Dynamic Gas Slippage: A Unique Dual-Mechanism Approach to the Flow of Gas in Tight Formations

Abstract

Summary A mathematical formulation, applicable to both numerical simulation and transient well analysis, that describes the flow of gas in very tight (k8<0.1md) porous media and includes a dual-mechanism transport of gas is developed. Gas is assumed to be traveling under the influence of a concentration field and a pressure field. Transport through the concentration field is a Knudsen flow process and is modeled with Fick's law of diffusion. Transport through the pressure field is a laminar process and is modeled with Darcy's law (inertial/turbulent effects are ignored). The combination of these two flow mechanisms rigorously yields a composition-, pressure-, and saturation-dependent slippage factor. The pressure dependence arises from treating the gas as a real gas. The derived dynamic slippage is most applicable in reservoirs with permeabilities 0.01 md. The results indicate that in reservoirs of this type, differences between recoveries after 10 years of production with the dynamic-slip and constant-slip approaches were as great as 10%, depending on the initial gas saturation. If an economic production rate is considered, differences as great as 30% can be expected. Introduction lt has been estimated that 400 to 1,000 Tcf [11l.3×1012 to 28.3×1012 m3] of natural gas are trapped in formations designated as "tight sands" (k8<0.1 md). Also, another 300 to 2,700 Tcf [8.5×1012 to 76.5×1012 m3] of natural gas may be trapped in other low-permeability formations, such as Devonian shales and coal seams. The application of Darcy's law to gas flow in these low-permeability formations requires a correction for the Klinkenberg effect (gas slippage across the capillary walls of the pore channels). This correction takes the form of a slippage factor, b, in the Klinkenberg equation:Equation 1 Klinkenberg2 made the following observations:Fig. 1, 2, and 3** show that the apparent permeability is approximately a linear function of the reciprocal mean pressure. This linear function, however, is an approximation, as becomes evident from Tables 5, 6, and 7*** wherein the value of constant b increases with increasing pressure.Even with an idealized pore system, the factor b cannot be expected to be constant, as the theory of Kundt and Warburg cannot be applied to the flow of gas through a capillary if the mean free path is no longer small compared with the radius of the capillary (i.e., deviations to be expected at reduced pressures).This change in the factor b however, will not be discussed here in detail. Rose3 and, more recently, Sampath and Keighin4 conducted gas flow experiments in cores partially saturated with water. Their results show that the slope of the line ka vs. 1/p (i.e., the slippage effect) decreases with increasing water saturation. During depletion, a gas reservoir undergoes changes (both in time and location) in pressure and saturation. The effect of slippage, therefore, varies throughout the life of the reservoir. To date, no detailed theoretical or experimental investigation has been conducted regarding the dynamic behavior of gas slippage. This is surprising because of the large reserves of gas trapped in tight formations. We have developed a dynamic slippage model that is similar to the approach of Adzumi5 for slip through capillary tubes. This approach, based on simultaneous flow resulting from viscous (Darcian) and diffusion (Fickian) flow processes, yields a pressure-, composition-, and saturation-dependent slippage factor. In this way, it is possible to build the time- and space-dependent character of the slippage phenomenon into the gas-transport equation in porous media.