The laminar boundary layer on a moving continuous flat surface with suction and injection is investigated for cases where the nonzero transverse velocity profile at the surface is such that similar solutions may be obtained. Under these conditions the thermal and concentration boundary layers are investigated for the case of constant temperature and concentration at the flat surface. Numerical solutions of the boundary layer momentum, energy, and diffusion differential equations are presented for a wide range of the injection parameter, f(0), at Prandtl and Schmidt numbers of 1, 10, and 100. Equations for the Nusselt and Sherwood numbers, and the dimensionless heat and mass transfer coefficients are derived. An asymptotic

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x, = VI,vacuum wave length of light frequency observed by particle incident frequency scattered frequency solid angle at aperture angle between incident and scattered wave LITERATURE CITED 1. 2.3. 4. 5.6.

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This paper was prepared for presentation at the 47th Annual Fall Meeting of the Society of Petroleum Engineers held in San Antonio, Tex., Oct. 8–11, 1972. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by who the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Introduction The oil industry has shown a great deal of interest in polymer solutions for use in recovery operations. A small amount of polymer in the injected water of flooding polymer in the injected water of flooding operations reduces the mobility of the driving phase. With the use of polymer solutions for phase. With the use of polymer solutions for recovery operations, the need has developed to understand and describe the mechanisms of non-Newtonian flow through porous media. Many investigators have studied the flow of fluids through porous media. Early investigators, such as Darcy and Kozeny, formulated equations for predicting pressure drop versus flow rate relationships for Newtonian fluids. In recent years, investigators have modified Darcy's equation to account for flow of purely viscous, non-Newtonian fluids. Some purely viscous, non-Newtonian fluids. Some workers have mentioned the effects of elasticity on flow through porous media. One rheological model has been reported where an empirical term is used to describe the effect of elasticity during flow through porous media. porous media. The present investigation shows how viscoelastic fluids behave while flowing through unconsolidated porous media. Details are given on the performance of various polymer solutions in a capillary flow device. polymer solutions in a capillary flow device. This instrument was used to determine if selected polymer solutions exhibited elastic properties. Rheological results are presented properties. Rheological results are presented for different polymer solutions. Flow results in porous media are given for these same polymer solutions. Rheological and flow results polymer solutions. Rheological and flow results are compared as a means of considering the effect of viscoelasticity on polymer solution flow through porous media. A modified form of Darcy's law is presented for use with viscoelastic liquids. SOLUTION CHARACTERIZATION Polymer solutions for this investigation were prepared as specified in Table 1. All polymers were mixed in distilled water. polymers were mixed in distilled water. Polymer was added slowly and the solution stirred Polymer was added slowly and the solution stirred gently by hand to avoid shear degradation. For the more concentrated solutions, the polymer addition period took up to one day. After polymer addition period took up to one day. After standing for a few days, solutions were filtered through a cotton cloth to remove any gelled material.

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tion, cm./sec. inder wall, cm./sec. wall, cm./sec. ur = molar average velocity perpendicular to the cylz), = molar average velocity parallel to the cylinder w = mass flow rate, g./sec. y = dimensionless longitudinal coordinate, z/r, z = length dimension, em.~y = distance between consecutive points in the y di-AS = distance between consecutive points in the s dip = viscosity, g.cm./sec. rection in the finite difference grid rection in the finite difference grid p = density, g./cc. Several methods that can be used to obtain solutions to the laminar boundary layer momentum, energy, and diffusion differential equations for moving continuous flat surfaces with suction and injection are presented. Results are obtained for a wide range of the injection parameter, f(O), a t Prandtl and Schmidt numbers of 1, 10, and 100. Those methods which permit hand calculation of the properties of interest are compared using the numerical solutions of the boundary layer differential equations as the exact solutions. The new integral method of Hanson and Richardson which gives results for the momentum thickness that deviate less than 2.2% from the exact values is recommended for predicting values of the momentum boundary layer parameters. The Von Karman-Pohlhausen method, which was modified to account for suction and injection, is most generally valid. This methad gives acceptable values of the transfer coefficients for heat, mass and momentum transfer for most of the values considered.