Starting with the commonly cited Forchheimer equation:
$$\tag{1} -\frac{dp}{ds}=\frac{\mu q}{kA}+\frac{\beta \rho q^2}{A^2}$$
multiplying through by density
$$\tag{2} -\rho \frac{dp}{ds}=\frac{\mu \rho q}{kA}+\frac{\beta \rho^2 q^2}{A^2}$$
noting that $\rho q = \dot m$
Dividing (2) through by viscosity:
$$\tag{3} -\frac{\rho}{\mu} \frac{dp}{ds}=\frac{\dot m}{kA}+\frac{\beta \dot m^2}{\mu A^2}$$
Here I see I cannot completely separate the pressure dependent variables ($\rho$ and $\mu$) as I have a viscosity term on the far right-hand side of (3).
Multiplying through by the differential $ds$ and then including the integration notation:
$$\tag{4} -\int_{p_b}^p \frac{\rho}{\mu} dp=\frac{\dot m}{kA}\int_0^L ds+\frac{\beta \dot m^2}{A^2} \int_0^L \frac{1}{\mu} ds$$
So here is my question: $\mu$ is a function of pressure and pressure varies as a function of length (or distance). Is it possible to solve for the integral on the far right-hand side of (4) noting the aforementioned fact? I would also like to know what this type of problem in called in mathematics, if there is such a term for this type of problem.
Pressure as a function of length (or distance) may be written as:
$$\tag{5} p_p (p_s)=p_p(p_0)+\frac{\dot m \mu}{kA}s$$
where
$$\tag{6} p_p(p)=\frac{\mu_i z_i}{p_i}\int_{p_b}^p \frac{p}{\mu z} dp$$
where $\mu_i$, $z_i$, and $p_i$ are the normalizing viscosity, real-gas z-factor, and pressure values, respectively, which are constants, selected as the values at the inlet of the porous medium where $s=0$.
Is it possible to invoke (5) into the $\int_0^L \frac{1}{\mu} ds$ term of (4) and perform the integration?
