Date: 202105
Local time of day: 1201-1800
Location: ZZZ.ARTCC, US
Flight conditions: IMC
Aircraft: Light Transport; Low Wing; 2 Turbojet Eng
Operator: Air Taxi
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb; Initial Climb
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Reporter organization: Air Carrier
Reporter function: Captain; Pilot Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Communication Breakdown
Anomaly: ATC Issue All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Fuel Issue; Inflight Event / Encounter Weather / Turbulence
Result: Flight Crew Requested ATC Assistance / Clarification
Contributing factors: Procedure; Weather
Primary problem: Procedure

Narrative: The flight from ZZZ to ZZZ1 was the third leg of our day. We had landed at XA00Z which was an hour advance of planned passenger's arrival. Our passengers soon notified us of late arrival upon which I checked to ensure we would not be affected by EDCT (Estimated Departure Time Clearance) and that weather wouldn't be a factor. I then obtained our flight clearance which included a full route meaning we weren't getting our shorter overwater route. That route would also avoid any of the approaching weather by a wide margin. I re-ran our flight plan to ensure we had adequate fuel as the new one was 110 NM longer and would slightly change winds. The original route required 3;570 lbs total including enroute burn; 45 min reserve; and alternate based on cruise altitude of FL400. No alternate was required though as destination weather was > 5000/5 for the entire day and projected to remain so. Still; we plan and carry alternate fuel anytime we have the ability to do so. I had rounded the final fuel load up to 4;000 lbs total as it's all too common flying in or out of ZZZ3 airspace and suddenly getting significantly different routing or held down below cruise altitude for a bit. That decision would prove critical in preventing a divert later in flight. We boarded passengers at XB30Z and were given taxi clearance to 30. Upon nearing the runway; we were told to taxi to the holding area and would start taxi again at XB50Z for a XB54Z release. Approaching XC00Z; we were cleared to taxi and then into position and hold where we sat for approximately 3 minutes before being cleared for takeoff. Shortly after checking in with departure; we were given a 250 kt speed restriction and then given multiple vectors 90 degrees off of route. During most of this we were held below 15;000 ft. After being cleared back onto our flight plan; we were step climbed 6 times until we reached FL 330 about 30 minutes into the flight. During this time we also had to request deviations for weather. A slight left deviation would've allowed us to safely clear all the weather even at the lower altitude; but all aircraft were being directed that right was the only choice. Based on what traffic we had directly in view and on TCAS; it didn't make sense as it looked clear with all possible conflicts being to the west. We attempted to work right but even with significant deviations directly westward we were going to be boxed in by cells. Both NEXRAD and our onboard radar indicated yellow to red echoes well above our current altitude. We and other aircraft at higher altitudes continued to request eastward deviations and were denied. We also repeatedly requested higher as that would've been a satisfactory way of being able to comply with what they needed; but again were denied. At this point; I told the ZZZ3 Center controller that we had no choice but to go left as there was no place we could safely penetrate the weather. They finally relented and after a short deviation that was within 2-3 NM east of the course line; we were around it and back on flight plan. I was prepared to declare pilot authority to deviate for safety of aircraft; passengers and crew had they not. Between the additional ground time; ATC vectors; deviations; and hold down; we had eaten significantly into the additional fuel and were showing below a 45 min reserve if we remained at the lower altitude. This was even after seeing the deteriorating fuel state early and slowing down significantly from cruise speed to M.65. After again requesting further climb and being denied; I declared minimum fuel. Instead of the controller acknowledging; we were shifted to the next controller. Upon checking in; we were still held down and told to expect some time before climb. I again interjected on the radio that we were minimum fuel which was acknowledged by the new controller by him stating; 'uh..ah...well; ok.' He then stated we wouldn't get any further climb until with ZZZ4 Center. By that time we would be over State andstill at FL330 just over an hour into flight. The PM and I evaluated the fuel state and knowing the potential ATC delays we could face in ZZZ1 area airspace despite the great weather that we may be facing a divert. I kept our home base of ZZZ5 and ZZZ6 in mind should that be required. We set 1;000 lbs as an acceptable; safe fuel state needed to both put us above minimum fuel and give us some additional in the event of further delays. After ensuring ZZZ4 Center knew of our fuel state; we were held down for nearly another 10 minutes and then stepped 4 times from FL330 to FL400. After reaching FL400; we pulled back to M.67 and finally saw 1;100 lbs overhead destination and decided to continue. I considered requesting FL430; but based on less tailwind and the increased burn to climb I decided it wouldn't be beneficial. Fuel state improved to 1;280 lbs as the winds shifted and thanks to a small shortcut by Center. During decent with ZZZ4 Center; we were turned roughly 120 degrees off the arrival and it wasn't until questioning the controller if he knew that we had declared that he stated no and turned us back towards ZZZ1. Even though I had never rescinded the previous declaration; it apparently had not been passed to some of the subsequent ATC sectors. I told the controller that we were currently above min fuel but would not continue to be with continued vectors away from destination. We were shortly passed off to the approach controller and landed uneventfully from the RNAV XX. Final fuel state at shutdown was 1;145 lbs.Our passengers being on time might have alleviated some of this as weather had not moved onto our route at that time. More fuel would've have also created more pad; but the additional alternate fuel that wasn't required by regulation or the situation plus additional contingency fuel should've been more than sufficient to prevent it. Getting held down well below planned cruise altitude until almost halfway through a 2.5hr flight is pretty uncommon but played the biggest role in this.




Synopsis: EMB-505 Captain reported being denied shorter weather avoidance route and ATC's denial to climb or deviate for weather; resulted in unplanned minimum fuel situation.