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Paraffin wax-oil gels are unique rheological fluids which undergo shear degradation starting at a deformation (shear strain level) of approximately 1%. Flow commencement in pipelines filled with wax-oil gels is a complex hydrodynamic process involving propagation of acoustic, diffusive, and rheological degradation pressure wave fronts. Dynamic simulation informed by qualified rheological relations provides useful insight into the physical nature of these flow processes. Eulerian simulations are presented which emulate known physical phenomena and essential characteristics of wax-gel flow dynamics. A constitutive rheological equation set accounts for deformation-driven reduction in yield stress and viscosity terms. No explicit time-dependent rheological parameters are utilized in the equations. Rheological yielding alters the nature of the dominant pressure wave from inherently diffusive towards self-sharpening. Axial stress localization effectively sequentializes the gel breakage process, quantified by reduced length of the pressure wave-front zone. Ultimately, axial stress localization allows flow in longer pipe segments, albeit with a concomitant time delay. Viscous behavior and yielding degradation behavior are shown to account for upward and downward concavity in transient axial pressure profiles, respectively. Overall, a unique synergy between gel compressibility and gel degradation is revealed. Deformation-coupled interaction between compressibility and degradation allows pressure propagation and subsequent sustained flow through a gel material which is otherwise immobile in the incompressible case.