Pressure-Transient Analysis of Fractal Reservoirs

Abstract

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.

Deliverability of Wells in Shallow Confined Aquifers of Non-Integer Spatial Dimensions

“…The connectivity index,  , which is related to the spectral density, [3], does not show up in the fractional model.…”

“…If the fracture network is self-similar and has some connectivity, then it is of fractal type (Chang and Yortsos [3]). Both descriptions lead to rock properties of single term power law expressions.…”

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

“…Chang and Yortsos [5] presented a generalization of the Warren and Root [14] model to describe a double porosity fractal system (dimensionless form):…”

“…According to Chang and Yortsos [5], the mathematical representation of the transfer of fluid from matrix to fracture is:…”

“…Chang and Yortsos [5] were the first authors adopting a fractal model to describe the response of a double porosity system (naturally fractured reservoir). Acuña et al [6] applied this model to naturally fractured reservoirs and found that the wellbore pressure is a power-law function of time.…”