Introduction to Modeling of Hydrogeologic Systems Using Fuzzy Differential Equations

The many simultaneously occurring processes in unsaturated-saturated heterogeneous soils and fractured rocks can cause field observations to become imprecise and incomplete. As a result, field observations can become inconsistent with deterministic and stochastic mathematical models used for predictions. The performance of a system that includes soil, rock, and a monitoring network can become uncertain because of vagueness or “fuzziness,” which is inherent in the system behavior, rather than being purely random. Fuzzy systems modeling, already widely used in such fields as engineering, physics, chemistry, and biology, is expected to become useful in simulating hydrogeologic system behavior. After presenting a hydrogeologic system as a fuzzy system, we derive a fuzzy-logic form of Darcy’s equation. Based on this equation, a fuzzy logic form of the parabolic-type partial differential equations is derived. The elliptic-type (Laplace equation) and the parabolic-type (Richards equation) partial differential equations were approximated using fuzzy variables and solved using the basic principles of fuzzy arithmetic. The results of fuzzy systems modeling are then compared with those obtained using deterministic models. The application of fuzzy ordinary and partial differential equations to various earth sciences problems, such as flow and transport in the subsurface, is an emerging area of research.