Multiple reflection and transmission phases in complex layered media using a multistage fast marching method

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1338-1350 ◽  
Author(s):  
N. Rawlinson ◽  
M. Sambridge
Keyword(s):  
Layered Media ◽  
Computational Domain ◽  
Grid Refinement ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Reflection And Transmission ◽  
Local Grid Refinement ◽  
Local Grid ◽  
Marching Method ◽  
Grid Based

Traditional grid‐based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only. However, later arrivals can be numerous and of greater amplitude, making them a potentially valuable resource for practical applications such as seismic imaging. The aim of this paper is to introduce a grid‐based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media. A finite‐difference eikonal solver known as the fast marching method (FMM) is used to propagate wavefronts from one interface to the next. By treating each layer that the wavefront enters as a separate computational domain, one obtains a refracted branch by reinitializing FMM in the adjacent layer and a reflected branch by reinitializing FMM in the incident layer. To improve accuracy, a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high. Several examples are presented which demonstrate the viability of the new method in highly complex layered media. Even in the presence of velocity variations as large as 8:1 and interfaces of high curvature, wavefronts composed of many reflection and transmission events are tracked rapidly and accurately. This is because the scheme retains the two desirable properties of a single‐stage FMM: computational speed and stability. Local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost.

Modeling Hydraulically Fractured Shale Wells Using the Fast-Marching Method With Local Grid Refinements and an Embedded Discrete Fracture Model

SPE Journal ◽  
2019 ◽  
Vol 24 (06) ◽  
pp. 2590-2608 ◽  
Author(s):  
Xu Xue ◽  
Changdong Yang ◽  
Tsubasa Onishi ◽  
Michael J. King ◽  
Akhil Datta–Gupta
Keyword(s):  
Computational Efficiency ◽  
Flow Simulation ◽  
Fracture Model ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Discrete Fracture Model ◽  
Discretization Schemes ◽  
Discrete Fracture ◽  
Local Grid ◽  
Marching Method

Summary Recently, fast–marching–method (FMM) –based flow simulation has shown great promise for rapid modeling of unconventional oil and gas reservoirs. Currently, the application of FMM–based simulation has been limited to using tartan grids to model the hydraulic fractures (HFs). The use of tartan grids adversely impacts the computational efficiency, particularly for field–scale applications with hundreds of HFs. Our purpose in this paper is to extend FMM–based simulation to incorporate local grid refinements (LGRs) and an embedded discrete fracture model (EDFM) to simulate HFs with natural fractures, and to validate the accuracy and efficiency of the methodologies. The FMM–based simulation is extended to LGRs and EDFM. This requires novel gridding through the introduction of triangles (2D) and tetrahedrons (2.5D) to link the local and global domain and solution of the Eikonal equation in unstructured grids to compute the diffusive time of flight (DTOF). The FMM–based flow simulation reduces a 3D simulation to an equivalent 1D simulation using the DTOF as a spatial coordinate. The 1D simulation can be carried out using a standard finite–difference method, thus leading to orders of magnitude of savings in computation time compared with full 3D simulation for high–resolution models. First, we validate the accuracy and computational efficiency of the FMM–based simulation with LGRs by comparing them with tartan grids. The results show good agreement and the FMM–based simulation with LGRs shows significant improvement in computational efficiency. Then, we apply the FMM–based simulation with LGRs to the case of a multistage–hydraulic–fractured horizontal well with multiphase flow, to demonstrate the practical feasibility of our proposed approach. After that, we investigate various discretization schemes for the transition between the local and global domain in the FMM–based flow simulation. The results are used to identify optimal gridding schemes to maintain accuracy while improving computational efficiency. Finally, we demonstrate the workflow of the FMM–based simulation with EDFM, including grid generation; comparison with FMM with unstructured grid; and validation of the results. The FMM with EDFM can simulate arbitrary fracture patterns without simplification and shows good accuracy and efficiency. This is the first study to apply the FMM–based flow simulation with LGRs and EDFM. The three main contributions of the proposed methodology are (i) unique mesh–generation schemes to link fracture and matrix flow domains, (ii) DTOF calculations in locally refined grids, and (iii) sensitivity studies to identify optimal discretization schemes for the FMM–based simulation.

3-D traveltime computation using the fast marching method

Geophysics ◽  
1999 ◽  
Vol 64 (2) ◽  
pp. 516-523 ◽  
Author(s):  
James A. Sethian ◽  
A. Mihai Popovici
Keyword(s):  
Point Source ◽  
Eikonal Equation ◽  
Initial Conditions ◽  
Plane Waves ◽  
Three Dimensions ◽  
Computational Domain ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Grid Points ◽  
Marching Method

We present a fast algorithm for solving the eikonal equation in three dimensions, based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagation wavefronts. We begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details. We then show examples of traveltime propagation through the SEG/EAGE salt model using point‐source and plane‐wave initial conditions and analyze the error in constant velocity media. The algorithm allows for any shape of the initial wavefront. While a point source is the most commonly used initial condition, initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another. The algorithm presented here is designed for computing first‐arrival traveltimes. Nonetheless, since it exploits the fast marching method for solving the eikonal equation, we believe it is the fastest of all possible consistent schemes to compute first arrivals.

Turbulence Modeling, Local Grid Refinement and Absorbing Boundary Conditions for Free-Surface Flow Simulations in Offshore Applications

2014 ◽  
Author(s):  
Arthur E. P. Veldman ◽  
Roel Luppes ◽  
Henri J. L. van der Heiden ◽  
Peter van der Plas ◽  
Bülent Düz ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Turbulence Modeling ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Grid Refinement ◽  
Wave Impact ◽  
Absorbing Boundary ◽  
Local Grid Refinement ◽  
Local Grid ◽  
Hydrodynamic Wave

To study extreme hydrodynamic wave impact in offshore and coastal engineering, the VOF-based CFD simulation tool ComFLOW is being developed. Recently, much attention has been paid to turbulence modeling, local grid refinement, wave propagation and absorbing boundary conditions. The turbulence model has to cope with coarse grids as used in industrial applications. Thereto a blend of a QR-model and a regularization model has been designed, in combination with a dedicated wall model. Local grid refinement is based on a semi-structured approach. Near refinement interfaces special discretization stencils have been designed. The computational domain is restricted to the close environment of the objects studied. To suppress unphysical reflections, special generating and absorbing boundary conditions have been designed. The combined performance of the new ingredients will be demonstrated with several applications. For validation, experiments have been carried out at MARIN.

A Generalized Fast Marching Method on Unstructured Triangular Meshes

2013 ◽  
Vol 51 (6) ◽  
pp. 2999-3035 ◽  
Author(s):  
E. Carlini ◽  
M. Falcone ◽  
Ph. Hoch
Keyword(s):  
Fast Marching Method ◽  
Triangular Meshes ◽  
Fast Marching ◽  
Marching Method
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