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Soil water characteristic curve (SWCC) has been an important role in hydraulic engineering, civil engineer and petroleum engineering, etc. Most of SWCC models neglected the film flow in the dry state, so that they cannot accurately describe the SWCC over entire range of water content. In this work, an alternative fractal model is proposed to predict the SWCC over entire range of water content by combining Campbell and Shiozawa model and Tao model. The proposed model can well predict twelve sets of experimental data, and its parameters, including the fractal dimension, the saturated volumetric water content, the matric suction at oven-dry condition, and the air-entry value, accord with theoretical value. The results show that there is a strong linear relationship between volumetric water content and matrix suction in log-log scale for different fractal pore-size distribution of soils. In addition, good agreement is obtained between the experimental data and the model predictions in all of the cases. Cited as : Jin, T., Cai, X., Chen, Y., Jiang, S., Wei, W. A fractal-based model for soil water characteristic curve over entire range of water content. Capillarity, 2019, 2(4): 66-75, doi: 10.26804/capi.2019.04.02

Large-scale hydraulic fracturing of horizontal wells is a key technology for shale gas extraction, during which tens of thousands of cubic meters of water could be injected into the formations for hydraulic fracturing. Water is imbibed into the shale pores under the action of capillary force and osmotic pressure. The relevant research has intensively been conducted worldwide on imbibition in porous media, including fractal modeling of the phenomenon. One significant drawback of the imbibition models proposed is their focus solely on the capillary force and pore fractal characteristics, ignoring the self-imbibition of clay pores under osmotic pressure. In this paper, the pore characteristics of shale are analyzed by field emission scanning electron microscopy (FSEM) and fractal analysis. A multi-pore fractal model is then developed for imbibition in shales, honoring capillary forces, osmotic pressure and pore structure. The results show that the morphology of organic and brittle pores tends to resemble an ellipse, while that of clay pores is similar to parallel fractures. It is observed that the imbibition capacity is more influenced by the fractal dimension of pore tortuosity rather than pore volume. In this regard, the effect of the maximum short axis of an elliptic pore on imbibition is more important that of the minimum short axis. The imbibition ability has a direct relationship with fluid viscosity. But there is a critical viscosity value, beyond which the imbibed volume flattens out. In the case of multiple-pore imbibition, clay pores demonstrate the fastest, longest and largest liquid imbibition process.

We present an experimental study aiming at the identification of the hydraulic conductivity in an aquifer which was packed according to four different configurations. The conductivity was estimated by means of slug tests, whereas the other parameters were determined by the grain size analysis. Prior to the fractal we considered the dependence of the conductivity upon the porosity through a power (scaling) law which was found in a very good agreement within the range from the laboratory to the meso-scale. The dependence of the conductivity through the porosity was investigated by identifying the proper fractal model. Results obtained provide valuable indications about the behavior, among the others, of the tortuosity, a parameter playing a crucial role in the dispersion phenomena taking place in the aquifers.

In this paper, we present the upgraded version of RockGIS, a stochastic program for the numerical simulation of rockfalls and their fragmentation, based on a fractal model. The code has been improved to account for a range of fragmentation scenarios, depending on the impact conditions. In the simulation, the parameters of the fractal fragmentation model that define the sizes of the generated fragments were computed at each impact according to the kinematic conditions. The performance of the upgraded code was verified and validated by real-scale rockfall tests performed in a quarry. The tests consisted of the release of 21 limestone blocks. For each release, the size and spatial distribution of the fragments generated by the impacts were measured by hand and from orthophotos taken via drone flights. The trajectories of the blocks and the resulting fragments were simulated with the code and calibrated with both the volume distribution and the runout distances of the fragments. Finally, as all the relevant rockfall parameters involved were affected by strong uncertainty and spatial variability, a parametric analysis was carried out and is discussed. This work has been carried out with the support of the Spanish Ministry of Economy and Competitiveness thanks to a fellowship to the first author (BES-2014-069795) and in the framework of the research project RockModels (Ref. BIA2016-75668-P, AEI/FEDER, UE). The collaboration of Canteras Hermanos Foj (Barcelona metropolitan area, Spain) is gratefully acknowledged.

The modern methods of materials (including cement matrix materials) design and testing impose the application of an approach appropriate to materials engineering. A quantitative description of the association between the properties of these materials and their structure is a necessity. What remains the scientific aim, however, is the clarification and description of the occurring phenomena by means of models mapping their actual behavior in the closest way possible. The article presents a cracking fractal model based on tests on the morphology of concrete fracture surfaces. The recorded fractal nature of the cracking of cement matrix materials enabled fractal geometry in the model development to be applied. Owing to the application of statistical analysis, together with an extensive base of data on the profile lines separated out of the real fracture surfaces of concrete, it was possible to develop a cracking fractal model. Not only does this model satisfy the condition of the equality of the fractal dimension of the real and model profile lines, it also offers the possibility of introducing an order to the apparently chaotic phenomena, such as the cracking process. An advantage and novelty of the model is that unlike the other authors’ proposals, there is a possibility of reaching an infinitely large number of solutions for model profile lines, which approximates the model to the real-life scenario. The results of fractal tests were supplemented with strength measurements, identifying concrete’s compressive and fracture toughness (determining the critical stress intensity factor KIcS). A connection between the fractal dimension and the investigated properties of concrete was demonstrated. A higher fractal dimension was observed in the profile lines separated out of the fracture surfaces of concretes of higher water−cement ratio. The advantages of the model include the simplicity and applicability in model studies on other materials of the cement matrix.

Skin, separating the vital organs of a human body, is a desirable route for drug delivery. However, the intact skin is normally permeable only for drug molecules with a low molecular weight. The stratum corneum (SC), being the outermost layer of the skin and the epidermis being the second – more permeable – layer of the skin, play an essential function in transdermal drug delivery. Physical and chemical methods of skin poration are used to enhance transdermal drug delivery. Each poration leads to an irregular system of pores which are connected with a system of micro-capillaries passing through the epidermis. Both the systems by their irregularity form a fractal porous matrix. Drugs administrated by this matrix can be either suspensions and solutions or creams and gels, therefore they have to be modelled as non-Newtonian fluids. To analyse the fluid flow through the porous matrix the model of the epidermis is assumed as gobbet-andmortar with the tortuous mortar of variable thickness and after transition from the mortar to the tube one considered classical and fractal capillary flows of selected non-Newtonian fluids. Fractal expressions for the flow rate, velocity and permeability of fluids flow in a porous matrix are derived based on the fractal properties of the epidermis and capillary model. Each parameter in the proposed expressions does not contain any empirical constant and has a clear physical meaning and the proposed fractal models relate the flow properties of considered fluids with the structural parameters of the epidermis as a porous medium. The presented analytical expressions will help understand some of the physical principles of transdermal drug delivery.

The first fractal model for the loading&ndash, unloading process between two cylindrical surfaces with friction is presented. The nonlinear relation between the real contact area and the contact load in different deformation stages are deduced for a load&ndash, unload cycle. The impacts of parameters in the model are discussed. The numerical results show that for a given dimensionless contact load, the dimensionless real contact area of the loading&ndash, unloading process of cylindrical contact surface with friction, as well as the differences of the dimensionless real contact area between the loading and unloading processes, increase with the increase of the loading interference and fractal dimension, decrease of the profile scaling parameter and curvature radius, or the substitution of external contact for internal contact.

Characterizing changes in the soil particle-size distributions (PSD) are a major issue in environmental research because it has a great impact on soil properties, soil management, and desertification. To date, the use of soil volume fractal dimension (D) is a feasible approach to describe PSD, and its calculation is mainly dependent on subdivisions of clay, silt, sand fractions as well as different soil particle-size classification (PSC) systems. But few studies have developed appropriate research works on how PSC systems affect the calculations of D. Therefore, in this study, topsoil (0&ndash, 5 cm) across nine forest density gradients of Pinus sylvestris var. mongolica plantations (MPPs) ranging from 900&ndash, 2700 trees ha&ndash, 1 were selected in the Mu Us sandy land, China. The D of soil was calculated by measuring soil PSD through fractal model and laser diffraction technique. The experimental results showed that: (1) The predominant PSD was distributed within the sand classification followed by clay and silt particle contents, which were far less prevalent in the study area. The general order of D values (Ds) was USDA (1993) &gt, ISO14688 (2002) &gt, ISSS (1929) &gt, Katschinski (1957) &gt, China (1987) &gt, Blott &amp, Pye (2012) PSC systems. (2) Ds were significantly positively related to the contents of clay and silt, and Ds were significantly negatively to the sand content. Ds were susceptible to the MPPs establishment and forest densities. (3) Ds of six PSC systems were significantly positive correlated, which indicated that they not only have difference, but also have close connection. (4) According to the fractal model and descriptions of soil fractions under different PSC systems, refining scales of clay and sand fractions could increase Ds, while the refining scale of silt fraction could decrease Ds. From the conclusions above, it is highly recommended that USDA (1993) and Blott &amp, Pye (2012) PSC systems be used as reliable and practical PSC systems for describing and calculating D of soil PSD.

In this paper, a simple and efficient fractal-based approach is presented for capturing the effects of initial void ratio on the soil-water characteristic curve (SWCC) in a deformable unsaturated soil. In terms of testing results, the SWCCs (expressed by gravimetric water content) of the unsaturated soils at different initial void ratios were found to be mainly controlled by the air-entry value (&Psi, a), while the fractal dimension (D) could be assumed to be constant. As a result, in contrast to the complexity of existing models, a simple and efficient model with only two parameters (i.e., D and &Psi, a) was established for predicting the SWCC considering the effects of initial void ratio. The procedure for determining the model parameters with clear physical meaning were then elaborated. The applicability and accuracy of the proposed model were well demonstrated by comparing its predictions with four sets of independent experimental data from the tests conducted in current work, as well as the literature on a wide range of soils, including Wuhan Clay, Hefei and Guangxi expansive soil, Saskatchewan silt, and loess. Good agreements were obtained between the experimental data and the model predictions in all of the cases considered.

The article examined the fractal model of organizational diagnosis of pathologies in project management development. The proposed fractal model based on the competency approach to project management and allows evaluating the pathology of the project-oriented organizations.

Understanding the changes in multiphase flow parameters caused by mineral dissolution-precipitation is required for multiple applications ranging from geological storage of CO2, enhanced geothermal energy production or ground water pollution. We present a physically-based theoretical model for describing the temporal evolution of porosity, saturated and relative permeabilities, retention curve and diffusion coefficient during rock dissolution by reactive fluids. The derivation of the model is based on the assumption that the pore structure of the rock can be represented by an ensemble of capillary tubes with fractal tortuosity and cumulative pore size distribution. Therefore, the model depends only on the minimum and maximum pore radii, the size of the representative elementary volume and the fractal dimensions of pore size and tortuosity, but do not need any other fitting parameters. Using this fractal description and known physical properties, we obtain analytical expressions for the hydrodynamic properties required by continuum (i.e., Darcy scale) multiphase flow models. Further, assuming periodic fluctuations in the radius of the pores, it is also possible to represent constrictivity and hysteresis. Finally, assuming a constant rate dissolution reaction it is possible to derive closed-form analytical expressions for the time evolution of porosity, retention curve, saturated and relative permeabilities and diffusion coefficient. Fil: Guarracino, Luis. Universidad Nacional de La Plata. Facultad de Ciencias Astronómicas y Geofísicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina Fil: Rötting, Tobias. Consejo Superior de Investigaciones Científicas; España. Instituto de Diagnóstico Ambiental y Estudios del Agua; España. Universidad Politécnica de Catalunya; España Fil: Carrera, Jesús. Consejo Superior de Investigaciones Científicas; España. Instituto de Diagnóstico Ambiental y Estudios del Agua; España

In cold regions, hydraulic conductivity is a critical parameter for determining the water flow in frozen soil. Previous studies have shown that hydraulic conductivity hinges on the pore structure, which is often depicted as the pore size and porosity. However, these two parameters do not sufficiently represent the pore structure. To enhance the characterization ability of the pore structure, this study introduced fractal theory to investigate the influence of pore structure on hydraulic conductivity. In this study, the pores were conceptualized as a bundle of tortuous capillaries with different radii and the cumulative pore size distribution of the capillaries was considered to satisfy the fractal law. Using the Hagen-Poiseuille equation, a fractal capillary bundle model of hydraulic conductivity for saturated frozen soil was developed. The model validity was evaluated using experimental data and by comparison with previous models. The results showed that the model performed well for frozen soil. The model showed that hydraulic conductivity was related to the maximum pore size, pore size dimension, porosity and tortuosity. Of all these parameters, pore size played a key role in affecting hydraulic conductivity. The pore size dimension was found to decrease linearly with temperature, the maximum pore size decreased with temperature and the tortuosity increased with temperature. The model could be used to predict the hydraulic conductivity of frozen soil, revealing the mechanism of change in hydraulic conductivity with temperature. In addition, the pore size distribution was approximately estimated using the soil freezing curve, making this method could be an alternative to the mercury intrusion test, which has difficult maneuverability and high costs. Darcy&rsquo, s law is valid in saturated frozen silt, clayed silt and clay, but may not be valid in saturated frozen sand and unsaturated frozen soil.

The metabolic theory of ecology (MTE) predicts the effects of body size and temperature on metabolism through considerations of vascular distribution networks and biochemical kinetics. MTE has also been extended to characterise processes from cellular to global levels. MTE has generated both enthusiasm and controversy across a broad range of research areas. However, most efforts that claim to validate or invalidate MTE have focused on testing predictions. We argue that critical evaluation of MTE also requires strong tests of both its theoretical foundations and simplifying assumptions. To this end, we synthesise available information and find that MTE's original derivations require additional assumptions to obtain the full scope of attendant predictions. Moreover, although some of MTE's simplifying assumptions are well supported by data, others are inconsistent with empirical tests and even more remain untested. Further, although many predictions are empirically supported on average, work remains to explain the often large variability in data. We suggest that greater effort be focused on evaluating MTE's underlying theory and simplifying assumptions to help delineate the scope of MTE, generate new theory and shed light on fundamental aspects of biological form and function.

The metabolic theory of ecology (MTE) predicts the effects of body size and temperature on metabolism through considerations of vascular distribution networks and biochemical kinetics. MTE has also been extended to characterise processes from cellular to global levels. MTE has generated both enthusiasm and controversy across a broad range of research areas. However, most efforts that claim to validate or invalidate MTE have focused on testing predictions. We argue that critical evaluation of MTE also requires strong tests of both its theoretical foundations and simplifying assumptions. To this end, we synthesise available information and find that MTE's original derivations require additional assumptions to obtain the full scope of attendant predictions. Moreover, although some of MTE's simplifying assumptions are well supported by data, others are inconsistent with empirical tests and even more remain untested. Further, although many predictions are empirically supported on average, work remains to explain the often large variability in data. We suggest that greater effort be focused on evaluating MTE's underlying theory and simplifying assumptions to help delineate the scope of MTE, generate new theory and shed light on fundamental aspects of biological form and function.

The soil water retention curve (SWRC) is a fundamental hydraulic property majorly used to study flow transport in soils and calculate plant-available water. Since, direct measurement of SWRC is time-consuming and expensive, different models have been developed to estimate SWRC. In this study, a fractal-based model was developed to predict SWRC. A wide range of soil textures (130 soil samples) was used to determine the fractal dimension of SWRC (D SWRC ). Moreover, the SWRC pedotransfer functions were established based on easily available soil properties such as particle size distribution and bulk density by applying multiple linear regression analysis. The measured D SWRC for 110 soil samples was considered for function parameterization and the remaining was used for model validation. The results illustrated that the D SWRC linearly correlates with clay and silt contents and soil bulk density (r 2 = 0.909). The SWRC can, therefore, be easily and concisely estimated by the proposed fractal-based functions. Keywords: Fractal model; Pedotransfer functions; Regression analysis; Soil water retention curve

This study presents an intermingled fractal model capable of simulating the porous microstructure of natural stones used in historical buildings. The developed model is aimed at predicting, by an analytical approach, the sorptivity of these materials. To verify the actual ability of the proposed method to predict stone sorptivity, in this study the intermingled fractal units model was applied to eight types of natural stones. The results are in very good agreement with sorptivity values obtained by experimental tests on the investigated stones. Compared to other analytical formulas proposed in the literature for predicting the sorptivity of porous materials, the newly proposed method better matches with the experimental results, and this can be attributed to that fact it takes into consideration the whole pore size distribution of the analyzed material, rather than the average pore radius only. Thanks to the proposed method, the sorptivity of natural stones used in historical buildings can be calculated from their pore size distribution determined by MIP, which, contrary to standard tests for sorptivity determination, only requires small and irregular samples.

A matemática fractal tem sido utilizada para caracterizar o transporte de água e solutos em meios porosos e também para simular características físicas e geométricas de meios porosos. O objetivo deste trabalho foi avaliar a correlação entre os parâmetros de infiltração de água sortividade e expoente de tempo (n) e os parâmetros dimensão fractal (D) e expoente de Hurst (H). Para isso, dez colunas horizontais foram simuladas em computador, sendo preenchidas com material de textura franca ou argilosa, puros ou em combinações de camadas alternadas dos dois materiais, seguindo a distribuição de um Conjunto de Cantor determinístico com dimensão fractal 0,63. As simulações de movimento horizontal de água foram realizadas no modelo numérico Hydrus 2D. Os parâmetros sortividade (S) e expoente de tempo (n) da equação de Philip foram estimados para cada simulação, utilizando o módulo de regressão não linear do software SAS®. A sortividade aumentou com o aumento de material de textura franca nas colunas, o que foi atribuído à dependência do parâmetro S no raio capilar. O parâmetro expoente de tempo estimado por regressão não linear mostrou valor menor que o valor empírico de 0,5 sugerido na literatura. A dimensão fractal estimada utilizando o expoente de Hurst foi 17,5 % menor do que a dimensão fractal do Conjunto de Cantor utilizado para gerar as colunas. Fractal mathematics has been used to characterize water and solute transport in porous media and also to characterize and simulate porous media properties. The objective of this study was to evaluate the correlation between the soil infiltration parameters sorptivity (S) and time exponent (n) and the parameters dimension (D) and the Hurst exponent (H). For this purpose, ten horizontal columns with pure (either clay or loam) and heterogeneous porous media (clay and loam distributed in layers in the column) were simulated following the distribution of a deterministic Cantor Bar with fractal dimension H" 0.63. Horizontal water infiltration experiments were then simulated using Hydrus 2D software. The sorptivity (S) and time exponent (n) parameters of the Philip equation were estimated for each simulation, using the nonlinear regression procedure of the statistical software package SAS®. Sorptivity increased in the columns with the loam content, which was attributed to the relation of S with the capillary radius. The time exponent estimated by nonlinear regression was found to be less than the traditional value of 0.5. The fractal dimension estimated from the Hurst exponent was 17.5 % lower than the fractal dimension of the Cantor Bar used to generate the columns.

Several studies have shown that fractal geometry is a tool that can replicate and investigate the nature of the materials and their physical properties. The Sierpinski carpet is often utilized to simulate porous microstructures. By using this geometric figure it is possible to study the influence of pore size distribution on deterministic fractal porous media. The determination of the thermal conductivity can be carried out using the electrical analogy. So, microstructure models have been converted in electrical fractal patterns. This fractal procedure is characterized by a close relationship with the actual microstructure and prevent papers has been validate it with experimental data in a series of former papers. In this work it is possible to show how thermal conductivity changes in relation to pore size distribution and geometric microstructure parameters.