Fractal Solutions for Understanding Complex Systems in Earth Sciences

We analyse two classes of methods widely diffused in the geophysical community, especially for studying potential fields and their related source distributions. The first is that of the homogeneous fractals random models and the second is that of the homogeneous source distributions called “one-point” distributions. As a matter of fact both are depending on scaling laws, which are used worldwide in many scientific and economic disciplines. However, we point out that their application to potential fields is limited by the simplicity itself of the inherent assumptions on such source distributions. Multifractals are the models, which have been used in a much more general way to account for complex random source distributions of density or susceptibility. As regards the other class, a similar generalization is proposed here, as a multi-homogeneous model, having a variable homogeneity degree versus the position. While monofractals or homogeneous functions are scaling functions, that is they do not have a specific scale of interest, multi-fractal and multi-homogeneous models are necessarily described within a multiscale dataset and specific techniques are needed to manage the information contained on the whole multiscale dataset.

The earth’s magnetic field is used to find the depth of anomalous sources as shallow as few metres to tens of kilometres. The deepest depths found from the magnetic field sometimes correspond to Curie depth, a depth in the crust where magnetic minerals lose their magnetic field due to increase in temperature. Estimation of depth from magnetic/aeromagnetic data generally assumes random and uncorrelated distribution of magnetic sources equivalent to white noise distribution. The white noise distribution is assumed because of mathematical simplicity and non-availability of information about source distribution, whereas from many borehole studies it is found that magnetic sources follow random and fractal distribution. The fractal distribution of sources found many applications in depth estimation from magnetic/aeromagnetic data. In this chapter Curie depth estimation from aeromagnetic data for fractal distribution of sources will be presented.

Nature is not random as we often assume for the simplicity of mathematical calculations. Scaling or power laws, also known as fractal behavior, are ubiquitous in nature, and the analysis of many physical properties of the earth shows fractal behavior. In this chapter attempt is made to integrate the geological understanding and fractal behavior of the faults to use this knowledge in practice. This understanding could potentially help to reduce the uncertainty and risk in the fault modelling and some of the properties associated with the faults such as transmissibility and shale gauge ratio. The study aims at understanding faults in hydrocarbon reservoirs, however, the concepts are universally valid and could be useful for the readers interested in seismology or mining-related studies as well.

Geophysical well-log data provide a unique description of the subsurface lithology, as they represent the depositional history of the subsurface formations, vis-à-vis the variation of their physical properties as a function of depth. However, a correct identification of depths to different lithostratigraphic units is possible only by using effective data analysis tools. In the present study, detrended fluctuation analysis (DFA) technique has been applied to gamma-ray log, sonic log and neutron porosity log of three different wells, A, B and C, located off the west-coast of India (i) to discuss the statistical characterization of different subsurface formation properties based on their fractal behavior and (ii) to identify the depths to the tops of formations by comparing the results of DFA with those of wavelet analysis. The DFA technique primarily facilitates to understand the intrinsic self-similarities in non-stationary signals like well-logs by determining the scaling exponent in a modified least-squares sense. In the present study, DFA was carried out in two ways using (i) non-overlapping window method to determine the global scaling exponent and (ii) overlapping window method to determine the local scaling exponent. In the non-overlapping window method, data segments of different windows, each having equal length, were first used to estimate the average fluctuations. The linear least-squares regression between the logarithm of average fluctuations and the logarithm of window lengths then defines the global scaling exponent. For gamma-ray logs of all the three wells, the non-overlapping window method shows two distinct ranges of global scaling exponents, in the ranges 0.5–1.0 and 1.0–1.6. While the former signifies the presence of persistent long-range power-law correlations, indicating the stochastic nature of the sedimentation pattern in the data, the latter indicates the existence of short-range correlations of non-stochastic nature but cease to be of power-law form. On the other hand, the sonic and neutron porosity logs of wells A and C show a single global scaling exponent value of greater than 1.0, signifying the non-stochastic nature of the interval transit time (primary porosity) and neutron porosity, respectively, in the entire data sequence as a function of depth. However, in case of well B, the sonic and neutron porosity logs show two distinct ranges of global scaling exponents, one in the range 0.5–1.0 and the other between 1.0 and 1.5, probably suggesting the effect of different diagenetic conditions in well B, compared to those in wells A and C. Choosing a particular window length and sliding it with unit shifts over the entire length of data for estimating the continuous variation of local scaling exponents as a function of depth defines the overlapping window method. This has been applied on all the log data sets of all the wells to generate the plots of variation of local scaling exponents as a function of depth. Comparison of such plots of variation of local scaling exponents of all the logs with the wavelet scalograms of respective logs revealed that the obtained depth estimates agree well with the known lithostratigraphy of the study region.

Porosity is a complex multivariate function which controls fluid flow through porous media. The variables essentially are the characteristics of pore structures such as type, size, shape and arrangement of pores; pore space connections; area of pores that is open for flow; tortuosity of the flow paths and composition of pores, etc. Predicting the rate of flow and the flow patterns of fluid in bulk geological formation is very important for many environmental problems and oil industry. Fracture networks are secondary porosity that is known to exist within the subsurface geology that is expected to influence the flow through geological heterogeneous irregular porous media. Many of them are relevant to the migration and entrapment of fluids within the reservoir. It shows the nature of symmetry in geological formation. However, no general framework exists to systematically study the fluid flow through the fractured subsurface geometry which is complex in nature. Since direct measurements of flow through complex permeable media are time consuming and require experiments that are not always feasible, an analytical model could be more useful. Therefore, the focus of this chapter is on the development of simple analytical models based on medium structural characteristics to explain the flow in natural fractures. The pattern of fracture heterogeneity in reservoir scale of natural geological formations can be viewed as spatially distributed self-similar tree structures. Application of fractal geometry is useful to define the porous structure, where fractal geometry is the study of mathematical shapes which display self-similar, meandering and tortuous porous media details. Based on the above understanding, a fractal model of continuum percolation is presented here, which quantitatively reproduces the flow path geometry by applying Poiseuille’s equation, where flow in fractures is driven by the pressure differences at the two ends of the path.

The conventional approach for fractal dimension (FD) estimation using box count method has been widely used in the analysis of imageries especially in the domain of earth systems modelling and has been known to provide insight into the complexities of the system as well as the dynamics of the processes involved. However, for heterogeneous imageries such as micrographs, etc., the information provided by estimated FD seems to be limited. The present work establishes this limitation in the use of FD (using HarFA 5.5 software) and extends the concept of fractal dimensioning into lower scale segregation levels and evaluating their differential scores. In this approach, fractal differential adjacency segregation (F-DAS) scores are estimated using MATLAB 14.0 for each of the image pixels (of SEM imageries) using the arithmetic means of the grey levels of the adjacency pixels enclosed by the box (used for counting in the conventional methods). The present analysis provides a better understanding of the variability of the system (in this case, adsorbents–adsorbate interactions), unexplored by qualitative analysis of SEM imageries as well as the functional groups using FTIR. This work provides systematic steps of estimation of F-DAS scores of any imagery, the assumptions underlying the approach as well as the scopes of its applications in analysis of various earth systems.

The multi-fractal scaling properties of seismograms are investigated in order to quantify the complexity associated with high-frequency seismic signals. The third-order MDFA (MDFA3) method is capable of characterising the multi-fractality of earthquake records associated with frequency- and scale-dependent correlations of small and large fluctuations within seismogram. These correlations are related to changes in waveform properties and hence are a measure of the heterogeneities of the medium at different scales, sensed by direct and converted phases in a seismogram with different amplitudes and phases. The non-linear dependence of generalised Hurst and mass exponent with order q confirms the multi-fractal nature of earthquake records. Amongst different types of earthquakes analysed, the multi-fractal properties are more pronounced for signals with distinct P-, S- and coda waves. The multi-fractal singularity spectrum parameters (maximum, asymmetry and width) are used to measure the frequency-dependent complexity of seismograms. The degree of multi-fractality decreases with increasing frequency, and is generally more for the time period windowing dominant seismic phases in the seismogram. Significant difference in spectrum width between the original record and its randomly shuffled surrogates demonstrates that the multi-fractality in earthquake records is predominantly due to long-range correlation of small and large fluctuations within seismogram, although its origin due to broad probability distribution cannot be completely ruled out, based on the values of scaling exponent (H _q  ≈ 0.5) and their weak q-dependency for the surrogates.

Fires represent one of the most critical issues in the context of natural hazards. Yearly, they affect large areas worldwide causing loss of biodiversity, decrease in forests, alteration of landscape, soil degradation, increase in greenhouse, etc. Most of these fires have anthropic causes, however there are natural factors, above all summer drought, that strongly influence fire ignition and spread. The investigation of the time dynamics of fires can be carried out considering the fire process per se or focusing on some signal whose variability can be informative of a fire occurrence. In the first case, fires are described by a stochastic point process, whose events are identified by spatial location, occurrence time and size of burned area, or amount of loss. In the second case, time-continuous signals are employed to reveal indirectly the occurrence of fires; one of the mostly used signals is the satellite normalized difference vegetation index (NDVI) that gives information about the “health” of vegetation and, thus, is suited to enhance the status of vegetation after a fire stress. In both cases, the concept of fractal can be used to qualitatively and quantitatively characterize the time dynamics of fires. Fractals are featured by power-law statistics, and, if applied to time series, can be a powerful tool to investigate their temporal fluctuations, in terms of correlation structures and memory phenomena. In the present review we describe fractal methods applied to fire point processes and satellite time-continuous signals that are sensitive to fire occurrences.