The top answer to Is there an equation for the strong nuclear force? explains that there is a coulomb's law-like formula:
$V(r)=−\frac{4}{3}\frac{α_s(r)ℏc}{r}+kr$
that describes how quarks and antiquarks interact with each other. The author points out that the $\alpha_s$ factor is dependent on the distance between quarks, so what is that dependency? What is $\alpha_s$ as a function of r?
The phenomenological potential you wrote is closest to the Richardson 1979 potential loosely inspired by one gluon exchange, where $\alpha_s$ decreases logarithmically with distance for small r, $$ \alpha_s \propto \frac{1}{\ln (1/r^2\Lambda^2)} $$ Λ being the QCD scale of ~200MeV or so.
Such potentials, including confinement linear terms and more functions with more adjustable parameters can be found in reviews, eqn (15a), such as A. Bykov, Dremin, & Leonidov (1984) Usp Fiz Nauk 143 321 ...