Mikhail Panfilov

Barodiffusive Extension of Three-phase Flow Model for NegSat Method and Regularization of Three-phase Instability

Two fundamental problems of three-phase compositional flow in underground reservoirs have been solved by introducing the similar technique of barodiffusive extension of the classical three-phase model. It consists of introducing of the pseudo barodiffusion terms that are proportional to the weighted sum of the gradients of phase pressures, due to which one can change the direction of the fluxes of individual chemical components. First of all, this technique enabled us to complete the method of negative saturations for three-phase flow, which was developed to describe the situations when various zones of reservoir contain different number of phases. The method consists of replacing the true fluid by a fictitious three-phase fluid having specific properties that satisfy the equivalence principle. Two fundamental problems, non resolved in preceding publications, concern (a) the replacement of a two-phase fluid by three phases, and (b) the extension to the case when overcritical zones appear. We have shown that the main difficulty in establishing the equivalence between two-phase and three-phase fluids consists of the non-colinear fluxes of chemical components in a two-phase flow. To reach the vectorial equivalence between fluxes, we have introduced the pseudo barodiffusion in the fictitious three-phase fluid. The barodiffusion coefficients and the directions of the fluxes result from the equivalence conditions in a unique way. The same technique provides the solution for the case when the flow contains the zones occupied by overcritical fluid. In the case of ideal mixing within the phases without capillarity, the flow equations can be converted to the system of conservation laws with respect to the saturations or total concentrations. However the uniform flow equations are non-classical due to the terms of pseudo barodiffusion. The analysis has revealed that the barodiffusion terms ensure the hyperbolic character of the system. Consequently, the well known physical instability that arises in three-phase flow due the loss of hyperbolicity, does not appear in our extended barodiffusive model. Thus, the introduction of the small barodiffusion is the way to suppress the appearance of three-phase instability. To ensure the numerical stability, we applied the monotone upwind high-order scheme for conservation laws with predictor-corrector. We have calculated several cases of miscible gas injection into the reservoir containing initially oil and water, and proved the good convergence of the result obtained compared to the simulations performed by Eclipse compositional and other techniques.

Non-Equlibrium Two-Velocity Effects in Gas-Condensate Flow through Porous Media

The flow of gas-condensate mixture in porous media is characterized by three properties which determine a significant non-equilibrium in transfers between gas and liquid: 1) a capillary and gravity coagulation of small liquid drops with forming large aggregates, 2) a high difference in diffusion coefficients for liquid and gas, which leads to a delay in establishing the liquid aggregate composition, and 3) a high difference between the liquid and gas mobility, which causes the dependence of the non-equilibrium parameters on the relative phase velocity. The objective of this study is to construct a closed non-equilibrium compositional flow model and to reveal various regimes of the non-equilibrium behaviour. The analysis was based on separating in time the capillary-gravity coagulation and the phase exchange. The coagulation was studied using the method of pore network modelling. An isolated liquid drop limited by two meniscus inside a pore can move in direction of the resulting capillary or gravity force. The drop motion is limited by percolation conditions saying that the gas from the neighbouring pores can not be displaced if it has no cluster connection to the medium exit. As the result we obtained the relations between the correlation length of the pore radii field or the permeability field and the scale of the liquid aggregate. At the next step we constructed a set of the averaged compositional flow models taking into account the phase non-equilibrium. The averaging was performed by two-scale asymptotic expansion method. Three regimes of the non-equilibrium were revealed. At a small relative phase velocity the macroscale exchange is diffusion limited, being described by a nonlocal integro-differential operator. Its kernel is calculated as the result of solution to a cell problem. A growth of the relative phase velocity determines an increasing influence of the rotational flow inside liquid aggregates on mass transfer between the phases. Such a diffusion-rotation regime is described in terms of the double relaxation model, presenting a second-order nonlinear kinetic differential equation. The diffusion and rotation relaxation times are calculated as the result of solution to a problem of gas flow around a macroscale liquid aggregate in a porous medium with mass transfer. The fluid flow was described by the Brinkman equations which allow the rotations, in contrast to the Darcy law. The “slip regime” arises at very high relative phase velocities, when the non-equilibrium degree becomes to decrease, however the system tends to a new equilibrium state in which the time of contact between liquid and gas appears to be much lower than the time of mass exchanges. The generalized kinetic model of the slip regime is obtained by homogenization technique. A number of examples are simulated for gas-condensate flow in the vicinity of a well, where the non-equilibrium degree appears to be the most significant. The influence of the capillary number, the Forchheimer effect and the velocity-dependent relative permeability on the non-equilibrium was analysed. We develop generalized relations which describes uniformly all the three non-equilibrium regimes, which may be used as a plug-in to the existing PVT or hybrid thermodynamic-hydrodynamic software, with providing a new option to simulate the non-equilibrium behaviour of multicomponent gas-condensate mixtures. The research is financed by the Schlumberger Moscow Research Center.