Abstract Local volume-averaging of the equations of continuity and of notion over a porous medium, is discussed. For steady-state flow such that inertial effects can be neglected, a resistance transformation is introduced which in part transforms the local average velocity vector into the local force per unit volume which the fluid exerts on the pore walls. It is suggested that for a randomly deposited, though perhaps layered, porous structure this resistance transformation is invertible, symmetric and positive-definite. Finally, for an isotropic porous structure [the proper values of the resistance transformation are all equal and are termed the resistance coefficient] and an incompressible fluid, the functional dependence of the resistance coefficient is discussed using the Buckingham-Pi theorem for an Ellis model fluid, a power-model fluid, a Newtonian fluid and a Noll simple fluid. Based on the discussion of the Noll simple fluid, a suggestion is made for the correlation and extrapolation of experimental data for a single viscoelastic fluid in a set of geometrically similar porous structures. Introduction Darcy's law, involving a parameter k termed the permeability, was originally proposed as a correlation of experimental data for the flow of an incompressible Newtonian fluid of viscosity mu moving axially with a volume flow rate Q through a cylindrical packed bed of cross-section A and length under the influence of a pressure difference [Ref. 1, p. 634), ........................................(1) Eq. 1 has suggested for isotropic porous media a vector form of Darcy's law, ........................................(2) A major difficulty of this equation has been that, since it was not derived, the average pressure P and average velocity z were undefined. Whitaker has recently derived a generalization of Eq. 2 appropriate to anisotropic porous media by taking a local average of the equation f motion. In his result, P and V are local surface averages of pressure and velocity, respectively. The object of this paper is to develop by a method considerably different from Whitaker's an extension of Darcy's law which is appropriate to viscoelastic fluids. [Viscoelastic is used here in the sense that the materials obey neither of the classical linear relations: Newton's law of viscosity and Hooke's law of elasticity. We being by discussing in the first section the problem, of local volume averaging of the equation of motion as opposed to the local surface averaging explained by Whitaker. In the second section a resistance transformation [the words transformation and tensor are used interchangeably here] is introduced to describe in part the force per unit volume which the fluid exerts on the pore walls; we discuss this transformation for randomly deposited, though perhaps layered, porous media. In the third and fourth sections we specialize to isotropic media and consider the functional dependence of the resistance parameter by means of the Buckingham-Pi theorem. In the third section we take up two simple empirical models which do not account for normal stress effects or the possible memory of the fluid. In the fourth section we consider the problem for the incompressible Noll simple fluid, currently believed to be a general description of a wide variety of memory fluids.

This paper was prepared for the Society of Petroleum Engineers Symposium on Mechanics of Rheologically Complex Fluids, to be held in Houston, Tex., Dec. 15–16, 1966. 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 whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon requested to the Editor of the appropriate journal, 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 discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Introduction This communication presents an extension of an earlier study by Christopher and Middleman of the applicability of a non-Newtonian generalization of the Blake-Kozeny equation to the laminar flow of non-Newtonian polymer solutions through porous media. The earlier work, done with dilute aqueous solutions of carboxymethylcellulose, indicated that the model put forth correlated data with an average error of 18 per cent and a standard deviation of the friction factor of 0.21 over a range of three orders of magnitude in a modified Reynolds number. A survey of the literature available at that time indicated some evidence of viscoelastic effects, in contrast with the conclusions drawn by Christopher and Middleman it was decided to extend the earlier study by examining a more elastic fluid, polyisobutylene dissolved in toluene. In addition, data were obtained for tubes packed with sand, and for tubes packed with binary mixtures of glass spheres. We still fail to see viscoelastic effects. The basic experimental techniques, and the methods of reducing and correlating the data, are as described by Christopher and Middleman and in more detail by Christopher and by Gaitonde. Polyisobutylene [PIB L-100, Enjay Co.] solutions in toluene were run at 25.0 deg C. Power-law parameters were obtained from capillary viscometry performed over a range of shear rates corresponding to the shear rates achieved in the porous medium. Table 1 shows the solution properties, including the zero-shear viscosities of the solutions and the viscosity average molecular weight of PIB L-100. CORRELATION OF DATA Fig. 1 shows the friction factor-Reynolds number correlation of the data obtained using tubes packed with glass spheres of a narrow size range. The data are correlated with an average error of 9.8, per cent and a standard deviation of 0.14 by ........................................(1) where ........................................(2) and ........................................(3)

This paper was prepared for the Society of Petroleum Engineers Symposium on Mechanics of Rheologically Complex Fluids, to be held in Houston, Tex., Dec. 15–16, 1966. 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 whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon requested to the Editor of the appropriate journal, 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 discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract This paper proposes a friction-factor correlation for turbulent flow of viscoelastic fluids in tubes. The correlation is a generalization of the Blasius form such that ........................................(1) where f is the friction factor and K is a parameter for the particular fluid. N'Re is the Reynolds number generalized from the Oswald-deWaele model ........................................(2) where D is tube diameter, V is bulk velocity, rho is density, K' and n' are the consistency variable and the flow behavior index, respectively. f" is the friction factor predicted from the correlation of Dodge and Metzner for a given value of n' and a modified Reynolds number [1 + NWs] N'Re where NW, is the Weissenberg number. The correlation fits existing viscoelastic data within an average of 8 per cent. Introduction There are many fluids which do not follow Newton's law of viscosity. These fluids are broadly classified as either purely viscous non-Newtonian or viscoelastic. Three approaches are used to develop predictive techniques for the flow of these complex fluids: the phenomenological approach, where one correlates semi-empirical functions with experimental data; the molecular or structural approach, where one tries to relate bulk behavior with the detailed dynamics of the structural unit; and the continuum approach, where one tries to generate models by placing general restrictions on the form of the rheological equations as prescribed by the rules of tensor analysis. Correlations have been tested for purely viscous non-Newtonian fluids against considerable data for smooth tubes. Three turbulent flow correlations exist for viscoelastic fluids; each correlation has been derived from an independent set of flow data. This paper presents a correlation for friction factors for fluids in turbulent, viscoelastic flow which fits the data as well as or better than the three above-cited correlations. The proposed correlation is a generalization of the Blasius correlation for the turbulent flow of Newtonian fluids.

Abstract This work was undertaken to define the conditions under which the Deborah number characteristic of the flow process may become large enough to cause significant deviations from the usual drag coefficient-Reynolds number relationships for purely viscous fluids flowing through porous media under non-inertial conditions. The analysis suggests that major [order-of-magnitude] effects may be expected to occur at Deborah number levels in the range 0.1 to 1.0. Experimental studies using a porous medium, having a permeability of 46.4 × 10-0 sq cm. support this analysis and define the critical value of the Deborah number at which viscoelastic effects are first found to be measurable. Prior studies, in which no effects attributable to fluid elasticity were found, are seen to have been confined to lower Deborah number levels. The influences of a high Deborah number level upon the uniformity of the flow and upon the distribution of the fluid residence times in the porous medium are considered briefly. Introduction While the pragmatic significance of studies of flows through porous media requires no discussion, there is additionally a very strong motivation for such studies from a strictly theoretical point of view: flows in this geometry provide an excellent opportunity for a study of the behavior of viscoelastic 25 fluids at high levels of the Deborah number. This dimensionless group, representing a ratio of time scales of the material and the flow process, may be defined as: ........................................(1) in which f1 denotes the relation time of the fluid under the conditions of interest in the problem under consideration and IId represents the second invariant of the deformation rate tensor. This latter term depicts the intensity or the magnitude of the deformation rate process, and the dimensionless group defined by Eq. 1 may be considered to represent the ratio of the size interval required for that fluid to respond to a change in imposed conditions of deformation rate as compared to the time interval between such changes. It is thus an index of the extent to which the velocity field is unsteady from a Lagrangian viewpoint [i.e., from, the viewpoint of an observer moving with a given fluid element as it proceeds its course or trajectory in a process], using the relaxation time of the fluid as a wait of time. For perfectly steady flows (e.g., under laminar flow conditions in a very long tube the Deborah number is identically zero; for highly unsteady processes it may be large. It has been shown elsewhere that quantitative mathematical descriptions of the properties of viscoelastic fluids rather generally predict a fluid-like response to be exhibited whenever the Deborah number is sufficiently low, and that the same materials will exhibit an essentially solid-like response whenever the Deborah number becomes large. In the case of dilute polymeric solutions in steady laminar shearing flows [NDeb = 01], the fluid-like response is, of course, well known and requires no further discussion. That the same materials may behave as elastic solids when deformed suddenly enough [NDeb large] may be demonstrated dramatically by impacting a blunt object suddenly upon a pool of such a "fluid": in this case the material may deform appreciably [sheets 6 to 20 in. in diameter are readily formed], but it retracts elastically to its initial configuration, rather than flowing or splashing as a Newtonian fluid does.

Abstract The lowest value [Re]' of the Reynolds number at which streamline flow breaks down has been studied experimentally for the flow of "ordinary" liquids [water, glycerol solution, salt solution, turpentine] and of relaxing liquids [solutions of Cellofas in water and polyacrylamide in water] through a straight tube and in beds of glass beads. The tube was 30.3-cm long and 0.368-cm I.D., two sizes of beads were used—0.206-cm and 0.114-cm diameter —packed in brass tubes of diameter 1.59 cm and length 15.2 cm,. Most experiments were made at 25 1C, but some have also been made at 15C and 35C. In both the tube and beads [Re]' is the same for all "ordinary" liquids, but [Re]' is between 6 and 12 per cent higher for the Cellofas solution, but only while it is degrading. In the case of the polyacrylamide solutions-departures from laminar flow, occur at a value of [Re]' which is about 9 per cent lower in the flow and about 32 per cent lower in porous bed flow; these departures in polyacrylamide flow are accompanied by pronounced drag reduction in the tube, but not in the porous beds. Lowering the temperature in the flow of polyacrylamide solutions lowers [Re]' in porous beds and increases drag reduction in the tubes. Introduction Jones and Williams showed that the value of the flow of gaseous carbon dioxide through tubes and through porous materials was different from the value of [Re]' in the flow of gaseous nitrogen through the same tube or porous material. The difference was attributed to the relaxation of a vibrational mode of the CO2 molecule either absorbing or emitting energy to the bulk motion of the gas, depending on the conditions of the experiment. As a consequence of the experiments and their interpretation, it was anticipated that very similar effects would occur in the flow of relaxing liquids through tubes and porous materials; preliminary experiments confirming these predictions are reported here. There were also theoretical reasons for expecting the onset of instabilities to be different in a relaxing liquid [see Thomas and Walters]; also, considerable drag reduction has been observed in the turbulent flow of such liquids through tubes [see Savins]. EXPERIMENTAL METHOD The flows of "ordinary" liquids [water, 5 per cent weight-weight glycerol solution in water, 5 per cent weight-weight salt solution in water, and turpentine] and of relaxing liquids [a solution of Cellofas in water .005 weight-weight, and solutions of polyacrylamide in water ranging from .05 to .005 per cent weight-weight] through a straight tube 30.3-cm, long and 0.368-cm I.D. have been studied. The flow of water and of the glycerol solution and that of the relaxing liquids have also been studied in beds of glass beads. Two sizes of beads have been used—0.206- and 0.114-cm diameter—packed in brass tubes of diameter 1.59 cm and length 15.2 cm. The porosities of the beds were 0.382 and 0.369, respectively. All experiments were thermostatted initially to 25C +1C. Some experiments have been done at 15C and 35C to study the effect of temperature. The mass rate of flow of liquid (V rho) and the pressure difference [Delta p] across the tube or porous bed were measured. The resistance Delta p to laminar flow through tubes is given by: ........................................(1)