Fractals

Spontaneous imbibition (SI) is a capillary-driven flow process, in which a wetting fluid moves into a porous medium displacing an existing non-wetting fluid. This process likely contributes to the loss of fracking fluids during hydraulic fracturing operations. It has also been proposed as a method for an enhanced recovery of hydrocarbons from fractured unconventional reservoirs. Numerous analytical and numerical approaches have been employed to model SI. Invariably, these idealize a fracture as the gap formed between parallel flat surfaces. In reality, rock fracture surfaces are rough over multiple scales, and this roughness will influence the contact angle and rate of fluid uptake. We derived an analytical model for the early-time SI behavior within a fracture bounded by parallel impermeable surfaces with fractal roughness assuming laminar flow. The model was tested by fitting it to experimental data for the SI of deionized water into air-filled rock fractures. Twenty cores from two rock types were investigated: a tight sandstone (Crossville) and a gas shale (Mancos). A simple Mode I longitudinal fracture was produced in each core by compressive loading between parallel flat plates using the Brazilian method. Half of the Mancos cores were fractured perpendicular to bedding, while the other half were fractured parallel to bedding. The two main parameters in the SI model are the mean separation distance between the fracture surfaces, $\bar{x}$, and the fracture surface fractal dimension $2\le D<3$. The $\bar{x}$ was estimated for each core by measuring the geometric mean fracture aperture width through image analysis of the top and bottom faces, while $D$ was estimated inversely by fitting the SI model to measurements of water uptake obtained using dynamic neutron radiography. The $\bar{x}$ values ranged from 45$\mu$m to 190$\mu$m, with a median of 93$\mu$m. The SI model fitted the height of uptake versus time data very well for all of the rock cores investigated; medians of the resulting root mean squared errors and coefficients of determination were 0.99mm and 0.963, respectively. Estimates of $D$ ranged from 2.04 to 2.45, with a median of 2.24. Statistically, all of the $D$ values were significantly greater than two, confirming the fractal nature of the fracture surfaces. Future research should focus on forward prediction through independent measurements of $D$ and extension of the existing SI model to late times (through the inclusion of gravity) and fractures with permeable surfaces.

The relationship between fracture aperture (maximum opening; $d_{\mathrm{\text{max}}}$) and fracture width ($w$) has been the subject of debate over the past several decades. An empirical power law has been commonly applied to relate these two parameters. Its exponent ($n$) is generally determined by fitting the power-law function to experimental observations measured at various scales. Invoking concepts from fractal geometry we theoretically show, as a first-order approximation, that the fracture aperture should be a linear function of its width, meaning that $n=1$. This finding is in agreement with the result of linear elastic fracture mechanics (LEFM) theory. We compare the model predictions with experimental observations available in the literature. This comparison generally supports a linear relationship between fracture aperture and fracture width, although there exists considerable scatter in the data. We also discuss the limitations of the proposed model, and its potential application to the prediction of flow and transport in fractures. Based on more than 170 experimental observations from the literature, we show that such a linear relationship, in combination with the cubic law, is able to scale flow rate with fracture aperture over $\sim$14 orders of magnitude for variations in flow rate and $\sim$5 orders of magnitude for variations in fracture width.

Foamed concrete possesses characteristics such as high strength-to-weight ratio and low density, and widely used to reduce dead loads on the structure and foundation, contributes to energy conservation, and lowers the labor cost during construction. In this paper, the objective is to propose prediction relation for the compressive strength of foamed concrete by fractal theory. A theoretical relation was derived for the compressive strength relating to porosity based on the fractal model for foamed concrete. The proposed relation stands out compared to empirical model since it employs easily measurable parameter, the fractal dimension of porous structure in foamed concrete. The fractal dimension of porous structure can be calculated from the scaling law of the compressive strength of foamed concrete. The fractal model for porous structure serves as a simple and effective tool for predicting the compressive strength of foamed concrete because of its ease in application. The prediction relation of the compressive strength developed in this paper is found to match well with the measured strength.

Most models for multiple fractured horizontal wells (MFHWs) in tight oil reservoirs (TORs) are based on classical simplified dual-porosity model that ignores the influence of imbibition, while the distribution of fracture system is heterogeneous, multi-scale and self-similar, which can be described by fractal dual-porosity model on fractal theory, and imbibition production is the important part of fracture system production. In this paper, a multi-linear fractal model (MFM) considering imbibition for MFHWs in TORs was established based on fractal theory and semi-analytical method. In this model, fractal theory was used to describe the heterogeneous, complex fracture network, and imbibition was considered by analogy of fluid crossflow law in fractured-porous dual media. And the approximate analytic solution was given by using the Laplace transformation and iteration method. The pressure responses in the domain of real time were obtained with Stehfest numerical inversion algorithms. The pressure transient and production rate were used to analyze, and sensitivity analysis of some related parameters were discussed. The results show that the fluid flow in MFHWs can be divided into nine main flow periods by analysis of type curves, and the fractal parameters of fracture system have great effect on the middle and later periods and imbibition influences the period of crossflow.

Lucaogou tight oil reservoir, located in the Junggar Basin, Northwest of China, is one of the typical tight oil reservoirs. Complex lithology leads to a wide pore size distribution (PSD), ranging from several nanometers to hundreds of micrometers. To better understand PSD and fractal features of Lucaogou tight oil reservoir, the experiment methods including scanning electron microscope (SEM), rate-controlled mercury injection (RMI) and pressure-controlled mercury injection (PMI) were performed on the six samples with different lithology. The results indicate that four types of pores exist in Lucaogou tight oil reservoir, including dissolution pores, clay dominated pores, microfractures and inter-granular pores. A combination of PMI and RMI was proposed to calculate the overall PSD of tight oil reservoirs, the overall pore radius of Lucaogou tight oil reservoir ranges from 3.6nm to 500$\mu$m. The fractal analysis was carried out based on the PMI data. Fractal dimension (Fd) values varied between 2.843 and 2.913 with a mean value of 2.88. Fd increases with a decrease of quartz content and an increase of clay mineral content. Samples from tight oil reservoirs with smaller average pore radius have stronger complexity of pore structure. Fractal dimension shows negative correlations with porosity and permeability. In addition, fractal characteristics of different tight reservoirs were compared and analyzed.

Fractal dimension is an important parameter in the evaluation of tight reservoirs. For an outcrop section of the Nenjiang formation in the Songliao Basin, China, the pore structure and pore fractal characteristics of shale parasequences were investigated using fractal theory. In addition, factors causing pore structure changes were analyzed using the results of low-temperature nitrogen adsorption and scanning electron microscope (SEM) experiments. Conducive to gas migration and secondary pores development such as dissolution, results showed that nanoscale pores dominated by fracture-like morphology and consequent good internal connectivity were observed in each pore size section within the target layer. Each parasequence is characterized by a sequential upward decrease of average pore size and an upward increase of total pore volume, with an increasing number of pores from 2nm to 50nm. Pores are isolated from each other, with poor connectivity and relatively complex composition of brittle minerals and clay minerals. Main components of the brittle minerals, quartz and feldspar, occur in 20–50% and higher clay mineral content ranging from 50% to 70%. In the parasequence cycle, clay mineral gradually decreases while the brittle mineral content increases. Fractal dimension is negatively correlated with clay mineral content and positively correlated with brittle mineral (quartz and feldspar) content. The fractal dimension calculated by the imaging method and the FHH method shows an upward increasing tendency in each of the parasequence cycles. This is as a result of different phenomena, varied sediment hydrodynamic forces leading to particle size differences and increased brittle minerals resulting in microcracks, therefore, the fractal dimension of the large pores (imaging method) increases upward in the parasequence. Simultaneously, with increased content and accompanied dissolution of brittle minerals causing an increase of small pores from base to top of the parasequence, the fractal dimension of the small pores (FHH method) grows.

To avoid the blindness of the overall de-noising method and retain useful low frequency signals that are not over processed, we proposed a novel audio magnetotelluric (AMT) signal-noise identification and separation method based on multifractal spectrum and matching pursuit. We extracted two sets of multifractal spectrum characteristic from AMT time-series data to analyze the singularity. We used a support vector machine approach to learn the multifractal spectrum characteristics in a sample’s library and generate a model of support vector machine to distinguish between sections with and without interference in the measured AMT data. The matching pursuit algorithm was used to separate only those sections identified as having interference. Experimental results showed that the proposed method can effectively identify interference in the EMTF mathematical model and measured AMT data. Sections without interference were accurately preserved and reconstructed AMT signals were close to the natural electromagnetic field. The resulting apparent resistivity-phase curve is more continuous and smooth, and effectively improves the quality of AMT data. Moreover, the proposed method provides more reliable AMT data for subsequent electromagnetic inversion.

The distribution of fractures is highly uncertain in naturally fractured reservoirs (NFRs) and may be predicted by using the assisted-history-matching (AHM) that calibrates the reservoir model according to some high-quality static data combined with dynamic production data. A general AHM approach for NFRs is to construct a discrete fracture network (DFN) model and estimate model parameters given the observations. However, the large number of fractures prediction required in the AHM process could pose a high-dimensional optimization problem. This difficulty is particularly challenging when the fractures form a complex multi-scale fracture network. We present in this paper an integrated AHM approach of NFRs to tackle these challenges. Two essential ingredients of the method are (1) a 2D fractal-DFN model constructed as the geological simulation model to describe the complex fracture network, and (2) a mixture of multi-scale parameters, built according to the fractal-DNF model, as an inversion parameter model to alleviate the high-dimensional optimization burden caused by complex fracture networks. A reservoir with a multi-scale fracture network is set up to test the performance of the proposed method. Numerical results demonstrate that by use of the proposed method, the fractures well recognized by assimilating production data.

Cretaceous Qingshankou Formation $({\mathrm{\text{K}}}_2\mathrm{\text{qn}})$ in the Songliao Basin, the largest petroliferous basin in East China, has favorable geological conditions for the shale oil reservoir. Although the first member of Qingshankou formation $({\mathrm{\text{K}}}_2{\mathrm{\text{qn}}}^1)$ has the maximum value of the organic matter content (total organic carbon (TOC)) comparing with other members, most of the layers are less than 1cm. This limits the application of surface seismic survey and logging methods. To explore “sweet spots” of the shale oil and to evaluate resource potential, it is of great significance to determine the thickness and distribution of silty beds/laminae in the thick shale system. In this study, the observed data of the outcrops, cores, and thin sections of ${\mathrm{\text{K}}}_2{\mathrm{\text{qn}}}^1$ are used to analyze the characteristics of reservoir space in silty laminae. Distribution model of silty beds/laminae was established by using fractal method. The results show that the silty beds/laminae have large particle size, poor sorting, and relatively developed reservoir space. The pore sizes in silty laminae vary greatly, and the diameters are mostly larger than 50nm. The connectivity between pores is good. Although the thickness of single silty bed/laminae mostly ranges from 1mm to 6mm, the average layers per meter (layers/m) of silty bed/laminae range from 2 to 64. This contributes the cumulative thickness 2.73–31.79% of the total thickness of the whole shale reservoir. According to the fractal analysis, the silty beds/laminae have the uniform fractal characteristics in the decimeter, centimeter, and millimeter levels. And the fractal dimension keeps the scale invariance. The layers and thickness of the thin silty laminae, calculated from Number–Size (NS) fractal model, are in good agreement with observations. This indicates that the fractal-based NS model is an effective method to estimate the parameters of the silty laminae, such as layer number, cumulative thickness, and the ratio of sand to formation. This method provides a reliable reference for the “sweet spot” exploration.

The one-dimensional modified Korteweg–de Vries equation defined on a Cantor set involving the local fractional derivative is investigated in this paper. With the aid of the fractal traveling-wave transformation technology, the nondifferentiable traveling-wave solutions for the problem are discussed in detail. The obtained results are accurate and efficient for describing the fractal water wave in mathematical physics.

To better understand the nanopore characteristics and their effects on methane adsorption capacity of shales, we performed fractal analysis of nine shale samples collected from the fifth member of Upper Triassic Xujiahe Formation in the Sichuan Basin, southwest China. ${\mathrm{\text{N}}}_2$ adsorption results show that shales have different adsorption characteristics at relative pressure of 0–0.5 and 0.5–1. Two fractal dimensions $D_1$ and $D_2$ were calculated using the Frenkel–Halsey–Hill (FHH) equation. Results show that the methane adsorption capacity increases with the increase of $D_1$ and $D_2$, of which $D_1$ has a more significant influence on adsorption capacity than $D_2$. Further studies indicate that $D_1$ represents the pore surface fractal characteristics caused by the irregularity of shale surface, whereas $D_2$ represents the pore structure fractal characteristics, which is mainly affected by shale components (e.g. TOC, clay minerals) and pore parameters (e.g. average pore diameter, micropores content). A higher $D_1$ corresponds to a more irregular pore surface, which provides more space for methane adsorption. While a higher $D_2$ indicates a more complex pore structure and a stronger capillary condensation action on the pore surface, which in turn enhances the methane adsorption capacity.

Due to the significant challenges in the measurements, evaluation of permeability coefficient for unsaturated soil is of immense importance for investigating the seepage and hydro-mechanical coupling problems of unsaturated soil. However, the predictions of existing typical models reveal significance divergence for permeability coefficient of unsaturated soils even under identical conditions. In particular, the existing models are greatly restricted in their practical application due to their complexity in the form of integral expressions that require significant computational effort. Here, a simplified unified model is presented to estimate the relative permeability coefficient. First, a fractal-form of soil–water characteristic curve (SWCC) is derived from fractal theory. Then, on the basis of the proposed SWCC models, the classical models (i.e. Childs and Collis-George (CCG) model, Burdine model, Mualem model and Tao and Kong model, respectively) for evaluating the permeability coefficient of unsaturated soil are converted to be presented in fractal forms. It is interestingly found that the fractal forms of these models are enormously similar. Based on these observations, a simplified unified fractal model for the relative permeability coefficient of unsaturated soil is proposed, where only two parameters (i.e. fractal dimension and air-entry value) are included, thereby significantly reducing the computational efforts. The detailed procedure for determining model parameters is elaborated. The accuracy of this model is verified by comparing its predictions with the experimental data for over 12 types of unsaturated soils. The results highlight that, compared with existing models, the proposed model would be much more efficiently used for estimating the relative permeability coefficient of unsaturated soils, thereby facilitating its application for investigating the associated seepage and hydro-mechanical coupling problems in practice.

Foam fluid has found wide applications in oilfield development, such as profile control, water plugging, gas channeling control, fracturing, and so on. As a non-Newtonian fluid, the successful application of foam is significantly influenced by its structure. The foam texture, however, is complex and irregular, and becomes even more complicated in porous media by the boundary effects. Therefore, the description of dynamic foam structure is crucial and a quantitative description method for foam fluid is worth exploring. In this paper, the fractal characteristics of foam in porous media are verified and combined with foam microdisplacement experiment, and the fractal rule of foam is found. The relationship between fractal dimension and pressure is also discussed. The results show that foam has dynamic fractal characteristics during transport in porous media and the box-counting fractal dimension ranges from 1 to 2. Furthermore, the dynamic change of foam fractal dimension during transport in porous media could be divided into three stages. In the first stage when no foam forms, the fractal dimension is about 2; in the second unsteady foam stage, the fractal dimension is reduced from 1.9 to 1.6; the last one is the steady stage and the fractal dimension is almost constant (about 1.6). Besides, the fractal dimension of foam fluid is closely related to displacement pressure. Low pressure corresponds to higher fractal dimension, and high pressure corresponds to lower fractal dimension. Pressure is negatively linearly correlated with fractal dimension. These results are expected to enrich the understanding of the foam dynamic characteristics in their advanced applications.

Pore structure in shale controls the gas storage mechanism and gas transport behaviors. Since nanoscale pores in shale matrix have fractal characteristics, fractal theory can be used to study its structure. In addition, fractal method has its own advantages to investigate nanopores in shale, especially for the heterogeneity and irregularity of nanopores in shale. In this work, fractal features of nanoscale pores and the implication on methane adsorption capacity of shale were investigated by employing low pressure nitrogen adsorption, scanning electron microscopy (SEM), and methane adsorption experiments. Frenkel–Halsey–Hill (FHH) model was also used to calculate the fractal parameters of nanoscale pores in shale. The results showed that nanoscale pores in 12 shale samples have obvious fractal features. All the fractal curves of these shale samples can be divided into two segments, which are cut off by $P/P_0=0.5$, and the fractal dimensions of these two segments vary from 2.48 to 2.92 $(D1)$ and 2.42 to 2.80 $(D2)$, respectively. Based on SEM images, it is found that self-similarity of organic pore systems in shales refers to two aspects. One is that relatively large-scale and small-scale pores have similar formation properties and types, which are of elliptical shape with rough surface. The other is that some small-scale pores are formed by rough surface of relatively large pores. The results also demonstrate that methane adsorption capacity of shale samples increase with increasing total organic carbon (TOC) contents. This is mainly because organic matter is rich in pores and has relatively large fractal dimension values. Larger fractal dimensions indicate rougher pore surfaces and could form more small-scale organic pores. These organic pores would provide more space for methane adsorption.

The fracture has great impact on the flow behavior in fractured reservoirs. Fracture traces are usually self-similar and scale-independent, which makes the fractal theory become a powerful tool to characterize fracture. To obtain three-dimensional (3D) digital rocks reflecting the properties of fractured reservoirs, we first generate discrete fracture networks by stochastic modeling based on the fractal theory. These fracture networks are then added to the existing digital rocks of rock matrixes. We combine two low-permeable cores as rock matrixes with a group of discrete fracture networks with fractal characteristics. Various types of fractured digital rocks are obtained by adjusting different fracture parameters. Pore network models are extracted from the 3D fractured digital rock. Then the permeability is predicted by Darcy law to investigate the impacts of fracture properties to the absolute permeability. The permeability of fractured rock is subject to exponential increases with fracture aperture. The relationship between the permeability and the fractal dimension of fracture centers is exponential, as well as the relationship between permeability and the fractal dimension of fracture lengths.