The nature of flow in porous media is determined by the interaction of the physical properties of the medium and fluids, and by the interplay of various forces involved in the displacement process. Identifying flow regions at a given reservoir operating condition is a key issue in forecasting reservoir performance and hence of optimizing operations. This work identifies dominant flow regions at various conditions. Three dimensionless groups I N gv M/(l + M) (gravity/viscous ratio), Ncv M/(l + M) (capillary/viscous ratio) and Rl2 (shape factor), are defined and used to resolve flow regions, The analysis shows that the relative magnitudes of forces involved in the system combined with the reservoir properties determine the flow region and fluid distribution in the medium.By choosing appropriate ranges for the dimensionless numbers, recovery processes can be specified from the general theory, and the boundaries to flow regions confirmed by comparison with existing experimental and simulation results. Three commonly studied flow systems have been investigated, which are miscible displacements without dispersion (Ncv ~ 0) in layered reservoirs, immiscible displacements (N gv ~ 0) in layered and homogeneous media, and flow in highly fractured reservoirs.
This paper examines normalized forms of Stone's two methods for predicting three-phase relative permeabilities. Recommendations are made on selection of the residual oil parameter, S om, in Method I. The methods are tested against selected published three-phase experimental data, using the plotting program called CPS-1 to infer improved data fitting. It is concluded that the normalized Method I with the recommended form for S om, is superior to Method II. Introduction Stone has produced two methods for estimating three-phase relative permeability from two-phase data. Both models assume a dominant wetting phase (usually water), a dominant nonwetting phase (gas), and an intermediate wetting phase (usually oil). The relative permeabilities for the water and gas are assumed to permeabilities for the water and gas are assumed to depend entirely on their individual saturations because they occupy the smallest and largest pores, respectively. The oil occupies the intermediate-size pores so that the oil relative permeability is an unknown function of water and gas saturation. For his first method, Stone proposed a formula for oil relative, permeability that was a product of oil relative permeability in the absence of gas, oil relative permeability in the absence of gas, oil relative permeability in the absence of mobile water, and some permeability in the absence of mobile water, and some variable scaling factors. He compared this formula with the experimental results of Corey et al., Dalton et al., and Saraf and Fatt. The formula is likely to be most in error at low oil relative permeability where more data are needed that show the behavior of residual oil saturation as a function of mixed gas and water saturations. In particular, the best value for the parameter S om that occurs in the model is not well resolved. In his second method, Stone developed a new formula and compared it against the data of Corey et al., Dalton et al., Saraf And Fatt, and some residual oil data from Holmgren and Morse. Stone suggested that his second method gave reasonable agreement with experiments without the need to include the parameter S om. If in the absence of residual oil data, S om = 0 is used in the first method, the second method is then better than the first method, although it tends to under predict relative permeability. Dietrich and Bondor later showed that Stone's second method did not adequately approximate the two-phase data unless the oil relative permeability at connate water saturation, k rocw, was close to unity. Dietrich and Bondor suggested a normalization that achieved consistency with the two-phase data when k rocw, was not unity. This normalization can be unsatisfactory because k roc an exceed unity in some saturation ranges if k rocw is small. More recently this objection has been overcome by the normalization of Method II introduced by Aziz and Settari. Aziz and Settari also pointed out a similar normalization problem with Stone's first method and suggested an alternative to overcome the deficiency. However, no attempt was made to investigate the accuracy of these normalized formulas with respect to experimental data. In the next section of the paper we review the principal forms of Stone's formulas, and introduce some new ideas on the use and choice of the parameter S om. Later we examine the first of Stone's assumptions that water and gas relative permeabilities are functions only of their respective saturations for a water-wet system. This involves a critical review of all the published experimental measurements. Earlier reviews did not take into account some of the available data. Last, we examine the predictions of normalized Stone's methods for oil relative permeability against the more reliable experimental results. It is concluded that the normalized Stone's Method I with the improved definition of S om is more accurate than the normalized Method II. Mathematical Definition of Three-Phase Relative Permeabilities We briefly review the principal forms of the Stone's formulas that use the two-phase relative permeabilities defined by water/oil displacement in the absence of gas, k rw = k rw (S w) and k row = k row (S w) and gas/oil displacement in the presence of connate water, k rg = k rg (S g) and k rog = k rog (S g). SPEJ p. 224
This paper presents extensions to Stone's Method 1 in the definition of the residual oil saturation (ROS) parameter, Sonn, in which linear, quadratic, and cubic forms are compared with measurements. The methods are also examined in terms of the system providing hyperbolic characteristics, and it is shown that small elliptic regions will usually occur in the saturation space. Some of the factors influencing the elliptic regions are analyzed and their significance is discussed.
Summary. An approximate viscous fingering model is derived with physically related parameters. Plausible assumptions are made physically related parameters. Plausible assumptions are made about some limits on finger fluid mixing, and the use of a fingering function is suggested. It is shown that for the horizontal linear problem, a hyperbolic partial-differential equation governs the problem, a hyperbolic partial-differential equation governs the solvent fractional flow behavior. Satisfactory agreement with classic miscible displacement experiments is demonstrated for a reasonable choice of parameters. when gravity is introduced into the equation, the same model gives adequate agreement with the rate dependency observed in a vertical displacement experiment. Recommendations are made concerning extension of the model for use in three-dimensional (3D) compositional simulations. Introduction It has long been recognized that miscible displacement with a low-viscosity fluid driving a more viscous oil will be an unstable process leading to development of viscous fingers. Although process leading to development of viscous fingers. Although laboratory experiments (see for example Perkins et al.) and perturbation theory have indicated the manner in which fingers perturbation theory have indicated the manner in which fingers will be initiated and grow, there is little direct evidence that fully developed finger growth will necessarily have reproducible stochastic properties on a macroscopic scale. Nevertheless, there remains an expectation that the fingering process should be amenable to being represented in a deterministic model whose parameters should be related to specific aspects of the physical phenomena involved. The position can be compared with the development of turbulence theory in ordinary fluid flow, although currently the equivalent analysis methods are much less developed for unstable miscible displacement in porous media. Two empirical models have been suggested by Koval and Todd and Longstaff to give a basis for computation of miscible displacement. Both models suffer from adoption of empiricism in which the principal parameters involved have little or no direct physical significance. The parameters can be fitted to simple one-dimensional (1D) laboratory experiments, for example, but translation of the same parameter values to a 3D reservoir is an uncertain undertaking. The work of Dougherty represented an attempt to set up a deterministic model with simplifying approximations and physically significant parameters, but the need to fit three different "diffusion" constants with rather large variations between experiments has not encouraged use of this model for general computation. The present paper examines a different and possibly more plausible set of simplifying assumptions than those adopted by plausible set of simplifying assumptions than those adopted by Dougherty, which results in a single hyperbolic equation in this analysis. The new model allows self-consistent approximations to be made for spatial variations of fingered fluid viscosities and densities that are necessary for calculating gravitational segregation rates. Suggestions are made for generalization of the model to 3D applications, which observe the vectorial nature of the fingering phenomenon. Because the basic physical approximations in this case phenomenon. Because the basic physical approximations in this case can be translated consistently to the 3D case, this model should be capable of giving more dependable results in field applications. Koval and Todd and Longstaff Models and Their Physical Limitations The Koval method assumes that the solvent fractional-flow behavior can be treated on a basis similar to immiscible displacement theory, with the relative permeabilities being proportional to solvent and oil volume fractions, respectively. It is assumed that there is some average effective solvent viscosity, independent of local solvent concentrations, that should be used in the mobility terms. Thus, FsK = ..............................(1) (1-C) 1 + C Koval was able to fit the effluent concentration behavior from a number of miscible displacement experiments by assuming = 0.22 and a one-fourth-power viscosity mixing law to give = + .................................(2) It is surprising that the results of this method do appear to give good agreement with measurements for a range of viscosity ratios using the fixed expression for Ms' but there is no physical justification for this success. The method provides no definition of the total effective mobility of the combined flow, which is needed for the pressure equation in a numerical simulator, nor is it clear that the effective solvent density to be used when gravitational segregation is to be computed should also be based on =0.22. The Todd and Longstaff method is based on somewhat similar assumptions, with relative permeabilities proportional to the respective component volume fractions; but in this case, effective viscosities of both solvent and oil are modified by use of the mixing rules, = ..................................(3) and .................................(4) where w is a mixing parameter and = + ............................(5) This then gives the fractional flow equation: FsTL = ..............................(6) SPERE P. 551
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