Box Dimension Research Articles

Mechanisms of Proppant Transport in Rough Fractures of Offshore Unconventional Reservoirs: Shale and Tight Sandstone

After hydraulic fracturing, unconventional reservoirs frequently encounter challenges related to limited effective proppant support distance and suboptimal proppant placement. Due to the strong heterogeneity of offshore reservoirs, which causes varying fracture roughnesses depending on different lithologies, a systematic study of the relationship between roughness and proppant transport could optimize operational parameters. This study incorporates the box dimension method for fractal dimension analysis to quantify roughness in auto-correlated Gaussian distributed surfaces created by true triaxial tests. Combined with the numerical analysis of (computational fluid dynamics) CFD-DEM (discrete element method) for bidirectional coupling, the laws of proppant deposition and transport processes within fractures with different roughnesses are obtained through comparative verification simulations. The results show that for rougher fractures of shale, the proppants are transported farther, but at JRC_52, (joint roughness coefficient), where there may be plugging in curved areas, there is a risk of near-well blockages. Compared to the smooth model, fluctuations in JRC_28 (tight sandstone) drastically increase turbulent kinetic energy within the fracture, altering particle transport dynamics. Moreover, smaller proppants (d/w ≤ 0.3) exhibit better transport capacity due to gravity, but the conductivity of the proppant is limited when the particles are too small. A d/w of 0.4 is recommended to guarantee transport capacity and proppant efficiency near the well. Additionally, proppants injected sequentially from small to large in shale fractures offer optimal propping effects, and can take advantage of the better transport capacity of smaller proppants in rough fractures. The large proppant (d/w = 0.8) is primarily deposited by gravity and forms a sloping sand bed, which subsequently ensures the aperture of the fractures. This research provides a fresh perspective on the influence of fracture roughness on proppant transport in offshore unconventional reservoirs and offers valuable considerations for the order of proppant injection.

Fractal dimension and its implication to mineral exploration: a case study from the Jonnagiri and Gadag gold deposits in India

Fractal dimensions characterize the spatial distribution pattern of mineral deposits, allowing the mineralization processes beneath and at the surface to be understood. This study considered a mineralized vein along the fracture/shear zone near the Dona Temple in the Jonnagiri gold deposit and borehole data from the Sangli Block in the Gadag gold deposit. The above-mentioned zones were considered using fractal dimensional analysis to constrain the fluid-flow pathways. The box-counting technique was used to measure the fractal dimension of the mineralized vein along the surface from a detailed to a regional scale. The fractal correlation integral method was used to provide the fractal dimension values from the gold assay (in g t −1 ). The box-counting fractal dimension values obtained varied from 1.07 to 1.36, whilst the fractal dimensions obtained from the fractal correlation integral method ranged from 0.03 to 1.97. Higher fractal values signify a fully matured interlinked fracture/fault network with steep fluid-flow pathways, while lower values indicate the converse condition. Further, the study showed that the fractal analysis of the mineralized vein along the fracture/shear zone appeared to correlate with the fault system rather than with the litho-contact. Economic gold deposition occurs mainly along a structure with a steep fractal gradient, which is the target area for economic gold exploration.

Effects of sinusoidal wall temperature on thermal dynamics and irreversibility around an inclined plate embedded in a square cavity

Abstract Effective thermal and flow control within complex geometries is essential for engineering applications. In this study, an in-depth examination of flow dynamics, entropy, and thermal regulation is undertaken within a square cavity featuring sinusoidal wall temperature. To introduce complexity, an inclined plate obstacle is strategically positioned within the cavity with an inclination angle of 45o, and the investigation spans three distinct scenarios: adiabatic, cold, and hot conditions. The initial physical model is developed by formulating a system of partial differential equations, which are then transformed into a dimensionless representation using relevant variables. Subsequently, the Galerkin method is employed for approximated analysis of the simplified fluid flow model, and the computational code is verified in tabular format. The embedded physical parameters are constrained to specific numerical values to ensure the convergence of the physical model in each scenario. The physical characteristics of isotherms, streamlines, Nusselt numbers, entropy, and Bejan numbers are investigated. Notably, the results demonstrate that the introduction of a cold inclined plate leads to peak values in generating the entropy and average heat transfer rates. When comparing the cold inclined plate to the heated inclined plate, an increase of approximately 20% in the average heat transfer rate and a 15% rise in the entropy generation rate was found for the cold inclined plate. Furthermore, the Bejan number showed a 10% decrease for the cold inclined plate compared to the heated inclined plate. Additionally, increasing the amplitude and wavenumber led to a rise in average heat transfer and entropy generation rates, with 25% and 30% increases, respectively.

Comparative Analysis of Repeatability in CT Radiomics and Dosiomics Features under Image Perturbation: A Study in Cervical Cancer Patients.

This study aims to evaluate the repeatability of radiomics and dosiomics features via image perturbation of patients with cervical cancer. A total of 304 cervical cancer patients with planning CT images and dose maps were retrospectively included. Random translation, rotation, and contour randomization were applied to CT images and dose maps before radiomics feature extraction. The repeatability of radiomics and dosiomics features was assessed using intra-class correlation of coefficient (ICC). Pearson correlation coefficient (r) was adopted to quantify the correlation between the image characteristics and feature repeatability. In general, the repeatability of dosiomics features was lower compared with CT radiomics features, especially after small-sigma Laplacian-of-Gaussian (LoG) and wavelet filtering. More repeatable features (ICC > 0.9) were observed when extracted from the original, Large-sigma LoG filtered, and LLL-/LLH-wavelet filtered images. Positive correlations were found between image entropy and high-repeatable feature number in both CT and dose (r = 0.56, 0.68). Radiomics features showed higher repeatability compared to dosiomics features. These findings highlight the potential of radiomics features for robust quantitative imaging analysis in cervical cancer patients, while suggesting the need for further refinement of dosiomics approaches to enhance their repeatability.

Residual strength of porous alumina ceramics and fractal characterization of their crack patterns after thermal shocks

Thermal shock behaviors of porous alumina samples with different porosities (4.6–16.0 %) and mean grain sizes (2.72–5.02 μm) were investigated under different quenching temperatures. The slit island method was used to analyze the structures of the pores. The effect of the pore structure on the thermal shock resistance of alumina ceramics was described quantitatively. The results revealed that the thermal shock resistance decreased when the fractal dimension of the pores increased. Alumina ceramic strips with a porosity and mean grain size of 4.6 % and 5.02 μm, respectively, were fabricated using raw powders with a mean particle size of ∼0.3 μm. These samples possessed the highest residual strength ratio (∼46.16 % at 600 °C) and critical temperature difference (∼262 °C). In addition, the fractal characterization of cracks on the surfaces of the alumina ceramics after thermal shock was carried out using the box-counting dimension method. In addition, we investigated the correlation between the mechanical properties and fractal dimensions of the surface cracks. The box-counting dimension increased with crack propagation and negatively correlated with the residual strength. Furthermore, the propagation rate of the surface cracks decreased as the temperature difference increased, resulting in a decrease in the growth rate of the fractal dimension.

Approximation with fractal radial basis functions

The article reports on the construction of a general class of fractal radial basis functions (RBFs) in the literature. The fractal RBFs is defined through fractal perturbation of a RBF through suitable choice of iterated function system (IFS). A fractal RBF may be smooth depending on the choice of the germ function and the IFS parameters. Characterizations of conditionally strictly positive definite and strictly positive definite fractal functions are studied using the definition of k-times monotonicity. Furthermore, error estimates and shape-preserving properties for the approximants Pfα defined through linear combination of cardinal fractal RBFs are investigated. Several examples are presented to illustrate the convergence of the operator Pfα across various parameters, highlighting the advantages of the fractal approximant Pfα over the corresponding classical operator Pf. Finally, estimates for the box dimension of the graphs of approximants derived from fractal radial basis functions are given.

On the box dimension of recurrent fractal interpolation functions defined with Matkowski contractions

On the box dimension of recurrent fractal interpolation functions defined with Matkowski contractions

Evaluation method of tunnel cracking disease under biased pressure based on enhanced image fractal features

This paper focuses on the tunnel cracking disease under biased pressure. Through model tests on three lining conditions, the study investigates the expansion process of surface cracks on the lining under biased pressure. Proposing an improved method for calculating fractal characteristics of lining crack disease based on box counting dimension. Quantifying crack disease through digital image processing to obtain two parameters: total width and fractal dimension. Discussed the relationship between quantification parameters of surface crack disease in linings and the service performance of the lining. 29 groups of crack disease quantization parameters obtained by model test were selected to establish the disease sample space, and the spatial data were automatically learned and grouped based on clustering method. Finally, a new diagnostic Index of Tunnel Lining Defect Index-Crack & Dimension (TLDI-C&D) and its disease classification criteria for the whole lining ring are proposed. The results indicate that under the biased pressure, regardless of the presence of pre-existing defects, the biased pressure region is the most severely diseased area in terms of cracking. Compared to the intact lining, the initial cracking loads for arch crown defects and arch waist defects are advanced by 33.1 % and 73.0 %, respectively. After the apparent cracks in the lining are fully formed, pre-existing defects no longer affect the speed and distribution of crack propagation.The larger the fractal dimension, the more complex the morphology of the corresponding crack disease. Under the same image area, the ratio of crack quantity to width has a greater impact on the fractal dimension. There is a corresponding relationship between quantification parameters of crack disease and the bearing performance of the lining structure, making them indicators for measuring the load-bearing capacity of the lining. The most reasonable grading level for lining crack disease under the biased pressure is four levels. The diagnostic index for tunnel lining cracks, TLDI-C&D is 1.0～2.2 (level I), 2.2～3.2 (level II), 3.2～4.3 (level III), and 4.3～5.5 (level IV). On-site measurements have proven the accuracy of this crack disease diagnostic evaluation method.

Preliminary Results of Developing Imaging Complexity Biomarkers for the Incidence of Severe Radiation Pneumonitis Following Radiotherapy in Non-Small Cell Lung Cancer Patients with Underlying Idiopathic Pulmonary Fibrosis.

Background: Idiopathic pulmonary fibrosis (IPF) has the potential to cause fatal pulmonary toxicity after radiotherapy and can increase the morbidity and mortality of non-small-cell lung cancer (NSCLC) patients. In this context, we aimed to develop imaging complexity biomarkers to predict the incidence of severe pulmonary toxicity in patients with NSCLC who have underlying IPF and are treated with radiotherapy. Methods: We retrospectively reviewed the medical records of 19 patients with NSCLC who had underlying IPF and were treated with radiotherapy at the Korea University Guro Hospital between March 2018 and December 2022. To quantify the morphometric complexity of the lung parenchyma, box-counting fractal dimensions and lacunarity analyses were performed on pre-radiotherapy simulation chest computed tomography scans. Results: Of the 19 patients, the incidence of grade 3 or higher radiation pneumonitis was observed in 8 (42.1%). After adjusting for age, sex, smoking status, histology, and diffusing capacity of the lung for carbon monoxide, eight patients with a lower fractal dimension showed a significantly higher hazard ratio of 7.755 (1.168-51.51) for grade 3 or higher pneumonitis than those with a higher fractal dimension. Patients with lower lacunarity exhibited significantly lower hazards in all models, both with and without adjustments. The lower-than-median lacunarity group also showed significantly lower incidence curves for all models built in this study. Conclusions: We devised a technique for quantifying morphometric complexity in NSCLC patients with IPF on radiotherapy and discovered lacunarity as a potential imaging biomarker for grade 3 or higher pneumonitis.

Melting heat transfer and entropy analysis of MHD Casson nanofluid flow through a stretchy surface with Joule heating and complete slip

Purpose The purpose of this study is to examine the melting heat transfer of magnetohydrodynamics Casson nanofluid flow with viscous dissipation, radiation, and complete slip effects on a porous stretching sheet. Since, the study of melting heat transfer has mesmerized the attention of scientists and engineers in the sense of its enormous uses in industrial processes, solidification, casting, and technology. Design/methodology/approach Bejan number and entropy are analyzed. Exploration of irreversibility is modeled using the thermodynamics second law. There is a discussion on thermophoresis and Brownian diffusion along with first-order chemical reactions. Adequate transformations are introduced to convert the controlling partial differential equations to ordinary differential equations. The three-phase Lobatto solvers (bvp5c) are used to obtain numerical solutions of the transmitted equations. Findings The effects of various factors on temperature, velocity, concentration, Bejan number and entropy rate are shown graphically. The velocity field is enhanced by increasing the melting heat parameter, and it declines for growing magnetic parameters. Temperature is decreased for increasing parametric values of melting heat, porous and Casson parameters. A 7% decrease in the Sherwood distribution is seen when we increase the Brownian motion parameter from 0.1 to 0.2. Similarly, an 11% decrement is found in the Nusselt distribution for increasing the Brinkman number from 0.5 to 1. Originality/value Entropy and Bejan number experience dual tendencies whenever the melting heat parameter increases. Nusselt number and skin friction experience the opposite behavior for the increasing values of melting parameter. Sherwood number decreases for the increasing values of melting parameter. The velocity profile is directly related to the melting parameter and inversely related to porous and magnetic parameters. Thermophoresis and Brinkman parameters boost the temperature profile and it is controlled by melting and porous parameters. Some notable fields where the present study is used inevitably are silicon wafering, geothermal energy recovery and semiconductor manufacturing.

L q -spectra of graph-directed planar non-conformal measures

We estimate the upper and lower Lq -spectra of measures generated by a class of graph-directed planar non-conformal iterated function systems. These systems are composed of C1+ϵ contractions with lower triangular pointwise Jacobian. The estimation provides a criterion for the existence of the Lq -spectra in certain case. As an application, we derive the box-counting dimension of planar self-affine sets generated by lower triangular matrices, whose geometric construction satisfies a finite overlapping type condition.

The Minkowski dimension of suitable weak solutions of the 3D co-rotational Beris-Edwards system

The Minkowski dimension of suitable weak solutions of the 3D co-rotational Beris-Edwards system

Damage evolution law of multi-hole blasting igneous rock and quantitative evaluation model of damage degree based on fractal theory and clustering algorithm

The geological phenomenon of igneous rock invading coal seam is widely distributed, which induces mining risk and affects efficient mining. The pre-splitting blasting method of igneous rock is feasible but difficult to implement accurately, resulting in unnecessary safety and environmental pollution risks. In this paper, the blasting model with penetrating structural plane and the multi-hole blasting model with different hole spacing were established based on the Riedel–Hiermaier–Thoma (RHT) damage constitutive to explore the stress wave propagation law under detonation. The damage cloud diagram and damage degree algorithm were used to quantitatively describe the spatio-temporal evolution of blasting damage. The results show that the explosion stress wave presents a significant reflection stretching effect under the action of the structural plane, which can effectively aggravate the presplitting blasting degree of the rock mass inside the structural plane. The damage range of rock mass is synchronously evolved with the change of blasting hole spacing. The blasting in the igneous rock intrusion area of the 21,914 working face is taken as an application example, and the damage degree of rock mass is reasonably evaluated by the box-counting dimension and K-means clustering method, which proves the effectiveness of the blasting scheme and provides reference value for the implementation of related blasting projects.

Widths and entropy numbers of the classes of periodic functions of one and several variables in the space $B_{q,1}$

Exact-order estimates are obtained for the entropy numbers and several types of widths (Kolmogorov, linear, trigonometric and orthowidth) for the Sobolev and Nikol'skii-Besov classes of one and several variables in the space $B_{q,1}$, $1 <q< \infty$. It is shown, that in the multivariate case, in contrast to the univariate, the obtained estimates differ in order from the corresponding estimates in the space $L_q$.

Effects of Alumina Bubble Addition on the Properties of Corundum-Spinel Castables Containing Cr2O3.

Purging plugs made of corundum-spinel castables containing Cr2O3 have been widely utilized in secondary refining process. However, their poor thermal shock resistance has greatly limited the improvement of their service life. Aiming to enhance their properties, we introduced alumina bubbles (ABs) to corundum-spinel castables, and the effects of the AB addition on the properties of the castables are studied in this manuscript. The results indicate that the apparent porosity, permanent linear change, cold strength, and hot strength all increased with an increasing AB amount. The thermal shock resistance of the samples with the AB addition was improved; the residual strength and residual strength ratio of the sample with 4 wt% ABs was the best. The effects of ABs on the tabular alumina aggregate distribution and relationship between the cold strength of the samples and the AB content was evaluated via the box dimension method. With the increments of AB content, the box dimension value of the tabular alumina within the samples significantly decreased, indicating that the tabular alumina aggregate distribution was related to the amount of ABs. In addition, the relationship between the box dimension and the strength was also established.

Talbot Effect for the Manakov System on the Torus

In this paper, the Talbot effect for the multi-component linear and nonlinear systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles is investigated. Firstly, for a class of two-component linear systems satisfying the dispersive quantization conditions, we discuss the fractal solutions at irrational times. Next, the investigation to nonlinear regime is extended, we prove that, for the concrete example of the Manakov system, the solutions of the corresponding periodic initial-boundary value problem subject to initial data of bounded variation are continuous but nowhere differentiable fractal-like curve with Minkowski dimension 3/2 at irrational times. Finally, numerical experiments for the periodic initial-boundary value problem of the Manakov system, are used to justify how such effects persist into the multi-component nonlinear regime. Furthermore, it is shown in the nonlinear multi-component regime that the interplay of different components may induce subtle different qualitative profile between the jump discontinuities, especially in the case that two nonlinearly coupled components start with different initial profile.

Analytical properties and the box-counting dimension of nonlinear hidden variable recurrent fractal interpolation functions constructed by using Rakotch's fixed point theorem

Rakotch contraction is a generalization of Banach contraction, which implies that in the case of using Rakotch's fixed point theorem, we can model more objects than using Banach's fixed point theorem. Moreover, hidden variable recurrent fractal interpolation functions (HVRFIFs) with Hölder function factors are more general than the fractal interpolation functions (FIFs), recurrent FIFs and hidden variable FIFs with Lipschitz function factors. We demonstrate that HVRFIFs can be constructed using the Rakotch's fixed point theorem, and then investigate the analytical and geometric properties of those HVRFIFs. Firstly, we construct a nonlinear hidden variable recurrent fractal interpolation functions with Hölder function factors on the basis of a given data set using Rakotch contractions. Next, we analyze the smoothness of the HVRFIFs and show that they are stable on the small perturbations of the given data. Finally, we get the lower and upper bounds for their box-counting dimensions.

On the Local Fourier Uniformity Problem for Small Sets

Abstract We consider vanishing properties of exponential sums of the Liouville function $\boldsymbol{\lambda }$ of the form $$ \begin{align*} & \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq H}\boldsymbol{\lambda}(m+h)e^{2\pi ih\alpha}\bigg|=0, \end{align*} $$ where $C\subset{{\mathbb{T}}}$. The case $C={{\mathbb{T}}}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset{{\mathbb{T}}}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C={{\mathbb{T}}}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2\pi ih\alpha }$ is replaced by a polynomial phase $e^{2\pi ih^{t}\alpha }$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $< 1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of “rigid” sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.

A novel numerical reconstruction method of coal based on characteristics of the cleat system

To effectively characterize the characteristics of the pore network of coal and to provide a realistic flow channel for flow simulation, a numerical reconstruction method of coal based on characteristics of the cleat system was proposed. Based on the real coal obtained through high-resolution micrometer CT, the cleat system characteristic parameters are defined and extracted. Then, a series of coals with different pore characteristics at the same scale and the same pore characteristics at different scales were reconstructed. Besides, the applicability of box dimension in calibrating the pore characteristics of coals was quantitatively analyzed. The results show that the coals reconstructed based on the method proposed in this paper can truly restore the pore characteristics of the real coal, especially the non-extension features of cleats; When the cleat system characteristic parameters remain consistent, the reconstructed coals maintain the same pore characteristics, with an average error of 3.37 % in the autocorrelation function, 0.78 % in the variation function and 1.94 % in the average percolation probability function. The box dimension can effectively identify the differences in pore characteristics of coals. At the same scale, as the distance between face cleats increases, the value of the box dimension of the reconstructed coals monotonically decreases, with a fitting slope of −0.22; At different scales, when the cleat system characteristic parameters remain the same, the value of the box dimension of the reconstructed coals is consistent, with an average relative error of 0.34 %. The research results provide a reasonable and effective flow channel for the microscopic/mesoscopic flow simulation of coalbed methane and provide a reference for the quantitative calibration of the pore characteristics of coals.

A hybrid decision-making technique based on extended entropy and trapezoidal fuzzy rough number

A hybrid decision-making technique based on extended entropy and trapezoidal fuzzy rough number