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**File Type**

This abstract presents a comparison of discontinuous Galerkin (DG) and staggered-grid finite difference (FD) methods for the time domain (TD) acoustic wave equations in heterogeneous media regarding the computation cost for a given level of accuracy. It is well known staggered-grid FD methods for waves in heterogeneous media are first-order schemes due the first-order interface misalignment error. DG and other finite element methods can remedy the interface error by using interfacefitting meshes. The results in this abstract suggest that the interface error of a staggered-grid FD method is accumulated as the opportunities of waves passing through interfaces increase. Thus for more complex models the DG method seems to be more efficient.

Numerical seismic modeling, duplicating the seismic survey procedure and generating synthetic seismograms, has many important applications. Various numerical methods are applicable to wave propagation problems. FD and DG methods are of our concerns in this abstract. Though a lot of effort has been paid to both methods, there are few works regarding careful comparison. DeBasabe and Sen (2007) compared the grid dispersion and stability criteria of FD methods, classical finite-element methods (FEM) and spectral FEM introduced by Komatitsch and Tromp (1999) for seismic modeling. DeBasabe and Sen (2009) presented a stability analysis of spectral FEM and DG methods with numerical evidences. Our goal in this work is to carefully compare the FD and DG methods within the context of reflection seismology, and come up with some useful conclusions which could provide a guideline for choosing the proper method for the proper problem. FD methods are widely used for solving wave propagation problems due to their desirable trade-off between the computation efficiency and accuracy, as well as their relatively easy implementation. However, staggered-grid FD methods for models of heterogeneous media lead to a first-order interface error. Brown (1984) first analyzed this first order error component of FD methods for an interface problem. Symes and Vdovina (2009) theoretically and numerically quantified the first order interface error for the second order in time and space staggered-grid FD method applied to the pressure-velocity formulation of the acoustic wave equations, and provided an explicit expression of a non-zero time shift of numerical solutions due to the first order interface error. The interface error in staggered-grid FD methods is unavoidable for models of heterogeneous media, because several grids are employed in staggered-grid FD methods and the misalignment with the material interface must occur for at least one grid. DG methods have drawn a lot of attentions in computational electromagnetic (Warburton, 1999; Hesthaven and Warburton, 2002; Cockburn et al., 2004; Cohen et al., 2006) and fluid dynamics communities (Cockburn and Shu, 1989; Bassi and Rebay, 1997; Giraldo et al., 2002) for decades.K¨aser et al. (2008) implemented DG methods using high performance computing for the application of seismic wave field modeling. Hesthaven and Warburton (2008) scrutinized DG methods together with other popular numerical methods for PDEs and outline the general properties of DG methods and conclude that DG methods possess every useful feature discussed.

Computational physics, discontinuous galerkin method, domain acoustic, elastic wave, equation, finite difference method, geophysics, Maxwell, method, propagation, Reservoir Characterization, reservoir description and dynamics, SEG Denver, seismic processing and interpretation, Upstream Oil & Gas, Warburton

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

algorithm, borehole, characterization, component, conductive patch, Conductor, deposit, formation evaluation, Gamsberg, Gamsberg deposit, line, loop, management and information, mesh, metals & mining, mineralization, model, reservoir description and dynamics, Response, SEG SEG Denver, solution, surface, Walker

SPE Disciplines:

algorithm, aperture, Artificial Intelligence, cable, Cable Extrapolation, direction, domain, extrapolation, filter, frequency, function, geometry, interpolation, prediction, Prediction error, Reservoir Characterization, reservoir description and dynamics, Response, sample, seismic processing and interpretation, side, spatial reasoning, trace, transform, Upstream Oil & Gas

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

Technology: Information Technology > Artificial Intelligence > Representation & Reasoning > Spatial Reasoning (0.34)

acquisition, acquisition geometry, Amundsen, cable, Comparison, geometry, Heidrun, node, node seismic, ocean bottom, ocean bottom seismic, receiver, receiver line, Reservoir Characterization, reservoir description and dynamics, SEG SEG Denver, seismic processing and interpretation, source, Statfjord, survey, Upstream Oil & Gas, volve

Oilfield Places:

- North America > United States > Gulf of Mexico > Mississippi Canyon > Block 806 > Deimos Field (0.99)
- Europe > Norway > Central North Sea > South Viking Graben > Block 16 > Volve Field (0.99)
- Europe > Norway > Norwegian Sea > Halten Bank Area > Heidrun Field (0.89)

Li, Yandong (Research Institute of Petroleum Exploration and Development, PetroChina Company Limited) | Li, Jinsong (Research Institute of Petroleum Exploration and Development, PetroChina Company Limited) | Zheng, Xiaodong (Research Institute of Petroleum Exploration and Development, PetroChina Company Limited)

channel, channel system, channel system delineation, component, decomposition, evolution, method, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, slice, spectral decomposition, stratal slice, system, upper jurassic formation, Upstream Oil & Gas, white arrow, wigner-ville distribution-based spectral decomposition

In this work we studied and developed an efficient method to handle the wave simulation in presence of topography based on curvilinear finite difference. Foremost, we derived the modified equations and the free surface condition on the continuous problem. We used afterwards optimized stencils and optimized selective filters adapted from aeroacoustics. The use of conventional grid allowed us to directly extend the non-centered stencils at the boundaries developed by Berland et al. (2007) to our problem in curvilinear coordinates.

With the increasing scarcity of oil, the oil exploration industry explores ever more complex and challenging geological features. The complexity can arise because of complex geological structures (salt ridges, weathered zones, foothills for e.g.). In the particular context of foothills, specific elastic wave propagation phenomena arise from the the presence of topography; scattering of waves on the topography; site effects caused by interferences that cause amplification of the wave amplitude; complicated propagation of ground-roll, including mode conversions between surface and volume waves. On the mathematical side, finite elements and finite volume methods are the most suitable approaches to handle accurately the free surface condition in presence of a general topography. For instance, Marfurt (1984) used a finite element scheme. More recently, Komatitsch and Vilotte (1998) introduced a spectral element scheme as a computionnaly efficient way of tackling this problem. However, when dealing with operational needs that require extensive means of computations, such as seismic feasability studies, the finite difference method is the most appropriate (Regone, 2007). To overcome the staircase approximation of the free surface, Hestholm and Ruud (1994) adapted the finite difference method to curvilinear coordinates, as initially introduced by Fornberg (1988).

General equations In a foothills context, only the top edge of the geological model is curved. Therefore, we consider a transformation that map only the free surface topography. We assume that (z0(x), x 2 [0,xmax]) is the equation of the topography, so that |z0(x)| remains smaller than the maximum depth H of the geological model. We introduce the coordinate transform that we use in this work as initially introduced by Fornberg (1988).

Space differencing We developed a new scheme for curvilinear finite difference based on conventional grid. While the staggered and the rotated staggered grids are the most used for Cartesian finite difference, their application is no more adapted in our problem. In the following we explain the major reasons that motivate this choice. Hestholm and Ruud (1994) extended the Cartesian staggered grid formulation to curvilinear coordinates. They further used optimized finite difference operators in order to reduce numerical dispersion. However the application of the staggered grid requires an extensive mean of computations. This prohibitive cost is mainly due to the computation of the spatial vertical derivatives and the interpolation operator needed to achieve the calculation at the desired point. Moreover, the use of an interpolation operator can decrease the overall accuracy of the scheme as quoted by Magnier et al. (1994).

computational, Computational physics, elastic wave, elastic wave equation, geophysical journal international, geophysics, method, modeling, presence, Reservoir Characterization, reservoir description and dynamics, SEG Denver, seismic processing and interpretation, seismic wave propagation, Simulation, surface topography, topography, Upstream Oil & Gas

facies, location, physics, porosity, quantify, Reservoir Characterization, reservoir description and dynamics, rock physics, rock physics model, sand shale, seismic processing and interpretation, sequence, sequence stratigraphy, soft-sand model, spatial, textural maturity, Trend, Upstream Oil & Gas, well

Oilfield Places:

- North America > United States > Texas > Powderhorn Field (0.99)
- Africa > Equatorial Guinea > Equatorial Guinea Offshore > Rio Muni Basin (0.99)
- Asia > Myanmar > Tertiary Basin (0.98)

INTRODUCTION

For the development of hydrocarbon as well as geothermal reservoirs fluid injections through a borehole into the surrounding rock are frequently used. Such operations accomplished for enhancing hydrocarbon recovery (Economides and Nolte, 2000) or for creation of Enhanced Geothermal Systems (Majer et al., 2007) are accompanied by microseismic activity. This is because large parts of the Earths crust are in a sub-critical state where small, local stress changes may trigger microearthquakes. Introducing the so-called SBRC-approach (seismicity-based reservoir characterization) and assuming that the pore-fluid pressure perturbation induced by a fluid-injection obeys the law of linear diffusion Shapiro et al. (2002) show that the order of magnitude of the field-scale hydraulic permeability tensor can be estimated from the spatio-temporal distribution of observed microseismicity. The estimation of the hydraulic transport properties is done with the concept of the so-called ’triggering front’ providing an approximate outermost envelope of the distances between event locations and the injection point r as function of the time t elapsed since beginning of injection. However from field and laboratory experiments one has understood that the assumption of linear pore-fluid pressure diffusion is not always valid (Shapiro and Dinske, 2009). Pore-fluid pressure can strongly impact fluid transport properties. Lie et al. (2009) show a significant influence of effective pressure on permeability of several tight gas sandstone samples as well as for a granite rock. Yilmaz et al. (1994) simulate 1-D pore-fluid pressure profiles with a pressure-dependent permeability. Based on laboratory studies their suggested model shows an exponential relation of permeability on pore-fluid pressure. On that basis Hummel and M¨uller (2009) simulate and analyse synthetic clouds of 1-D and 2-D microseismicity based on nonlinear pore-fluid pressure diffusion. We use the triggering front (equation 1) to obtain heuristic estimates of hydraulic diffusivity from microseismic clouds.

analysis, diffusion, diffusivity, equation, estimate, exponential, flow in porous media, Fluid Dynamics, front, good agreement, hydraulic diffusivity, hydraulic transport, injection, interaction, microseismicity, nonlinear fluid-rock, permeability, profile, property, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, Shapiro, spatiotemporal pore-fluid, synthetic cloud, Upstream Oil & Gas

SPE Disciplines:

Biot, bulk modulus, chalk, coefficient, elastic moduli, equation, equivalent pore radius, flow in porous media, Fluid Dynamics, frame, frequency, Gassmann, mechanical strength, mechanism, permeability, Pore Fluid, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, shear modulus, Upstream Oil & Gas, vacuum, water, water weakening, weakening

Thank you!