Abstract
The reservoir is described as a “supercritical cluster”; that is, an aggregate of conductive elements that comprises a “backbone” of connected pores or fractures that span the zone of interest, and also a collection of “sub-critical clusters” or “dangling ends” joined to the backbone to a limited extent. The scheme resembles the usual fracture and matrix-blocks setting but both backbone and sub-clusters are of the same material and share similar petrophysical properties. Whereas the backbone is a homogeneous porous medium, the sub-critical clusters behave as fractal porous media. The backbone-cluster type of flow has been observed in laboratory experiments. The sub-critical clusters were approximated as linear fractal media characterized by static and dynamic fractal exponents and also by porosity and permeability of the compound medium. One of the ends of the linear clusters is closed and the other is joined to the backbone, where the mainstream occurs. A new solution was developed for that problem. The Laplace transform in time and space was used in the mathematical scheme. The theory developed was applied to field cases of interference between wells in aquifers. The matches of computed and observed dynamic pressures show fair fits.
Highlights
The main characteristics of fractal media are two exponents that are defined in deterministic fractals. df is the fractal dimension: the exponent that links a scalar factor applied to a regular fractal object and the number of elements that result from that scale-up
Validation of the formulation was made by comparing its results with the analytical normal results with exponents a 1⁄4 2; b 1⁄4 1
The presented approach was validated with comparisons to the normal case with integer exponents; 2
Summary
The main characteristics of fractal media are two exponents that are defined in deterministic fractals. df is the fractal dimension: the exponent that links a scalar factor applied to a regular fractal object and the number of elements that result from that scale-up. We adopt that link between static and dynamic exponents It is essential for the application of fractal principles to natural materials that random or stochastic fractals, representative of natural objects, preserve the properties of deterministic fractals, the fractal in Figure 3b may behave to the one in Figure 3a in actual flow processes, (Ben-Avraham and Havlin, 2000). The solution, Equation (3), does not account for interference tests (Sahimi, 1993; Camacho Velazquez et al, 2008) It has been criticized for limiting diffusion as only space dependent but time independent (Sahimi, 1993; Leveinen, 2000). A cluster of bounding elements becomes connected across a region of interest when the probability of an element of area being a conducting pore or bond, is greater than a fraction that depends on the underlying type of lattice, (see page 11 table 2.1 in Sahimi (1994); ben Avraham and Havlin, 2000). A turbiditic system (Chen et al, 2012) with communicating channels and appended lobes may be approximated by the BB-SC description – in this case with different properties
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