Rate Dependence of Unstable Waterfloods
- P. Sigmund (U. of Calgary) | H. Sharma (U. of Calgary) | D. Sheldon (Petro-Canada Resources Ltd.) | K. Aziz (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- May 1988
- Document Type
- Journal Paper
- 401 - 409
- 1988. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex), 5.5 Reservoir Simulation, 5.4.9 Miscible Methods, 7.2.2 Risk Management Systems, 4.1.2 Separation and Treating, 5.3.1 Flow in Porous Media, 5.4.1 Waterflooding, 1.2.3 Rock properties
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Summary. viscous fingering is part of the flow mechanisms that are operative in waterflooding and in a wide range of EOR methods. This work presents a laboratory and computer model analysis of the viscous-fingering presents a laboratory and computer model analysis of the viscous-fingering dynamics that develop when a less mobile fluid is immiscibly displaced by a more mobile fluid in a permeable medium.
Physical experiments of horizontal fluid displacements conducted in a rectangular bead pack show that the amplitude/frequency character of the wave-like fingers that form depends on flow rate and mobility ratio. The nature of these wave-like features is shown for two viscosity ratios and several displacement rates. Spatial frequency domain analyses of finger shapes at fixed time intervals were conducted on digitized records of the laboratory experiments. The results of these analyses indicate that fluctuations comprising the fingered zone have a wave number corresponding to a maximum growth rate. The analyses suggest that the amplitude of flow fluctuations can be characterized by a root mean square (RMS) growth rate that is linear in time. Finite-difference solutions of a set of two-dimensional (2D) flow-in-porous-media equations were made to exhibit similar frontal instability. The linear growth rate of the mixing zone suggests that fractional flow relationships can provide an adequate and practical method of representing space-averaged two-phase flow variations practical method of representing space-averaged two-phase flow variations for many reservoir engineering applications.
A rate dependency of displacement processes can result because of the growth of wave-like fingers that result from some initial perturbation in fluid velocity or interface position. Such disturbance perturbation in fluid velocity or interface position. Such disturbance may occur as a result of local inhomogeneities in saturation or flow resistance caused by variations in either fluid mobility or permeability, or both. Depending on the nature of fluid and rock permeability, or both. Depending on the nature of fluid and rock heterogeneities and the rates of mixing, velocity disturbances mapropagate and grow or may dampen and become insignificant down-stream from the initiation point.
Linear stability theory has been used to describe the details of frontal instability around a sharp interface in more than one dimension. Saffman and Taylor published a condition for the stability of an immiscible interface moving in a gravitation field between two closely spaced glass plates (Hele-Shaw cell). The derived relationships consider the onset of instability and the notion of wavelength of largest instability. Chuoke et al. considered the effects of capillary pressure across the interfacial discontinuity in terms of an effective interfacial tension (IFT)/curvature product. This permitted calculation of a critical wavelength and a maximum-growth permitted calculation of a critical wavelength and a maximum-growth wavelength. Their results suggested that fingers with wavelengths less than the critical wavelength would decay, while those having wavelengths close to that of the maximum growth would grow. Peters and Flocks later modified the Chuoke et al. theory to Peters and Flocks later modified the Chuoke et al. theory to consider the effect of boundaries on the range of wavelengths that could be accommodated by a porous medium. Many of these aspects of viscous fingering have recently been well reviewed by Homsy. A second category of study has focused on methods of lumping the 2D and three-dimensional (3D) detail of fingering processes into an equivalent average property in a single dimension. Experiments have confirmed that for miscible floods in stable modes, mixing zones grow with the square root of time, and that upon becoming unstable, they grow linearly with time. The Koval and Todd-Longstaff models are often used to account for flow representations of fingers. Fayers recently presented a method of analysis that includes the notion of a volume fraction of the displacing fluid that is fingerable. The analysis included a determine a tion of flow velocities and relative size of fingered and nonfingered fractions from the Buckley-Leverett equations. All these models use empirical correlations to account for mixing to lessen the highly unfavorable fractional flow relationships between displacing and resident fluids that were predicted on the basis of original mobility ratios. For immiscible flow, Skaugen has suggested that the physical development of viscous fingers can be described by a model physical development of viscous fingers can be described by a model that incorporates an averaging or lumping of cross-sectional water fingers into an equivalent average saturation that fits into a higherorder Buckley-Leverett/Welge format. At present, the relationship between the detailed wave-like structure of viscous fingers and equivalent single-dimension transition zones is a problem that is important in both miscible-flood and waterflood applications.
In this work, we examine the relationship between finger wavelength, rates of finger growth, displacement velocity, and viscosities of resident and displacing fluids by means of experiments in two-phase flow systems. A method of treating the observed data in the spatial frequency domain to characterize the apparently random front shapes that occurred during the unstable displacements is presented. An analysis of the front behavior generated as a response to a random flow disturbance in a linear 2D, two-phase flow simulator shows some of the limitations one might expect with such a model. Consideration of the results indicates that Buckley-Leverett fractional flow relationships are, for many purposes, adequate for modeling the average flow properties across a fingered cross section.
Observation of Flow Fluctuations in Physical Experiments
Immiscible flow experiments were performed in a flow model consisting of a glass bead pack contained in the recess between two plexiglass plates (Fig. 1). A uniform plate separation of 3.2 mm plexiglass plates (Fig. 1). A uniform plate separation of 3.2 mm [0.13 in.] was attained by machining a 406 x 254-mm [16x 10-in.] recessed rectangular area between the plates. Distribution grooves were machined at both ends of the lower plate and were filled with glass fiber. Connections at either end allow injection and withdrawal to or from the distribution grooves.
A constant-flowrate Sage syringe was used to inject fluids into the inlet distribution groove of the flow model. Production of the incompressible fluids was collected in a burette open to the atmosphere. The flow model was a random pack of spherical glass beads. The character of the packed model is summarized in Table 1, and Table 2 summarizes the properties of the fluid pairs used in the experiments. In all cases, the initial state of fluid saturation was 100% mineral oil.
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