An inhibitive water-based polymer mud system and method for using the system in drilling and in stabilizing wellbores is disclosed for use in water sensitive formations as an alternative to oil-based muds or water-based muds comprising ferro-chrome lignosulfonates. The system comprises a fresh water or salt water base thinned or dispersed with a sulfonated acrylic copolymer having a hybrid/graft lignosulfonate multipolymer structure containing carboxylate and sulfonate functional groups with synthetic polymer side chains covalently linked to a base lignosulfonate material, having a molecular weight in the range of about 1,000 to about 15,000, and having a high anionic charge density. This system is effective and has stable rheology over a broad pH range, even at a near neutral pH of 8.0. The drilling fluids do not contain heavy metals and are rheologically tolerant to contaminants such as cement, anhydrite and sodium and temperatures as high as about 400° F.