Numerical modeling of thermo-hydro-mechanical coupling processes in porous media
Thermo-hydro-mechanics (THM) is a branch of mechanics aimed to predict how deformable porous media behave, while heat transfer and fluid transport simultaneously occur in the pores filled by liquid and/or gas. Understanding these multi-physical responses is important for a wide spectrum of modern engineering applications, such as tissue scaffolding, geothermal heating, mineral exploration and mining, hydraulic fracture, energy piles, tunneling with frozen soil and nuclear waste storage and management. The major difficulty encountered for modeling this coupling physical processes is that the thermal and pore-fluid diffusions, and the deformation of solid skeleton may take place in profoundly different spatial (micron vs. kilometer) and temporal scales (second vs decades). Bundling all the physical processes in simulations while maintaining accuracy and robustness is therefore a difficult task.In the last three decades, a considerable progress has been made for deriving mathematical theories and implementing computer models to replicate the fully coupled thermo-hydro-mechanical processes. This journal club article will focus on the derivation and implementation of the thermo-hydro-mechanical model for numerical simulations. Any feedback or critique are greatly appreciated.
1. Monolithic vs. operator-splitting schemes
The governing equation of the thermo-hydro-mechanical problem can be obtained from the balance principles. The displacement, pore pressure, Darcy’s velocity and temperature are usually the primary solution field stored at the nodal points. If the whole set of governing equations are solved simultaneously, then the solver is called monolithic. The monolithic solver is particularly important for strongly coupling problems in which passing parameters among fluid and solid solvers may not be sufficient (Preisig & Prevost, 2011).For instance, a monolithic small strain finite element code, FRACON, has been developed by (Nguyen & Selvadurai,1995). In this code, the balance of linear momentum and mass are fully coupled, while thermal transport may affect the solid deformation and pore-fluid diffusion but not vice versa. Li, Liu and Lewis, introduce a co-rotational FEM formulation and incorporate plasticity into THM model to model the non-isothermal elastoplastic responses of porous media at large strain in (Li,2005). Recent work by Preisig and Prevost employed a fully coupled THM simulator to compare the numerical solutions against the field data in a case study for carbon dioxide injection at In Salah, Algeria (Preisig & Previst, 2011). Kolditz et al., 2012 introduces an open-source project OpenGeoSys, which takes advantage of an object-orient framework and provides software engineering tools such as platform-independent compiling and automated benchmarking for developers.
In addition to the monolithic finite element scheme, attempts have been made to sequentially coupled multiphase flow and geomechanical simulators by establishing proper feedback and information exchange mechanisms. One such example is TOUGH-FLAC, which links flow simulator TOUGH2 with a small strain finite difference code FLAC (Rutqvist,2011). Klar et al. 2013 employ an explicit time-marching scheme to simulate the fully coupled thermal, and pore-fluid diffusions and the path dependent responses of the solid skeleton. This sequential coupling approach is an attractive alternative to the monolithic approach, as it is easier to implement and maintain flow and solid simulators separately. In addition, the operator-splitting approach, if used properly, can offer tremendous saving on memory and CPU times. The operator splitting approach also enables one to assign different time steps to different physical processes. For instance, in reactive-diffusion simulations, it is common to have the reaction component updating with a several order finer time step than the diffusion counterpart. However, proper communication must be established to ensure the correctness and numerical stability of numerical solutions (Jha & Juanes, 2007, Preisig, 2011, Klar et al. 2013, Sun et al., 2013a, Sun et al. 2013b). As noted in (Jha & Juanes,2007), the sequential coupling scheme used to link the fluid and solid simulators may have profound impact on the efficiency, stability and accuracy of the numerical solutions (Schrefler et al.,1995, Schrefler et al.,1997).
If the fluid and solid simulators use different grids or meshes (eg. finite volume for fluid and finite element for solid), then a proper data projection scheme is required to transfer information from Gauss points and nodes of the solid mesh to the fluid mesh and vice versa (Goumiri & Previst, 2011). For large scale parallel simulations, the sequential couplings must be carefully designed to avoid causing bottleneck due to the difference in solver speed. This can be a significant problem if either the solid or fluid solver runs only in serial.
2. Mixed finite elements (e.g. Taylor-Hood, Raviart-Thomas) vs. stabilized procedures
As noted in Zienkiewicz et al., 1999, Wan, 2002, Mira et al., 2003, Truty & Zimmermann, 2006, Simoni et al., 2008, White & Borja, 2008, Preisig & Prevost , 2011, Sun et al, 2013a, Sun et al. 2013b Borja, 2013, numerical stability is often a major challenge for poromechanics models. Due to the lack of inf-sup condition (Babuvska, 1973, Brezzi et.al, 1985, Bathe, 2001, Bochev et al. 2006), pore pressure and temperature fields may exhibit spurious oscillation patterns and/or checkerboard modes if the displacement, pore pressure and temperature are spanned by the same set of basis function. While these spurious oscillations are less severe at the drained/isothermal limit, they may intensify when small time step is used or when materials are near undrained/adiabatic limit. From a mathematical viewpoint, these non-physical results are due to the kernel (null space) of the discrete gradient operator being non-trivial. This non-trivial kernel makes it possible to have certain spatially oscillating pore pressure and temperature fields that have no impact on the solid deformation in a numerical simulation.
To cure the numerical instability due to the lack of inf-sup condition, previous researches have established a number of techniques that employ different basis functions to interpolate displacement and pore pressure and obtain stable solutions. For instance, Zienkiewicz and coworkers Zienkiewicz et al. 1999, and Borja & Alarcon 1995 used Taylor-Hood finite element (quadratic basis functions for displacement and linear basis functions for pore pressure) to satisfy inf-sup condition and maintain numerical stability for isothermal hydromechanics problems.
On the other hand, Jha & Juanes 2007 have shown that linear displacement combined with pore fluid velocity in the lowest-order Raviart-Thomas space, and piecewise constant pore pressure may also lead to stable solutions for isothermal poromechanics problems. Nevertheless, inf-sup stable mixed finite element models require multiple meshes for displacement, pore pressure and/or fluid velocity. As a result, they require additional programming effort to pre- and post-processing data and maintain the more complex data structure.
To avoid the complications of using multiple meshes for each solution field, an alternative is to use one single equal-order finite element mesh with stabilization procedures. Many stabilization procedures have been proven to be able to eliminate the spurious oscillation modes without introducing extra diffusion for small strain isothermal poromechanics problems. For instance, White & Borja 2008 employed a polynomial projection scheme originated from Dohrmann & Bochev, 2004 to simulate slip weakening of a fault segment. A numerical example in this paper shows that spurious oscillation may persist in the inf-sup stable finite elements (e.g. quadratic-displacement/linear-pore-pressure pair) if the diffusivity is very low for a given time step size. Perhaps the major drawback for the stabilized finite element approach is that it is often necessary to find the stabilization parameter that just give the right amount of stabilization effect without over-diffusing the solution. Recently, my collaborators and I have attempted to address this issue for isothermal poromechanics problem in the large deformation regime, where the stabilization term is adaptively adjusted to avoid over-diffusion (Sun 2013a and Sun et al. 2013b). The stabilization term is derived by solving a small strain one-dimensional deformation-diffusion problem. By determining the critical value of the diffusivity that leads to complex valued growth/decay rate, the optimal value for the stabilization parameter can be estimated. For the three-field thermo-hydro-mechanical problem, the situation is more complicated. Since the thermal convection-diffusion and pore fluid diffusion may reach steady-state in profoundly different rates, it is unclear how to stabilize both the pore pressure and temperature fields without over-diffusing one or both of them.
3. Length scale and bifurcations
Unlike single-phase materials, porous media are multiphase in which solid, liquid(s) and gas(s) can all occupy fractions of volume of the representative elementary volume. As a result, the deformation of the solid skeleton are influenced by the heat transfer and pore fluid diffusions. One direct consequence is that the diffusion will introduce rate dependence to the mechanical responses. This rate dependence is sometime considered by some as a localization limiter which regularizes the governing equations when deformation band formed in numerical simulations. Zhang, et al (1999) analyzed the characteristic equation of a one-dimensional wave propagation problem and found that a length scale may be derived for compressive wave even when softening occurs for solid skeleton, but this is not the case for shear wave. Abellan & de Borst (2005) conducted a similar dispersion analysis and found that the physical length scale disappear in short wavelength limit. This result is supported by numerical simulations in which the strain of a softening bar composed of saturated porous media is found to be mesh dependence. In other words, using diffusion alone as a mean to circumvent mesh dependence is not productive, according to both Zhang et al (1999) and Abellan & de Borst (2005). Nevertheless, a mesh independent post-bifurcation response may still be obtained in numerical simulations, if a length scale can be introduced through other means (e.g. grain rotation, gradient dependence, nonlocal plasticity or damage model, rate dependent solid constituent…etc).
A similar observation is made by Bazant (2010) for models that couple multiple spatial scales. The author observed that incorporating sub-scale simulations to macroscopic simulations alone does not introduce length scale for the softening materials. This treatment merely move the strain localization phenomenon one scale down. Recent multiscale discrete element/finite element models proposed by Guo & Zhao (2014) and by Nguyen et al. (2014) both confirm that the macroscopic finite element responses are mesh dependence in the post-bifurcation regime even though the macroscopic responses are inferred from grain-scale simulations.
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