Convection in Fractured Reservoirs Numerical Calculation of Convection in a Vertical Fissure, Including the Effect of Matrix-Fissure Transfer
Authors: Peaceman D.W. 
Published: December 2007

Abstract
A fractured reservoir undergoing pressure depletion, evolution of gas at the top of the oil zone leads to an unstable density inversion in the fissures. The resulting convection brings heavy oil into contact with matrix blocks containing light oil, and results in the transfer of dissolved gas between matrix and fissure in the undersaturated region of the oil zone. To provide a better understanding of this process, numerical solutions were obtained to the differential equations that describe convection in a vertical fissure and include the matrix-fissure transfer. The numerical procedure is an extension of the method of characteristics for miscible displacement problems in two dimensions. The effect of gas evolution in the upper portion of the oil zone is also included in the numerical model. Calculations for a vertical fissure of rectangular shape, with an initial sinusoidal perturbation of an in verse density gradient, show an initial exponential growth of the perturbation that agrees well with that predicted from mathematical theory. Calculations predicted from mathematical theory. Calculations for a vertical fissure having a similar, but slightly tilted, rectangular shape and with an initial, correspondingly tilted, inverse density gradient, show that the effect of the matrix-fissure transfer parameters on the time for overturning can be parameters on the time for overturning can be correlated quite well by curves obtained from mathematical perturbation theory. The most realistic calculations were carried out for the same vertical fissure having a slightly tilted rectangular shape, with declining pressure at the apex and gas evolution included in the gassing zone. The relative saturation-pressure depression in the fissure below the gassing zone can be characterized as increasing as the square of the time, following an incubation period. For practical ranges of the matrix-fissure period. For practical ranges of the matrix-fissure transfer parameters investigated, it is concluded that saturation-pressure depression will be significant. A preliminary correlation for this Pb depression was obtained. The fissure thickness was found to affect the Pb depression significantly.

Introduction
Most fractured reservoirs of commercial interest are characterized by the existence of a system of high-conductivity fissures together with a large number of matrix blocks containing most of the oil. It has been recognized for some time that analysis of the behavior of a fractured reservoir must involve an understanding of the performance of single matrix blocks under the various environmental conditions that can exist in the fissures. However, it has only recently been recognized that a similar need exists for a better understanding of convective mixing taking place within the oil-filled portion of the fissure system. In a fractured reservoir undergoing pressure depletion, gas will be evolved at all points of the reservoir where the oil pressure has declined below the original saturation pressure. This depth interval is referred to as the gassing zone. Because of the high conductivity of the fracture, the gas in the fissures will segregate rapidly from the oil before reaching the producing wells and most, if not all, of it will join the expanding gas cap. At a sufficient depth, however, the oil pressure will still be higher than the saturation pressure, and the oil there remains in an undersaturated condition. In the gassing zone, gas evolves from the oil in both the fissures and the matrix. The oil left behind in the fissures within the gassing zone contains less dissolved gas and is heavier than the oil below it in the undersaturated zone. This density inversion can result in considerable convection within the highly conductive fissures. As a result of this convection, heavy oil containing less gas is transported downward through the fissures, placing it in contact with matrix blocks containing lighter oil with more dissolved gas. Transfer of dissolved gas from matrix to fissure takes place owing to molecular diffusion through the porous matrix rock; and even more transfer takes place owing to local convection within the matrix block induced by the density contrast between fissure and matrix oil.
SPEJ
p. 281