Comparison of WHAK and MAC weathering formulations with our geologic carbon cycle model for a planet with Earth-like hypsometry and instellation (Bond albedo, α = 0.3). Mass-dependent water ingassing is assumed, and escape processes are included in these calculations. The WHAK model is denoted by shades of orange for varying initial water inventories, and the MAC model is denoted by shades of blue. Under the WHAK weathering formulation, the system maintains habitable temperatures even for very small initial surface water reservoirs. In contrast, the MAC model demonstrates a runoff dependency with runaway warming at low water inventories.