The permeability of single fractures is commonly approximated by the cubic law assumption, which is however only valid under the condition of a single phase laminar flow between parallel plates. Departure from cubic law are related to many features like aperture fluctuations due to fracture surface roughness, relative shear displacement, the amount of flow exchange between the matrix and the fracture itself, etc. In order to quantify constitutive relationships among the aforementioned aspects, we have conducted a flow-through experiment with a porous rock sample (Flechtinger sandstone) containing a single macroscopic fracture. Based on this experiment, we obtained range of variations of intrinsic rock parameters, permeability and stress-strain relationships of the combined matrix-fracture system under hydrostatic loading. From the measured deformation of the matrix-fracture system, we derived the evolution in the mechanical aperture of the fracture. In order to quantify the processes behind the laboratory observations, we carried out coupled hydro-mechanical simulations of the matrix-fracture system. Navier–Stokes flow was solved in the 3-dimensional open rough fracture domain, and back-coupled to the Darcy flow and the poroelastic behaviour of the rock matrix. The results demonstrate that the elastic behaviour and the related permeability alteration of the fracture domain could be captured by the numerical simulation. Furthermore, the stress-strain values obtained in the vicinity of the fracture asperities suggest that inelastic deformation develops at low mechanical load. An attempt was made to quantify the inelastic deformation by using the failure envelope obtained by laboratory experiments (whether tensile, shear, compaction, or a combination of those). However, change in permeability observed in the experiments are significantly larger than that in the simulation showing the importance of plastic deformation during opening and closure of the fracture and its impact on the cubic law approximation.

Flow along fractures or in fractured systems becomes increasingly important in the context of faulted or Enhanced Geothermal Systems (EGS), shale gas recovery, as well as the stability of underground constructions, such as tunnels or nuclear waste repositories. Fault zones and natural fracture networks are increasingly considered as main reservoir targets, as for example the geothermal exploitation in the Southern German Molasse Basin

It has been nowadays recognised that the sustainability of the hydro-mechanical performance of fractures is crucial for engineering applications. Field observations indicate that pore pressure changes due to production and injection or fracture relaxation when the well is not operated, will alter the hydro-mechanical behaviour and therefore the fracture permeability

A quantification of the stress and pressure distributions along the fracture surfaces and their alteration due to induced flow and mechanical loading is still missing in the present literature. Furthermore, the processes responsible for permeability changes such as microscopic tensile or shear failure, compaction or fracture mineralisation, as well as their interactions are still poorly understood. To address these issues, numerical simulation of the composite matrix-fracture system under mechanical loading has become increasingly important.
Generally, such simulations mostly consider flow within the fracture only

In this study we aim at improving the current understanding on these topics by presenting in a first stage results of flow through experiments under varying effective pressure conditions of a porous rock samples (Flechtinger sandstone) containing a single macroscopic fracture. In a second stage, we make use of these experimental results to carry out a numerical investigation of the hydro-mechanical behaviour of the matrix-fracture system to analyse stress and fluid pressure distribution and potential failure mechanism.

Experiments have been carried out in a MTS tri-axial cell

Experimental setup and Flechtinger sandstone samples (SBT6-BE) containing a shear-displaced tensile fracture.

The results indicate a permanent damage expressed in terms of an overall reduction of the rock permeability and fracture closure (aperture) when increasing confining pressure (Fig.

To represent the geometrical features of the matrix-fracture system, a fracture surface scan (Fig. ^{®} white light scanner (fringe projection, optical, non-contact, monochromatic (LED), resolution: 2048

FE-Mesh of the fracture

To analyse the stress, pressure and velocity field in a matrix-fracture system, an adequate physical model of the involved processes and their coupling is required. The Multiphysics Object Oriented Simulation Environment (MOOSE) in combination with GOLEM

Definition of 3-D domains, all physical surfaces, and interfaces between the fracture and the surrounding rock matrix.

In order to derive a solution for the matrix-fracture system, the following initial and boundary conditions were chosen. The initial condition for the rock matrix is: pore pressure and displacement field equal to zero. Furthermore, the fluid pressure and the velocity inside the fracture are set to zero. In addition to the boundary conditions applied at the interface, we imposed an inflow of 0.16 m s

Material properties of the rock matrix used for the numerical simulation.

Due to solving Navier–Stokes equation for incompressible fluids, an almost instantaneous steady state pressure field in the fracture was achieved which diffused into the rock matrix. The time required for pore pressure diffusion to the matrix and for establishing a steady state pressure field is about 0.1 s simulation time (Fig.

Pressure field in the matrix-fracture system after 0.0001

Velocity field in the matrix-fracture system after 0.0001

After 10 s of fluid flow, a load ramp in negative

Stress, strain, and displacement in the matrix-fracture system after after applying 17 MPa normal stress at the top surface.

This deformation led to a closure of the fracture and an increase in the average inlet fluid pressure from 4728 to 4778 Pa. With an inlet fracture area of 1.3 mm

Even if only elastic rheology is introduced in the model, the calculated stress-strain state was compared to failure envelopes obtained by laboratory experiments in order to detect possible limit of the elastic behavior. Such failure criteria are power-law functions of the mean stress acting on the plane of failure

Although true tri-axial strength data in various rocks are best modelled by power-law relationships between octahedral shear stress and mean effective normal stress, it has been suggested that they can be approximated by a linear relationship

The material parameters

Normalized failure criterion:

In order to generate the FE-Mesh, the scan of one single fracture surface was used. Indeed, scanning only one fracture surface and duplicating it, has the advantage of providing the simplest model for generating a perfect tensile fracture. For such a perfect tensile fracture, the influence of surface roughness and surface orientation on the fracture aperture can be neglected. Fracture aperture was then introduced by imposing both shear offset along the fracture plane and normal offset perpendicular to the fracture plane. The generated fracture volume can be used to test the presented method but requires some improvement to represent real fracture opening. To obtain a more realistic fracture open volume, either both surfaces of the fracture should be scanned and recombined or two synthetic self-affine surfaces should be generated

In real fractures, mineralisation inside the fracture volume

The current numerical simulation solves for the dynamic pressure due to the viscous term of the fluid. In contrast, the absolute pressure acting on the fracture surfaces is essential in fracture mechanics. For a better approximation of the hydro-mechanical processes in the matrix-fracture system, both, the dynamic, as well as the absolute pressure, must be considered. Such an implementation would allow for a simulation of fluid pressure and stress conditions as applied in the laboratory experiments which will make direct comparison more effective. Furthermore, imposed effective stresses during the laboratory experiments were between 30 and 60 MPa. In the numerical simulation only a normal stress of 17 MPa at the top surface was applied. This will be valid to simulate the general trend of deformation but provides some limitations to extrapolate the numerical results to the laboratory analogue.

For calculating the fracture permeability, we assumed laminar flow conditions and applied Darcy's law.
Although the flow rate

These numerical obtained permeability values should be compared to the laboratory results. For the laboratory experiments no information about local fluid velocity and local pressure gradients are available and only the permeability on the macroscopic scale of the matrix-fracture system was determined. Therefore, we calculated the numerical obtained permeability of the overall system by imposing the same pressure boundaries for the rock matrix, as well as for the fracture. This would result in a constant pressure boundary at the inlet (matrix and fracture) and at the outlet. For the overall system the cross-sectional area

The laboratory experiments indicate that the permeability alteration is due to elastic and inelastic deformation. Elastic deformation only causes 13 % to 19 % of the total deformation. This elastic deformation leads to a permeability recovery of 19 % to 30 % (see Sect.

Due to the size of the simulation results (5 GB) the data are not publicly accessible. The simulation input files can be requested by eMail: guido.bloecher@gfz-potsdam.de.

The laboratory experiments were conducted by CK with scientific support of HM. The nuermical simulation was performed by GB and related/required code development were carried out by MC and AJ. Linkage between the laboratory experiments and numerical simulation were derived by JS and GB.

The authors declare that they have no conflict of interest.

This article is part of the special issue “European Geosciences Union General Assembly 2019, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2019, Vienna, Austria, 7–12 April 2019.

The authors acknowledge the financial support by the Federal Ministry for Economic Affairs and Energy of Germany (BMWi) in the project RESALT (project no. 0324244C).

The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Christopher Juhlin and reviewed by two anonymous referees.