Abstract. The Budyko hypothesis (BH) is an effective approach to investigating long-term water balance at large basin scale under steady state. The assumption of steady state prevents applications of the BH to basins, which is unclosed, or with significant variations in root zone water storage, i.e., under unsteady state, such as in extremely arid regions. In this study, we choose the Heihe River basin (HRB) in China, an extremely arid inland basin, as the study area. We firstly use a calibrated and then validated monthly water balance model, i.e., the abcd model, to quantitatively determine annual and monthly variations of water balance for the sub-basins and the whole catchment of the HRB, and find that the roles of root zone water storage change and that of inflow from upper sub-basins in monthly water balance are significant. With the recognition of the inflow water from other regions and the root zone water storage change as additional possible water sources to evapotranspiration in unclosed basins, we further define the equivalent precipitation (P_e) to include local precipitation, inflow water and root zone water storage change as the water supply in the Budyko framework. With the newly defined water supply, the Budyko curve can successfully describe the relationship between the evapotranspiration ratio and the aridity index at both annual and monthly timescales, whilst it fails when only the local precipitation being considered. Adding to that, we develop a new Fu-type Budyko equation with two non-dimensional parameters (ω and λ) based on the deviation of Fu's equation. Over the annual timescale, the new Fu-type Budyko equation developed here has more or less identical performance to Fu's original equation for the sub-basins and the whole catchment. However, over the monthly timescale, due to large seasonality of root zone water storage and inflow water, the new Fu-type Budyko equation generally performs better than Fu's original equation. The new Fu-type Budyko equation (ω and λ) developed here enables one to apply the BH to interpret regional water balance over extremely dry environments under unsteady state (e.g., unclosed basins or sub-annual timescales).