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Practical Applications of Space- Time Fractional Diffusion in Naturally Fractured Reservoirs

SPE JOURNAL 2025年第4期30卷 1970-1983页

作者： Chen, Chih Chu, Wei-Chun Bermudez, Leonardo Kappa Engn Paris France PTA RTA Consulting Dubai U Arab Emirates Hillwood Int Energy Dallas TX USA

This study introduces a radial space-time fractional-diffusion (ST-FD) model, now adapted for Laplace space. This model incorporates dual-porosity factors and, when applied to well tests in naturally fractured reservoirs, addresses the effects of rock heterogeneity in fractured or fissured formations by solving a multi-index ST-FD equation through the Laplace transformation. For the first time, pressure distributions in cylindrical coordinates for heterogeneous fractured rocks with space-and time-dependent properties are presented. These findings are novel and unreported in existing literature, highlighting the utility of this approach in analyzing fractured reservoirs. The method developed in this paper was applied to five example well tests from fractured carbonate fields in northern Iraq, demonstrating the practical application of this methodology. Two key models commonly used in the industry to study naturally fractured reservoirs are the Warren and Root (1963) dual-porosity model and de Swaan (1976) model. The Warren and Root model assumes a pseudosteady-state flow between the matrix and fractures, while de Swaan's model addresses transient flow between these components. When applied to pressure buildup (BU) responses in fractured reservoirs from northern Iraq that exhibit a power-law pattern, these traditional models could lead to misinterpretations, attributing the behavior to boundary effects such as faults. In contrast, using the continuous time random walk framework, specifically the ST-FD model developed by Raghavan and Chen (2021, 2023), and now modified in this paper with solutions in Laplace space, provides a more accurate interpretation of these pressure responses. The analysis of real-world data using this modified ST-FD model reveals that the power-law behavior in pressure BU responses is linked to the reservoir's complex heterogeneity as shown in five example well tests presented here. The effectiveness of the modified ST-FD model in Laplace space

关键词： complex reservoir superdiffusion example 1 well test fractured reservoir heterogeneity log-log plot fractional diffusion geologist flow capacity

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Box Height-Independent Differential Bar Cumulation (DBC) for 3D Raster Surface Fractal Dimension Analysis

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JOURNAL OF GEOVISUALIZATION AND SPATIAL ANALYSIS 2025年第1期9卷 1-19页

作者： Malleswar, S. D. Isoda, Yuzuru Nakaya, Tomoki Tohoku Univ Grad Sch Sci Dept Earth Sci Sendai Japan Tohoku Univ Grad Sch Environm Studies Dept Frontier Sci Adv Environm Sendai Japan

The box-counting method is the predominant technique for calculating fractal dimensions (FD) for raster data. However, past studies focused primarily on measuring the FD of 2D binary raster. While fields like image analysis have advanced with methods such as differential box counting (DBC) to measure the FD of 3D surfaces, this approach has yet to be applied to geographical raster data, mainly due to challenges like various data depth formats, choice of box height, and type of distribution of raster values within data. In this study, we address these challenges, particularly focusing on selecting an appropriate box height which is also a methodological issue even for images. The difficulties in DBC, such as too-small and too-large box height problems, appear on finer and coarser scales of log-log plots, respectively, and there are optimal scales that represent the accurate FD somewhere in the middle. We proposed key modifications to the conventional DBC method theoretically to reduce the number of inappropriate scales due to the difficulties in DBC. To identify the effective scales in log-log plot, we introduced a novel visualization technique which can be used to detect subtle variations in log-log plots by visual inspection, aiding in the identification of box height problems and related difficulties. We applied and compared both the conventional and modified DBC method to two empirical datasets, population data and digital elevation model data, across the USA. Our results demonstrate that the log-log plots display greater consistency, and the FD values converge to a larger value as the box height decreases. The enhanced log-log plots from the results showed that there is no too-small box height problem at all, and the box height is smaller the better. Based on the empirical results, we proposed a derived method that eliminates the need for selecting a specific box height, addressing all three difficulties of DBC identified in this study. Since the derived method

关键词： Fractal dimension Raster analysis Differential box-counting method log-log plot Scale invariance

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Enhancing resilience analysis of power systems using robust estimation

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RELIABILITY ENGINEERING & SYSTEM SAFETY 2019年 186卷 134-142页

作者： Shen, Lijuan Tang, Loon Ching Natl Univ Singapore Dept Ind Syst Engn & Management Singapore 117576 Singapore

It has been well-recognized that the distribution of the blackout size of a power grid system has a heavy tail. The power-law distribution is a popular model for the heavy-tail phenomenon, and it is widely used in power system disruptions. However, there are significant reporting errors in the disruption data reported in public available databases, such as the database of the Electric Disturbance Events (OE-417) maintained by the US Department of Energy. Traditional inference techniques such as the maximum likelihood estimation can be sensitive to such contaminated data due to the reporting errors. In this paper, we propose a robust estimation procedure for the power-law distribution based on the minimum distance estimation method. A comprehensive simulation is used to evaluate the performance of the proposed method, and compare the performance with the existing maximum likelihood method. It is found that the proposed method outperforms the existing maximum likelihood method in the presence of contaminated data. We apply the proposed method to the blackout data from Jan 2002 to Aug 2016 based on the OE-417 database.

关键词： Heavy-tailed Maximum likelihood Minimum distance estimation log-log plot Disruption

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Understanding reproductive allometry in turtles: A slippery "slope"

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ECOlogY AND EVOLUTION 2019年第20期9卷 11891-11903页

作者： Iverson, John B. Lindeman, Peter V. Lovich, Jeffrey E. Earlham Coll Dept Biol Richmond IN 47374 USA Edinboro Univ Penn Dept Biol & Hlth Sci Edinboro PA USA US Geol Survey Southwest Biol Sci Ctr Flagstaff AZ 86001 USA

Measures of reproductive output in turtles are generally positively correlated with female body size. However, a full understanding of reproductive allometry in turtles requires logarithmic transformation of reproductive and body size variables prior to regression analyses. This allows for slope comparisons with expected linear or cubic relationships for linear to linear and linear to volumetric variables, respectively. We compiled scaling data using this approach from published and unpublished turtle studies (46 populations of 25 species from eight families) to quantify patterns among taxa. Our results suggest that for log-log comparisons of clutch size, egg width, egg mass, clutch mass, and pelvic aperture width to shell length, all scale hypoallometrically despite theoretical predictions of isometry. Clutch size generally scaled at similar to 1.7 to 2.0 (compared to an isometric expectation of 3.0), egg width at similar to 0.5 (compared to an expectation of 1.0), egg mass at similar to 1.1 to 1.3 (3.0), clutch mass at similar to 2.5 to 2.8 (3.0), and pelvic aperture width at 0.8-0.9 (1.0). We also found preliminary evidence that scaling may differ across years and clutches even in the same population, as well as across populations of the same species. Future investigators should aspire to collect data on all these reproductive parameters and to report log-log allometric analyses to test our preliminary conclusions regarding reproductive allometry in turtles.

关键词： clutch mass regression clutch size egg mass log-log plot pelvic aperture scaling

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The Spurious Deflection on log-log Superposition-Time Derivative plots of Diagnostic Fracture-Injection Tests

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SPE RESERVOIR EVALUATION & ENGINEERING 2017年第4期20卷 1045-1055页

作者： McClure, Mark McClure Geomech Palo Alto CA 94301 USA

log-log superposition-time derivative plots are used to identify flow regimes in well tests with variable rate. The use of superposition time adjusts for the effect of the prior rate history, and (under some conditions) shows what the transient would have looked like if the test had been performed at a constant rate. In this report, I show that if these plots are used to interpret shut-in transients from diagnostic fracture-injection tests (DFITs), the superposition-time derivative has an upward deflection that does not represent actual reservoir or transient behavior. I review mathematical properties of the superposition-time derivative. I derive equations for the pressure transient in a simplified model DFIT in which closure does not occur. I show that the onset of late-time impulse flow is controlled by injection volume and formation, wellbore, and fracture properties, not the duration of injection (as implied by the definition of superposition time). log-log superposition-time derivative plots of DFITs exhibit a slope of 3/2 at intermediate time. However, pressure change never scales with a 3/2 power of time. One form of the G-function superficially resembles a superposition-time function constructed by summing constant-rate solutions with 3/2 power scaling. However, this is not a mathematically or physically valid interpretation. The 3/2 power arises from a spatial integration of the Carter leakoff solution. There is not a mathematical, physical, or practical justification for plotting DFIT pressure-time data in a way that creates a 3/2 slope. I conclude by providing a field example and practical recommendations for DFIT interpretation.

关键词： Superposition time log-log plot DFIT G-function

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Reply to "Comment on 'Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function' by A. Mignan" by S. Hainzl and A. Christophersen

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SEISMOlogICAL RESEARCH LETTERS 2016年第5期87卷 1134-1137页

作者： Mignan, Arnaud Swiss Fed Inst Technol Inst Geophys NO H66Sonneggstr 5 CH-8092 Zurich Switzerland

Hainzl and Christophersen (2016) recently suggested that the approach used by Mignan (2016) to visualize the Omori law in a log-log plot using the complementary cumulative density function (CCDF) was misleading. They showed that one should use a temporal upper bound t(max) = max(t(obs)), with t(obs) the occurrence time of a set of observed aftershocks, instead of t(max) -> infinity, as used by Mignan. They found that both the Omori law and stretched exponential function (SEF) are undistinguishable on a corrected CCDF log-log plot but that the Omori law is preferred based on maximum-likelihood estimations (MLEs). I first clarify the rationale for using the CCDF log-log plot for comparison of the SEF with a power law but verify that the comparison indeed becomes misleading when the data sequence is incomplete. However, I then show that MLEs obtained for the Omori law are ambiguous, because this function may win versus the SEF, even if the true relaxation process follows an SEF (subject to early data incompleteness). This debate highlights that statistics alone may not be enough to choose between the stretched exponential and the Omori law and that physical considerations should be added to the discussion.

关键词： CENTRAL CLOUDY DYSTROPHY OF FRANCOIS EXPONENTIAL FUNCTIONS log-log plot logarithmic scale IL17RD gene data sequence

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Reduced Dataset for Allan Variance Analysis

Reduced Dataset for Allan Variance Analysis

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IEEE International Conference on Recent Trends in Electronics, Information & Communication Technology

作者： Himsikha Hazarika Bagubali A Alex Noel Joseph Raj Vipan Kumar Vinod Karar Shashi Poddar Department of embedded technology VIT University Vellore India Dept. of Optical Devices & Systems CSIR-Central Scientific Instruments Organisation Chandigarh India

ISBN: (纸本)9781509037056

The stochastic errors present in the inertial sensors are most commonly analysed using Allan variance technique. The traditional Allan variance technique requires large time for computing noise parameters wherein the cluster size is varied in an arithmetic sequence. In this paper, redundant data removal and K-means clustering based techniques have been adopted to reduce the size of dataset. The reduced dataset decreases the overall time taken to compute the Allan variance. The least square fitting has been applied on the log-log plot of Allan deviation for obtaining the bias instability and angular random walk of an inertial sensor. The proposed scheme of redundancy removal is found to have a promising performance at the cost of reduced accuracy. However, clustering based data reducing is not found to work suitable. The dataset for noise characterization has been obtained from an ADIS16448 sensor suite by logging the raw data offline.

关键词： Allan variance Angular random walk Bias instability Stochastic errors Allan Variance Arithmetic Sequence Dataset Random walk log-log plot Noise parameters inertia sensor

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2-Octylaminopyridine assisted solvent extraction system for selective separation of palladium(II) ion-pair complex from synthetic mixtures and real samples

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ANALYTICAL METHODS 2015年第6期7卷 2497-2504页

作者： Suryavanshi, Vishal J. Pawar, Rupali R. Anuse, Mansing A. Mulik, Ganpatrao N. Shivaji Univ Dept Chem Analyt Chem Lab Kolhapur 416004 Maharashtra India Balwant Coll Dept Chem Vita 415311 Maharashtra India

Herein, 2-octylaminopyridine (2-OAP) amine is used in the liquid-liquid extraction of palladium(II) from aqueous salicylate media. It is found that pH does not affect the metal extraction process in the pH range of 1.8-2.4. Palladium(II) complex formation was verified by a log-log plot, whereas from the slope analysis, the ion pair complex stoichiometry 1 : 1 (metal : ligand) and 1 : 3 (metal : acid) was confirmed. The extraction system was optimized for the choice of diluent (xylene) as well as for the effective stripping agent (5.0 M ammonia). The selectivity for palladium(II) was found to be maintained when the organic phase (0.03 M 2-OAP in xylene) is used to extract the palladium(II). The extraction saturation capacity of palladium(II) was determined from 0.02 M sodium salicylate solution with 10 mL 0.03 M 2-OAP, and its experimental value exceeds 2.5 mg under the experimental conditions. Stripping of palladium(II) from the loaded organic phase was performed with 5.0 M ammonia. The extraction with varying parameters like equilibrium time, temperature, diluents, concentration of extractant, loading capacity, regeneration power and stability of extraction was also studied. The extraction behavior of associated metals, namely, noble metals and base metals, has been investigated, and the conditions for binary separation involving these metal ions have been optimized. Based on the extraction protocol, palladium(II) has been recovered from real samples.

关键词： Xylene metal ions Solvent extraction variable parameter log-log plot mining system BASE METAL Liquid-Liquid Extraction experimental value MOMENT MAGNITUDE

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Synthesis and Dichromate Extraction Study of Calixarene Derivative

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POLYCYCLIC AROMATIC COMPOUNDS 2014年第2期34卷 103-114页

作者： Bhatti, Asif Ali Memon, Shahabuddin Memon, Najma Univ Sindh Natl Ctr Excellence Analyt Chem Jamshoro 76080 Pakistan

The present article demonstrates synthesis of 25,27-bis-N-(N,N-diethyl-2-aminoethyl-amidomethoxy)-26,28-dihydroxycalix[4]arene (4), its characterization and extraction ability toward the oxoanions such as HCr2O7-/Cr2O72 -. The extraction efficiency was evaluated by using Pederson extraction method and data was analyzed through log-log plot analysis. The extraction was performed at varying pHs, i.e., pH 2-7. The analysis of results reveals that ligand 4 is a good extractant for dichromate at pH 3. The stoichiometric calculations obtained from Job's plot and log-log plot analyses reveals that it forms ligand:anion complex in 1:2 ratio.

关键词： log-log plot calix[4]arene extraction chromium characterization

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Determining the Strength of Coarse-Grained AlON and Spinel

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JOURNAL OF THE AMERICAN CERAMIC SOCIETY 2014年第2期97卷 592-600页

作者： Swab, Jeffrey J. Pavlacka, Robert Gilde, Gary Kilczewski, Steve Wright, Jared Harris, Donovan US Army Res Lab Mat & Mfg Sci Div Ceram & Transparent Mat Branch Aberdeen MD 21005 USA Oak Ridge Inst Sci & Educ Oak Ridge TN USA Data Matrix Solut Herndon VA USA Bowhead Sci & Technol LLC Belcamp MD USA

The strength of two coarse-grained (grain size>200m) cubic ceramics, a magnesium aluminate spinel (MgAl2O4) and an AlON , along with a fine-grained (1.5m) MgAl2O4, was determined by conducting a series of four-point and equibiaxial flexure tests on specimens of different sizes. Weibull strength size scaling revealed a linear relationship on a log-log plot between average flexure strength and effective specimen area for the fine-grained spinel, but a nonlinear relationship for both coarse-grained materials. Initial fractography showed that each material had a single flaw population limiting the strength over the entire specimen size range, which does not account for the nonlinear size scaling relationship in the two coarse-grained materials. However, further fractography revealed that in both materials there was an initial flaw and a critical flaw. The former appears to be machining/polishing damage that started the fracture process while the latter was a cleaved grain in AlON or a cracked grain boundary in the HP/HIP spinel that lead to fracture of the specimen. The difference between the initial and critical flaw size coupled with a detailed analysis of the strength as a function of test specimen thickness accounted for the nonlinear strength size scaling relationship. As a result, strength values obtained using thin test specimens can lead to an erroneous strength prediction for large components made of these ceramics. The implication of these findings is that strength tests must be conducted using appropriately thick specimens to obtain a representative strength value. If appropriately thick specimens cannot be tested, then fractography must be conducted to determine the flaw size. If the flaw size is sufficiently large, compared with the specimen thickness, then the strength must be adjusted according to a stress field correction factor to obtain a more accurate strength value.

关键词： LARGE-COMPONENT spinel Fractography critical defect size log-log plot coarse-grained materials flexural strength Strength tests flaw size

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