Not sure if this is what you mean, but this ends up being pretty common for empirically derived dimensional constants within equations. As a random example from my field, if you look at Table 2 from this paper you'll see that they're reporting the values and units of a variety of constants for a set of equations to estimate how the rate of rock erosion within a river varies as a function of water discharge (among other things). In there, there are various constants with somewhat nonsensical units, e.g., k_e has units m^(2.5) s^(2) kg ^(-1.5), k_w has units m^(-0.65) s^(0.55), and k_t has units of m^(-7/3) s^(-4/3) kg.

So what does these physically mean? Not much really. These are empirical constants and the units are in effect almost like dummy values to ensure that actual physical values have meaningful units. Take the k_w value for example. This is a constant in the equation w_b = k_w * Q_b^(omega_b), which is relating the bankfull width of a river (w_b) to the banfull discharge (Q_b) as a powerlaw. You'd get the value of k_w and omega_b by fitting observed values of width (in meters) and discharge (in m^(3)/s) and thus the units on k_w will depend on the best fit power law exponent (omega_b) and be appropriate to make sure that when you raise discharge in m^(3)/s to that power and multiply by k_w, you get width in meters.