Uncertainty is a structural and inevitable characteristic of all hydrological processes arising from the essential complexity of the related natural systems. The quantification of uncertainty in the geologic properties of the subsurface is of major interest in subsurface flow applications, and necessary to account for the risk within planning and decision- making.

Typically, uncertainty of a priori geologic parameters, for example, the permeability of the subsurface, significantly affects the flow predictions. A common strategy to reduce uncertainty in the modeling is to incorporate data, typically corrupted by noise, that arise from the flow response to the given geologic scenario. Because of the prior uncertainty in geological information, and because of the noise in the data, this assimilation of data into subsurface flow models is naturally framed in a Bayesian fashion.

Figure 1: Illustration of spatial distribution of permeability in complex geology (using FEFLOW software, a development of DHI-WASY, http://www.dhigroup.com).

### Bayesian inverse problem

Within the framework of the industrial project we study an inverse problem arising in groundwater flow simulations. The inverse problem consists in identifying the per- meability of the subsurface and the uncertainty associated with it, from hydraulic head measurements based on a steady Darcy model of groundwater flow.

A parameterisation of permeability fields leads to a parameter identification problem for a finite number of unknown parameters determining the geometry (e.g., for layered structures) and physical values of permeabilities. Since the parameters of the model to be estimated are uncertain all relevant information my be obtained via their probability density functions. Bayesian framework provides a **quantication of the uncertainty **via the posterior distribution of unknown parameters given noisy data and prior distribution of parameters.

### Computational approaches

For numerical experiments, efficient computational approaches to explore the posterior of the unknown parameters can be applied, for instance

- Markov Chain Monte Carlo (MCMC)

- Kalman Filter (KF): Ensemble KF, PCE-based KF

These numerical algorithms provide estimates of the permeability and the uncertainty associated with it, and only require forward model evaluations.

### Numerical experiment

In the following numerical experiment, the inverse problem of identifying the per- meability in one-layer model from synthetic data is considered. The parametrisation of the random log permeability field via the Karhunen-Loeve (KL) expansion leads to a parameter identification problem for the KL coefficients.

Figure 2: The ”true” permeability represented by KL expansion with 9 KL modes (left), the corresponding solution of Darcy equation (middle) and measurement locations (right).

Figure 3: Posterior densities of the unknown param. (the first 4 KL coef.) identified by the Ensemble KF method.

Synthetic data is generated by using the true permeability in Darcy equation and specified by 36 measurement locations as depicted in Figure 2. Figure 3 shows the prior distribution and updates of the Ensemble KF method for posterior densities of unknown parameters. The resulting posterior densities depicted in red concentrate around the truth. In contrast to deterministic approach application of the Bayesian framework pro- vides a reasonable estimation of the truth alongside with an accurate estimate of its uncertainty.

### Project framework

The project ”Efficient mathematical methods for model calibration and uncertainty quantification in environmental simulations” is funded by the Investitionsbank Berlin within the framework of the program for support of research, innovation and technolo- gies (ProFIT). It is a joint project between Weierstrass Institute for Applied Analysis and Stochastics (WIAS) and DHI-WASY GmbH located in Berlin.

**Con****tact**

N*ataliya Togobytska *WIAS Mohrenstraße 39 10117 Berlin Nataliya.Togobytska@wias-berlin.de

*Zahra Lakdawala *DHI-WASY GmbH Volmerstrae 8 12489 Berlin zla@dhigroup.com

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