013 .

[14]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[15]  B. M. Dobratz,et al.  LLNL Explosives Handbook, Properties of Chemical Explosives and Explosive Simulants , 1985 .

[16]  R. Higdon,et al.  Initial-boundary value problems for linear hyperbolic systems , 1986 .

[17]  P. Smereka,et al.  A Remark on Computing Distance Functions , 2000 .

[18]  Joseph B. Keller,et al.  Damping of Underwater Explosion Bubble Oscillations , 1956 .

[19]  Peter D. Lax,et al.  A Random Choice Finite-difference Scheme for Hyperbolic Conservation Laws , 2015 .

[20]  P. Roe.  CHARACTERISTIC-BASED SCHEMES FOR THE EULER EQUATIONS , 1986 .

[21]  Estimation of bubble time period for air-gun clusters using potential isosurfaces , 2013 .

[22]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[23]  Kenneth J. Ruschak,et al.  MODELING ARTIFICIAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLOW , 2005 .

[24]  R. Courant,et al.  On the Partial Difference Equations, of Mathematical Physics , 2015 .

[25]  Khoon Seng Yeo,et al.  The simulation of compressible multi-medium flow. I. A new methodology with test applications to 1D gas–gas and gas–water cases , 2001 .

[26]  K. Thompson.  Time-dependent boundary conditions for hyperbolic systems, II , 1990 .

[27]  Antoni Ziolkowski,et al.  Marine seismic sources: QC of wavefield computation from near‐field pressure measurements , 1997 .

[28]  M. Heikal,et al.  Numerical Simulation of Cavitation Flows Based on Their Hydrodynamic Similarity , 2006 .

[29]  Rainald Löhner,et al.  On the computation of multi-material flows using ALE formulation , 2004 .

[30]  P. Helluy,et al.  Comparison and validation of compressible flow simulations of laser-induced cavitation bubbles , 2009 .

[31]  J. Blake,et al.  Growth and collapse of a vapour cavity near a free surface , 1981, Journal of Fluid Mechanics.

[32]  Hrvoje Jasak,et al.  A pressure-based, compressible, two-phase flow finite volume method for underwater explosions , 2013 .

[33]  G. W. Hedstrom,et al.  Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems , 1979 .

[34]  Furen Ming,et al.  Numerical simulation of column charge underwater explosion based on SPH and BEM combination , 2013 .

[35]  A comprehensive method for evaluating the design of air guns and air gun arrays , 1984 .

[36]  M. Liou.  A Sequel to AUSM , 1996 .

[37]  J. R. C. King,et al.  Boundary conditions for simulations of oscillating bubbles using the non-linear acoustic approximation , 2015, J. Comput. Phys..

[38]  Oliver V. Atassi,et al.  Implementation of nonreflecting boundary conditions for the nonlinear Euler equations , 2008, J. Comput. Phys..

[39]  Wolfgang Wagner,et al.  International Equations for the Saturation Properties of Ordinary Water Substance , 1987 .

[40]  G. Taylor.  THE VERTICAL MOTION OF A SPHERICAL BUBBLE AND THE PRESSURE SURROUNDING IT , 1943 .

[41]  Marcus J. Grote,et al.  Exact Nonreflecting Boundary Conditions for the Time Dependent Wave Equation , 1995, SIAM J. Appl. Math..

[42]  Martin Landrø,et al.  High-frequency signals from air-gun arrays , 2011 .

[43]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[44]  A. Colagrossi,et al.  Numerical simulation of interfacial flows by smoothed particle hydrodynamics , 2003 .

[45]  P. Colella.  A Direct Eulerian MUSCL Scheme for Gas Dynamics , 1985 .

[46]  A. Harten.  ENO schemes with subcell resolution , 1989 .

[47]  F. Millero,et al.  Compressibility of water as a function of temperature and pressure , 1973 .

[48]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[49]  High‐speed photography of the bubble generated by an airgun1 , 1996 .

[50]  Weixing Yuan,et al.  Modeling and computation of unsteady cavitation flows in injection nozzles , 2001 .

[51]  Hiroshi Terashima,et al.  A front-tracking/ghost-fluid method for fluid interfaces in compressible flows , 2009, J. Comput. Phys..

[52]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[53]  R. Teyssier,et al.  Two-dimensional versus Three-dimensional Supernova Hydrodynamic Instability Growth , 2000, The Astrophysical Journal.

[54]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[55]  Yi Sui,et al.  Bubble dynamics in a compressible liquid in contact with a rigid boundary , 2015, Interface Focus.

[56]  Boo Cheong Khoo,et al.  The ghost fluid method for compressible gas-water simulation , 2005 .

[57]  T. Hagstrom.  Radiation boundary conditions for the numerical simulation of waves , 1999, Acta Numerica.

[58]  Regiane Fortes Patella,et al.  Constraints on equation of state for cavitating flows with thermodynamic effects , 2011, Appl. Math. Comput..

[59]  SYSTEM APPROACH TO AIR-GUN ARRAY DESIGN * , 1973 .

[60]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[61]  W. Sutherland.  LII. The viscosity of gases and molecular force , 1893 .

[62]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[63]  Boo Cheong Khoo,et al.  Ghost fluid method for strong shock impacting on material interface , 2003 .

[64]  Anton Ziolkowski,et al.  A Method for Calculating the Output Pressure Waveform from an Air Gun , 1970 .

[65]  B. Leer,et al.  Flux-vector splitting for the Euler equations , 1997 .

[66]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[67]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[68]  Daniel Theodore Johnson,et al.  Understanding air‐gun bubble behavior , 1994 .

[69]  Paul Glaister,et al.  An approximate linearised Riemann solver for the Euler equations for real gases , 1988 .

[70]  David Gottlieb,et al.  A Mathematical Analysis of the PML Method , 1997 .

[71]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[72]  G. Taylor.  The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[73]  D. Givoli.  Non-reflecting boundary conditions , 1991 .

[74]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[75]  A. Shima,et al.  Mechanisms of impulsive pressure generation and damage pit formation by bubble collapse , 1986, Journal of Fluid Mechanics.

[76]  S. Otto,et al.  Comparison of methods for modelling the behaviour of bubbles produced by marine seismic airguns , 2004 .

[77]  S. Wilkerson.  A Boundary Integral Approach for Three-Dimensional Underwater Explosion Bubble Dynamics , 1992 .

[78]  Eleuterio F. Toro,et al.  On Riemann solvers for compressible liquids , 1998 .

[79]  Richard Saurel,et al.  Treatment of interface problems with Godunov-type schemes , 1996 .

[80]  J. Sethian,et al.  A Fast Level Set Method for Propagating Interfaces , 1995 .

[81]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .

[82]  R. Fedkiw,et al.  Coupling an Eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method , 2002 .

[83]  Boo Cheong Khoo,et al.  The simulation of compressible multi-medium flow: II. Applications to 2D underwater shock refraction , 2001 .

[84]  Andrea Prosperetti,et al.  Vapour-bubble growth in a superheated liquid , 1978, Journal of Fluid Mechanics.

[85]  L. Hatton,et al.  Computer modelling of clustured airguns , 1990 .

[86]  Anton Ziolkowski,et al.  The pressure wave field of an air gun array , 1984 .

[87]  Kun Xu,et al.  Dissipative mechanism in Godunov‐type schemes , 2001 .

[88]  R. Fedkiw,et al.  The Ghost Fluid Method for Viscous Flows , 2010 .

[89]  Boo Cheong Khoo,et al.  A Real Ghost Fluid Method for the Simulation of Multimedium Compressible Flow , 2006, SIAM J. Sci. Comput..

[90]  M. Holt,et al.  Glimm's method applied to underwater explosions , 1981 .

[91]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[92]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[93]  Roger Sollie,et al.  Source Signature Determination By Inversion , 1992 .

[94]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[95]  W. K. Soh,et al.  Behaviour of an underwater explosion bubble near a rigid boundary: Theory and experiment , 1994 .

[96]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[97]  P. Lax,et al.  Systems of conservation laws , 1960 .

[98]  Anton Ziolkowski,et al.  The signature of an air gun array: Computation from near‐field measurements including interactions , 1982 .

[99]  M. Landrø,et al.  An alternative method for modeling close-range interactions between air guns , 2014 .

[100]  Jang-Hyuk Kwon,et al.  On the dissipation mechanism of Godunov-type schemes , 2003 .

[101]  Tim Colonius,et al.  MODELING ARTIFICIAL BOUNDARY CONDITIONS FOR COMPRESSIBLE FLOW , 2004 .

[102]  A. Pishevar,et al.  An adaptive ALE method for underwater explosion simulations including cavitation , 2010 .

[103]  Xiangyu Hu,et al.  An interface interaction method for compressible multifluids , 2004 .

[104]  C. H. Cooke,et al.  Continuous front tracking with subcell resolution , 1991 .

[105]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[106]  G. Hatsopoulos,et al.  Bubble-growth calculation without neglect of interfacial discontinuities. , 1967 .

[107]  R. Lindsay.  On the Pressure Developed in a Liquid During the Collapse of a Spherical Cavity (1917) , 1970 .

[108]  P. Spelt.  A level-set approach for simulations of flows with multiple moving contact lines with hysteresis , 2005 .

[109]  John R. Blake,et al.  Acoustic cavitation: the fluid dynamics of non–spherical bubbles , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[110]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[111]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[112]  John E. Field,et al.  Shock-induced collapse of single cavities in liquids , 1992, Journal of Fluid Mechanics.

[113]  Alexandre J. Chorin,et al.  Random choice solution of hyperbolic systems , 1976 .

[114]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

[115]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[116]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[117]  W.,et al.  Time Dependent Boundary Conditions for Hyperbolic Systems , 2003 .

[118]  J. Bérenger.  Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves , 1996 .

[119]  Nikolaus A. Adams,et al.  A conservative interface method for compressible flows , 2006, J. Comput. Phys..

[120]  A. Ziolkowski,et al.  Measurement of air-gun bubble oscillations , 1998 .

[121]  Gordon Poole,et al.  Shot-to-Shot Directional Designature Using Near-Field Hydrophone Data , 2013 .

[122]  Bgp Offshore.  Modeling air gun signatures in marine seismic exploration considering multiple physical factors , 2010 .

[123]  Les Hatton,et al.  The Signature of an Air Gun Array: Computation From Near-Field Measurements Including Interactions-Practical Considerations , 1982 .

[124]  D. Tritton,et al.  Physical Fluid Dynamics , 1977 .

[125]  J. Blake,et al.  Approximate methods for modelling cavitation bubbles near boundaries , 1990, Bulletin of the Australian Mathematical Society.

[126]  Martin Landrø,et al.  Temperature effects on airgun signatures , 1993 .

[127]  Wai-Sun Don,et al.  An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws , 2008, J. Comput. Phys..

[128]  Thomas Hagstrom,et al.  Accurate Boundary ConditiOns for Exterior Problems in Gas Dynamics , 1988 .

[129]  Non-spherical core collapse supernovae. I. Neutrino-driven convection, Rayleigh-Taylor instabilities, and the formation and propagation of metal clumps , 2003, astro-ph/0302239.

[130]  Martin Landrø,et al.  Modelling of GI gun signatures , 1992 .