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The coiled tubing injector head is the core component of the coiled tubing oil and gas well operation equipment, which directly determines the coiled tubing operation capability, and its chain drive system pulls the coiled tubing in or out the wellbore. However, the unsteady chain drive and the wear between the chain and sprocket are common problems which are the main factors leading to slipping, pipe sliding and bites of coiled tubing in engineering practice. Based on the standard roller chain and sprocket, the mechanical model of the engagement contact between the injector head chain and the sprocket teeth is established to analyze the influence of chain tension force, chain tension angle and chain pitch on the position of contact points and tooth surface support reaction force, and the accuracy of the contact mechanical model is verified by Ansys simulation which lays a theoretical foundation for the design and safety of the injector. The research shows that when the tension force of the chain increases and the loosening force remains unchanged, the engaging-in position of the chain on the tension side is basically unchanged, the reaction force of the tooth surface support increases, and the engaging-out position of the chain rollers is close to the bottom of the sprocket teeth. However, when the tension force approaches the loosening force, there is a sudden increase in the support reaction force of the engaging-out contact position. The increase of the angle between the tension side and the vertical direction of the chain has little effect on the engaging-in position and the tooth surface support reaction force, the engaging-out position of the chain rollers is close to the bottom of the sprocket teeth. The engaging in and out points of the chain are respectively away from the direction of the bottom of the sprocket teeth and the contact position is improved when the chain is extended.