A generalized solution for transient radial flow in hierarchical multifractal fractured aquifers

Abstract

[1] An analytical solution in the Laplace domain is derived for modeling anomalous pressure diffusion during pumping tests in aquifer displaying hierarchical fractal fracture networks. The proposed solution generalizes all of the analytical models for fractal flow published previously by combining multifractal diffusion and nested multiporosity with transient exchanges, interface skin effects, and well storage effect. Solutions are derived for fracture-delimited blocks with planar, cylindrical, and spherical shapes, as well as with any fractional dimensional shapes. Any combinations of these shapes can be defined in order to model a large range of situations. Within each permeability level, the fractal properties of the fracture network can be specified and the fractal dimension can be distinct from the shape dimension of the block.

Unified fractional differential approach for transient interporosity flow in naturally fractured media

“…In what follows, the analysis will focus on spherically shaped porous matrix blocks of radii R i with symmetrical distribution of the fluid inside. In many cases such blocks are also used to approximate flow through a naturally fractured medium with cubic or other bounded shapes of matrix blocks [14,27,31,[43][44][45][46]. In the case of spherical matrix blocks, the source/sink term is…”

“…Derivation of transient interporosity flow models for other geometries of matrix blocks encountered many technical difficulties; and as a result most authors tried to concentrate on identical, uniform and regular shapes of matrix blocks. In the analysis of irregular or nontypical matrix blocks, idealization models with approximation of these matrix blocks with regular shaped matrix blocks in the form of slabs, cylinders or spheres are usually considered [14,27,31,[43][44][45][46]. Due to difficulties in formulating transient interporosity flow models for irregular shapes, no analytical studies have addressed the question of the comparison of such models for regular (slabs, cylinders, spheres) and irregular matrix block shapes.…”

Influences of Aquifer Properties on Flow Dimensions in Dolomites

“…A recent attractive approach is the use of fractal methods, which have also been used since their introduction (Mandelbrot 1983) in many fields of geology (Turcotte 1992) as well as in recent studies of flow and transport in fractured rocks (Bonnet et al 2001;Doughty and Karasaki 2002;Lods and Gouze 2004), for analyses of water flow in fractures with the introduction of the fractional flow dimension (Barker 1988;Acuna and Yortsos 1995), in tracer tests with fractional flow dimensions (van Tonder et al 2002;Walker et al 2006), in further applications of fractal diffusion in fractures (Ben-Avraham and Havlin 2000;de Dreuzy and Davy 2007), and in the latest generalized models of pumping tests in aquifers displaying hierarchical fractal fracture sets accounting for anomalous pressure diffusion (Lods and Gouze 2008). A recent attractive approach is the use of fractal methods, which have also been used since their introduction (Mandelbrot 1983) in many fields of geology (Turcotte 1992) as well as in recent studies of flow and transport in fractured rocks (Bonnet et al 2001;Doughty and Karasaki 2002;Lods and Gouze 2004), for analyses of water flow in fractures with the introduction of the fractional flow dimension (Barker 1988;Acuna and Yortsos 1995), in tracer tests with fractional flow dimensions (van Tonder et al 2002;Walker et al 2006), in further applications of fractal diffusion in fractures (Ben-Avraham and Havlin 2000;de Dreuzy and Davy 2007), and in the latest generalized models of pumping tests in aquifers displaying hierarchical fractal fracture sets accounting for anomalous pressure diffusion (Lods and Gouze 2008).…”

Scale dependency of fractional flow dimension in a fractured formation

“…Some models were proposed to describe the behavior of fracture systems (e.g. Barker, 1988;Chang and Yortsos, 1990;Acuna and Yortsos, 1995;Lods and Gouze, 2008). Barker (1988) developed a generalized radial flow (GRF) model for hydraulic tests in fractured formations by regarding the dimension of the flow as a parameter.…”

“…Simulated annealing (SA) is one of the major representatives of these optimization methods. The theory of SA was developed by Metropolis et al (1953). They introduced a simple algorithm to incorporate the idea of the behavior of a particle system in thermal equilibrium into numerical calculations of equation state.…”