Real-Time Modeling of Capillary Pressure Measurements
Authors: van Lingen P.P. , van Kruijsdijk C.P.J.W 
Published: December 2007

Abstract
Capillary pressure measurements are generally performed with a controlled injection-rate, or with a controlled external capillary pressure. Because of the viscous forces in the porous medium, these measurements will show some degree of error. Accurate measurement requires a slow-time consuming-displacement process. The uncertainty in the measured data is often not calculated.
We focus on the controlled external capillary pressure method. The measurement sequences consist of separate cycles. Each cycle is initiated by a change in the external capillary pressure, followed by an equilibration period. We have developed a real time modeling technique to optimize the measurement method. We apply this technique in a fully automated experimental set-up.

Introduction
Various authors have published work on the on-line analysis of capillary pressure measurements. Kokkedee measured capillary pressure by means of continuous injection in a modified porous plate experiment. The measured data were analyzed through on-line history matching using a numerical simulator. In turn, the results were used to control the pump. He reports that the method is well suited to determine relative permeabilities close to the residual saturations.
Using centrifuge experiments, O'Meara et. al. and Nordtveldt et. al. apply numerical history matching techniques to simultaneously evaluate the relative permeability- and capillary pressure curves.
Jennings et. al. applied numerical history matching in a modified porous plate experiment, in which the phase pressures were applied by means of fluid columns. They report that the proposed technique is relatively quick, and is well suited to determine the small relative oil permeabilities required for the prediction of gravity drainage.
Scope of this Work. In this work, we focus on capillary pressure measurements where the external-boundary-phase pressures are controlled. We derive analytical formulae which describe the saturation response to a step-wise change in the boundary oil- and water pressure. The analytical formulation enables us to make a trade-off between the accuracy, resolution, and measurement time of capillary pressure curves. Additionally, we obtain information on phase relative permeabilities. We apply the described methodology in an automated experimental set-up, in which we measure the capillary pressure curves of glass-bead packs.

Experimental Set-up and Procedure
Fig. 1 shows a schematic diagram of the experimental set-up. The porous sample is placed in a holder with an oil-wet filter placed on the top, and a water-wet filter placed on the bottom. Through the filters, the pressure of the water- and oil column are exerted onto the porous medium. The external pressure difference between the fluid-phases is measured by a differential pressure transducer. The average saturation is calculated from the weight of the oil reservoir on the balance.
Operation. The set-up is operated by a personal computer, which periodically initiates a step-wise change in the height of the water column, and measures the saturation response. The measured data are analyzed in real-time. Based on this analysis, the computer starts a new measurement cycle by changing the height of the water-column.

Saturation Response to a Step-Wise Change of the External Phase Pressures
We define the porous medium as the porous sample including the two semi-permeable filters. P. 425