On the Inverse Problem of the Estimation of Multiphase Flow Functions

摘要

An inverse method is described for simultaneously obtainingrnrelative permeability and capillary pressure curves fromrnmeasurements taken during an unsteady state oil-water flowrnexperiment at the laboratory. This inverse method algorithmrnhas four basic components: 1) an IMPES finite-differencernnumerical simulator of the flow through the core; 2) functionalrnrepresentations -power law models or piece-wise linearrnsplines- of relative permeability and capillary pressure curvesrnin terms of a set of adjustable parameters found by minimizingrnan objective function; 3) the objective function, formed by thernsum of the square of the differences between thernexperimentally measured and the numerically simulated data;rn4) the Quasi-Newton Approximation for the Least-SquaresrnProblem (QNA) technique to minimize that objective function.rnThe method is tested with synthetic and experimental data.rnThe QNA optimization technique performs better than otherrnmethods of the Newton type. Besides, it is simpler tornimplement than the genetic and optimal control algorithms.rnThe accuracy of the relative permeability and capillaryrnpressure estimations depends on the objective functionrndefinition. That definition, in turn, depends of the differentrnkinds of available measurements (flow rates, pressure drop,rnsaturation profiles at different times) and of the functionalrnrepresentations (power functions or piece-wise linear splines)rnof relative permeability and capillary pressure curves.rnAutomatic history matching of a laboratory two-phaserndisplacement to obtain relative permeabilities and capillaryrnpressure is a classical ill-posed inverse problem. The solutionrnof the lack of uniqueness and convergence of the methodrncannot be warranted for all cases by the introduction of morerndata (i.e. saturation profiles), more refined functionalrnrepresentations (piece-wise linear splines) or betterrnoptimization techniques.