Transient pressure in conduit

John

I am new to the CFD. I have used Darcy law mainly in my previous works. From Darcy law it's evident that if the pressure gradient increases, the flow rate should increase. However, I have difficulty in the Navier Stokes equations. I have solve the Navier Stokes equations in a conduit using SIMPLER method of patankar. The fluid is initially at rest and a constant volumetric flow is injected at the inlet and the outlet is kept at constant pressure. I see for the transient times that the pressure gradient increases and then decreases and reaches the stady stae (say 4pa/m initially and the 0.8pa/m at steady state) my mass balance is conserved, Everything looks ok, except the fact that the pressure gradient does no increase gradually. Do you think I have done some thing wrong? Is there any reference so I can compare my transient pressure and velocity values with?