The representer method for two-phase flow in porous media

The representer method is applied to a one-dimensional two-phase flow model in porous media; capillary pressure and gravity are neglected. The Euler-Lagrange equations must be linearized, and one such linearization is presented here. The representer method is applied to the linear system iteratively until convergence, though a rigorous proof of convergence is out of reach. The linearization chosen is easy to calculate but does not converge for certain weights; however, a simple damping restores convergence at the cost of extra iterations. Numerical experiments are performed that illustrate the method, and quick comparison to the ensemble Kalman smoother is made.