Interaction of Natural Convection and Gravity Segregation in Oil/Gas Reservoirs

Abstract

Summary. The distribution of chemical components in reservoirs can be significantly affected by diffusional segregation caused by gravity. This segregation, however, can be opposed by convective mixing. Parallel-flow solutions are presented that shed light on the interaction of these two processes. The solutions are used to explain results of numerical calculations of convection-diffusion in model rectangular reservoirs. Introduction and Schematic Solution Since the time of Gibbs, a static multicomponent fluid subject to a gravitational field has been known to diffuse toward a state that has composition gradients. In thermodynamic equilibrium, the mole fractions of the fluid's denser components decrease with increasing height. The Gibbs sedimentation equations, (1) can be solved for these equilibrium gradients, given an equation of state (EOS). Muskat and Sage and Lacey first discussed the sedimentation equations in the context of oil/gas reservoirs. They used idealized EOS's to yield analytic solutions of Eq. 1. Schulte was the first to make realistic calculations. He applied the Soave-Redlich-Kwong and Peng-Robinson EOS's to multicomponent oil/gas columns and concluded that the mole fractions of the heavier components could vary significantly over the column height. Recently, Riemens et al. and Hirschberg convincingly applied the theory to explain observations made in several oil/gas fields. The chief problem with the assumption of diffusional equilibrium is that a reservoir might be naturally convective. After all, both horizontal and vertical temperature gradients are measured in almost all reservoirs. There has been particular interest recently in modeling the natural state of geothermal reservoirs, where large vertical temperature gradients and pressure gradients measurably different from the hydrostatic give unmistakable evidence for Rayleigh-Benard convection. In more typical reservoirs, the vertical temperature gradient by itself is usually insufficient to excite convection. Any horizontal density gradient, however, will produce horizontal convection. Assuming horizontal fluid homogeneity, the maximum fluid particle velocity (which occurs at the bottom and top of the reservoir) for a given horizontal temperature gradient is of the order. For a temperature gradient of about 2 degrees F/mile [0.7 degrees C/km], typical values for the reservoir height, permeability, and other properties yield a particle velocity on the order of 3 ft/yr [1 m/a]. A fluid particle could thus make a round trip in a reservoir in some 104 to 105 years. This is orders of magnitude faster than the geological time scale of reservoir development and appears to be quite fast enough for convection to have an important influence on the distribution of the reservoir's chemical components. The primary effect of such convection obviously would be to mix the reservoir fluid-i.e., to make it homogeneous. The question is whether gravitational diffusion is strong enough to overcome this mixing. The vertical diffusion time scale in a reservoir for a component i is h2/ki. From diffusion and tortuosity data in Katz, Ki for the lighter components in gas caps can be 10 – 6 ft2 /Sec [10(-7) M2/S]. For h=300 ft [90 m], this yields a diffusion time scale of about 104 years, close enough to the convective time scale to indicate the likelihood of significant segregation. Diffusion times in liquid can be much longer, however. For example, diffusion time scales for heavier components (C12+) in oil can be more like 10(6) years. As things turn out, however, an additional mechanism can strengthen the effectiveness of diffusion. The assumption made for our convective-time-scale calculation that the fluid could be treated as homogeneous in the horizontal direction is incorrect. Instead, horizontal concentration gradients that impede convection can build up. Any convection caused by the horizontal temperature gradient would drive bottom fluid toward the hot (light) end and top fluid toward the cold (heavy) end. But, because of gravitational diffusion, the bottom fluid is richer in dense components than the top, and thus convection in an initially horizontally homogeneous fluid would cause a net flux of the denser components toward the hot end, If the horizontal temperature gradient is weak enough, a horizontal composition gradient that nullifies the effect of the temperature gradient can develop. The convection almost completely stops and the fluid eventually achieves nearly complete diffusional equilibrium. If the horizontal temperature gradient is too strong, however, the buildup of heavy components at the hot end is weak (because they are rapidly swept back by returning fluid) and the reservoir stays strongly convective with only weak gravity segregation. To illustrate this mathematically, we introduce a very simplified system of flow equations. These have been discussed previously in Ref. 10. Using the Boussines approximation and ignoring horizontal diffusion, after non dimensionalization (which will be given later), we can approximate the flow equations for a two-component fluid in a horizontal reservoir by (2a) (2b) (2c) (2d) and (2e) The nondimensional upper and lower diffusion boundary condition is (3) The system has two nondimensional parameters, the imposed horizontal temperature gradient, Tx, and C's equilibrium concentration gradient, T.T is negative because C is the concentration fraction of the denser component. Steady-state parallel-flow solutions to Eqs. 2 and 3 are (4a) and (4b) where Cx is the assumed (constant) horizontal gradient of C. Eqs. 4 can serve as approximations to steady reservoir convection if they meet the constraint of no horizontal flux of C; = 0. This leads to a cubic equation that fixes Cx: (5) For less than ()1/2, this yields just one solution, Cx=Tx. This solution shows the congregation of the heavy component toward the fluid's hot end. Diffusion is dominant because px =0 and the fluid is static. From Eq. 4a, the fluid is completely gravity segregated. A convective steady state is not possible unless Tx >(10)1/2 Then Cx is generally small and the fluid is mixed vertically. P. 233^