# Applications of Data Assimilation Methods on a Coupled Dual Porosity Stokes Model

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## Abstract

Porous media and conduit coupled systems are heavily used in a variety of areas such as groundwater system, petroleum extraction, and biochemical transport. A coupled dual porosity Stokes model has been proposed to simulate the fluid flow in a dual-porosity media and conduits coupled system. Data assimilation is the discipline that studies the combination of mathematical models and observations. It can improve the accuracy of mathematical models by incorporating data, but also brings challenges by increasing complexity and computational cost. In this paper, we study the application of data assimilation methods to the coupled dual porosity Stokes model. We give a brief introduction to the coupled model and examine the performance of different data assimilation methods on a finite element implementation of the coupled dual porosity Stokes system. We also study how observations on different variables of the system affect the data assimilation process.

## Keywords

Data assimilation Dual porosity Stokes equation Multiphysics## 1 Introduction

Hou et al. [6] has proposed the Coupling of dual porosity flow with free flow as a replacement of the widely used Stokes-Darcy family. The proposed model has a better representation than the traditional Stokes Darcy model in modeling fractured porous media with large conduits. Potential applications of this model include petroleum extraction, hydrology, geothermal systems, and carbon sequestration. A finite element implementation of this model using FEniCS has been developed and studied by the authors [8]. Data assimilation is the discipline that studies the combination of mathematical models and observations. In this paper, we will apply data assimilation methods to the implementation of the coupled model to improve the accuracy of the model predictions [4, 9].

In Sect. 2, we give an introduction to the mathematical model of the coupled dual porosity Stokes model proposed by Hou et al. [6]. In Sects. 3 and 4 we illustrate the applications of data assimilation methods on the coupled dual porosity Stokes model. We set up a data assimilation context from our model in Sect. 3. We present the numerical results based on synthetic data in Sect. 4. In Sect. 5 we draw conclusions and discuss future works.

## 2 A Coupled Dual Porosity Stokes Model

Equation (6) represents the no mass exchange condition between the matrix subsystem in \(\varOmega _d\) and the conduit. This is an assumption based on of the huge difference in permeabilities between the matrix and the microfracture subsystems. Equation (7) imposes conservation of mass exchange between the conduit and the microfracture subsystem on the interface. Equation (8) balances the two forces on the interface: the kinetic pressure in the microfracture subsystem and the normal component of the normal stress in the free flow. Equation (9) is the empirical Beavers-Joseph interface condition [3], which claims that the tangential component of the normal stress incurred by the free flow along the interface is proportional to the difference of the tangential component of flow velocities at two sides of the interface.

## 3 A Data Assimilation Problem Based on the Coupled Model

In order to apply data assimilation methods to the coupled dual porosity Stokes model, we first convert the dual porosity Stokes model into a discrete dynamical system, and define the observations on it.

Similarly, the Dirichlet boundary conditions will also cause singularities as they do not depend on previous boundary values. We remove all Dirichlet boundary values from the state variable \(v_t\) using the same technique.

We base the dynamical model on a two dimensional dual porosity Stokes model shown in Fig. 2. Let \(\varOmega =[-0.5,0.5]\times [0,1]\) be a shifted unit square, Open image in new window , and Open image in new window . The interface is Open image in new window . The domain is partitioned uniformly into \(\frac{1}{16}\times \frac{1}{16}\) squares.

Also, we let \(\varDelta t = 0.01\), \(\xi _t\sim \fancyscript{N}(\mathbf {0},5\pmb I)\), \(v_0^*\sim \fancyscript{N}(\mathbf {0},100\pmb I)\). The large variance of \(v_0^*\) indicates that we have little knowledge about the initial condition.

## 4 Numerical Results

We run the model against the three dimensional variational method (3DVAR), the strong constraint four dimensional variational method (s4DVAR) with a time window with length \(0.04\), the extended Rauch-Tung-Striebel smoother (ExtRTS) [15], the extended Kalman Filter (ExtKF) [10], the ensemble Kalman Filter (EnKF) [5, 7] with 100 particles, and ensemble Rauch-Tung-Striebel smoother (EnRTS) [13] with 100 particles. Note that since we have a linear data assimilation problem, the extended methods ExtRTS and ExtKF are just the Rauch-Tung-Striebel smoother (RTS) and the Kalman Filter (KF). We also use a baseline filtering method *Forward* that only uses the mathematical model \(\varvec{\varPsi }\) and ignores all data. It starts at \(v_0^*=\mathbf {0}\) and then applies \(\varvec{\varPsi }^*\) to get an approximation for \(v_t^*\). Since the model is linear, we expect an optimal solution by ExtKF for filtering and ExtRTS for smoothing.

All numerical experiments were run with the data assimilation package DAPPER [14] on the Teton computer cluster at the Advanced Research Computing Cluster (ARCC) at the University of Wyoming.

Average root mean square error for filtering (rmse_f) and elapsed time

Forward | 3DVAR | ExtKF | EnKF | |
---|---|---|---|---|

rmse_f | 0.4717 | 0.2824 | 0.2604 | 0.2651 |

elapsed time | 5 s | 1 s | 5 s | 56 s |

Average root mean square error for smoothing (rmse_s) and elapsed time

s4DVAR | ExtRTS | EnRTS | |
---|---|---|---|

rmse_s | 0.3 | 0.1907 | 0.2033 |

elapsed time | 121 s | 32 s | 62 s |

The error of different data assimilation methods over time are shown in Figs. 3 and 4. Since Forward, 4DVAR, ExtKF, and EnKF all start with an initial guess \(\tilde{v_0}=\mathbf {0}\), they all have the same predictions at \(t=0.01\). This is why they all have the same error at \(t=0.01\) for forecasting as shown in Fig. 3. The predictions are made every 0.01 time units. ExtKF has a smaller forecasting error than all the other methods except for 3DVAR. Our 3DVAR implementation utilizes all true states to approximate the background covariance \(B_t\). The exposure to the true states enables the 3DVAR implementation to surpass the theoretical optimal solution from the Kalman Filter. EnKF has a result very similar to that of ExtKF. In EnKF, the calculations of mean and variance of the states are approximated using the Monte Carlo method. Since the states follows a Gaussian process, the approximations converge to the truths as the number of particles increases. We can also see in Fig. 4 that by utilizing all observations, the smoothing error at \(t=0.01\) is reduced by half, comparing to the forecasting error in Fig. 3. Note that the Kalman Smoother ExtRTS achieves the best result at all time, and the ensemble Kalman Smoother EnRTS has a very similar result as ExtRTS, but consumes much more computation time as shown in Table 2.

*Forward*also has a decreasing error with respect to time. This is caused by the characteristics of our dynamical system. Because of the essential boundaries in our coupled model, solutions to the PDE system with different initial conditions all converge to each other as \(t\rightarrow \infty \). This can also be explained by the linear dynamical system. Consider a linear dynamical system with \(\varvec{\varPsi }\)(\(v_t\)) = \(Mv_t\) where Open image in new window . Then \(\varvec{\varPsi }^{(n)}(v_t)\) \(\rightarrow \mathbf {0}\) as \(t\rightarrow 0\).

We can see from Fig. 7 that the data on the flow pressure \(p_m\) in the matrix subsystem in the dual porosity subdomain provides most of the information while the other two variables provide little improvement over the *Forward* baseline method, which uses no observation at all. This behavior exists in all our test models with different boundary conditions, source terms and geometries. This phenomenon needs further investigation. Here we conclude that in our limited test cases, observations on \(p_m\) provide significant information about the true states while observations on \(p_f\) and \({u}\) do not.

## 5 Conclusions and Future Work

In this paper, we introduced the coupled dual porosity Stokes model. We set up a data assimilation problem based on the coupled model and applied different data assimilations to solve the problem. Due to the linearity of the coupled dual porosity Stokes model, the Kalman Filter and the Kalman Smoother achieve optimal solutions for filtering and smoothing, respectively, as expected. From our numerical experiments we have seen that observations of pressures in the matrix subsystem contain most of the useful information for data assimilation.

Future work includes exploring different data assimilation methods on the nonlinear coupled dual porosity Navier-Stokes model, applying data assimilation methods with experiment data and investigating the reason behind the uneven distribution of information in different variables.

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