Step 1: Recall why the pseudo pressure transform is used at all:
For a highly compressible real gas, both the viscosity \(\mu\) and the gas deviation factor \(Z\) vary strongly with pressure, so the usual gas diffusivity equation in terms of \(p\) alone is nonlinear. The pseudo pressure substitution \[ m(p) = 2\int_{p_{0}}^{p} \frac{p'}{\mu(p')Z(p')}\, dp' \] is chosen precisely so that, after the substitution, the diffusivity equation for \(m\) behaves much like the simple linear liquid diffusivity equation.
Step 2: Use dimensional and structural reasoning to fix the form of \(dm/dp\):
Since \(m(p)\) is built as twice the integral of \(p'/(\mu Z)\) with respect to \(p'\), the fundamental theorem of calculus says that differentiating this integral with respect to its upper limit simply reproduces the integrand at that limit, multiplied by the constant factor of 2 that sits outside the integral. That is, \[ \frac{dm}{dp} = \frac{2p}{\mu(p)Z(p)} \] Nothing about the lower limit \(p_{0}\) survives the differentiation because it is a fixed constant.
Step 3: Convert the pressure derivative into a time derivative:
In the reservoir, pressure is a function of both space and time, \(p(x,t)\), and pseudo pressure inherits that dependence only through \(p\). Treating \(m\) as a composite function of \(t\) via \(p\), the chain rule for partial derivatives gives \[ \frac{\partial m}{\partial t} = \frac{dm}{dp}\cdot \frac{\partial p}{\partial t} = \frac{2p}{\mu Z}\cdot \frac{\partial p}{\partial t} \]
Step 4: Match against the answer choices and confirm:
This expression has \(p\) to the first power in the numerator and the product \(\mu Z\) in the denominator, multiplied by \(\partial p/\partial t\), with an overall factor of 2. Only option (A) has exactly this structure, so it is the correct choice; the remaining three options are dimensionally or structurally inconsistent with the derivation.
Final Answer:
\[ \boxed{\dfrac{\partial m}{\partial t} = \left(\dfrac{2p}{\mu Z}\right)\dfrac{\partial p}{\partial t}\ \text{(Option A)}} \]