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1. An Implementation of a Coupled Dual-Porosity-Stokes Model with FEniCS

   Hu, X.; Douglas, C.C. (June 2019, Lecture notes in computer science)

   Porous media and conduit coupled systems are heavily used in a variety of areas. A coupled dual-porosity-Stokes model has been proposed to simulate the fluid flow in a dual-porosity media and conduits coupled system. In this paper, we propose an implementation of this multi-physics model. We solve the system with the automated high performance differential equation solving environment FEniCS. Tests of the convergence rate of our implementation in both 2D and 3D are conducted in this paper. We also give tests on performance and scalability of our implementation. 

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

   Hu, Xiukun; Douglas, Craig C. (July 2020, Lecture notes in computer science)

   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 stud- ies 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. 

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3. Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

   https://doi.org/10.1002/nme.6158  

   Al Mahbub, Md. Abdullah; He, Xiaoming; Nasu, Nasrin Jahan; Qiu, Changxin; Zheng, Haibiao (July 2019, International Journal for Numerical Methods in Engineering)

   Summary In this paper, we propose and analyze two stabilized mixed finite element methods for the dual‐porosity‐Stokes model, which couples the free flow region and microfracture‐matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual‐porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. 

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4. Evaluation of Dual Domain Mass Transfer in Porous Media at the Pore Scale

   https://doi.org/10.1111/gwat.13328  

   Dorchester, Leland; Day‐Lewis, Frederick_D; Singha, Kamini (June 2023, Groundwater)

   Abstract Dual‐porosity models are often used to describe solute transport in heterogeneous media, but the parameters within these models (e.g., immobile porosity and mobile/immobile exchange rate coefficients) are difficult to identify experimentally or relate to measurable quantities. Here, we performed synthetic, pore‐scale millifluidics simulations that coupled fluid flow, solute transport, and electrical resistivity (ER). A conductive‐tracer test and the associated geoelectrical signatures were simulated for four flow rates in two distinct pore‐scale model scenarios: one with intergranular porosity, and a second with an intragranular porosity also defined. With these models, we explore how the effective characteristic‐length scale estimated from a best‐fit dual‐domain mass transfer (DDMT) model compares to geometric aspects of the flow field. In both model scenarios we find that: (1) mobile domains and immobile domains develop even in a system that is explicitly defined with one domain; (2) the ratio of immobile to mobile porosity is larger at faster flow rates as is the mass‐transfer rate; and (3) a comparison of length scales associated with the mass‐transfer rate (L_α) and those associated with calculation of the Peclet number (L_Pe) showL_Peis commonly larger thanL_α. These results suggest that estimated immobile porosities from a DDMT model are not only a function of physically mobile or immobile pore space, but also are a function of the average linear pore‐water velocity and physical obstructions to flow, which can drive the development of immobile porosity even in single‐porosity domains. 

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5. A Three‐Field Formulation for Two‐Phase Flow in Geodynamic Modeling: Toward the Zero‐Porosity Limit

   https://doi.org/10.1029/2023JB027469  

   Lu, Gang; May, Dave_A; Huismans, Ritske_S (December 2023, Journal of Geophysical Research: Solid Earth)

   Abstract Two‐phase flow, a system where Stokes flow and Darcy flow are coupled, is of great importance in the Earth's interior, such as in subduction zones, mid‐ocean ridges, and hotspots. However, it remains challenging to solve the two‐phase equations accurately in the zero‐porosity limit, for example, when melt is fully frozen below solidus temperature. Here we propose a new three‐field formulation of the two‐phase system, with solid velocity (v_s), total pressure (P_t), and fluid pressure (P_f) as unknowns, and present a robust finite‐element implementation, which can be used to solve problems in which domains of both zero porosity and non‐zero porosity are present. The reformulated equations include regularization to avoid singularities and exactly recover to the standard single‐phase incompressible Stokes problem at zero porosity. We verify the correctness of our implementation using the method of manufactured solutions and analytic solutions and demonstrate that we can obtain the expected convergence rates in both space and time. Example experiments, such as self‐compaction, falling block, and mid‐ocean ridge spreading show that this formulation can robustly resolve zero‐ and non‐zero‐porosity domains simultaneously, and can be used for a large range of applications in various geodynamic settings. 

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