1D Simulation of Polymer Flooding Including the Viscoelastic Effect of Polymer Solution

Abstract

Summary Simple simulation models are constructed to predict the performance of 1D polymer flooding. In the models, two phases of oil and polymer solution were assumed to be immiscible with each other. Because the displacing fluid was non-Newtonian, the Buckley-Leverett equation could be modified and a new approach developed to calculate fractional-flow curves. The rheological behavior of polymer solution was modeled with an Ellis type model and a viscoelastic model. To verify the models, two ID flooding experiments were carried out on 2.8-cm-diameter, 47-cm-long, unconsolidated cores packed with glass beads (70/100 mesh). Porosities of the cores are about 37% and permeabilities are around 26 µm2. Two white mineral oils of viscosities 25 and 60 mPa·s, and a 200-ppm polyacrylamide solution were used. In. each experiment, polymer flooding was done after waterflooding. Initial water saturation was controlled to be almost the same at the start of each flood. The calculated polymer-flooding performances were compared with experimental data. The Ellis model predicted earlier breakthrough of polymer solution and lower oil recoveries than the experimental data. On the other hand, the viscoelastic model predicted fractional-flow curves, oil recovery performances, and breakthrough times of the experiments very well. The viscoelastic effect of polymer solution is thought to play an important role in the improvement of oil recovery. Introduction Polymer solutions are used in many operations in the petroleum industry, and their viscoelastic effects are generally recognized.1,2 A polymer solution behaves as a pseudoplastic fluid in tube flow, but in flow through porous media, it shows a viscoelastic effect. In the latter case, polymer molecules are distorted by the flow through tortuous and converging/diverging pore channels. If the molecules do not have enough time to relax and change their configurations, a polymer solution will show an elastic response in addition to the viscous one.3 In many simulations of polymer flooding, polymer solution is not assumed to be a viscoelastic fluid but a Newtonian or a pseudoplastic one.4-6 This assumption is valid when the flow rate in pore space is so slow that the viscoelastic effect of polymer solution can be neglected. In fact, our experiments show that the viscoelastic effect is very small at low shear rates below about 2 seconds−1. Polymer viscoelasticity must be considered, however, in high-shear-rate regions, such as the vicinity of injectors and producers, and in reservoirs having strongly heterogeneous pore geometry.7 In this paper, we construct simple simulation models of 1D polymer flooding. Two kinds of models are used that differ according to the treatment of viscous terms; in one model, rheological behavior is approximated by the Ellis model, and in the other model, elastic viscosity is included. In the following sections, the former model is called the Ellis model and the latter the viscoelastic model. 1D polymer-flood experiments are carried out with glass-beadpacked cores. Experimental results are compared with the model calculations. Ellis Model The equation that describes the flow of a non-Newtonian fluid in porous media is derived by combining a rheological model for the fluid and a physical model for the porous medium. In this study, the Kozeny-Carman model is used for the porous medium. Non-Newtonian viscosity, ?, of a polymer solution is described by the Ellis model: Equations 1 and 2 The modified Darcy equation for a non-Newtonian fluid is expressed by Darcy viscosity, µ', defined in Eq. 3. Equation 3 Absolute permeability, k, is given by the Kozeny-Carman model as follows.8 Equation 4 Combination of the modified Darcy equation, the Kozeny-Carman model, and the Ellis model leads to the following relationship that calculates the shear rate, ?a, in porous media. Equation 5 Because the Darcy viscosity of a polymer solution in Eq. 3 cannot be expressed explicitly as a function of Darcy velocity, u, it is computed by solving Eqs. 6 and 7 numerically. Equations 6 and 7 where ssw=average shear stress at the wall of flow channels in porous media. Equation 8 The derivation is described in more detail in Ref. 8. Viscoelastic Model The Kozeny-Carman model regards a porous medium as a bundle of tortuous tubes with an average hydraulic radius. Therefore, it cannot express the elastic resistance of polymer solution flowing through tortuous and converging/diverging channels. To represent this resistance, we extend the Darcy viscosity of Eq. 3 to include elastic viscosity, µ'elas, in addition to viscous viscosity, µ'vis, as Eq. 9: Equation 9 Littmann9,10 suggested a similar idea. In this equation, the viscous viscosity is calculated by Eqs. 6 and 7, as mentioned in the description of the Ellis model. The elastic viscosity is derived by the following procedures.