the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
RHEA v1.0: Enabling fully coupled simulations with hydrogeomechanical heterogeneity
José M. Bastías Espejo
Andy Wilkins
Gabriel C. Rau
Philipp Blum
Realistic modelling of tightly coupled hydrogeomechanical processes is relevant for the assessment of many hydrological and geotechnical applications. Such processes occur in geologic formations and are influenced by natural heterogeneity. Current numerical libraries offer capabilities and physics couplings that have proven to be valuable in many geotechnical fields like gas storage, rock fracturing and Earth resources extraction. However, implementation and verification of the full heterogeneity of subsurface properties using highresolution field data in coupled simulations has not been done before. We develop, verify and document RHEA (Real HEterogeneity App), an opensource, fully coupled, finiteelement application capable of including elementresolution hydrogeomechanical properties in coupled simulations. To extend current modelling capabilities of the Multiphysics ObjectOriented Simulation Environment (MOOSE), we added new code that handles spatially distributed data of all hydrogeomechanical properties. We further propose a simple yet powerful workflow to facilitate the incorporation of such data to MOOSE. We then verify RHEA with analytical solutions in one and two dimensions and propose a benchmark semianalytical problem to verify heterogeneous systems with sharp gradients. Finally, we demonstrate RHEA's capabilities with a comprehensive example including realistic properties. With this we demonstrate that RHEA is a verified opensource application able to include complex geology to perform scalable, fully coupled, hydrogeomechanical simulations. Our work is a valuable tool to assess challenging realworld hydrogeomechanical systems that may include different levels of complexity like heterogeneous geology and sharp gradients produced by contrasting subsurface properties.
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The complexity of processes occurring in a fluidsaturated deformable porous medium and their importance to a wide range of subsurface applications presents a major challenge for numerical modelling especially when including realistic heterogeneity. Example applications in geoengineering that inherently require coupling of hydrogeomechanical processes are the interaction between pressure, flow and fracturing of rocks (Atkinson, 2015; Weng, 2015; Berre et al., 2019); land surface subsidence caused by the extraction of Earth resources (Peng, 2020; Ye et al., 2016); underground gas storage (Yang et al., 2016; Tarkowski, 2019); and mass movement (Zaruba and Mencl, 2014; Haque et al., 2016; Gariano and Guzzetti, 2016). Even though the fundamental mathematical description of coupled hydrogeomechanical processes has reached general consensus (Cheng, 2016; Wang, 2017), realistic modelling of such processes requires a precise description of the underground.
Heterogeneity is ubiquitous across scales and strongly affects the mechanical properties as well as the movement of fluids through the subsurface. For instance, the hydraulic conductivity of fractures within a porous rock is often orders of magnitude greater than that of unfractured rock, so that fine spatial discretisation around fractures is needed in certain numerical models, resulting in expensive computational demands (Morris et al., 2006; Eaton, 2006). As a result, the development of coupled hydrogeomechanical models generally requires simplifying or averaging heterogeneity, i.e. homogenising (Blum et al., 2005, 2009). Recent research has identified the need to improve modelling of coupled hydrogeomechanical systems (Lecampion et al., 2018; Grigoli et al., 2017; Birkholzer et al., 2019), and particularly also the importance of introducing highresolution details to improve the accuracy of numerical simulations (McMillan et al., 2019). However, integrating spatially distributed material properties to numerical tools is not trivial because the shape of geological formations can consist of complex geometries produced by natural processes acting over a long period.
Terzaghi (1923) first described the elastic interactions between a porous medium and a fluid occupying its pore space, and the unidirectional system's dynamic responses to external forces. Biot (1941) later generalised this theory to three dimensions giving rise to the well known theory of consolidation or poroelasticity, also termed Biot theory. Since the 1970s, a large number of numerical libraries have been developed, optimised and applied to a diverse range of poroelastic applications (Bear and Verruijt, 1987; Verruijt, 1995; Cundall and Hart, 1993; Boone and Ingraffea, 1990). Notable is the work of Verruijt (2013), who designed a number of numerical solvers for typical one and twodimensional poroelastic problems.
Well known subsurface simulation libraries are concisely reviewed in the following. Since the number of subsurface simulation codes is vast, we only included platforms that are relevant to modelling spatially distributed heterogeneity. For an exhaustive list of codes the reader is referred to White et al. (2018). Current subsurface hydrogeomechanical simulation codes can be classified based on the numerical solution scheme and modelling approach of the coupled physics. For example, sequential coupling solves for the hydraulic and geomechanical variables independently and in sequence. Notable examples are geomechanics models based on TOUGH (Transport Of Unsaturated Groundwater and Heat) (Pruess et al., 1999; Xu et al., 2006; Lei et al., 2015; Lee et al., 2019). These consist of different libraries to solve for coupled thermohydromechanical (THM) applications relying on the numerical capabilities provided by TOUGH. The libraries differ in their fundamental equations, numerical solution methods and discretisation schemes (Rutqvist, 2017). Although sequential codes allow flexible and efficient code management in conjunction with reasonable computational costs, they tend to perform poorly in tightly coupled processes, since transient interaction between variables may not be computed accurately (Kim et al., 2011; Beck et al., 2020). However, sequential coupling combined with iterative schemes can significantly improve the numerical accuracy. In such implementations, feedback between variables occurs by transferring hydraulic variables to the geomechanics implementation, followed by returning the calculated stress and strain back into the flow problem for the next iteration (Beck et al., 2020). The numerical stability of such iterative methods is discussed by Kim et al. (2011) and Mikelić and Wheeler (2013). Another massively parallel subsurface flow package is PFLORTRAN, an opensource, multiscale and multiphysics code for subsurface and surface processes (Hammond et al., 2014). PFLORTRAN solves nonisothermal multiphase flow, reactive transport and geomechanics in a porous medium. It has previously been applied to simulate hydrogeomechanical systems (Lichtner and Karra, 2014).
Another concept is to solve the hydrogeomechanical equations as a fully coupled system (i.e. all equations are solved simultaneously). This is often performed using an implicit timestepping scheme, which has unconditional numerical stability and high accuracy but is computationally expensive. This approach has proven to be useful in geoengineering applications (Nghiem et al., 2004; Hein et al., 2016; Pandey et al., 2018). Various fully coupled hydrogeomechanical libraries have been developed and released. Proprietary software such as COMSOL (Holzbecher, 2013) has been used intensively in geomechanical applications, in particular for modelling of coastal aquifers (Zhao et al., 2017). COMSOL can import material data from text files into simulations. However, an automatic interpolation between neighbouring materials is performed automatically and may lead to undesired results. More recently, Pham et al. (2019) included geomechanical and poroelastic capabilities into the proprietary groundwater modelling environment FEFLOW (finiteelement flow). MRST is an opensource code developed within the proprietary software MATLAB for fast prototyping of new tools in reservoir modelling (Lie, 2019). While MRST is not a simulator itself, it supports multiphase flow with THM physics. MRST has been used for hydrogeomechanical problems, such as fracture rocks (Zhao and Jha, 2019) as well as several other subsurface applications (Garipov et al., 2018; Ahmed et al., 2017; Edwards et al., 2017). Notably also, two opensource Python codes have been developed. The first is the FEniCS project (Haagenson et al., 2020; Alnæs et al., 2015), while the second is called Porepy and was specifically developed to simulate THM processes in rock fractures (Keilegavlen et al., 2017, 2021). Despite the fact that Pythonbased coding offers the advantage of highlevel programming within a relatively friendly user interface, these codes are designed to facilitate rapid development of features that cannot be properly represented by standard simulation tools rather than general multiphysics problems. Other fastprototyping novel codes include Dang and Do (2021), Tran and Jha (2020), Reichenberger (2003), Martin et al. (2005), and Frih et al. (2012).
An additional option is OpenGeoSys (OGS), a well known opensource library to solve multiphase and fully coupled THM physics (Kolditz et al., 2012). The code is well documented and features several examples in different subsurface areas. Further, different developers are constantly contributing new features to the source code (Graupner et al., 2011; Kosakowski and Watanabe, 2014; Li et al., 2014). Similarly, DuMux is a free, opensource, fully coupled numerical simulator for multiphase flow and transport in a porous medium, (Flemisch et al., 2011). It is based on the Distributed Unified Numeric Environments (DUNE), a C++based ecosystem that solves finiteelement (FE) models based on PETSc. DuMux is well known for its strong focus on multiphase flow and transport in a porous medium. Its recent release adds extra features and facilitates physics coupling, such as Navier–Stokes models (Koch et al., 2021). From our experience, however, users without some computational background and experience in programming in C++ or Python as well as using a debugger may require a significant amount of time to take full advantage of the features that OGS and DuMux offer. Furthermore, to our best knowledge we are unaware of a peerreviewed verification of these codes that includes fully distributed hydraulic and geomechanical heterogeneity.
The multiphysics coupling framework MOOSE (Multiphysics Object Oriented Simulation Environment) (Permann et al., 2020) offers a unique environment where users can couple different physical processes in a modular approach. Within the objectoriented ecosystem of MOOSE, each physical process (or its partial differential equation, PDE) is treated separately as an individual MOOSE object, and coupling is performed by the backend routines of MOOSE. The MOOSE numerical scheme is based on the FE method. It offers clean and effective numerical PDE solvers as well as mesh capabilities with a uniform approach for each class of problem. This design enables easy comparison and use of different algorithms (for example, to experiment with different Krylov subspace methods, preconditioners, or truncated Newton methods) which are under constant development. MOOSE enables the user to focus on describing the governing equations while the underlying numerical technicalities are taken care of by the system.
We have found that mastering the basic concepts of the MOOSE workflow requires a steep learning curve. However, it requires minimum C++ coding skills, which facilitates the learning experience from users that do not necessarily have a computer science background. Once the basics are mastered the benefits are significant; for example an experienced user can easily modify the source code to add desired features such as multiscale physics, nonlinear material properties, complex boundary conditions or even basic postprocessing tools with only a few lines of code.
An example of MOOSE's capabilities in simulating coupled processes in a porous medium was illustrated by Cacace and Jacquey (2017), who developed a MOOSEbased application named GOLEM. It was optimised to model threedimensional THM processes in fractured rock (Freymark et al., 2019). Another cuttingedge implementation is PorousFlow, an embedded MOOSE library to simulate multiphase flow and thermal–hydraulic–mechanical–chemical (THMC) processes in a porous medium (Wilkins et al., 2020). PorousFlow has been verified and applied to simulate a number of complex and realistic systems, for example shallow geothermal systems (Birdsell and Saar, 2020), CO_{2} sequestration (Green et al., 2018) and groundwater modelling with plastic deformation (Herron et al., 2018). However, it has not yet been extended and verified for the simulation of spatial heterogeneity of mechanical parameters. In other words, despite its ability to handle spatially distributed heterogeneity of permeability and porosity, it does not support spatially distributed heterogeneity of mechanical properties such as bulk and Young's moduli. To the best of the authors' knowledge no existing opensource numerical tool is able to integrate full heterogeneity including all hydrogeomechanical parameters representative of complex geologic formations.
The aim of this paper is therefore to develop, verify and illustrate a novel and generic workflow for modelling fully coupled hydrogeomechanical problems allowing the inclusion of hydraulic and geomechanical heterogeneity inherent to realistic geological systems. This was achieved by extending the current capabilities of the native MOOSE physical modules, namely PorousFlow and Tensor Mechanics. We call this workflow RHEA (Real HEterogeneity App). RHEA is based on MOOSE's modular ecosystem and combines the capabilities of PorousFlow and Tensor Mechanics with material objects that are newly developed in our work and provide the novel ability to allocate spatially distributed properties at element resolution in the mesh. By integrating new C++ objects, we modified the underlying MOOSE code within PorousFlow and Tensor Mechanics. To streamline preprocessing efforts arising from this improvement, we developed a Pythonbased, automated workflow which uses a standard data format to generate input files that are compatible with the material objects in MOOSE format. Finally, we verified the correctness of RHEA with a newly developed, analytical benchmark problems allowing vertical heterogeneity and illustrated its performance using a sophisticated 2D example with distributed hydraulic and mechanical heterogeneity. In this work, we first describe the workflow required to compile a RHEA app, formulate a modelling problem and run a simulation. We then compare RHEA's simulation results with one and twodimensional analytical solutions and propose a benchmark semianalytical solution to validate RHEA's performance when sharp gradients are present. Finally, we apply RHEA to a complicated twodimensional problem with centimetrescale heterogeneities demonstrating its capabilities. We anticipate that our work will lay the foundation for accurate numerical modelling of hydrogeomechanical problems allowing full spatial heterogeneity.
Modelling of coupled hydrogeomechanical processes requires solving the equations describing fluid flow in a deformable porous medium. The coupled processes can be described physically in a representative elementary volume (REV) by a balance of fluid, mass and momentum, where local equilibrium of thermodynamics is assumed and macroscopic balance equations are considered to be the governing equations. In this section, the governing equations for hydrogeomechanical processes in a fully saturated porous medium with liquid fluid are presented on the basis of Biot's theory of consolidation. In the pore pressure formulation, the field variables are the liquidphase pressure p_{f} and the displacement vector u. The material parameters can be spatially variable but remain independent of time. Permeability and elastic parameters are described as tensors, whereas the Biot coefficient is a scalar.
Fluid flow within a deformable and fully saturated porous medium is described by the continuity equation
where α is the Biot coefficient, ε_{kk} is the volumetric strain, Q_{f} is a fluid sink or source term, and M is the Biot modulus of the porous medium (the reciprocal of the storage coefficient). In Biot's consolidation theory, the Biot modulus is defined as
where ϕ, K_{f} and K_{s} represent the porosity, fluid and solid bulk modulus respectively. As Darcy flow is assumed, the fluid discharge q_{d} can be expressed as a momentum balance of the fluid like
where v_{f} and v_{s} are the fluid and solid matrix velocities respectively, k is the permeability tensor, μ_{f} is the dynamic viscosity of the fluid, ρ_{f} is the density of the fluid, and g is the gravitational acceleration vector.
The mechanical model is defined via momentum balance in terms of the effective Cauchy stress tensor ${\mathit{\sigma}}^{\mathbf{\prime}}(x,t)$ as
where 𝕀 is the ranktwo identity tensor. The mass of fluid per volume of porous medium is expressed as the sum of the phases
where ρ_{s} is the solid density. The elastic strain can be expressed in terms of displacements with the relation
The effective stress is related to elastic strains by the generalised Hooke's law:
where ℂ_{ijkl} is the elastic compliance tensor.
Together, Eqs. (1) to (7) constitute the coupled system that represents hydrogeomechanical systems with linear elastic deformation.
As a derivative of the MOOSE framework, RHEA enables access to a wide array of options to fine tune a simulation. Solver options such as numerical schemes and adaptive timestepping as well as general PETSc options are available. By default, RHEA uses a firstorder fully implicit time integration (backward Euler) for unconditional stability and solves the coupled equations simultaneously (full coupling) (Kavetski et al., 2002; Manzini and Ferraris, 2004; Gaston et al., 2009). RHEA also allows operator splitting to implement loose coupling, i.e. solving the fluid flow while keeping the mechanics fixed, then solving the mechanics while keeping the fluid flow fixed. While this can be executed on separate meshes with different timestepping schemes; this feature is not explored in the current article (Martineau et al., 2020).
Explicit time integration (with full or loose coupling) and other schemes such as Runge–Kutta are available in MOOSE and RHEA, but stability limits the timestep size, so these are rarely used in the type of subsurface problems handled by RHEA. By default, MOOSE and RHEA use linear Lagrange finiteelements (tetrahedra, hexahedra and prisms for 3D problems, triangles and quads for 2D problems), but higherorder elements may be easily chosen if desired (Hu, 2017).
RHEA does not implement any numerical stabilisation for the fluid equation to eliminate overshoots and undershoots; however, fluid volume is conserved at the element level (Cacace and Jacquey, 2017). Although not explored in this study, RHEA's fluid flow may be extended to multiphase, multicomponent flow with highprecision equations of state, as well as finitestrain elastoplasticity (Wilkins et al., 2020).
RHEA is an opensource simulation workflow and tool specifically developed to allow fully coupled numerical simulations in a saturated porous medium with spatially distributed heterogeneity in hydraulic and geomechanical properties. We built RHEA as a derivative of MOOSE, the massively parallel and opensource FE simulation environment for coupled multiphysics processes (Gaston et al., 2009; Permann et al., 2020). MOOSE offers virtually unlimited simulation capabilities covering a wide spectrum of applications. This is based on a workflow where the end user does not need to know the details of the FE implementation. To achieve that, MOOSE utilises the libMesh library, a framework capable of manipulating multiscale, multiphysics, parallel and meshadaptive FE simulations (Kirk et al., 2006). While the numerical methods, solvers and routines are executed by PETSc libraries (Balay et al., 2019), MOOSE is designed to allow the user to interact and control these two libraries without having to do any complex programming. Instead, the user frames the problem simply through an input file with unique syntax.
We found that learning how to perform numerical simulations based on the MOOSE framework is not a trivial task. Our aim is to further develop modelling capabilities while simplifying the complexity of the problem through an easytofollow workflow accompanied by a visual summary. The RHEA workflow can be summarised as follows:
 Step 1 – RHEA compilation:

The user creates the RHEA application following the structure outlined in Fig. 1. In other words, the user creates an executable file which is able to model fully coupled hydrogeomechanical systems in a heterogenous medium. We accomplished this through new MOOSEbased materials functions able to allocate data in each element of the mesh based on a pregenerated input file. Furthermore, we integrated the multiphysics of Sect. 2 to RHEA by adding the PorousFlow (Wilkins et al., 2020) and Tensor Mechanics modules that are part of the MOOSE framework. Once RHEA is downloaded, the user can access the necessary files to build RHEA and can even access those files to modify the physics. This procedure is generic for any new MOOSE application. The core components of any MOOSE app such as RHEA are as follows (Fig. 1).
 Block 1 – Kernels:

The kernels (or partial differential equation terms) describing the physics are implemented in their weak form (Jacob and Ted, 2007). In the MOOSE ecosystem, PDEs are represented by one simple line of code; this is highlighted with a cyan rectangle in Fig. 1. This straightforward way of describing complex multiphysics constitutes the most powerful feature of MOOSE.
 Block 2 – Material properties:

Values, including spatially distributed values, can be prescribed for each of the materials appearing in the kernels.
 Block 3 – Kernel coupling:

The user can couple different physics by including different kernels in its model, or by creating new kernels.
This dynamic procedure allows flexible creation of the RHEA application or any MOOSEbased application requiring minimal knowledge of C++ programming skills.
 Step 2 – Preparation of material properties:

The spatially distributed data is formatted to the structure required by the RHEA app compiled in Step 1. We implemented this with a custom Python script that imports and formats the original CSV or VTK data set into a RHEAcompatible data structure. Within RHEA, the hydrogeomechanical material properties are field properties, which means that each value in the data set has to be allocated to a respective mesh element. Therefore, when the mesh is generated, the discretisation has to match the number of data points of the data set. That way, each property value is represented within the simulation. Note that if this is not done correctly, RHEA may assign undesired property values. This is because RHEA will automatically linearly interpolate any values provided to the mesh. Thus, if the initial mesh discretisation does not match the usersupplied samples, interpolated values are assigned which could lead to undesired results.
 Step 3 – Simulation setup:

To define the numerical model, a RHEA script has to be created in the standard MOOSE syntax. The script consists of an array of systems that describe the mesh, physics, boundary conditions, numerical methods and outputs. A short example along with brief system descriptions is illustrated in Fig. 1. The blocks consist of MOOSE functions that are written and designed in a generic manner and independently of the nature of the problem; this way the blocks can be recycled and reused. The spatially distributed material properties can be imported into the Function system and subsequently be stored in the AuxVariable system to be assigned as material property in the Materials block.
In summary, numerical simulations of hydrogeomechanical problems with spatially distributed material properties can be performed by calling RHEA's executable file (created in Step 1), using the simulation control script (created in Step 3) which contains the necessary instructions, as well as reading in the spatially distributed material properties (created in Step 2).
To test if RHEA accurately solves the differential equations stated in Sect. 2 and if boundary conditions are correctly satisfied, four different tests were developed. The proposed tests use predefined material properties that were imported into RHEA using the workflow presented in Sect. 3. The tests were designed to gradually build up complexity and cover the typical spectrum of consolidation problems. In two of the examples, RHEA's performance in simulations with sharp gradients is tested. First, a onedimensional consolidation problem where the hydraulic conductivity varies in 4 orders of magnitude between layers and a second twodimensional example with realistic heterogeneity in which the hydraulic conductivity of the geological facies varies over 6 orders of magnitude.
The first test, the classical Terzaghi's problem, is used as a basic benchmark of the hydromechanical coupling in RHEA. In later sections, we illustrate the full potential of RHEA when simulating spatially heterogeneous systems in one and two dimensions. The four verification scenarios are described in the following subsections. All numerical solutions were calculated using an 8core Intel i73770 CPU at 3.40 GHz with 32 GB DDR4 RAM memory, and the results were stored on a hard disk drive.
4.1 Terzaghi's problem
In the onedimensional consolidation problem, also known as Terzaghi's problem (Terzaghi, 1923), a single load q is applied at t=0 on the top of a fully saturated homogeneous sample with the height L. The system is only drained at the top, where the pressure of the fluid is assumed to be p=0 for t>0. At the moment of loading, t=0, the undrained compressibility of the solid increases the pressure of the sample. For t>0, the system is allowed to drain and the consolidation process begins.
In the absence of sources and sinks, Eq. (1) is reduced to the basic storage equation as
where the product ρ_{f}⋅g was written as γ_{f} and represents the volumetric weight of the fluid. Equation (3) is used to couple the fluid discharge q_{d}. From Hook's law, assuming onedimensional deformation, the vertical strain equals the volume change
where m_{v} is the confined compressibility of the porous medium
and K_{s} and G_{s} are the bulk and shear moduli of the porous medium respectively. Substituting Eq. (9) into the storage equation (Eq. 8), the general differential equation for onedimensional consolidation is obtained:
For t>0, the total load q is kept constant and the total stress σ_{zz} is also constant. Consequently, Eq. (11) reduces to
Since the system is undrained at t=0, the initial condition can be established from Eq. (11) as
The boundary conditions at the top and bottom of the sample are
and
The analytical solution of the problem is well known and reads as follows (Wang, 2017; Cheng, 2016; Verruijt, 2018):
For this example, the height of the sample was set to 100 m, the hydraulic conductivity is $\mathrm{1}\times {\mathrm{10}}^{\mathrm{4}}$ m s^{−1}, the porosity is 0.2, the Biot coefficient is 0.9, the bulk modulus is 8.40×10^{7} Pa and the shear modulus is 6.25×10^{7} Pa. The performance and consistency of RHEA on the consolidation problem is shown as pore pressure versus depth profiles at discrete times in Fig. 2a. A comparison of the analytical and RHEA's solution reveals excellent agreement, thereby verifying the numerical solution. The total time for computing 101 time steps was 1.92 s.
4.2 Layered Terzaghi's problem
The objective of this test is to investigate the performance of RHEA when heterogeneity and sharp gradients are present. The consolidation experiment of the previous section is performed on a sample with multiple layers of contrasting properties. For simplicity, porosity and mechanical parameters are assumed to be homogeneous. Since the total load q is constant for t>0, Eq. (11) reduces to Eq. (12) across n layers as follows
which describes the consolidation in each layer. Here, ${z}_{i\mathrm{1}}\le z\le {z}_{i}$ is the depth of the sample, p_{f,i} and k_{i} are the fluid pressure and permeability of the solid in each layer i, respectively. The contact between layers is assumed to be perfect; i.e. the boundary conditions at the layers are represented by equivalent matching fluid pressure as
The sample is drained at the top, whereas the bottom remains undrained
and
The fluid pressure produced by the external load starts to dissipate when t>0, but at different rates depending on the consolidation coefficient of the layer. The height of the sample is 100 m, and 10 layers are equally distributed along the sample with 10 m height. To represent sharp gradients, the selected hydraulic conductivities have a difference of 4 orders of magnitude between layers, $\mathrm{1}\times {\mathrm{10}}^{\mathrm{4}}$ m s^{−1} and $\mathrm{1}\times {\mathrm{10}}^{\mathrm{8}}$ m s^{−1}. The high and lowpermeability layers are alternating. The porosity is set to 0.2, the Biot coefficient is 0.9, the bulk modulus is 8.40×10^{7} Pa and the shear modulus is 6.25×10^{7} Pa.
A stepbystep semianalytical solution of the diffusion problem in a layered sample was derived by Hickson et al. (2009). To solve this problem in RHEA, a mesh of 100 elements was used with a time step of 1×10^{4} s. The total time for computing 701 time steps was 13.8 s. A comparison between the analytical solution and RHEA's numerical simulation is shown in Fig. 2b. In the layers with high hydraulic conductivity, the consolidation process occurs rapidly, leading to faster pore pressure dissipation (vertical pore pressure profile) and therefore also faster water movement. In contrast, the consolidation process is slower in the lowconductivity layers with slower pore pressure dissipation and water movement.
4.3 Plane strain consolidation
To evaluate the performance of RHEA for twodimensional heterogeneity, a consolidation problem with plane strain is developed. The twodimensional consolidation caused by a uniform load over a circular homogeneous area can be represented by the storage equation (Eq. 1) in a twodimensional case as
where ε represents the volumetric strain. Including two equilibrium equations, in terms of total stress, gives
and
The total stress is related to the effective stress through
and
The analytical solution can be found by expressing the equilibrium (Eqs. 22 and 23) in terms of the displacement components u_{x} and u_{z} using Hooke's law as
and
and
Here, the assumed plane strain is the y axis, i.e. u_{y}=0. Replacing Hooke's law in the plane strain (Eqs. 26 to 28) with the effective stress balance (Eqs. 22 and 23) combined with Eqs. (24) and (25) leads to a complete system of equations as
and
where the elastic strain is
The boundary conditions are represented by a constant load in an area of width 2a, applied at t=0. The system is allowed to drain for t>0 as
and
and
When the sample is loaded, a confined pore pressure is generated which starts to drain instantaneously through the borders of the system. A semianalytical solution in the Fourier domain and Laplace transform for the given equation system and boundary conditions is presented in Verruijt (2013). The height and the width of the sample are 10 m. The load is applied on the surface of the sample between −1 and 1 m. The hydraulic conductivity is $\mathrm{1}\times {\mathrm{10}}^{\mathrm{5}}$ m s^{−1}, the porosity is 0.2, the Biot coefficient is 0.9, the bulk modulus is 8.40×10^{7} Pa and the shear modulus is 6.25×10^{7} Pa. To solve this problem, a coarse mesh was defined, and MOOSE's native mesh adaptivity system was employed to automatically generate a finer resolution in areas where the pore pressure gradients are steep. This significantly reduces the computational time when compared with using a fine mesh throughout. The total simulation time was 312.6 s for 101 time steps and 10 000 elements with 20 502 nodes.
The results are illustrated in Fig. 3. Figure 3a shows a cross section of the sample as contour plot. Figure 3b shows a pore pressure profile with depth at the centre of the sample x=0. Excellent agreement between the analytical solution and the simulated solution by RHEA is observed.
4.4 Modelling realistic geology
The last example aims to study and illustrate the performance of RHEA with a real data set. This example illustrates how to generate input files using the developed workflow and demonstrates the potential of RHEA for simulating increased spatial complexity and sharp gradients. While the Herten analogue is a 3D data set, the example was reduced to two dimensions to facilitate presentation. However, simulations in three dimensions are also possible and can be done using the presented workflow in unmodified form. The 2D consolidation problem was solved with RHEA, integrating the multifacies realisations and material properties of the Herten analogue (Bayer et al., 2015a). Although the data set does not contain geomechanical subsurface properties, the hydraulic conductivity varies over 6 orders of magnitude, which provides sufficient proof of RHEA's capabilities.
4.4.1 Herten aquifer data set description
Realistic modelling relies not only on accurate data concerning material parameters, but also on appropriate spatial distribution of such parameters (Houben et al., 2018; Irvine et al., 2015; Kalbus et al., 2009). Typically, distributed material parameters are generated by stochastic random fields based on an a priori statistical distribution (Vanmarcke et al., 1986). Although random fields have proven to be useful, they do not capture the usual continuity of material parameters (Strebelle, 2002). Consequently, the use of highresolution data, such as “aquifer analogues”, is preferred (Alexander, 1993; Zappa et al., 2006). Aquifer analogues consist of centimetreresolution data obtained from detailed investigation of geological formations at outcrops. Although aquifer analogues are rare, they have been widely used in different subsurface fields (Höyng et al., 2015; Beaujean et al., 2014; Finkel et al., 2016). The Herten analogue is a well known and rigorously generated 2D outcrop (Bayer et al., 2015a). It consists of a fluvial braided river deposit from the southeast of Germany, which represents one of the most important drinking water resources in central Europe. Its architecture consists of sedimentary facies, and its body of unconsolidated gravel and well sorted sand. The dimensions of the 2D outcrop are 16 m wide by 7 m high, and it features horizontal and vertical data resolution of $\mathrm{5}\times {\mathrm{10}}^{\mathrm{2}}$ m for hydraulic conductivity and porosity. Hence, the 2D cross section has a total of 4480 measurements points. The corresponding hydraulic conductivity k ranges from $\mathrm{6}\times {\mathrm{10}}^{\mathrm{7}}$ to $\mathrm{1.30}\times {\mathrm{10}}^{\mathrm{1}}$ m s^{−1}, and porosity ϕ from 0.17 to 0.36 (Fig. 4a). To represent spatial distribution of mechanical properties, typical values of bulk and shear moduli for gravel and sand were assumed to be linearly correlated with the porosity of the aquifer: similar trends have been reported in previous studies (Mondol et al., 2008; Hardin and Kalinski, 2005; Hicher, 1996). Representative geomechanical moduli can be found in soil mechanics literature as shown in Table 1. The elastic tensor is assumed to be isotropic in this example; hence elastic moduli are related via (Wang, 2017; Cheng, 2016)
where E_{s} and ν_{s} denote the Young's modulus and Poisson's ratio of the solid material respectively. The result is that the bulk moduli vary between 6.70×10^{7} and 1.70×10^{8} Pa, whereas the shear moduli range between 3.0×10^{7} and 3.50×10^{8} Pa, as shown in Fig. 4a. RHEA does not require the mechanical moduli to be related to the hydraulic properties in the way we have described in this particular example.
Subramanian (2011)Subramanian (2011)Look (2014)Das (2019)Xu (2016)Lade (2001)Lade (2001)Kulhawy and Mayne (1990)4.4.2 Problem and model description
The twodimensional consolidation is described by Eqs. (21), (29) and (30). A constant load at the top of the sample is applied at t>0, which generates a confined pore pressure. After that, the system is allowed to drain through the top boundary and is subjected to the normal stress. The sample's bottom and sides are impermeable to the fluid, and subject to roller boundary conditions.
For this simulation, a quadrilateral mesh was generated with the mesh generator system of MOOSE. The mesh has 44 800 elements and 44 940 nodes, which matches the data set resolution. Since the material properties of the data set differs in orders of magnitude, the mesh adaptivity system of MOOSE was used to ensure accurate results. At each time step the 30 % of elements with the highest pore pressure gradient were refined, which reduces the local error at contrasting facies. Hence the mesh is refined in each time step. At the end of the simulation, the number of nodes had grown to 708 548 and the number of elements to 631 615. The total simulation time was 0.49 h for 70 time steps and 44 341 elements with 44 800 nodes.
4.4.3 Simulation results
The pore pressure profiles depicted in Fig. 5 illustrate how the physical heterogeneity of the crossbedded data set strongly influences the fluid flow through the sample. The effect of the centimetre resolution of the data set can be studied when the initial load is applied at t=0. Since the sample is not yet allowed to drain, confined pore pressure is generated which depends on the geomechanical characteristics of the sand and gravel. In facies where the soil is highly compressible, the generated pore pressure is also relatively high since the total load is shared between the fluid and the soil. In contrast, in facies that have higher elastic moduli values the confined pore pressure is relatively low. This effect is nicely shown in Fig. 5a. At time t>0 the top of the sample is allowed to drain. The effect of the highly permeable units made of poorly sorted and well sorted gravel is shown in Fig. 5b. The top facies of the aquifer consist of a highpermeability soil ($k=\mathrm{1.30}\times {\mathrm{10}}^{\mathrm{1}}$ m s^{−1}), which is divided by a thin lowpermeability layer ($k=\mathrm{6.10}\times {\mathrm{10}}^{\mathrm{5}}$ m s^{−1}); the latter causes contrasting pore pressure profiles. Similar permeability effects have been discussed before (Choo and Lee, 2018; Peng et al., 2017; Kadeethum et al., 2019). The influence of the temporal and spatial scales on the consolidation process is shown in Fig. 5c and d. It can be observed that the process occurs rather quickly and is strongly influenced by the lowpermeability facies. This example demonstrates that RHEA can solve complex and realistic heterogenous hydrogeomechanical coupled problems.
In this paper we develop and verify Real HEterogeneity App (RHEA): a numerical simulation tool that allows fully coupled numerical modelling of hydrogeomechanical systems. Moreover, RHEA can easily include the full heterogeneity of parameters as it occurs in real subsurface systems. RHEA is based on the powerful Multiphysics ObjectOriented Simulation Environment (MOOSE) opensource framework. Furthermore, we provide an easytounderstand workflow which explains how to compile the application and run a customised numerical simulation. Despite its simplicity, the workflow combines all the technical advantages provided by MOOSE and its well established framework. The latter allows the development and use of stateoftheart and massively scalable applications backed by the unconditional support of a growing community.
Beyond unlocking the ability to include the full heterogeneity of hydrogeomechanical parameters in simulations, our contribution provides examples to verify future numerical codes. Additionally, a semianalytical benchmark problem is proposed to verify the performance of numerical code when heterogeneity and sharp gradients are present.
Our example simulations illustrate that the subsurface hydrogeomechanical properties, in particular permeability (or transmissivity), play a key role in the consolidation process. Although this insight is valuable, it can lead to an oversimplification when models assume transmissivity varies heterogeneously while mechanical parameters are assumed to be homogeneous. This approach can lead to biased results in systems where different geologic formations are present. For example, land surface subsidence is a process that can occur due to anthropogenically induced decreases in subsurface pore pressure causing progressive consolidation and slow downward percolation across the layers within the subsurface. This process depends on the spatial distribution of the geomechanical properties, in particular those of clay layers within the subsurface. RHEA could be used to increase our understanding of the spatial and temporal evolution of land surface subsidence. Our newly developed workflow enables such advanced numerical simulations.
RHEA has the potential to advance our understanding of realworld systems that have previously been oversimplified. Further, RHEA offers the integration of highresolution data sets with sophisticated numerical implementations. Potential numerical instabilities caused by highly heterogeneous systems (i.e. settings with sharp gradients) are handled automatically by combining adaptive meshing capabilities with implicit time stepping. While this work demonstrates RHEA's capabilities for twodimensional problems, this can easily be extended to threedimensional simulations. In that case, a threedimensional mesh that is representative of the spatially distributed hydraulic and geomechanical properties of any available data set can be generated. The tasks follow the data formatting workflow and simulation control as described in Sect. 3.
Our current work focuses on hydrogeomechanical coupling of heterogeneous systems. However, RHEA could potentially be extended to also include thermal processes. While it would allow fully coupled simulations of thermal–hydraulic–mechanical (THM) systems including spatially distributed heterogeneities, verification will require the development of more advanced analytical solutions, a task that is, however, beyond the scope of this contribution.
The code and examples presented in this study are available at a Zenodo repository: https://doi.org/10.5281/zenodo.4767832 (Bastías Espejo and Wilkins, 2021). The continuous development of RHEA code is maintained at the GitHub repository https://github.com/josebastiase/RHEA (last access: 15 October 2021). The Herten analogue data set is available on https://doi.org/10.1594/PANGAEA.844167 (Bayer et al., 2015b).
JMBE developed RHEA and the analytical solutions used for verification, made the figures and tables, and wrote the first manuscript draft. GCR closely supervised JMBE. AW provided JMBE with technical support. PB reviewed the manuscript and provided suggestions.
The authors declare that they have no conflict of interest.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We would like to thank Mauro Cacace and two anonymous reviewers for their efforts and thoughtful comments that have helped to improve our paper.
This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 424795466).
The article processing charges for this openaccess publication were covered by the Karlsruhe Institute of Technology (KIT).
This paper was edited by Sergey Gromov and reviewed by Mauro Cacace and two anonymous referees.
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