Recently, several well testing models have been developed for reservoirs in which the natural fracture networks are described by using fractal distributions. The objective of this study was to extend the concept of fractal distribution to study the effect of temperature gradient on the transient pressure behaviour of a composite reservoir. A composite reservoir model is made up of two or more concentric circular regions of homogeneous rock and fluid properties. It was assumed that the mobility (or diffusivity) ill the oil bank can be approximated by a power law relationship with distance. Applications of this well testing model were demonstrated by simulating the transient pressure behaviour of a falloff test conducted at a steam or air injection well The results suggested that, depending on the size of the swept zone and the severity of wellbore storage effect, the first semilog straight line indicating pseudoradial flow ill the swept zone might not be observable. In addition, since the rock and fluid properties in the oil bank were not constant, it was not possible to develop a second semilog straight line as a resultof pseudoradial flow in the reservoir. However, the fractal characteristics of the reservoir properties could be determined from the slope of the log-log straight line in a graph of pressure against time using the data from a falloff test.
The concept of fractal geometry, developed by Mandelbrot14, suggests that structures which appear to be completely random can be described within a geometric mathematical framework and offers many possibilities in scientific applications. The class of structures treated in this study is limited to those that exhibit self-similar geometrical properties which means that the structure looks the same when observed under various scales of measurement. Mandelbrot was the first to determine that many structures in nature exhibit this self-similar geometrical properties. However, such structures are not exactly self-similar over all length scales and are sometimes referred to as statisticalfractals.
Many geological properties affecting the flow of fluids in porous media are known to demonstrate fractal characteristics. Garrison et al8 presented fractal distributions of pore sizes natural fractures and lithologic unit thickness for a number of sandstone and dolomite rock samples. Hewett10 also reported that porosity and permeability distributions in rock formation demonstrated fractal characteristics and introduced the use of fractal interpolation technique to describe the heterogeneities in a reservoir. Many more examples of natural fractals in geology can be found in the books by Mandelbrot14, Feder7and Peitgen17
Another example of a natural fractal is the much studied percolating network. This fractal network is thought to describe various physical structures such as gels, polymers, crystal growth and the coagulation of smoke particles20 It also forms the basis for studies of multiphase flow of liquids in a porous media which is of great interest in reservoir engineering studies. Displacement of a fluid in a porous medium by another fluid with a much lower viscosity usually results in a very unstable displacement from with the development of so called viscous fingers. The fractal characteristics of viscous fingers have been studied by Maloy15, Chen5 and others13.