Simultaneous Estimation of Relative Permeabilities and Capillary Pressure

Abstract

Summary An automatic numerical method is proposed for simultaneous determination ofrelative permeabilities and capillary pressure from the results of a singletwo-phase flow experiment in pressure from the results of a single two-phaseflow experiment in a range of velocities representative of field flowconditions. The experiment is a standard imbibition or drainage displacementwith an imposed difference of pressure or flow rate. The experimental data usedare the outlet production, the pressure drop between the two faces, and localsaturation profiles at different times along the core. Relative permeabilitiesand capillary pressure are estimated with a least-squares technique. Use of ahigh-order numerical scheme significantly reduces the numerical diffusion. Theoptimal control theory is used to solve the minimization problem. Thecorresponding FORTRAN code is generated by a symbolic problem. Thecorresponding FORTRAN code is generated by a symbolic program. The validity ofthe method first is demonstrated with program. The validity of the method firstis demonstrated with simulated data, then is tested in two laboratoryexperiments one imbibition and one drainage. Good agreement is obtained forboth situations. Introduction Simulation of multiphase flow in porous media requires knowledge of relativepermeabilities. Such data can be obtained by laboratory experiments, eithersteady- or unsteady-state methods. Steady-state methods are time-consuming, andseveral days are needed to acquire a complete set of relative permeabilities. Thus, these methods are used less commonly than unsteady-state methods. Unsteady-state methods are limited to displacements where the assumptionunderlying the Buckley-Leverett theory is satisfied; capillary pressure must beignored. Another limitation of these methods is that relative permeabilitiesare not calculated in the full saturation range. permeabilities are notcalculated in the full saturation range. To improve these unsteady-statemethods, analytical and adjustment techniques have been developed to takecapillary pressure into account and to obtain relative permeabilities for theoverall saturation range. Analytical methods provide relative permeabilities bydifferentiation of Darcy's law. Islam and Bentsen used saturation and pressureprofiles. Acquisition of the pressure data requires special laboratoryequipment. In Civan and Donaldson's method, gravity effects are not taken intoaccount. Adjustment methods infer relative permeabilities from pressure dropand production data obtained during laboratory experiments. Differentsimulations of a reservoir model are compared to calculate a single set ofrelative permeabilities by history matching of production and pressure data. This can be done automatically with a nonlinear least-squares approach to matchthe pressure drop and production history. The different automatic pressure dropand production history. The different automatic adjustment methods currentlyavailable may or may not account for capillary pressure and require one or moreexperiments. Chavent et al proposed an automatic adjustment method. Twoexperiments are needed for the interpretation: one with a high flow rate, whichallows capillary pressure to be ignored, and one with a low flow rate to takecapillary effects into account. The experimental data required are the pressuredrop and the cumulative volume of fluid produced as functions of time. Kerigand Watson estimated relative permeabilities from one displacement experiment. The data measured are the pressure drop across the core sample and thecumulative volume of displaced phase recovered. The relative permeabilities arerepresented by cubic spline. Yang and Watson again used the previous method, adding a Bayesian-type performance index to incorporate prior estimates ofrelative permeability curves into an automatic history-matching algorithm. Themethod was tested on hypothetical waterfloods, and good results were obtained. In Ref. 10, the capillary pressure was ignored, but the same method also wasused to estimate relative Permeability and capillary pressure curvessimultaneously. Permeability and capillary pressure curves simultaneously. Capillary pressure curves usually are obtained from laboratory tests withvarious techniques (porous plate, centrifugation, etc.). We describe in thispaper a method for the simultaneous estimation of relative permeabilities andcapillary pressure. This method incorporates the following new features withrespect to previous work.It is now possible to measure saturation profilesexperimentally along the core sample by gamma ray attenuation technique orcomputerized tomography scanning in addition to the pressure drop and theproduction data. pressure drop and the production data.A new scheme withslope limiters and implicit discretization for capillary diffusion has beendeveloped. This scheme has low numerical dispersion and good stability andclosely approaches the entropic solution.Development time and reliability of the code are improved by use of a symbolic program to generate thecorresponding FORTRAN code from the input of the discretized equations and forthe error function. As in previous work the optimal control theory is used tosolve the minimization problem in the least-squares method. Compared with Richmond and Watson's approach, our method uses additional measurements ofsaturation profiles. These additional data make the optimization problem easierto solve, especially for the estimation of capillary pressure. The basic modelsused to describe the different laboratory experiments studied are discussed, and then the data available from these experiments are presented. The thirdpart explains the parameter estimation method, and the fourth part describesthe parameter estimation method, and the fourth part describes the differentparameter representations. Finally, different test cases are presented. Basic Models The equation and boundary conditions used in the code are outlined in thissection. The, variation of water saturation inside the core is simulated with a1D, two-phase incompressible model based on Darcy's law, together with variousboundary conditions enabling the simulation of the following laboratoryexperiments.Forced drainage with given injection rate or given pressuredrop. The corresponding saturation boundary conditions pressure drop. Thecorresponding saturation boundary conditions are no flow for the wetting phaseat the injection end and no flow for the nonwetting phase at the production endunless the wetting-fluid saturation falls to the value at which the capillarypressure vanishes, after which both wetting and nonwetting phases pressurevanishes, after which both wetting and nonwetting phases are allowed to flow(unilateral boundary condition).Gravity drainage where fluids are flowingonly under gravity and capillary forces. Saturation boundary conditions are noflow for the wetting phase at the injection end and a given wetting-fluidsaturation at the production end, the imposed value being that of the initialsaturation profile.Forced imbibition with given injection rate or given pressure drop. pressure drop.