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.

The effect is studied of capillarity-driven viscous flow through macroscopic liquid films during the isothermal drying of porous materials. A mathematical model that accounts for viscous flow in a 2-D pore network, through both the liquid films and the bulk liquid

Typical models for the representation of naturally fractured systems generally rely on the double‐porosity Warren‐Root model or on random arrays of fractures. However, field observations have demonstrated the existence of multiple length scales in a variety of naturally fractured media. Present models fail to capture this important property of self‐similarity. We first use concepts from the theory of fragmentation and from fractal geometry to construct numerically a network of fractures that exhibits self‐similar behavior over a range of scales. The method is a combination of fragmentation concepts and the iterated function system approach and allows for great flexibility in the development of patterns. Next, numerical simulation of unsteady single‐phase flow in such networks is described. It is found that the pressure transient response of finite fractals behaves according to the analytical predictions of Chang and Yortsos (1990) provided that there exists a power law in the mass‐radius relationship around the test well location. Finite size effects can become significant and interfere with the identification of the fractal structure. The paper concludes by providing examples from actual well tests in fractured systems which are analyzed using fractal pressure transient theory.

We study experimentally and theoretically the downward vertical displacement of one miscible fluid by another lighter one in the gap of a Hele-Shaw cell at sufficiently high velocities for diffusive effects to be negligible. Under certain conditions on the viscosity ratio, M, and the normalized flow rate, U, this results in the formation of a two-dimensional tongue of the injected fluid, which is symmetric with respect to the midplane. Thresholds in flow rate and viscosity ratio exist above which the two- dimensional flow destabilizes, giving rise to a three-dimensional pattern. We describe in detail the two-dimensional regime using a kinematic wave theory similar to Yang & Yortsos (1997) and we delineate in the (M, U)-plane three different domains, characterized respectively by the absence of a shock, the presence of an internal shock and the presence of a frontal shock. Theoretical and experimental results are compared and found to be in good agreement for the first two domains, but not for the third domain, where the frontal shock is not of the contact type. An analogous treatment is also applied to the case of axisymmetric displacement in a cylindrical tube.

We study invasion percolation in the presence of viscous forces, as a model of the drainage of a wetting fluid from a porous medium. Using concepts from gradient percolation, we consider two different cases, depending on the magnitude of the mobility ratio M . When M is sufficiently small, the displacement can be modeled by a form of gradient percolation in a stabilizing gradient, involving a particular percolation probability profile. We develop the scaling of the front width and the saturation profile, in terms of the capillary number. In the opposite case, the displacement is described by gradient percolation in a destabilizing gradient and leads to capillary-viscous fingering. This regime is identified in the context of viscous displacements and in general differs from diffusion-limited aggregation, which also describes displacements at large M . Constraints for the validity of the two regimes are developed. Limited experimental and numerical results support the theory of stabilized displacement. The effect of heterogeneity is also discussed. ͓S1063-651X͑97͒08812-0͔PACS number͑s͒: 47.55. Mh, 0.5.40.ϩj, 47.55.Kf

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A mathematical model is formulated for the general class of problems that involve the transport of stable particulate suspensions in porous media. The porous medium is represented by a network of pore bodies (sites) and pore throats (bonds). Population balances for the species responsible for particle retention and permeability reduction are written in terms of the various mechanisms of particle capture and reentrainment. Rates of capture and release are evaluated using appropriate physical models. We specifically concentrate on mass transfer limited processes. The effective-medium theory is suitably formulated to determine the fluid flow distribution in the network and to calculate the permeability. The network representation of the porous medium together with the population balances and the rates of deposition and release provide a consistent model that finds application in filtration and fines migration processes.

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