Mass Fractal Dimension Research Articles

Glass-like Relaxation Dynamics during the Disorder-to-Order Transition of Viral Nucleocapsids.

Nucleocapsid self-assembly is an essential yet elusive step in virus replication. Using time-resolved small-angle X-ray scattering on a model icosahedral ssRNA virus, we reveal a previously unreported kinetic pathway. Initially, RNA-bound capsid subunits rapidly accumulate beyond the stoichiometry of native virions. This is followed by a disorder-to-order transition characterized by glass-like relaxation dynamics and the release of excess subunits. Our molecular dynamics simulations, employing a coarse-grained elastic model, confirm the physical feasibility of self-ordering accompanied by subunit release. The relaxation can be modeled by an exponential integral decay on the mean squared radius of gyration, with relaxation times varying within the second range depending on RNA type and subunit concentration. A nanogel model suggests that the initially disordered nucleoprotein complexes quickly reach an equilibrium size, while their mass fractal dimension continues to evolve. Understanding virus self-assembly is not only crucial for combating viral infections, but also for designing synthetic virus-inspired nanocages for drug delivery applications.

Structural features of rice starch-protein system: Influence of retrogradation time and quick-freezing temperature

This work investigated the effect of retrogradation time (0 h, 2 h, 4 h, 6 h, 8 h) and freezing temperature (−20 °C, −32 °C, −80 °C) on the muti-scale structures of the rice starch-protein system of quick-frozen wet rice noodles. The Relative crystallinity and porosity of the rice starch-protein system increased with increasing retrogradation time. However, while longer retrogradation does lead to an improvement in relative crystallinity, it also results in significant damage to the microstructure. When the retrogradation time was 6 h, the microstructure of the rice starch-protein system was less damaged and the quality was better. The mass fractal dimension and relative crystallinity of the rice starch-protein system exhibited an increase as the freezing temperature was decreased from −20 to −80 °C. Additionally, the retrogradation degree of starch decreased, the size of ice crystals decreased, and the disruption of microforms was reduced. The muti-scale structures of the rice starch-protein systems were similar when quick-frozen at temperatures of −32 and −80 °C. Therefore, the optimal treatment method for practical production is to quick-freeze at −32 °C and age for 6 h to obtain high-quality quick-frozen wet rice noodles.

Physicochemical quality improvement of dried rice noodles by direct heat-moisture treatment during the drying process

In view of the advantage of heat-moisture treatment (HMT) in modifying the properties of starchy materials, the purpose of this study was to explore the effects of directly applying HMT during the drying process on the properties of dried rice noodles. As attained to prescribed moisture content (20%–35%), the rice noodles were subjected to HMT at 110 °C for 2 h. By HMT, the dried rice noodles exhibited greater structural orderliness and thermal resistance. The relative crystallinity increased from 8.73% to 11.50% and the short-range order (the absorbance ratio of 1047 cm−1/1022 cm−1) increased from 0.57 to 0.72. On the contrary, the lamellar thickness, the radius of gyration, and the mass fractal dimension and pasting viscosities were lowered. Then, HMT induced dried rice noodles with restricted starch leaching during rehydration and reduced cooking loss. HMT increased microstructure homogeneity and texture properties of the cooked rice noodles. Notably, the heat-moisture treated rice noodles at 20% moisture content showed a significant higher cooking yield, about threefold of extensibility, and 1.68 folds of chewiness than the control. Our findings, if generally applicable to the processing of rice noodles, could offer a streamlined approach to increasing the quality of dried rice noodles.

Effects of Stir-Frying and Heat-Moisture Treatment on the Physicochemical Quality of Glutinous Rice Flour for Making Taopian, a Traditional Chinese Pastry.

Taopian is a traditional Chinese pastry made from cooked glutinous rice flour. The effects of heat-moisture treatment (110 °C, 4 h; moisture contents 12-36%, w/w) on the preparation of cooked glutinous rice flour and taopian made from it were compared with the traditional method of stir-frying (180 °C, 30 s). The color of heat-moisture-treated (HMT) flours was darker. HMT flours exhibited a larger mean particle size (89.5-124 μm) and a greater relative crystallinity of starch (23.08-42.92%) and mass fractal dimension (1.77-2.28). The flours exhibited water activity in the range of 0.589-0.631. Although the oil-binding capacity of HMT flours was largely comparable to that of stir-fried flours, HMT flours exhibited a lower water absorption index. Accordingly, the taopian produced with HMT flours exhibited a lower brightness, accompanied by a stronger reddening and yellowing. In addition, more firmly bound water was observed in the taopian produced with HMT flour. The taopian made with HMT flour with a moisture content of 24% exhibited moderate hardness, adhesiveness and cohesiveness and received the highest score for overall acceptability (6.80). These results may be helpful to improve the quality of taopian by applying heat-moisture treatment in the preparation of cooked glutinous rice flour.

Analysis on Fractal-Permeability Characteristics of Construction Waste Filling Body in Gob

The mine application of construction waste permeability filling body is an effective way to avoid the collapse of overlying rock strata caused by coal mining, to solve the questions of surface resource occupation and pollute the ecological environment, and to promote regional hydrogeological balance. Based on the infiltration test on twelve groups of non-cemented and cemented samples with different particle size ratio, the variation characteristics of permeability coefficient of non-cemented and cemented construction waste samples with different fine content and cement were analyzed. The porosity of non-cemented samples and cemented ones was obtained, the relationships of fine content and porosity, permeability coefficient were analyzed, the relationships of fractal dimension and porosity, permeability coefficient on non-cemented samples and cemented ones were discussed. The results showed that fine content was negatively correlated to the permeability coefficient of non-cemented samples and non-cemented ones. The porosity decreased with fine content increasing, the porosity of non-cemented samples ranged from 5.007~7.029%, and the one of cemented samples was 3.230~6.508%. Fractal dimension of non-cemented samples and cemented ones were fine aggregate> medium coarse aggregate> coarse aggregate, and the fractal dimension of non-cemented samples was less than the cemented ones, the porosity and the mass fractal dimension could well represent the relationship between the permeability and fine content.

Rock fragmentation mechanism of PDC cutter from the insight of cutting chips

To meet the global energy needs and demands, obtaining oil and gas resources from deep strata plays an increasingly important role in energy supply. However, it is difficult to drill since the special properties of rocks in deep formation. To further improve the breaking efficiency of polycrystalline diamond compact (PDC) bit in deep hard rock, this paper conducted several groups of experiments on cutting granite with PDC cutters. By changing the cutting depth, rake angle and linear velocity, the relationship between the mass proportion, fractal dimension and roughness index of cuttings and the mechanical specific energy (MSE) is explored, and the following conclusions are obtained: When cutting depth HC exceeds 0.8 mm, massive chips with particle size r > 2 mm begin to form and the brittle failure begins to occur, and the probability of brittle failure will increase with the further increase of cutting depth. With the increase of rake angle, more energy will be used to form new surfaces of cutting chips, resulting in a decrease in the roughness index and an increase in MSE. More massive chips will be formed with the increasing linear velocity, but MSE only decrease slightly. The variation of the fractal MSE (FMSE) that based on the fractal dimension of cutting chips is extremely consistent with the MSE. The relevant conclusions in this paper can provide some reference for the structural design of PDC bit, so as to improve the drilling efficiency in deep hard rock.

Clay Tailings Flocculated in Seawater and Industrial Water: Analysis of Aggregates, Sedimentation, and Supernatant Quality.

High-molecular-weight anionic polyacrylamide was used to analyze the effect of kaolin on the structure of particle aggregates formed in freshwater and seawater. Batch flocculation experiments were performed to determine the size of the flocculated aggregates over time by using focused beam reflectance measurements. Sedimentation tests were performed to analyze the settling rate of the solid-liquid interface and the turbidity of the supernatant. Subsequently, a model that relates the hindered settling rate to the aggregate size was used to determine the mass fractal dimension (Df). Flocculation kinetics revealed that greater amounts of kaolin generated larger aggregates because of its lamellar morphology. The maximum size was between 10 and 20 s of flocculation under all conditions. However, the presence of kaolin reduced the settling rate. The fractal dimension decreased with the increase in the kaolin content, resulting in the formation of irregular and porous aggregates. By contrast, factors such as the flocculation time, water quality, and quartz size had limited influences on the fractal dimension. Seawater produced a clearer supernatant because of its higher ionic strength and precoagulation of particles. Notably, the harmful effect of clays in seawater was reduced.

Study on surface erosion threshold and erosion mass of bentonite applied in nuclear waste repository based on fractal structure characteristics

Bentonite exhibits good self-sealing performance and low permeability after hydration, and therefore is selected as a buffer material for high-level radioactive waste repositories, isolating the canisters containing nuclear waste from the external environment. The bentonite cushion directly contacts the surrounding rock and generates gels at the contact interface with fracture water. Surface erosion occurs once the shear stress exerted by the flowing water exceeds the erosion threshold of the bentonite gel, resulting in gel particle detachment from the soil–water interface and transport through fissure. The mass loss of the buffer by erosion will weaken its functionality as designed and directly endanger the long-term safety of the repository. The erosion behavior of bentonite is related to the structure of the gel formed by bentonite hydration. In this paper, the X-ray small-angle scattering experiments were used to prove that the gel structure has fractal characteristics and to obtain the mass fractal dimension. Based on the fractal structure of the bentonite gel, the erosion threshold that determines whether erosion will occur was quantified as a function of the solid volume fraction of the gel by taking the inter-particle attractive bond as the primary source of resistance. Furtherly, the cumulative erosion mass model that evaluates the extent of erosion was constructed to reveal the relationship between the cumulative erosion mass and cumulative water flow after the erosion has achieved stability. The proposed relations are verified by the experimental results from published literature and are of great significance for judging whether erosion will occur and evaluating the degree of erosion.

Analysis of TBM tunneling performance based on mass fractal dimension of rock chips

The size of the rock chips and its particle size distribution can fully reflect the rock-breaking efficiency of the TBM, but the index that can comprehensively characterize the particle size distribution of rock chips has not yet been found to quantitatively evaluate the TBM tunneling efficiency. Based on the fractal theory, this study uses the mass fractal dimension to quantitatively analyze the particle size distribution of rock chips, and explores its relationship with TBM tunneling performance index. Field sieving tests were carried out on rock chips from two tunnel projects. According to the sieving test results, the mass fractal dimension of the rock chips is calculated as well as the correlation between it and common TBM performance evaluation indexes, average tunneling thrust is discussed. The research results show that the mass distribution of TBM rock chips has significant fractal and self-similar characteristics, the higher the content of large-size rock chips, the lower the content of small-size rock chips, the smaller the fractal dimension value. Simultaneously, under the same surrounding rock conditions, the mass fractal dimension of rock chips is positively correlated with the logarithm of specific energy and negatively correlated with the logarithm of coarseness index, which indicates that the smaller the fractal dimension, the higher the efficiency of rock breaking. It is demonstrated that the interval of TBM thrust force corresponding to smaller specific energy, larger coarseness index and smaller fractal dimension is the same, which can be determined as the optimal thrust interval.

The Impact of Long-Term Mulched Drip Irrigation on Soil Particle Composition and Salinity in Arid Northwest China

The evaluation of soil particle composition and salt dynamics is essential for promoting the sustainable development of oasis agriculture in arid regions under long-term mulched drip irrigation (MDI). In this study, we employed the space-for-time substitution method to investigate the long-term effects of MDI on soil particle composition and salinity. Additionally, seven fields, with MDI durations ranging from 0 to 16 years, were selected to represent the primary successional sequence though time in Northwest China. Soil samples were collected from three soil depths (0–30 cm, 30–60 cm, and 60–100 cm) and then analyzed in the laboratory for soil particle composition and salt content. Our findings demonstrated that influenced by the depth of mechanical cultivation and the maximum wetting front depth, the long-term application of MDI significantly altered both the structure of soil layers and the composition of soil particles after 8 years. Soil sand content and soil salinity gradually decreased, whereas the content of soil silt and clay increased with increasing MDI duration throughout 0–100 cm soil depth. Furthermore, the rates of soil desalination stabilized after 10 years of MDI application, with desalination levels exceeding 90% in the 0–100 cm soil layer. Additionally, the soil mass fractal dimension (Dm) exhibited an upward trend across 0–100 cm soil depth. The changes in soil particle composition indirectly influenced the variations in Dm and salt content. Our study demonstrated that long-term application of MDI effectively mitigated soil salinity, changed soil structure, and ultimately enhanced soil quality and cotton yield.

Effects of wheat protein on hot-extrusion 3D-printing performance and the release behaviours of caffeic acid-loaded wheat starch

In this study, the effects of wheat protein (WP) on the hot-extrusion 3D-printing (HE-3DP) performance of wheat starch (WS) gels, as well as effects of such gels on the encapsulation of caffeic acid, were investigated for the first time. The HE-3DP results show that the addition of WP can reduce print-line width and improve printing accuracy and fidelity, and the best printing results were achieved when using gels with 10 % WP. The rheological results show that WP reduced the gels' linear viscoelastic region (LVR), yield stress (τy), flow stress (τf) and consistency factor (K) but increased their structural recovery rate, which facilitated smooth extrusion during 3D printing and, thus, improved printing accuracy. The analysis of X-ray diffraction and small-angle X-ray scattering indicates that adding WP to WS could increase the mass fractal dimension and lead to denser gel network structures. The results regarding release kinetics demonstrate that the maximum release of caffeic acid from gels decreased by 28 % with the addition of WP, indicating slow-release behaviour. This study provided valuable information about processing wheat products via 3D printing.

Preparation of porous-structured flat potato starch noodles with gelatin for shortening cooking time

Although dried flat starch noodles have been widely popular in Asian countries, the problem of long rehydration time needs to be solved. In this study, dried flat potato starch noodles with porous structure were constructed by adding gelatin to shorten the cooking time. Compared with pure flat potato starch noodles, the rehydration time of the dried flat potato starch noodles supplemented with 15% gelatin was reduced by 50.38%, from 11.75 ± 0.18 min to 5.83 ± 0.21 min. The results of scanning electron microscopy showed that the pore number of dried flat potato starch noodles increased with an increase in gelatin concentration, and the average diameter of pores ranged from 80.56 μm to 273.64 μm. The results of small-angle X-ray scattering showed that the mass fractal dimension value decreased from 1.769 (control noodles) to 1.308 (noodles with 20% gelatin), indicating that the starch aggregate structure was loosened due to the addition of gelatin. This loose structure was conducive to the diffusion of water in the cooking process of dried flat potato starch noodles. These findings indicated that the addition of gelatin, especially at the appropriate concentration, is effective in improving the cooking quality of noodles.

Development of composite film based on polyvinyl alcohol with binary nanoreinforcements for active packaging of fresh-cut apples

To develop a sustainable and environmentally-friendly preservation packaging for fresh-cut fruits and vegetables (FFV), this study synthesized the nanohybrid LDH@PCA using layered double hydroxide (LDH) loaded with p-coumaric acid (PCA). Subsequently, the nanohybrid and lignin nanoparticles (LPs) were integrated into a polyvinyl alcohol (PVA) matrix to prepare a binary nanocomposite packaging material named PVA-LPs-LDH@PCA. In this case, LDH@PCA and LPs were homogeneously dispersed inside PVA, relying on hydrogen bonding at 3 % (w/w) and 5 % (w/w), respectively, while electrostatic attraction also played a role. The reinforcement of the binary nanomaterials led to a significant increase in the radius of gyration and mass fractal dimension of the microregions within the PVA films. Additionally, the tensile strengths were 66 %, 32 %, and 8 % higher than those of the PVA, PVA-LPs, and PVA-LDH@PCA films, respectively. Moreover, the binary nanocomposite packaging exhibited enhanced water vapor, oxygen, UV barrier, and thermal stability, while synergistically improving its antioxidant activity compared to the aforementioned films. Remarkably, PCA in PVA-LPs-LDH@PCA demonstrated sustained release properties, consistent with Avrami and Arabolic diffusion models, for up to 55 h. Packaging of fresh-cut apples with PVA-LPs-LDH@PCA effectively maintained the texture, appearance, and nutritional quality. Transcriptomic and metabolomic analyses revealed that this packaging inhibited cell wall degradation, maintained cell membrane integrity, stimulated scavenging of reactive oxygen species, and inhibited polyphenol oxidase, attenuating the adverse effects of water loss stress on fresh-cut apples. These findings suggest that PVA binary nanocomposite films hold great potential as a sustainable solution for freshness preservation in FFV packaging.

Experimental study on the diffusion characteristics of nonaqueous reactive expansive polymers in sand and gravel media under dynamic water conditions

Grouting is an effective method to solve water inrush from sand and gravel. However, due to the self-expandability of nonaqueous reactive expandable polymers, the diffusion behaviour of polymers in injected media becomes more complex, which brings great uncertainty to the selection of grouting parameters in grouting practice. For this reason, a series of diffusion experiments of nonaqueous reactive expansive polymers in sand and gravel media under dynamic water conditions were carried out using a homemade experimental device. The effects of grouting parameters such as dynamic pump pressure, mass fractal dimension, and grouting volume on the effective diffusion length and water plugging rate of the slurry were revealed. The results show that (1) the diffusion behaviour of nonaqueous reactive expansive polymers in sand and gravel media can be summarized as edge dispersion, downstream diffusion, and expansion diffusion. (2) The diffusion consolidated bodies of nonaqueous reactive expansive polymers mostly exhibit a hemispherical or columnar shape with a wide top and a narrow bottom. (3) The greater the pressure of the moving water pump is, the smaller the effective diffusion length of the slurry. Under the same grouting volume, with increasing mass fractal dimension, the effective diffusion length of the slurry first increases and then decreases. (4) Under the conditions of high dynamic water pump pressure and low mass fractal dimension, increasing the grouting volume has a better water plugging effect.

Performance and Solution Structures of Side-Chain-Bridged Oligo (Ethylene Glycol) Polymer Photocatalysts for Enhanced Hydrogen Evolution under Natural Light Illumination.

Converting solar energy into hydrogen energy using conjugated polymers (CP) is a promising solution to the energy crisis. Improving water solubility plays one of the critical factors in enhancing the hydrogen evolution rate (HER) of CP photocatalysts. In this study, a novel concept of incorporating hydrophilic side chains to connect the backbones of CPs to improve their HER is proposed. This concept is realized through the polymerization of carbazole units bridged with octane, ethylene glycol, and penta-(ethylene glycol) to form three new side-chain-braided (SCB) CPs: PCz2S-OCt, PCz2S-EG, and PCz2S-PEG. Verified through transient absorption spectra, the enhanced capability of PCz2S-PEG for ultrafast electron transfer and reduced recombination effects has been demonstrated. Small- and wide-angle X-ray scattering (SAXS/WAXS) analyses reveal that these three SCB-CPs form cross-linking networks with different mass fractal dimensions (f) in aqueous solution. With the lowest f value of 2.64 and improved water/polymer interfaces, PCz2S-PEG demonstrates the best HER, reaching up to 126.9µmolh-1 in pure water-based photocatalytic solution. Moreover, PCz2S-PEG exhibits comparable performance in seawater-based photocatalytic solution under natural sunlight. In situ SAXS analysis further reveals nucleation-dominated generation of hydrogen nanoclusters with a size of ≈1.5nm in the HER of PCz2S-PEG under light illumination.

Multifractal approach for a biological porous media: Human dentin case.

A multifractal characterization to human dentin porosity is made. Micrographs of human dentin samples gotten from scanning electron microscopy (SEM) were studied in order to characterize porosity. We got the generalized dimensions (multifractal moments) for pore space and matrix skeleton on gray scale and binary images for samples of both gender. For the case, we found that superficial porosity is linked to mass fractal dimension in an approximately linear relationship as for binary images. In addition, probability density distribution (PDD) for pore diameters is found a normal PDD type. Other quantities of interest like mean, standard deviation, maximum and minimum pore diameters, voit ratios, and percentage of chemical composition are reported. RESEARCH HIGHLIGHTS: SEM sample, images and procedure for multifractal analysis are detailed. Micro patterns in grayscale are multifractal and in binary scale are monofractals. Superficial porosity is relate to mass fractal dimension approximately linearly. From the porosity values, voit ratios are determined. Pores diameters obey a normal PDD.

Monitoring effects of hydrodynamic cavitation pretreatment of sodium oleate on the aggregation of fine diaspore particles through small-angle laser scattering

Hydrodynamic cavitation (HC) enhanced fine particle aggregation could be largely due to the generation of tiny bubbles and their role in bridging particles. However, the lack of adequate characterizations of aggregates severally limits our further understanding of the associated aggregation behaviors. In this study, the aggregation of fine diaspore particles was comparatively investigated in sodium oleate (NaOl) solutions with and without HC pretreatment through the small-angle laser scattering (SALS) technique in a shear-induced aggregation (SIA) system. Results showed that HC pretreatment caused the formation of bulk nanobubbles (BNBs), which significantly modified the particle interactions and thereby modified the size and mass fractal dimension (Df) of aggregates under different SIA conditions. Although HC pretreatment did not noticeably alter the gradual change trend of aggregate size and structure characteristics under specific variables, BNBs bridging facilitated the aggregation process towards the diffusion-limited cluster aggregation model, resulting in the formation of larger but looser aggregates. This effect was more pronounced under relatively high NaOl concentrations. Apart from BNBs, the aggregation was also affected by cavitation bubbles formed during shear cavitation, which was more significant under high stirring intensity conditions (i.e., 1800 rpm) than the low stirring intensity conditions (i.e., 600 rpm).

Aggregate-Breaking Mechanism Response to Polyacrylamide Application of Purple Soils in Southwestern China Using Le Bissonnais Method

Polyacrylamide (PAM) is a water-soluble polymer with strong cohesiveness and a strong water absorption capacity, and it has been widely used to modify soil structural stability. However, little information is available on the impact of PAM application on the aggregate-breaking process of purple soils in hilly areas of southwestern China. Therefore, the current study aimed to examine the influence of PAM application on the aggregate stability of purple soil in terms of different breakdown mechanisms at different hillslope locations. Three disruptive tests employing the Le Bissonnais method (FW, fast-wetting sieving; SW, slow-wetting sieving; and WS, wet-stirring sieving) were used to determine the mean weight diameter (MWD), geometric mean diameter (GMD), and mass fractal dimension (D) of the soil aggregates, and soil erodibility factor (K) was calculated as an index of soil anti-erodibility. Overall, the major aggregate-breaking mechanism for purple soils was the following: SW (differential swelling) > WS (mechanical breakdown) > FW (slaking). The content of water-stable aggregates (>0.25 mm) obviously rose after PAM application, with the most significant influences shown under FW. A significant difference in MWD was observed between PAM application and without polyacrylamide application (CK) under WS (p < 0.05). However, there was a significant difference in GMD between PAM and CK (p < 0.05) under FW and SW. In comparison with CK, D value in PAM under FW and SW was significantly reduced, mainly at the slope locations of 0 and 20 m. A descending order of FW, WS, and SW was found on the basis of K value at different slope locations. These findings contribute to improved understanding of proper application of soil amendments to control soil and water loss in purple soils.

Effect of thermal modification on the stress relaxation behavior and microstructure of the cell wall

The stress relaxation behavior and cell wall microstructure of sugi were evaluated after thermal modification. Stress relaxation is observed and has a broad relaxation spectrum, implying various relaxation mechanisms. The relaxation was analyzed using a stretched exponential function, namely, the Kohlrausch–Williams–Watts (KWW) function, which contains two parameters. Moreover, the structure of the amorphous phase in the cell wall was examined by small-angle X-ray scattering (SAXS) analysis using the mass fractal dimension. The variation in the relaxation spectrum reduced, and the specific relaxation time increased by thermal modification at 220 °C. The mass fractal dimension in SAXS increased owing to modification, indicating that the structure of the cell wall includes some defects between cellulose microfibrils. The mass fractal dimension was related to the relaxation parameter of the KWW function. Considering the change in crystallinity, the amorphous phase in the cell wall decomposed and condensed by thermal decomposition, which caused a longer relaxation time. Thus, the KWW function may be used to evaluate the stress relaxation behavior of wood, and the mass fractal dimension in SAXS can indicate the amorphous structure in the cell wall.

Analytical solutions of fractal and anomalous diffusion models for pressure and rate transient analysis

In this study, we examine the modeling of the naturally fractured reservoirs based on the fractal, Metzler, and Raghavan anomalous diffusion models which are different from Darcy flux law. The governing equations of these models are presented for a vertical well producing at either a constant rate (CR) or constant bottomhole pressure (CBHP) production in an infinite-acting reservoir. New analytical and early- and late-time asymptotic approximate solutions were derived for the finite-wellbore problem for the Raghavan anomalous model. The approximate solutions identify the fractal and anomalous diffusion parameters that influence behaviors of the early- and late-time pressure and rate transient data. The method of Laplace transformation is used to find the solutions in dimensionless forms which are then inverted to the real-time domain using a numerical inversion algorithm. In addition, we compare the solutions for the fractal, Metzler anomalous, and Raghavan anomalous models and delineate the similarities and differences among these solutions. We show the effect of fractal parameters on diffusion models, the relationship between the fractal model, and Raghavan and Metzler anomalous models, in terms of dimensionless pressure and dimensionless pressure derivative responses in the reservoir. We show that the “dimensionless” pressure defined by Raghavan is not a dimensionless quantity unless the Euclidean dimension d=2 (or for a fractal structure with the mass fractal dimension df=2). To the best of our knowledge, in the literature, there exists no study providing a detailed review and comparing the applicability of each diffusion model on real field well-test data as well as providing new analytical solutions and early- and late-time asymptotic approximate solutions for the anomalous diffusion for both the CR and CBHP cases for the finite-wellbore problem. Hence, this study fills this gap. In addition, we apply the three diffusion models to an interference well test data from the Kizildere geothermal reservoir in Turkey by using the ensemble smoother with multiple data assimilation (ES-MDA) method. The results show that the fractal model in terms of the goodness of fit (root-mean-square error – RMSE) best matches the field test data.