WP 2: Pathways through pressure-driven percolation


crack simulation At CAU Kiel, three samples under different stress boundary conditions undergo the fracking process using the true triaxial apparatus. According to the pre-existing mechanical stress distribution, the fracking pattern under applied hydraulic pressure is unique for each stress distribution.

IfG is exploring the opportunities presented by new software techniques to model the desiccation processes of clay.

Modelling: Difference of liquids and gases in pressure driven percolation

Salt caverns are used to store natural gas and mineral gas. When they are abandoned, they are usually filled with brine and sealed. The natural convergence of the caverns leads to an increase of the pressure, which can supersede the minimal principal stress in the contour. The cracks that form provide an additional accessible volume, to which the fluid reacts with a change of pressure. As the bulk moduli of liquids and gases are very different, there corresponding reaction is different, too. We modelled this process for an experiment on lab scale, and could reproduce the expected responses.


Experiment: Closure and healing of cracks in Opalinus clay

We conducted experiments on samples of the sandy facies of Opalinus clay, to study the long term behavior of cracks under isostatic stress conditions. It turned out that the clay was so fragile, that already the preparation of the samples induced enough microscopic damage to measure a significant permeability. A gas pressure of 0.5 bar was applied to one side of the probe, and the gas flow was measured for several days under various stresses. The decay rates agree with the expectation, that the closures take place faster under higher stresses.


 In the geotechnical/geomechnical laboratory of CAU Kiel, the fluid driven percolation tests are performed. The cubic Opalinus claystone samples are prepared in the side dimension of 43 mm and with the drilled cavity length and diameter of 20 and 8 mm, respectively. The samples embedded layering orientations are shown in Fig. 1, where in the first case the applied oil pressure is perpendicular to the layering orientations and in the second case it is parallel to the layering orientations. The syringe pump is used to pump the pressurized hydraulic oil (up to 517 Bar) into the sample and with gradually increasing the pressure the fracking process is carried out. In the first setup, the hydraulic fracking is initiated at 23 MPa and the clear flow paths through the embedded layering surfaces is observed. similarly, for the second test setup, the hydraulic fracking is initiated at 10 MPa through the embedded layering surfaces. Fig. 2 illustrates the borehole pressure evolution with flow volume change obtained from the experimental data. The initial volume of the pump is around 265 mL.

fib1a fig1b

                    (a)                                                            (b)

Fig. 1. The fracking paths through the Opalinus claystone for a (a) 1st stress, and (b) 2nd stress configuration

fig2a fig2b

                    (a)                                                            (b)

Fig.2. The borehole pressure vs. flow volume for an Opalinus claystone (a) 1st stress configuration, and (b) 2nd stress configuration

CAU Kiel

The fluid-driven percolation experiments on cubical clay and salt samples with dimensions of 43 x 43 x 43 mm are conducted under laboratory conditions at CAU Kiel. The hydraulic pump filled with hydraulic oil can reach a pressure of up to 700 bar. The pressure-controlled technique is considered, and the corresponding flow rates are stored. Three samples under different stress boundary conditions undergo the fracking process using the true triaxial apparatus. The existing P and S-wave ultrasonic sensors on pistons are used to measure the material property change using the analytical relation existing in literature. According to the pre-existing mechanical stress distribution, the fracking pattern under applied hydraulic pressure is unique for each stress distribution. In order to visualize the fracking path, X-Ray computed tomography (XRCT) scans are made at the University of Stuttgart from samples before and after fracking. To keep the cracks open, a mixture of metal powder with a maximum particle dimension of 30 micrometers is added to the fluid mixture.

201909_WP2 CAU figure1 Fig. 1: The pump used for applying the hydraulic pressure (left), the preparation of the sample in the true triaxial apparatus (right, top), and the prepared cubic sample with the drilled cylindrical cavity (right, bottom)


Modelling: Shrinking of clay and crack opening due to desiccation

When clay is exposed to air, for example on the surface of walls of drifts, it tends to develop cracks because its water content evaporates. The pore pressure in the material is reduced, leading to tensile stresses which eventually open up cracks. Recently, Itasca added a feature of bulk pore pressure to its 3DEC program in addition to the usual discrete element fluid knot feature, which could open up the possibility to model and analyze such processes. We are currently exploring the possibility by simulating a block of saturated clay with a constant evaporation rate at the top. As expected, we see the creation of cracks:

Sept 2019 fig 2 Fig. 2: Opening of cracks (color-coded displacement magnitude) on a block of clay because of evaporation of water

Up to now, only qualitative results have been obtained. It remains to be explored whether the model is also able to function with more realistic parameter sets.

At CAU Kiel, the gas and fluid pressure-driven percolation in clay and salt stones are experimentally and numerically investigated. Using the true triaxial device in the laboratory of CAU Kiel, a fluid-driven percolation test on a cubic sample is carried out. By controlling the three principal stresses individually, the frack paths and change of flow rate are determined.
Fig1 Fig. 1: The prepared cubic claystone sample (43 x 43 x 43 mm) (left) and the true triaxial device (right)
The crack initiation and propagation, change of permeability, change of flow rate upon cracking and during the healing process and developed frack paths are all captured with the hydro-mechanical dual lattice model (Grassi 2009). The conduct elements are the fluid pathways throughout the discretized medium.
Fig2 Fig. 2: The conduct elements shown with black lines
Fig3 Fig. 3: The 3D Hydro-mechanical LEM simulation of fluid-driven percolation is salt stone. The fracking surfaces are shown in red (left) the developed water pressures in the medium (right)
P. Grassl (2009). A lattice approach to model flow in cracked concrete. Cement & Concrete Composites 31 (2009) 454–460.

Experiments on Pressure-driven Percolation and Sealing/Healing of cracks

Up to now, three experiments have been carried out to study the sealing and healing of cracks in rock salt. Boreholes were drilled to the center of cylindrical samples and fitted with a metal pipe so that gas pressure could be applied to a small area at the center.



Initially, all samples were placed under isostatic stress of 50 MPa for one day to consolidate them. For the actual experiments, the isostatic stresses were then changed to 10 MPa, 30 MPa, and 50 MPa, respectively. A gas pressure at the center of the samples was then applied and increased in small steps until a gas flow was detected. In all three cases, a gas flow was detected before the percolation threshold was reached (usually a few bar below), which indicates that the samples suffered micro-fractures during preparation. After the last increase of the pressure, the flow rate was monitored for 24 to 60 hours under constant conditions. The corresponding permeabilities were:


In all three cases, a reduction of the permeability was observed, showing a narrowing of the pathways due to creep.

Further analysis and a numerical model for WP 2.2 (see following article: Numerical Model of Fractures) are work in progress.

Numerical Modeling of Fractures

The IfG worked on several Model Exercises (ME) that were agreed on between the project partners of GeomInt.

The first ME was a simple three-point bending test to compare how the various methods employed at the different institutions describe fracture mechanics. We used the discrete element method 3DEC (Itasca Consulting Group) to set up two models of a granite bar:

For each, the force and the crack mouth opening displacement (CMOD) were recorded as well as the acoustic emissions.

Fig2 Force-CMOD behaviour for the two samples Fig3 Locations of acoustic emissions for the first sample

The results agree very well with an experiment done by Tarokh et al. (Int. J. Fract. 204, p 191 (2017)).

In a second ME, pressure-driven percolation in a cubic sample of rock salt in an anisotropic true-triaxial stress state was simulated. The purpose was to study the orientation of the resulting fractures as a function of the stress state and to compare the performance of the various methods employed by the project partners. Here again we used 3DEC:

Fig4 Left: Experimental situation, right: Model results. As the minimal principal stress is in the vertical direction, a horizontal frac is produced. Fig5 Left: Experimental situation, right: Model results. As the minimal principal stress is in the horizontal direction, a vertical frac is produced.


Development of numerical methods: Quantitative treatment of pressure driven percolation and the  difference between liquids and gas                 

Simple model system Simple model system

Pressure profiles Pressure profiles

Pressure-driven percolation in a cylindrical sample: In a homogeneous sample and stress state, the fluid  expands in an isotropic pattern.

Isotropic distribution


BGR has tested the implementation of the new implemented X-FEM LIE approach in OGS-6. With this approach, it is possible to simulate the hydraulic stimulated deformation behavior of an existing crack. This includes opening and closing, shear deformation and change of crack permeability.

crack deformation Exemplary calculation of crack deformation, induced by fluid injection. Simulated with OGS-6.

In order to determine the flow pathways through pressure-driven percolation, the experimental result provided from our project partner IfG Leipzig is used. In their provided test a cylindrical sample is under radial confinement pressure of 50 MPa and gas pressure is gradually increased till the gas flow rate initiated in the medium (Figure 1).

Fig1 Figure 1: The cylindrical sample and cutoff to apply gas pressure and observe the change of flow rate with fracture propagation (IfG Leipzig)

With application of lattice element method (LEM) and using the dual lattice method to define the conduct elements as flow paths (Lij), the hydro model is coupled with previously developed mechanical model.

Fig2 Figure 2: The dual lattice model for coupled hydro-mechanical lattice model

The gas flow rate change with developed pressures and fracture aperture is determined using,


where kL is the permeability, ei’j’ is the aperture width in 2D lattice model, Lij is the conduct element length, μ is the viscosity and Pi and Pj are the gas pressure in i and j conduct points.

According to the differential equation of the non-stationary flow problems, the flow in a single conduct element (i,j) is,


where kM and CM are the conductivity and capacity matrix, respectively and P is the pressure potential with two degrees of freedom.


With solving the aforementioned equation the applied forces on conduct points (i,j) as well as lattice points (i‘,j‘) are determined and inserted in mechanical model. The change of aperture width, ei’j’, with crack opening is determined and transferred to flow model. The LEM boundary condition is shown in figure 3. The generated lattice model has 20,000 Voronoi cells and randomness factor of 0.5. The model has around 50,000 elements. The fracture flow path in the tip of the hole is shown in Figure 4. In general, the fracture pathway is in horizontal direction.

The comparison of experimental results from IfG Leipzig and LEM simulation results from CAU Kiel is shown in Figure 5. The fracture initiation is at gas pressure of 39.6 MPa, which corresponds to propagation of gas flow into medium. In LEM, the gas pressure is gradually increased, 0.1 MPa steps, till 45 MPa. The healing of the fractures is not captured with LEM, which need to be investigated.

Fig3 Figure 3: The LEM boundary condition for simulation of pressure-driven percolation Fig4 Figure 4: The LEM result of flow pathway in saltstone under pressure-driven percolation Fig5 Figure 5: Comparison of experimental and LEM simulation results of pressure-driven percolation in cylindrical saltstone.

The IfG set up a three-dimensional model to test several approaches for the propagation of fluids by pressure driven percolation. Pressure driven percolation is defined as the opening of intercrystalline grain boundaries, caused by fluid pressure, that leads to the creation of interconnected flow paths. A library of subroutines has been set up, with several numerical functions that can describe the advancement of the fluid front. Currently numerical simulations are under way to study the influence of the compressibility of the fluid, with a focus on the difference between gases and liquids.

Within AP2, the UFZ team implemented cohesive zone models for hydraulically stimulated fracturing as well as specific hydraulic fracture flow laws into numerical methods in OpenGeoSys using enriched finite element spaces. Simultaneously, a phase field model for pressure driven pathway generation has been implemented and is being tested.

Currently, comparisons between both methods are ongoing regarding the quantification of key physical quantities such as fracture volume or percolation/fracture initiation pressure.

The flow path in salt stones is due to pre-existing or developed discontinuities in the microstructure. The mechanical loads can result in new flow paths as well as propagation of pre-existing microcracks and eventually increase of the hydraulic conductivity of the salt stones. In this regard, in CAU Kiel laboratory, extensive experimental tests under coupled thermo-hydro-mechanical (THM) conditions will be conducted. The aim is to investigate the change of THM property of salt stones due to developed microcracks. Besides the laboratory tests, the numerical study is also conducted. While using in-house developed Lattice Element Method (LEM) and the experimental results, a new numerical method for determining the change of hydraulic conductivity under THM processes will be developed. According to our research results in CAU Kiel, the developed LEM is able to capture the developed fracture path and it‘s effect on mechanical and thermal properties of the rock samples.

fig1 Figure 1: The transform of (a) microscopic image of cemented rock material into (b) developed lattice element while using the image processing toolbox of MATLAB. [1]

fig2 Figure 2: The comparison of the determined experimental thermal conductivities of rock sample (Fig 1) with LEM while applying different confinement pressures. [1]

fig3 Figure 4. Variation of shear moduli of two rock samples at 20°C with the change in applied hydrostatic pressure up to 101 MPa. The local elemental failure is considered and the thermal strain is not considered for the failure. [2]


[1] A. S. Sattari, Z. H. Rizvi, H. B. Motra, F. Wuttke (2017). Meso-scale modeling of heat transport in a heterogeneous cemented geomaterial by lattice element method. Granular Matter. 19:66. (DOI 10.1007/s10035-017-0751-4).
[2] Zarghaam H. Rizvi, Amir S. Sattari, Hem B. Motra, Frank Wuttke (2017). Effective parameter evaluation in heterogeneous cemented geomaterials subjected to high temperature, pressure and constrained deformations, Second International Symposium on Coupled Phenomena in Environmental Geotechnics, University of Leeds, UK.

Prof. Dr. Ing Frank Wuttke, M.Sc. Amir Shoarian Sattari, M.Sc. Zarghaam Rizvi (Institute of Geoscience, CAU Kiel)