why displacement gradient is a unitless quantity?

Answer 1

Because a displacement gradient is the ratio of two quantities that have dimensions of "length" (as long as one is dealing with spatial gradients, and not some other sort of gradient, e.g., temporal). That is, a displacement gradient has units of (meters of displacement) per (meter). The length dimensions cancel, and you are left with a unitless quantity.

Edited to comment on answer that follows:

"Radians" are not a true unit. Anything measured in radians is actually a unitless quantity. Wikipedia has a good example of why this must be so: consider the Taylor expansion of the sine function -- sin(x) = x - x^3/3! + x^5/5!...., with x measured in radians. If radians were an actual unit, this sum would be meangless because you would be trying to add terms that have different units (radians - radians^3 + radians^5...etc.). See source.

Answer 2

Note that apparently unitless quantities may actually have the dimension of radians (or reciprocal of radians). The displacement gradient (length/length as answer 2 notes) may actually be visualizable as an angle.

Answer 3

Beacause it's a parameter in Archie's equation (n). It is related to the influence of insulating fluids on the shape and continuity of the electrically conductive solutions occupying pore volumes.