The inverse problem of capillary imbibition involves determination of the capillary geometry from the measurements of the time-varying meniscus position. This inverse problem is known to have multiple solutions, and to ensure a unique solution, measurements of imbibition kinematics in both directions of the capillary are required. We here present a closed-form analytical solution of the inverse problem of determining the axially varying radius of a capillary from experimental data of the meniscus position as a function of time. We demonstrate the applicability of the method for solving the inverse capillary imbibition problem for two cases, wherein the data for imbibition kinematics are obtained (i) using numerical simulations and (ii) from published experimental work. In both cases, the axially varying capillary radius predicted by the analytical solution agrees with the true capillary radius. In contrast to the previously proposed iterative methods for solving the inverse capillary imbibition problem, the analytical method presented here yields a direct solution. This analytical solution of the inverse capillary imbibition problem can be helpful in determining the internal geometry of micro- and nano-porous structures in a non-destructive manner and design of autonomous capillary pumps for microfluidic applications.
In reservoir simulations, model parameters such as porosity and permeability are often uncertain and therefore better estimates of these parameters are obtained by matching the simulation predictions with the production history. Bayesian inference provides a convenient way of estimating parameters of a mathematical model, starting from a probable range of parameter values and knowing the production history. Bayesian inference techniques for history matching require computationally expensive Monte Carlo simulations, which limit their use in petroleum reservoir engineering. To overcome this limitation, we perform accelerated Bayesian inference based history matching by employing polynomial chaos (PC) expansions to represent random variables and stochastic processes. As a substitute to computationally expensive Monte Carlo simulations, we use a stochastic technique based on PC expansions for propagation of uncertainty from model parameters to model predictions. The PC expansions of the stochastic variables are obtained using relatively few deterministic simulations, which are then used to calculate the probability density of the model predictions. These results are used along with the measured data to obtain a better estimate (posterior distribution) of the model parameters using the Bayes rule. We demonstrate this method for history matching using an example case of SPE1CASE2 problem of SPEs Comparative Solution Projects. We estimate the porosity and permeability of the reservoir from limited and noisy production data.
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