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Physical systems are characterized by inherent symmetries, one of which is encapsulated in theunits of their parameters and system states. These symmetries enable a lossless order-reduction, e.g.,via dimensional analysis based on the Buckingham theorem. Despite the latter's benefits, machinelearning (ML) strategies for the discovery of constitutive laws seldom subject experimental and/ornumerical data to dimensional analysis. We demonstrate the potential of dimensional analysis to significantlyenhance the interpretability and generalizability of ML-discovered secondary laws. Ournumerical experiments with creeping fluid flow past solid ellipsoids show how dimensional analysisenables both deep neural networks and sparse regression to reproduce old results, e.g., Stokes law fora sphere, and generate new ones, e.g., an expression for an ellipsoid misaligned with the flow direction.Our results suggest the need to incorporate other physics-based symmetries and invariancesinto ML-based techniques for equation discovery. 

The ability to quantify evapotranspiration (ET) is crucial for smart agriculture and sustainable groundwater management. Efficient ET estimation strategies often rely on the vertical‐flow assumption to assimilate data from soil‐moisture sensors. While adequate in some large‐scale applications, this assumption fails when the horizontal component of the local flow velocity is not negligible due to, for example, soil heterogeneity or drip irrigation. We present novel implementations of the ensemble Kalman filter (EnKF) and the maximum likelihood estimation (MLE), which enable us to infer spatially varying ET rates and root water uptake profiles from soil‐moisture measurements. While the standard versions of EnKF and MLE update the predicted soil moisture prior to computing ET, ours treat the ET sink term in Richards' equation as an updatable observable. We test the prediction accuracy and computational efficiency of our methods in a setting representative of drip irrigation. Our strategies accurately estimate the total ET rates and root‐uptake profiles and do so up to two‐orders of magnitude faster than the standard EnKF.

 

Multiscale heterogeneity and insufficient characterization data for a specific subsurface formation of interest render predictions of multi‐phase fluid flow in geologic formations highly uncertain. Quantification of the uncertainty propagation from the geomodel to the fluid‐flow response is typically done within a probabilistic framework. This task is computationally demanding due to, for example, the slow convergence of Monte Carlo simulations (MCS), especially when computing the tails of a distribution that are necessary for risk assessment and decision‐making under uncertainty. The frozen streamlines method (FROST) accelerates probabilistic predictions of immiscible two‐phase fluid flow problems; however, FROST relies on MCS to compute the travel‐time distribution, which is then used to perform the transport (phase saturation) computations. To alleviate this computational bottleneck, we replace MCS with a deterministic equation for the cumulative distribution function (CDF) of travel time. The resulting CDF‐FROST approach yields the CDF of the saturation field without resorting to sampling‐based strategies. Our numerical experiments demonstrate the high accuracy of CDF‐FROST in computing the CDFs of both saturation and travel time. For the same accuracy, it is about 5 and 10 times faster than FROST and MCS, respectively.

 

Identification of a heterogeneous conductivity field and reconstruction of a contaminant release history are key aspects of subsurface remediation. These two goals are achieved by combining model predictions with sparse and noisy hydraulic head and concentration measurements. Solution of this inverse problem is notoriously difficult due to, in part, high dimensionality of the parameter space and high computational cost of repeated forward solves. We use a convolutional adversarial autoencoder (CAAE) to parameterize a heterogeneous non‐Gaussian conductivity field via a low‐dimensional latent representation. A three‐dimensional dense convolutional encoder‐decoder (DenseED) network serves as a forward surrogate of the flow and transport model. The CAAE‐DenseED surrogate is fed into the ensemble smoother with multiple data assimilation (ESMDA) algorithm to sample from the Bayesian posterior distribution of the unknown parameters, forming a CAAE‐DenseED‐ESMDA inversion framework. The resulting CAAE‐DenseED‐ESMDA inversion strategy is used to identify a three‐dimensional contaminant source and conductivity field. A comparison of the inversion results from CAAE‐ESMDA with physical flow and transport simulator and from CAAE‐DenseED‐ESMDA shows that the latter yields accurate reconstruction results at the fraction of the computational cost of the former.

 

Water‐hammer equations are used to describe transient flow in pipe networks. Uncertainty in model parameters, initial and boundary conditions, and location and strength of a possible leak renders deterministic predictions of this system untenable. When deployed in conjunction with pressure measurements, probabilistic solutions of the water‐hammer equations serve as a tool for detecting leaks in pipes. We use the method of distributions to obtain a probability density function (PDF) for pressure head, whose dynamics are described by the stochastic water‐hammer equations. This PDF provides a prior distribution for subsequent Bayesian data assimilation, in which data collected by pressure sensors are combined with this prior to obtain a posterior PDF of the leak location and size. We conduct a series of numerical experiments with uncertain initial velocity and measurement noise to ascertain the robustness and accuracy of the proposed approach. The results show the method's ability to identify the leak location and strength in a water transmission main.

 

Evapotranspiration is arguably the least quantified component of the hydrologic cycle. We propose two complementary strategies for estimation of evapotranspiration rates and root water uptake profiles from soil‐moisture sensor‐array data. One is our implementation of ensemble Kalman filter (EnKF); it treats the evapotranspiration sink term in the Richards equation, rather than soil moisture, as the observable to update. The other is a maximum likelihood estimator (MLE) applied to the same observable; it is supplemented with the Fisher information matrix to quantify uncertainty in its predictions. We use numerical experiments to demonstrate the accuracy and computational efficiency of these techniques. We found our EnKF implementation to be two orders of magnitude faster than either the standard EnKF or MLE, and our MLE procedure to require an order of magnitude fewer iterations to converge than its counterpart applied to soil moisture. These findings render our methodologies a viable and practical tool for estimation of the root water uptake profiles and evaporation rates, with the MLE technique to be used when the prior knowledge about evapotranspiration at the site is elusive.