Reconstruction of rock discontinuities is crucial in rock engineering. This study proposes a novel optimization approach for three-dimensional reconstruction of rock discontinuities using the simulated annealing (SA) algorithm. A new perturbation mechanism and an objective function based on cumulative fracture intensity (CFI) were developed to constrain the reconstruction of discrete fracture network (DFN) models using logging data, thereby reducing the uncertainty in DFN model generation. The proposed method was first tested using synthetic fracture networks to evaluate the influence of key algorithmic and model parameters on reconstruction performance. The results indicated that the cooling rate (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\alpha }_{T}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation>) had the most significant impact on reconstruction performance, followed by the initial temperature (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T$</EquationSource> <EquationSource Format="MATHML"><math> <mi>T</mi> </math></EquationSource> </InlineEquation>) and the number of iterations per temperature (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">${N}_{iter}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mrow> <mi mathvariant="italic">iter</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>). For the initial values of DFN model parameters, we found that the fractal dimension (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">${D}_{{f}_{initial}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <msub> <mi>f</mi> <mrow> <mi mathvariant="italic">initial</mi> </mrow> </msub> </msub> </math></EquationSource> </InlineEquation>) and the normalized constant (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\alpha }_{initial}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mrow> <mi mathvariant="italic">initial</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>) had a more pronounced influence on the reconstruction results than the length exponent (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">${a}_{initial}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">initial</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>). The optimal initial ranges for these parameters were identified as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2.2\le {D}_{{f}_{initial}}\le 2.6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2.2</mn> <mo>≤</mo> <msub> <mi>D</mi> <msub> <mi>f</mi> <mrow> <mi mathvariant="italic">initial</mi> </mrow> </msub> </msub> <mo>≤</mo> <mn>2.6</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$2.0\le {\alpha }_{initial}\le 3.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2.0</mn> <mo>≤</mo> <msub> <mi>α</mi> <mrow> <mi mathvariant="italic">initial</mi> </mrow> </msub> <mo>≤</mo> <mn>3.0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$1.8\le {a}_{initial}\le 2.2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.8</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow> <mi mathvariant="italic">initial</mi> </mrow> </msub> <mo>≤</mo> <mn>2.2</mn> </mrow> </math></EquationSource> </InlineEquation>. Using initial values within these ranges achieved the optimal reconstruction results. The method was then validated with logging data from a 200-m-deep borehole at the Olkiluoto site, in which 686 fractures were identified. The reconstructed model achieved a 99.32% match in CFI values compared with the measured data, confirming the accuracy and robustness of the algorithm. However, since the exact 3D geometries of discontinuities are not directly observable, the validation is partial and relies on 1D logging constraints.Highlights<UnorderedList Mark="Bullet"> <ItemContent> A new approach is proposed for reconstructing 3D subsurface rock discontinuity structures. </ItemContent> <ItemContent> An objective function based on CFI values and a novel perturbation mechanism are developed. </ItemContent> <ItemContent> The full implementation of DFN reconstruction from logging data is presented in detail. </ItemContent> </UnorderedList>

Three-Dimensional Reconstruction of Rock Discontinuities: A Logging-Constrained Optimization Approach Based on Simulated Annealing Algorithm

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发表日期：

2026年3月23日

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Chao Huang,
Qiucai Zhang,
Shengyang Feng

摘要

Reconstruction of rock discontinuities is crucial in rock engineering. This study proposes a novel optimization approach for three-dimensional reconstruction of rock discontinuities using the simulated annealing (SA) algorithm. A new perturbation mechanism and an objective function based on cumulative fracture intensity (CFI) were developed to constrain the reconstruction of discrete fracture network (DFN) models using logging data, thereby reducing the uncertainty in DFN model generation. The proposed method was first tested using synthetic fracture networks to evaluate the influence of key algorithmic and model parameters on reconstruction performance. The results indicated that the cooling rate ( ${\alpha }_{T}$ $α_{T}$ ) had the most significant impact on reconstruction performance, followed by the initial temperature ( $T$ $T$ ) and the number of iterations per temperature ( ${N}_{iter}$ $N_{iter}$ ). For the initial values of DFN model parameters, we found that the fractal dimension ( ${D}_{{f}_{initial}}$ $D_{f_{initial}}$ ) and the normalized constant ( ${\alpha }_{initial}$ $α_{initial}$ ) had a more pronounced influence on the reconstruction results than the length exponent ( ${a}_{initial}$ $a_{initial}$ ). The optimal initial ranges for these parameters were identified as $2.2\le {D}_{{f}_{initial}}\le 2.6$ $2.2 \leq D_{f_{initial}} \leq 2.6$ , $2.0\le {\alpha }_{initial}\le 3.0$ $2.0 \leq α_{initial} \leq 3.0$ , and $1.8\le {a}_{initial}\le 2.2$ $1.8 \leq a_{initial} \leq 2.2$ . Using initial values within these ranges achieved the optimal reconstruction results. The method was then validated with logging data from a 200-m-deep borehole at the Olkiluoto site, in which 686 fractures were identified. The reconstructed model achieved a 99.32% match in CFI values compared with the measured data, confirming the accuracy and robustness of the algorithm. However, since the exact 3D geometries of discontinuities are not directly observable, the validation is partial and relies on 1D logging constraints.

Highlights

A new approach is proposed for reconstructing 3D subsurface rock discontinuity structures.

An objective function based on CFI values and a novel perturbation mechanism are developed.

The full implementation of DFN reconstruction from logging data is presented in detail.