On the Inverse Problem of the Estimation of Multiphase Flow Functions

Abstract

AbstractAn inverse method is described for simultaneously obtaining relative permeability and capillary pressure curves from measurements taken during an unsteady state oil-water flow experiment at the laboratory. This inverse method algorithm has four basic components: 1) an IMPES finite-difference numerical simulator of the flow through the core; 2) functional representations-power law models or piece-wise linear splines-of relative permeability and capillary pressure curves in terms of a set of adjustable parameters found by minimizing an objective function; 3) the objective function, formed by the sum of the square of the differences between the experimentally measured and the numerically simulated data; 4) the Quasi-Newton Approximation for the Least-Squares Problem (QNA) technique to minimize that objective function. The method is tested with synthetic and experimental data.The QNA optimization technique performs better than other methods of the Newton type. Besides, it is simpler to implement than the genetic and optimal control algorithms. The accuracy of the relative permeability and capillary pressure estimations depends on the objective function definition. That definition, in turn, depends of the different kinds of available measurements (flow rates, pressure drop, saturation profiles at different times) and of the functional representations (power functions or piece-wise linear splines) of relative permeability and capillary pressure curves. Automatic history matching of a laboratory two-phase displacement to obtain relative permeabilities and capillary pressure is a classical ill-posed inverse problem. The solution of the lack of uniqueness and convergence of the method cannot be warranted for all cases by the introduction of more data (i.e. saturation profiles), more refined functional representations (piece-wise linear splines) or better optimization techniques.