Diffusion on fractals

Stochastic dynamics from the fractional Fokker-Planck-Kolmogorov equation: large-scale behavior of the turbulent transport coefficient

The formulation of the fractional Fokker-Planck-Kolmogorov (FPK) equation [Physica D 76, 110 (1994)] has led to important advances in the description of the stochastic dynamics of Hamiltonian systems. Here, the long-time behavior of the basic transport processes obeying the fractional FPK equation is analyzed. A derivation of the large-scale turbulent transport coefficient for a Hamiltonian system with 11 / 2 degrees of freedom is proposed in connection with the fractal structure of the particle chaotic trajectories. The principal transport regimes (i.e., a diffusion-type process, ballistic motion, subdiffusion in the limit of the frozen Hamiltonian, and behavior associated with self-organized criticality) are obtained as partial cases of the generalized transport law. A comparison with recent numerical and experimental studies is given.

Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution

The nonlinear diffusion equation partial delta rho/delta t=D Delta rho(nu) is analyzed here, where Delta[triple bond](1/r(d-1))(delta/delta r)r(d-1-theta) delta/delta r, and d, theta, and nu are real parameters. This equation unifies the anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [ theta>(1-nu)d], "normal" diffusion [theta=(1-nu)d] and superdiffusion [theta<(1-nu)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.

Percolation in sign-symmetric random fields: topological aspects and numerical modeling

The topology of percolation in random scalar fields psi(x) with sign symmetry [i.e., that the statistical properties of the functions psi(x) and -psi(x) are identical] is analyzed. Based on methods of general topology, we show that the zero set psi(x)=0 of the n-dimensional (n>/=2) sign-symmetric random field psi(x) contains a (connected) percolating subset under the condition |nablapsi(x)| not equal0 everywhere except in domains of negligible measure. The fractal geometry of percolation is analyzed in more detail in the particular case of the two-dimensional (n=2) fields psi(x). The improved Alexander-Orbach conjecture [Phys. Rev. E 56, 2437 (1997)] is applied analytically to obtain estimates of the main fractal characteristics of the percolating fractal sets generated by the horizontal "cuts," psi(x)=h, of the field psi(x). These characteristics are the Hausdorff fractal dimension of the set, D, and the index of connectivity, straight theta. We advocate an unconventional approach to studying the geometric properties of fractals, which involves methods of homotopic topology. It is shown that the index of connectivity, straight theta, of a fractal set is the topological invariant of this set, i.e., it remains unchanged under the homeomorphic deformations of the fractal. This issue is explicitly used in our study to find the Hausdorff fractal dimension of the single isolevels of the field psi(x), as well as the related geometric quantities. The results obtained are analyzed numerically in the particular case when the random field psi(x) is given by a fractional Brownian surface whose topological properties recover well the main assumptions of our consideration.

Practical Application of Fractal Pressure Transient Analysis of Naturally Fractured Reservoirs

Summary

Pressure transient tests in naturally fractured reservoirs often exhibit non-uniform responses. Various models are available to explain such nonuniformity. However, the relevance of these models is often not justified on a geologic basis. Fractal geometry provides a method to account for a great variety of such transients under the assumption that the network of fractures is fractal. The theoretical basis for this method was presented in [7] and was verified numerically in [2]. This paper presents an application to real well tests in various fractured reservoirs. The physical meaning of the fractal parameters is presented in the context of well testing. Behaviors similar to the finite conductivity fracture model and to spherical flow are presented and explained by the alternative of fractal networks. A behavior that can be mistakenly interpreted as a double porosity case is also analyzed.

Introduction

It is well known that pressure transients of wells in naturally fractured reservoirs often lack a similar response. In fact, individual wells at different locations in the same reservoir often exhibit qualitatively different pressure responses. While the traditional double porosity model has been considered as the standard tool for the analysis of such tests, it is also common knowledge that the expected behavior of a parallel line, in a semilog plot of pressure vs. time, is frequently not observed. To reconcile such differences, various explanations are typically advanced, including effects of wellbore storage, short test duration and boundary effects.

While research in pressure transients of fractured systems has considerably advanced in the past decades, it has its underpinnings on the classical notion that naturally fractured systems are characterized by a few (usually two) distinct scales that delineate the fracture network and the embedded matrix. Variations on this approach, including randomly generated fracture networks, triple-porosity systems, etc., although adding complexity, still obey the general premise that the network of fractures is dense and space filling, namely that it is of Euclidean geometry.

Pressure Transient Analysis of Fractal Reservoirs

Summary

We present a formulation for a fractal fracture network embedded into a Euclidean matrix. Single-phase flow in the fractal object is described by an appropriate modification of the diffusivity equation. The system's pressure-transient response is then analyzed in the absence of matrix participation and when both the fracture network and the matrix participate. participate. The results obtained extend previous pressure-transient and well-testing methods to reservoirs of arbitrary (fractal) dimensions and provide a unified description for both single- and dual-porosity systems. provide a unified description for both single- and dual-porosity systems. Results may be used to identify and model naturally fractured reservoirs with multiple scales and fractal properties.

Introduction

Fractured reservoirs have received considerable attention over the past few decades. Naturally fractured reservoirs typically are represented by the two-scale (fracture/matrix) model of Warren and Root. The fracture network is assumed to be connected and equivalent to a homogeneous medium of Euclidean geometry. Alternatives must be sought, however, for reservoirs with multiple property scales and a non-Euclidean fracture network. Fractal geometry is a natural candidate for the representation of such systems.

Naturally and artificially fractured systems (e.g., carbonate reservoirs and stimulated wells) have been actively investigated. The following key concepts are typically applied in conventional models.

Premise 1.

There are two media (matrix/fracture network) with two distinctly different flow-conductivity (permeability) and storage (porosity) scales.

Premise 2.

The matrix is a Euclidean object (i.e., of dimension D = 2 for cylindrical-symmetry reservoirs) within which the fracture network is embedded. The fracture network is also Euclidean with dimension D = 2 in the dual-porosity case, or D = 1 in the single-fracture case.

Premise 3.

The matrix is not interconnected; thus fluid flow to and from wells occurs only through the perfectly connected fracture network.

These premises are reflected in the pressure-transient response models. Thus, the dual-porosity system exhibits the asymptotic behavior, pertinent to flow in a system with D = 2 and cylindrical symmetry, while the single-fracture system response is at early times and at later times, suggesting linear (D = 1) and bilinear (D = 3/2) flow geometry, respectively. Note with the singular exception of 2D cylindrical geometry, the asymptotic pressure response generally is the power-law type .

Although various improvements and modifications of the original model have been proposed (see Ref. 4 for a rigorous analysis), they all pertain to the well-ordered but rather restricted structure described above. Recognizing the need for further extension, Abdassah and Ershaghi recently proposed a triple-porosity model that relaxes Premise 1 by considering an proposed a triple-porosity model that relaxes Premise 1 by considering an additional scale. While this incremental approach may be adequate in several cases, it is less applicable to systems exhibiting a large number of different scales, poor fracture connectivity, and disordered spatial distribution. An alternative formulation for these systems is desirable.

Several naturally fractured reservoirs share many such features, notably a large variability in scales and fracture density and extent. These features are induced by the fracturing process in conjunction with the initial brittleness of the material. While such relations are actively researched, evidence increasingly points out that fracturing processes may lead to the creation of fractal objects. Examples range from Monte Carlo simulations to field observations and modeling. Fractal properties have been variously assigned to fracture perimeter, fracture-system mass, or the fracture-size density function.