LR55 DEP FROM DET TAXIING TO RWY 15; CROSSED RWY 33 WITHOUT ATC CLRNC.

Narrative

THE INCIDENT OCCURRED ON THE SECOND LEG OF A 3 LEG PART 135 CHARTER. WE DROPPED 5 PAX OFF AT THE FBO AT DET; MI. WE WERE RETURNING EMPTY TO RDG; PA. UPON COMPLETING OUR BEFORE TAXI CHKLIST; WE DISCUSSED THE TAXI RTE TO THE EXPECTED DEP RWY (RWY 15). WE RECEIVED A TAXI CLRNC TO TAXI TO RWY 15; CLRED TO CROSS RWY 25. AS WE COMMENCED LEAVING THE RAMP; I ASKED THE SIC; WHO HAD THE TAXI DIAGRAM; HOW TO PICK UP OUR TAXI RTE. HE ADVISED ME TO MAKE A L TURN. AS WE PROCEEDED TO CROSS THE END OF RWY 33; WE BOTH LOOKED TO CLR FINAL BOTH WAYS. ABOUT THE SAME TIME; WE REALIZED IT WAS THE WRONG WAY. I CONTINUED TO TAXI CLR OF RWY 33 AND ADVISED GND. CONTRIBUTING FACTORS INCLUDED FATIGUE FROM AN EARLY WAKE-UP. THE CHARTER BROKER DELAYING THE DEP FROM DET; IN HOPE OF SELLING THE EMPTY LEG. A LOSS OF SITUATIONAL AWARENESS. UPON RETURNING TO RDG AND BEING ADVISED TO CALL DET TWR; I LEARNED FROM THE CTLR THAT 'UNAUTH RWY XINGS' HAPPEN CONSISTENTLY AT DET. THERE ARE NO 'HOT SPOT' TAXI DIAGRAM MARKINGS AND THERE ARE NO ADDITIONAL RWY HOLDING LIGHTS IN PLACE. THE CLOSE PROX OF THE FBO RAMP TO THE INTERSECTING RWYS IS CONFUSING. POSSIBLY; MARKINGS THAT ARE MORE OBVIOUS SHOULD BE CONSIDERED. POSSIBLY; THE EXPANSE OF THE CONCRETE NON MOVEMENT AREA COULD BE PAINTED. IN REVIEWING THE TAXI DIAGRAM; IT ALLUDES ME HOW WE COULD HAVE MADE THE MISTAKEN TURN. OUR TAXI RTE APPEARS OBVIOUS. I SHOULD HAVE ASKED FOR A PROGRESSIVE TAXI TO GET US GOING IN THE RIGHT DIRECTION. ANOTHER CONTRIBUTING FACTOR WAS BEING UNFAMILIAR WITH THE FIELD. FIRST TIME OPERATING THERE FOR THE BOTH OF US.