Borehole Acoustic Inversion in Anisotropic Formations
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Solution Overview
Problem
Simulating borehole acoustic properties in anisotropic formations of the Earth's crust is computationally exhaustive, making real-time analysis during drilling challenging due to the complexity required to capture physical mechanisms accurately.
Innovation Solution
A method that formulates a geometric model in cylindrical coordinates, represents field variables as Fourier series expansions of π-periodic harmonics, and numerically solves three-dimensional wave equations, allowing for efficient computation and inversion of elastic properties, even in cases with material anisotropy that violates axis-symmetry conditions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If full 3-dimensional finite element modeling is used to simulate borehole acoustic problems in anisotropic formations, then the accuracy and capability to capture physical mechanisms is improved, but the computational complexity and resource requirements become prohibitive for real-time logging while drilling applications
Solution Approach 1:
The patent transforms the 3D wave propagation problem in cylindrical coordinates into a 2D problem by applying Fourier series expansion in the circumferential direction. This dimensionality reduction converts the computationally intensive 3D finite element model into a more efficient 2D model that retains accuracy for anisotropic formations while reducing computational complexity suitable for real-time logging applications.
2Adaptability or versatility
If standard three dimensional isoparametric finite elements are used to model layer by layer variations in elastic parameters and anisotropy, then the ability to capture complex physical mechanisms is improved, but the computational time and resources become exhaustive and prohibitive
Solution Approach 1:
The patent reduces the 3D computational problem to 2D by utilizing the cylindrical geometry and applying Fourier series in the circumferential direction. This allows the model to handle layer-by-layer anisotropic variations efficiently without requiring exhaustive 3D finite element computations, thus improving productivity while maintaining adaptability to complex formation properties.
Solution Approach 2:
The patent changes the mathematical representation of the wave equation by applying Fourier series expansion in the circumferential direction and using cylindrical coordinates. This parameter transformation converts the complex 3D anisotropic problem into a more tractable 2D problem that can be solved efficiently while preserving the ability to model layer variations and anisotropy.
3Reliability
If complex computational models are used to accurately simulate acoustic wave propagation in anisotropic formations, then the reliability of inversion results is improved, but the computational time increases making real-time analysis difficult
Solution Approach 1:
The patent applies Fourier series expansion in the circumferential direction to transform the 3D wave propagation problem into a 2D problem. This dimensionality reduction maintains the reliability of inversion results by preserving the essential physics of anisotropic wave propagation while significantly reducing computational time to enable real-time logging while drilling applications.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach enables high computational efficiency for real-time analysis of logging data while drilling, facilitating the determination of physical properties like elastic properties, velocity, density, and porosity, even in formations with varying anisotropy, by reducing the dimensionality of integrals and leveraging π-periodicity in the circumferential direction.
Implementation Method 1
the one or more field variables are represented as respective Fourier series expansions of π-periodic harmonics in the circumferential direction
Data Source
AI summary
A method of simulating a borehole acoustic response in an anisotropic formation of the crust of the earth includes formulating a geometric model of the formation. The geometric model includes a plurality of layers definable in a cylindrical coordinate system defined by an axial direction normal to each of the layers, a radial direction relative to the axial direction, and a circumferential direction relative to the axial direction. The method also includes formulating a computational model of wave propagation in the formation. The computational model includes field variables and a wave equation describing a behavior of the field variables. The field variables are represented as respective Fourier series expansions of Tr-periodic harmonics in the circumferential direction. The method also includes numerically solving the computational model.