Estimation of permeability field from sparse measurements of local permeability and water table fluctuations
The permeability of a porous medium is the most important parameter that governs the flow and transport inside the reservoirs (of groundwater, oil & gas, geothermal etc.). Furthermore it is spatially heterogeneous. In practice, permeability of soil cores, which are collected from a few discrete locations, is measured in a laboratory. Afterwards, it is interpolated elsewhere by means of kriging. Sometimes pumping tests are conducted to estimate permeability. However the conventional pumping and injection tests cannot reveal in practice the spatial variability of permeability. We have developed an integrated approach where the permeability field is estimated based on the combination of permeability measured at few locations and time series of water table fluctuations at some observation wells during water injection. The estimation of permeability from water table fluctuation was solved via a stochastic inverse approach, namely the Ensemble Kalman Filter (EnKF). For EnKF, multiple stochastic realizations conditioned to direct measurements of permeability were generated by K-L expansion. We modeled the water table variation due to injection by solving Richards equation for 3-D unsaturated and saturated subsurface flow. The time varying water table change at various observation wells were assumed to be available for assimilation after different time intervals. Our results indicate that around the injection well, the estimated permeability was very close to true permeability and standard deviation was quite small. The RMSE of log permeability decreased to 0.6 from the initial value around 1. Estimation of permeability was performed considering different correlation length, mean and standard deviation of the heterogeneous field than the actual values. The results were robust even for choosing 50% larger standard deviation and correlation length as the initial guess for generating the ensemble than those of the true field. The estimated permeability fields were compared for different spatial interval of well. We observed that the ensemble mean of log permeability field was very poor when the well spacing was more than three times the correlation length of the permeability field.