Y.C. Yortsos

Pressure-transient analysis of fractal reservoirs

The authors 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 systems pressure-transient response is then analyzed in the absence of matrix participation and when both the fracture network and the matrix 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. Results may be used to identify and model naturally fractured reservoirs with multiple scales and fractal properties.

Application of Fractal Geometry to the Study of Networks of Fractures and Their Pressure Transient

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.

A theoretical analysis of vertical flow equilibrium

AbstractThe assumption of Vertical Equilibrium (VE) and of parallel flow conditions, in general, is often applied to the modeling of flow and displacement in natural porous media. However, the methodology for the development of the various models is rather intuitive, and no rigorous method is currently available. In this paper, we develop an asymptotic theory using as parameter the variable

The dynamics of in-situ combustion fronts in porous media

R_{{L}} = L/H\sqrt {k_{{V}} /k_{{H}} }

Theory of multiple bubble growth in porous media by solute diffusion

. It is rigorously shown that the VE model is obtained as the leading order term of an asymptotic expansion with respect to 1/RL2. Although this was numerically suspected, it is the first time that it is theoretically proved. Using this formulation, a series of special cases are subsequently obtained depending on the relative magnitude of gravity and capillary forces. In the absence of strong gravity effects, they generalize previous works by Zapata and Lake (1981), Yokoyama and Lake (1981) and Lake and Hirasaki (1981), on immiscible and miscible displacements. In the limit of gravity-segregated flow, we prove conditions for the fluids to be segregated and derive the Dupuit and Dietz (1953) approximations. Finally, we also discuss effects of capillarity and transverse dispersion.

Dispersion driven instability in miscible displacement in porous media

Abstract The drying of liquid-saturated porous media is typically approached using macroscopic continuum models involving phenomenological coefficients. Insight on these coefficients can be obtained by a more fundamental study at the pore- and pore-network levels. In this paper, we present a model based on a pore-network representation of porous media that accounts for various processes at the pore-scale. These include mass transfer by advection and diffusion in the gas phase, viscous flow in liquid and gas phases and capillary effects at the gas–liquid menisci in the pore throats. We consider isothermal drying in a rectilinear horizontal geometry, with no-flow conditions in all but one boundary, at which a purge gas is injected at a constant rate. The problem is mainly characterized by two dimensionless parameters, a diffusion-based capillary number, Ca , and a Peclet number, Pe , in addition to the various geometrical parameters of the pore network. Results on the evolution of the liquid saturation, the trapped liquid islands and the drying rate are obtained as a function of time and the dimensionless parameters. The importance of trapped liquid islands on screening mass transfer to the continuous liquid cluster is emphasized. For fixed parameter values, the drying front does not in general obey invasion percolation rules. However, as drying progresses, and depending on the relative magnitude of the capillary and Peclet numbers, a transition to a percolation-controlled problem occurs. Effects of capillarity and mass transfer on saturation profiles and drying rates are discussed. The results are then used to discuss upscaling to continuum models.

On the upscaling of reaction-transport processes in porous media with fast or finite kinetics

The sustained propagation of combustion fronts in porous media is a necessary condition for the success of in situ combustion for oil recovery. Compared to other recovery methods, in situ combustion involves the complexity of exothermic reactions and temperature-dependent chemical kinetics. In the presence of heat losses, the possibility of ignition and extinction also exists. In this paper, we address some of these issues by studying the properties of forward combustion fronts propagating at a constant velocity in the presence of heat losses. We extend the analytical method used in smoldering combustion [7], to derive expressions for temperature and concentration profiles and the velocity of the combustion front, under both adiabatic and non-adiabatic conditions. Heat losses are assumed to be relatively weak and they are expressed using two modes: 1) a convective type, using an overall heat transfer coefficient; and, 2) a conductive type, for heat transfer by transverse conduction to infinitely large surrounding formations. In their presence we derive multiple steady-state solutions with stable low and high temperature branches, and an unstable intermediate branch. Conditions for self-sustaining front propagation are investigated as a function of injection and reservoir properties. The extinction threshold is expressed in terms of the system properties. An explicit expression is also obtained for the effective heat transfer coefficient in terms of the reservoir thickness and the front propagation speed. This coefficient is not only dependent on the thermal properties of the porous medium but also on the front dynamics.

Asymptotic solutions of miscible displacements in geometries of large aspect ratio

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 suciently high velocities for diusive eects 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 twodimensional 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 dierent 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 rst 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.