The hysteretic linear reservoir—a new Preisach model

The upper layers of the ground form a three-phase system containing air, water and particles of soil. The water content varies from almost zero (oven dry) to full (saturated) after which rainwater ponds on the surface of the ground. In the unsaturated zone, hysteresis in the contact angle of the air–water interface attached to soil particles is believed to be responsible for the hysteresis observed in the flow of soil–water at laboratory scale. As the flow of water changes direction, the air–water–soil contact angle flips. Conceptual models of hydrological processes at larger scales have, until now, ignored this hysteresis. Using one of the simplest conceptual models in hydrology—the single linear reservoir—this paper shows how rate-independent hysteresis can be inserted into conceptual models of flows of water at large scale. The porous matrix of the soil in a given area, such as a river catchment or basin, is the reservoir of soil–water. In the linear reservoir model the instantaneous rate of outflow in a river draining the reservoir is proportional to the volume, or mass, of water in the soil-reservoir. Hysteresis may be introduced most easily by making the coefficient of proportionality a hysteron jumping between two values. A differential equation stating that water-mass is conserved closes the model. Making the coefficient of proportionality equal to the weighted output of a set of hysterons arranged in parallel results in a new Preisach model—the hysteretic linear reservoir. This opens up the prospect of better prediction of hydrologic flows in the environment, such as floods. The problems of prediction and identification using synthetic data remain to be solved before application to real data.