A Direct Inverse Model to Determine Permeability Fields from Pressure and Flow Rate Measurements

The determination of the permeability field from pressure and flow rate measurements in wells is a key problem in reservoir engineering. This paper presents a Double Constraint method for inverse modeling that is an example of direct inverse modeling. The method is used with a standard block-centered finite difference method. With an a priori grid block permeability field as input, two forward runs are made: the first is constrained with the measured pressures; the second is constrained with the measured flow rates. We calculate the pressures in the grid block centers from the first run, while from the second run we calculate the fluxes through the faces between the grid blocks. Substitution of these pressures and fluxes into Darcy’s law then yields the transmissibilities at the faces and hence the permeabilities in the grid blocks. In this way the “hard” data (measured pressures and flow rates) are always honored while the “soft”, geological data can be incorporated at the discretion of the geologist. Using a synthetic example, we demonstrate the method and compare the results with another method: Ensemble Kalman Filtering. The two methods agree within the scope of their applicability. The Double Constraint method focuses initially on determining spatial distributions of the permeability field for single-phase, steady state flow. For history matching an extension is required to non-steady state, two-phase flow conditions, which is already possible with EnKF. We are currently investigating the possibility of combining the two methods, whereby the strengths of the two methods could be fully exploited.