Correspondence
Theory of Fluid Motion in Porous Media
n the December 1969 issue of INDUSTRIAL AND ENGINEERING CHEMISTRY, Stephen Whitaker presented an article entitled ‘(Advances in Theory of Fluid Motion in Porous Media.” T h e author assumes that the anisotropic nature of a porous medium can be represented by a single orientation vector w. Using the principle of material-frame indifference, he obtains the permeability tensor in the form
I
R,j
= B(1)6,,
+
Jq2,x,x,
(11
where a is a unit vector lying in the direction of 0. This form, however, appears to lack generality in that it does not admit three distinct permeabilities along the three principal axes--i.e., it cannot describe a general orthotropic material although the article gives the impression that Equation 1 does apply to general orthotropic materials. T h e fact that Equation 1 does not admit three distinct permeabilities in the direction of the principal axes is easily seen by referring Equation 1 to a coordinate system aligned with the principal axes of the porous medium. K is then represented by a diagonal matrix. I n order for the off-diagonal terms Kl2 and K13 to be zero, only one of the three components of a can a t most be different from zero. Suppose this nonzero component is XI. Then K takes the form
Curtis A . Chase, Jr. is with Shell Development Co., Exploration and Production Research Center, P. 0 . Box 481, Houston, Tex. 77001. AUTHOR
and we see that in the direction of at least two of the principal axes the permeabilities are always equal. A permeability tensor of this typc can only characterize a transversely isotropic porous material, not a n orthotropic material, which is the more general form of anisotropy. The difficulty conies from the fact that in three dimensions a symmetric second-order tensor is defined by six independent quantities and hence cannot be characterized by a single vector, which depends only upon three independent quantities. Thus, the assumption that anisotropy can be characterized by a single vector is too restrictive. If one wants to speak of fundamental quantities that characterize anisotropy, the basic quantities that have physical meaning and are easily visualized are the orientation of the principal axes of the porous material, and the permeability in the direction of each principal axis. Determination of these quantities defines a symmetric second-order tensor. One can then easily evaluate the six independent components of this tensor referred to any arbitrary coordinate system by coordinate transformation from the principal axes coordinate system. Apart from the above comments, Dr. Whitaker in a personal note agrees that Equation 5-30 is incorrect and should be written :
Also the equation following Equation 5-30 is not only incorrect but misleading, as pointed out above, for it implies that the theory is valid for generally anisotropic materials. Curtis A. Chase, Jr. VOL. 6 2
NO. 1 2
D E C E M B E R 1970
83
