We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.

This study presents an innovative fuzzy inverse method with the finite-element scheme for estimating the unknown time-varying load inputs on a three-dimensional (3D) spatial truss structural system. The finite-element scheme is employed to discretize the problem in space, allowing multidimensional problems of various geometries to be treated. This method is based on the fuzzy Kalman Filter (FKF) technology and the fuzzy weighting recursive least square method (FWRLSM). The fuzzy Kalman filter measures the system responses at two distinct nodes in the 3D spatial truss structure. The fuzzy weighting recursive least square method is derived using the residual innovation sequence to compute the input loads. The proposed method's superiority is demonstrated using several typical simulation cases that vary with different estimator and the distinct levels of the initial process noise covariance and the measurement noise covariance. The results show that this method has great stability and accuracy.