The invention relates to a modular drilling rig system comprising multiple components, which system is transfigurable between a transport mode and an operational mode, and comprises: a. a drilling rig mast which is positionable in a vertical orientation; b. a base structure adapted to be positioned on a surface near the well center to support the drilling rig mast thereon, wherein the base structure comprises two elongated members that are connected to each other by a cross beam; c. a displacement system to displace the drilling rig with respect to the surface in a horizontal direction; wherein the displacement system is configured to displace the two elongated members of the base structure together in order to displace the drilling rig, and is configured such that the two elongated members can be displaced with respect to each other when the two elongated members are not connected to each other.