Computing Pressure Front Propagation Using the Diffusive-Time-of-Flight in Structured and Unstructured Grid Systems via the Fast-Marching Method

SPE Journal ◽  
2021 ◽  
pp. 1-21
Author(s):  
Hongquan Chen ◽  
Tsubasa Onishi ◽  
Jaeyoung Park ◽  
Akhil Datta-Gupta
Keyword(s):  
Finite Difference ◽  
Analytical Solutions ◽  
Eikonal Equation ◽  
Front Propagation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Reservoir Properties ◽  
Grid Systems ◽  
Marching Method ◽  
Diffusive Time

Summary Diffusive-time-of-flight (DTOF), representing the travel time of pressure front propagation, has found many applications in unconventional reservoir performance analysis. The computation of DTOF typically involves upwind finite difference of the Eikonal equation and solution using the fast-marching method (FMM). However, the application of the finite difference-based FMM to irregular grid systems remains a challenge. In this paper, we present a novel and robust method for solving the Eikonal equation using finite volume discretization and the FMM. The implementation is first validated with analytical solutions using isotropic and anisotropic models with homogeneous reservoir properties. Consistent DTOF distributions are obtained between the proposed approach and the analytical solutions. Next, the implementation is applied to unconventional reservoirs with hydraulic and natural fractures. Our approach relies on cell volumes and connections (transmissibilities) rather than the grid geometry, and thus can be easily applied to complex grid systems. For illustrative purposes, we present applications of the proposed method to embedded discrete fracture models (EDFMs), dual-porositydual-permeability models (DPDK), and unstructured perpendicular-bisectional (PEBI) grids with heterogeneous reservoir properties. Visualization of the DTOF provides flow diagnostics, such as evolution of the drainage volume of the wells and well interactions. The novelty of the proposed approach is its broad applicability to arbitrary grid systems and ease of implementation in commercial reservoir simulators. This makes the approach well-suited for field applications with complex grid geometry and complex well architecture.

On: “3-D traveltime compuation using the fast marching method” (J. A. Sethian and A. M. Popovici, GEOPHYSICS, 64, 516–523).

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 682-682
Author(s):  
Fuhao Qin
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Marching Method ◽  
Difference Solution

The Sethian and Popovici paper “3-D traveltime computation using the fast marching method” that appeared in Geophysics, Vol. 64, 516–523, discussed a method to solve the eikonal equation for first arrival traveltimes which was called the “fast marching” method. The method, as the authors demonstrated, is very fast and stable. However, their method is very similar to the method discussed by F. Qin et al. (1992), entitled “Finite difference solution of the eikonal equation along expanding wavefronts,” Geophysics, Vol. 57, 478–487. F. Qin et al. first proposed the “expanding wavefront” method for solving eikonal equation in the 60th Ann. Internat. Mtg. of the SEG in 1990.

Development and Application of Fast Simulation Based on the PSS Pressure as a Spatial Coordinate

2021 ◽  
Author(s):  
Kenta Nakajima ◽  
Michael King
Keyword(s):  
Eikonal Equation ◽  
Time Of Flight ◽  
Fast Marching Method ◽  
Reservoir Model ◽  
Fast Marching ◽  
Flow Equations ◽  
Spatial Coordinate ◽  
Single Well ◽  
Marching Method ◽  
Diffusive Time

Abstract Recent studies have shown the utility of the Fast Marching Method and the Diffusive Time of Flight for the rapid simulation and analysis of Unconventional reservoirs, where the time scale for pressure transients are long and field developments are dominated by single well performance. We show that similar fast simulation and multi-well modeling approaches can be developed utilizing the PSS pressure as a spatial coordinate, providing an extension to both Conventional and Unconventional reservoir analysis. We reformulate the multi-dimensional multi-phase flow equations using the PSS pressure drop as a spatial coordinate. Properties are obtained by coarsening and upscaling a fine scale 3D reservoir model, and are then used to obtain fast single well simulation models. We also develop new 1D solutions to the Eikonal equation that are aligned with the PSS discretization, which better represent superposition and finite sized boundary effects than the original 3D Eikonal equation. These solutions allow the use of superposition to extend the single well results to multiple wells. The new solutions to the Eikonal equation more accurately represent multi-fracture interference for a horizontal MTFW well, the effects of strong heterogeneity, and finite reservoir extent than those obtained by the Fast Marching Method. The new methodologies are validated against a series of increasingly heterogeneous synthetic examples, with vertical and horizontal wells. We find that the results are systematically more accurate than those based upon the Diffusive Time of Flight, especially as the wells are placed closer to the reservoir boundary or as heterogeneity increases. The approach is applied to the Brugge benchmark study. We consider the history matching stage of the study and utilize the multi-well fast modeling approach to determine the rank quality of the 100+ static realizations provided in the benchmark dataset against historical data. The multi-well calculation uses superposition to obtain a direct calculation of the interaction of the rates and pressures of the wells without the need to explicitly solve flow equations within the reservoir model. The ranked realizations are then compared against full field simulation to demonstrate the significant reduction in simulation cost and the corresponding ability to explore the subsurface uncertainty more extensively. We demonstrate two completely new methods for rapid reservoir analysis, based upon the use of the PSS pressure as a spatial coordinate. The first approach demonstrates the utility of rapid single well flow simulation, with improved accuracy compared to the use of the Diffusive Time of Flight. We are also able to reformulate and solve the Eikonal equation in these coordinates, giving a rapid analytic method of transient flow analysis for both single and multi-well modeling.

BOAT-SAIL VORONOI DIAGRAM AND ITS APPLICATION

2009 ◽  
Vol 19 (05) ◽  
pp. 425-440 ◽  
Author(s):  
TETSUSHI NISHIDA ◽  
KOKICHI SUGIHARA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Voronoi Diagram ◽  
Water Surface ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Marching Method ◽  
Generalized Voronoi Diagram

A new generalized Voronoi diagram, called a boat-sail Voronoi diagram, is defined on the basis of the time necessary for a boat to reach on water surface with flow. A new concept called a boat-sail distance is introduced on the surface of water with flow, and it is used to define a generalized Voronoi diagram, in such a way that the water surface is partitioned into regions belonging to the nearest harbors with respect to this distance. The problem of computing this Voronoi diagram is reduced to a boundary value problem of a partial differential equation, and a numerical method for solving this problem is constructed. The method is a modification of a so-called fast marching method originally proposed for the eikonal equation. Computational experiments show the efficiency and the stableness of the proposal method. We also apply our equation to the shortest path problem and the simulation of the forest fire.

3D Facial Shape Reconstruction from a Single Frontal Image

2014 ◽  
Vol 989-994 ◽  
pp. 3544-3547
Author(s):  
Qian Ma
Keyword(s):  
Eikonal Equation ◽  
Shape From Shading ◽  
Shape Reconstruction ◽  
Fast Marching Method ◽  
Improved Method ◽  
Fast Marching ◽  
Facial Shape ◽  
Shape Invariant ◽  
Marching Method

In this paper, we propose an improved method for reconstruct 3D facial shape from a single frontal image. We use improved fast marching method to solve the Eikonal equation which can obtained from the method of shape from shading (SFS). In order to overcome the concave-convex ambiguity problems inherent to SFS, we find out the concave region and recover concave into the convex keeping the relative shape invariant to reconstruct the accuracy facial shape.

PyKonal: A Python Package for Solving the Eikonal Equation in Spherical and Cartesian Coordinates Using the Fast Marching Method

2020 ◽  
Vol 91 (4) ◽  
pp. 2378-2389
Author(s):  
Malcolm C. A. White ◽  
Hongjian Fang ◽  
Nori Nakata ◽  
Yehuda Ben-Zion
Keyword(s):  
Open Source ◽  
Eikonal Equation ◽  
Heterogeneous Media ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Cartesian Coordinates ◽  
Source Codes ◽  
High Level ◽  
Marching Method ◽  
Python Package

Abstract This article introduces PyKonal: a new open-source Python package for computing travel times and tracing ray paths in 2D or 3D heterogeneous media using the fast marching method for solving the eikonal equation in spherical and Cartesian coordinates. Compiled with the Cython compiler framework, PyKonal offers a Python application program interface (API) with execution speeds comparable to C or Fortran codes. Designed to be accurate, stable, fast, general, extensible, and easy to use, PyKonal offers low- and high-level API functions for full control and convenience, respectively. A scale-independent implementation allows problems to be solved at micro, local, regional, and global scales, and precision can be improved over existing open-source codes by combining different coordinate systems. The resulting code makes state-of-the-art computational capabilities accessible to novice programmers and is efficient enough for modern research problems in seismology.

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