Portal vein thrombosis is common in the cirrhotic population, though not a medical emergency. You can let the sun set on one while contemplating the risks and benefits of anticoagulation.
It can cause pain, or worsening of ascites in the cirrhotic patient (in the non-cirrhotic it should not cause ascites since it is pre-sinusoidal). If you suspect its duration is less than one month, and if there are no contraindications to anticoagulation (ie. recently banded varices, active alcohol abuse) then begin treatment. Remember, you don’t want to see propagation of a PVT into the superior mesenteric vein, which could preclude transplant candidacy.