It is not a replacement, but an assumption of proportionality based on observation.
Most transport equations take on a characteristic form : there are terms for advection, diffusion etc .... This allows anology between terms that appear in derived transport equations and the known form of the equation.
The derivation of the mixture fraction variance equation from first principles leads to an analogy between terms containing D grad(f) that appear in the derivation, and the expected form of the Mixture Fraction Variance scalar dissipation, D grad(f"), .
You will notice that the expression for Chi in the documentation is at the stoichiometric value, and that the documentation of equation (8-45) notes that the fluctuation in Chi is ignored at stoichiometric. Therefore:
Chi_st = 2D (grad(f))^2 = C (e/k) f"^2, where C is a constant of proportionality which may vary according to operating conditions and fuel.
This is then extended to all values of Mixture fraction when interpolating in the flamelet tables but will be most correct near the operating point (stoichiometric) of most devices, or equilbrium.