R.P. Srivastava

Introduction of Fractals

The classical geometry deals with objects of integer dimensions. Zero-dimensional points; one-dimensional lines; two-dimensional planes, like squares; and three-dimensional solids, such as cubes, make up the world as we have previously understood it, but many natural phenomena like a length of coast line and the shape of clouds are better described with a dimension partway between two whole numbers. To describe such irregular shapes, fractal geometry was coined, which describes the noninteger or fractal dimensions. So while a straight line has a dimension of one, a fractal curve like a rocky coast line will have a dimension between one and two depending on how much space it takes up as it twists and curves. The more that fractal fills a plane, the closer it approaches to two dimensions. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the two dimensions, while a rough surface composed of many medium-sized hills would be close to the three dimensions (Peterson, 1984). Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it. The essence of fractal theory lies in fractional dimensions.Abstract The classical geometry deals with objects of integer dimensions. Zero-dimensional points; one-dimensional lines; two-dimensional planes, like squares; and three-dimensional solids, such as cubes, make up the world as we have previously understood it, but many natural phenomena like a length of coast line and the shape of clouds are better described with a dimension partway between two whole numbers. To describe such irregular shapes, fractal geometry was coined, which describes the noninteger or fractal dimensions. So while a straight line has a dimension of one, a fractal curve like a rocky coast line will have a dimension between one and two depending on how much space it takes up as it twists and curves. The more that fractal fills a plane, the closer it approaches to two dimensions. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the two dimensions, while a rough surface composed of many medium-sized hills would be close to the three dimensions (Peterson, 1984). Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it. The essence of fractal theory lies in fractional dimensions.

Introduction of Fractals: Application to Gravity and Magnetic Data

The classical geometry deals with objects of integer dimensions. Zero-dimensional points; one-dimensional lines; two-dimensional planes, like squares; and three-dimensional solids, such as cubes, make up the world as we have previously understood it, but many natural phenomena like a length of coast line and the shape of clouds are better described with a dimension partway between two whole numbers. To describe such irregular shapes, fractal geometry was coined, which describes the noninteger or fractal dimensions. So while a straight line has a dimension of one, a fractal curve like a rocky coast line will have a dimension between one and two depending on how much space it takes up as it twists and curves. The more that fractal fills a plane, the closer it approaches to two dimensions. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the two dimensions, while a rough surface composed of many medium-sized hills would be close to the three dimensions (Peterson, 1984). Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it. The essence of fractal theory lies in fractional dimensions.Abstract The classical geometry deals with objects of integer dimensions. Zero-dimensional points; one-dimensional lines; two-dimensional planes, like squares; and three-dimensional solids, such as cubes, make up the world as we have previously understood it, but many natural phenomena like a length of coast line and the shape of clouds are better described with a dimension partway between two whole numbers. To describe such irregular shapes, fractal geometry was coined, which describes the noninteger or fractal dimensions. So while a straight line has a dimension of one, a fractal curve like a rocky coast line will have a dimension between one and two depending on how much space it takes up as it twists and curves. The more that fractal fills a plane, the closer it approaches to two dimensions. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the two dimensions, while a rough surface composed of many medium-sized hills would be close to the three dimensions (Peterson, 1984). Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it. The essence of fractal theory lies in fractional dimensions.

Chapter 1 - Introduction of Fractals: Application to Gravity and Magnetic Data

The classical geometry deals with objects of integer dimensions. Zero-dimensional points; one-dimensional lines; two-dimensional planes, like squares; and three-dimensional solids, such as cubes, make up the world as we have previously understood it, but many natural phenomena like a length of coast line and the shape of clouds are better described with a dimension partway between two whole numbers. To describe such irregular shapes, fractal geometry was coined, which describes the noninteger or fractal dimensions. So while a straight line has a dimension of one, a fractal curve like a rocky coast line will have a dimension between one and two depending on how much space it takes up as it twists and curves. The more that fractal fills a plane, the closer it approaches to two dimensions. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the two dimensions, while a rough surface composed of many medium-sized hills would be close to the three dimensions (Peterson, 1984). Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it. The essence of fractal theory lies in fractional dimensions.Abstract The classical geometry deals with objects of integer dimensions. Zero-dimensional points; one-dimensional lines; two-dimensional planes, like squares; and three-dimensional solids, such as cubes, make up the world as we have previously understood it, but many natural phenomena like a length of coast line and the shape of clouds are better described with a dimension partway between two whole numbers. To describe such irregular shapes, fractal geometry was coined, which describes the noninteger or fractal dimensions. So while a straight line has a dimension of one, a fractal curve like a rocky coast line will have a dimension between one and two depending on how much space it takes up as it twists and curves. The more that fractal fills a plane, the closer it approaches to two dimensions. So a fractal landscape made up of a large hill covered with tiny bumps would be close to the two dimensions, while a rough surface composed of many medium-sized hills would be close to the three dimensions (Peterson, 1984). Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it. The essence of fractal theory lies in fractional dimensions.

Chapter 3 - Fractal Inversion

Inversion of seismic data plays a vital role in reservoir characterization. High-resolution inversion methods provide models with higher resolution than those built from the conventional stacked seismic traces (e.g., Rowbotham et al., 2003; Sen, 2006). Such models are good for reservoir characterization and model building. Edited well logs provide the most accurate and the best vertical resolution of geophysical information of a subsurface reservoir. Though vertical resolution of well logs is very good, horizontal resolution is poor because of sparse availability and the small extent of lateral depths of investigation. One approach to obtain a high-resolution 3D description of a reservoir is to use geostatistical interpolation, extrapolation, or simulation (Hass and Dubrule, 1994) using available well-log data at a few sparse locations. Accuracy of such models is dictated by the number and spatial distribution of the wells over a reservoir. Therefore, seismic data being the most continuous information available (although at a lower vertical resolution), a stochastic inversion of seismic data that integrates seismic data and well-log data can add great value in reservoir characterization (Francis, 2006a) as it combines seismic data with well logs and uses vertical resolution from well logs and good horizontal resolution from seismic data into the estimated model. The fusion of seismic and well-log data is possible using seismic inversion, which converts seismic data information into petrophysical properties such as acoustic impedance and shear impedance (e.g., Dimri, 1992; Russell and Hampson, 2006; Srivastava and Sen, 2010; Vedanti and Sen, 2009). Merging seismic data directly with well-log data is difficult because they have a different range of scale/frequency of measurement compared to well logs and also their recording is basically in a different domain, viz., seismic data in the time domain, whereas logs are recorded with depth.

Chapter 4 - Seismic Reservoir Monitoring

Abstract Seismic reservoir monitoring is normally done by comparing the results of repeated seismic reflection surveys over the same area. This technique is known as time lapse or four-dimensional (4D) seismic, where the fourth dimension is time. 4D reservoir monitoring involves acquisition, processing, and interpretation of repeated seismic data over a field to monitor the changes occurring in the reservoir either due to production of hydrocarbons or enhanced oil recovery/improved oil recovery (EOR/IOR)-related changes, such as injection of water/gas, steam, polymer, etc., into the reservoir. EOR has been defined in Chapter 6 , citing real field examples. The basic principle of time-lapse seismic reservoir monitoring is to subtract out the first-order static geology part to produce clear images of the time-variant fluid flow changes. The reference survey in 4D is termed as baseline and repeated surveys are termed as monitor surveys. This technique helps to identify unswept oil and gas zones in the reservoirs. The idea behind its application is to generate a time-lapse difference dataset, which should be close to zero, except in the reservoir where changes have occurred. The reservoir properties may change due to change in saturation of rock fluids under the effect of production or EOR/IOR. For example, if oil is being produced under water flooding conditions, then saturation of both oil and water changes in the reservoir. This will lead to a change in the seismic response of the reservoir. Apart from saturation, the reservoir may undergo pressure changes either due to production or an EOR/IOR process. Change in pressure will lead to changes in seismic velocities. In general, the expected time-lapse change in the reservoir is small; hence, good quality control is required for 4D monitoring.

Chapter 5 - Reservoir Geophysics: Some Basic Concepts

Abstract Reservoir geophysics plays an important role to bridge the two important disciplines of science, viz., geophysics and reservoir engineering. In the current scenario, these two fields are so intertwined that anyone involved in exploration and exploitation of natural hydrocarbons needs to understand the basic concepts of geophysics and reservoir engineering. This chapter is aimed at providing a brief overview of hydrocarbon reservoirs and the properties of the rocks and fluids associated with such reservoirs. Also, fractal behavior of some of the physical properties is discussed. The chapter will serve as a basic guide to those who have no background in reservoir geophysics or petroleum engineering.

Fluid Flow and Recovery

Abstract In the previous chapter, we discussed the reservoir properties and numerical reservoir simulation in detail; however, the bases of numerical reservoir simulation are fluid flow and the material balance equations, which determine the flow of fluids in the reservoir and govern the recovery factor. These equations are used by reservoir engineers for interpreting and predicting reservoir performance.

Structural Modeling Using Fractals

Abstract Geologists provide models of the Earths subsurface based on the exposed outcrops and other geological inputs. Providing a reliable model of the Earths subsurface in terms of structural setting and property variations is the main objective of the investigations by earth scientists. With the advent of sophisticated geophysical techniques, it is possible to integrate the geophysical observations with the geological models to constrain them better and evolve a more realistic model. Several scattered studies (Yu and Li, 2001) have shown that the rock matrix follows fractal behavior from micro- to mega-scale. Geological models can be made using discrete objects, voxels, or surfaces that honor geological and geophysical observations. We present a geophysical modeling technique based on Voronoi tessellations, which honors available geophysical data, viz. horizons from seismic and physical properties derived from well logs, seismic data, and other kinds of geophysical data.