Realistic modelling of tightly coupled hydro-geomechanical 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 high-resolution field data in coupled simulations has not been done before. We develop, verify and document RHEA (Real HEterogeneity App), an open-source, fully coupled, finite-element application capable of including element-resolution hydro-geomechanical properties in coupled simulations. To extend current modelling capabilities of the Multiphysics Object-Oriented Simulation Environment (MOOSE), we added new code that handles spatially distributed data of all hydro-geomechanical 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 semi-analytical 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 open-source application able to include complex geology to perform scalable, fully coupled, hydro-geomechanical simulations. Our work is a valuable tool to assess challenging real-world hydro-geomechanical systems that may include different levels of complexity like heterogeneous geology and sharp gradients produced by contrasting subsurface properties.

The complexity of processes occurring in a fluid-saturated 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 geo-engineering that inherently require coupling of hydro-geomechanical processes are the interaction between pressure, flow and fracturing of rocks

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

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

Another concept is to solve the hydro-geomechanical equations as a fully coupled system (i.e. all equations are solved simultaneously). This is often performed using an implicit time-stepping scheme, which has unconditional numerical stability and high accuracy but is computationally expensive. This approach has proven to be useful in geo-engineering applications

An additional option is OpenGeoSys (OGS), a well known open-source library to solve multi-phase and fully coupled THM physics

The multi-physics coupling framework MOOSE (Multiphysics Object Oriented Simulation Environment)

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 multi-scale physics, non-linear material properties, complex boundary conditions or even basic post-processing 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

The aim of this paper is therefore to develop, verify and illustrate a novel and generic workflow for modelling fully coupled hydro-geomechanical 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 pre-processing efforts arising from this improvement, we developed a Python-based, 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 two-dimensional analytical solutions and propose a benchmark semi-analytical solution to validate RHEA's performance when sharp gradients are present. Finally, we apply RHEA to a complicated two-dimensional problem with centimetre-scale heterogeneities demonstrating its capabilities. We anticipate that our work will lay the foundation for accurate numerical modelling of hydro-geomechanical problems allowing full spatial heterogeneity.

Modelling of coupled hydro-geomechanical 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 hydro-geomechanical 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 liquid-phase pressure

Fluid flow within a deformable and fully saturated porous medium is described by the continuity equation

The mechanical model is defined via momentum balance in terms of the effective Cauchy stress tensor

Together, Eqs. (

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 time-stepping as well as general PETSc options are available. By default, RHEA uses a first-order fully implicit time integration (backward Euler) for unconditional stability and solves the coupled equations simultaneously (full coupling)

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 time-step size, so these are rarely used in the type of subsurface problems handled by RHEA. By default, MOOSE and RHEA use linear Lagrange finite-elements (tetrahedra, hexahedra and prisms for 3D problems, triangles and quads for 2D problems), but higher-order elements may be easily chosen if desired

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

RHEA is an open-source 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 open-source FE simulation environment for coupled multi-physics processes

Visual illustration of the steps required to create RHEA, generate distributed material properties files and write a simulation script.

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 easy-to-follow workflow accompanied by a visual summary. The RHEA workflow can be summarised as follows:

The user creates the RHEA application following the structure outlined in Fig.

The

Values, including spatially distributed values, can be prescribed for each of the materials appearing in the kernels.

The user can couple different physics by including different kernels in its model, or by creating new kernels.

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 RHEA-compatible data structure. Within RHEA, the hydro-geomechanical 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 user-supplied samples, interpolated values are assigned which could lead to undesired results.

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.

In summary, numerical simulations of hydro-geomechanical 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.

The first test, the classical Terzaghi's problem, is used as a basic benchmark of the hydro-mechanical 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 8-core Intel i7-3770 CPU at 3.40 GHz with 32 GB DDR4 RAM memory, and the results were stored on a hard disk drive.

In the one-dimensional consolidation problem, also known as Terzaghi's problem

In the absence of sources and sinks, Eq. (

For this example, the height of the sample was set to 100

The lines represent the analytical solution whereas the dots represent the RHEA solution.

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

The sample is drained at the top, whereas the bottom remains undrained

The fluid pressure produced by the external load starts to dissipate when

A step-by-step semi-analytical solution of the diffusion problem in a layered sample was derived by

To evaluate the performance of RHEA for two-dimensional heterogeneity, a consolidation problem with plane strain is developed. The two-dimensional consolidation caused by a uniform load over a circular homogeneous area can be represented by the storage equation (Eq.

When the sample is loaded, a confined pore pressure is generated which starts to drain instantaneously through the borders of the system. A semi-analytical solution in the Fourier domain and Laplace transform for the given equation system and boundary conditions is presented in

The results are illustrated in Fig.

The solution of the consolidation problem in plane strain by RHEA is shown in a sample 10 m in width and height.

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 multi-facies realisations and material properties of the Herten analogue

Realistic modelling relies not only on accurate data concerning material parameters, but also on appropriate spatial distribution of such parameters

Facies architecture and properties of the Herten aquifer analogue.

Typical elastic properties of sand and gravel.

The two-dimensional consolidation is described by Eqs. (

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

The pore pressure profiles depicted in Fig.

Sequence of snapshots of the consolidation process and pore pressure variation in the aquifer with time.

In this paper we develop and verify Real HEterogeneity App (RHEA): a numerical simulation tool that allows fully coupled numerical modelling of hydro-geomechanical 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 Object-Oriented Simulation Environment (MOOSE) open-source framework. Furthermore, we provide an easy-to-understand 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 state-of-the-art and massively scalable applications backed by the unconditional support of a growing community.

Beyond unlocking the ability to include the full heterogeneity of hydro-geomechanical parameters in simulations, our contribution provides examples to verify future numerical codes. Additionally, a semi-analytical 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 hydro-geomechanical 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 real-world systems that have previously been oversimplified. Further, RHEA offers the integration of high-resolution 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 two-dimensional problems, this can easily be extended to three-dimensional simulations. In that case, a three-dimensional 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 hydro-geomechanical 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:

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 open-access 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.