A new stochastic particle-tracking approach for fractured sedimentary formations


Corresponding author: M. Willmann, Institute of Environmental Engineering, ETH Zurich, Wolfgang-Pauli-Str. 15, CH-8093 Zurich, Switzerland. (willmann@ifu.baug.ethz.ch)


[1] Explicit particle-tracking simulations in both fractures and surrounding sediments are conceptually complex and difficult to implement. This is mainly due to the difference in the nature of the velocity fields in fractures and the matrix. The major problems are (1) to avoid particles in the matrix jumping over fractures and (2) to control particle behavior at the interfaces between fractures and the matrix. We developed a particle-tracking method that models advection and diffusion explicitly in both fractures and the matrix. Analogously to most flow simulations, we conceptualize transport by two separate domains, the fracture and the matrix domains, which exchange particles between them. No a priori assumptions on transport mechanisms in the matrix have to be made. Each individual particle step stops at an interface between two matrix cells, between two fracture cells, or between the matrix and a fracture. Mass exchange at the interface from fractures to the matrix is controlled by the fracture aperture, the matrix flux perpendicular to the fracture plane, the flux within the fracture, and a diffusive component. The method is developed for fractured sedimentary rocks where the advective fluxes in the matrix can be as large as the fracture fluxes. However, it could also be applied to fractured crystalline rocks where the matrix contribution is smaller. Finally, some simple applications to a fractured shale formation in northern Switzerland are included to illustrate the usefulness of this method to investigate transport in fractured sedimentary formations.

1. Introduction

[2] Most research on transport in fractured media has focused on crystalline rocks [e.g., National Research Council, 1996; Neuman, 2005; Davy et al., 2006; Bourbiaux, 2010]. Those formations are characterized by such small flow velocities in the background matrix that successfully transport calculations focused on the fractures only. The matrix contribution is either completely ignored, and only particles within a discrete fracture network (DFN) are calculated, or the matrix contribution is approximated as a delay/retention mechanism such as matrix diffusion [Neretnieks, 1980; Carrera et al., 1998]. Matrix diffusion leads to a power law tail of −3/2 in the breakthrough curves, which was often observed in crystalline rocks [e.g., Hadermann and Heer, 1996]. However, even for crystalline rocks, different slopes of breakthrough curve tails were observed [Becker and Shapiro, 2000], indicating that other effects like advective heterogeneity [e.g., Zinn et al., 2004; Willmann et al., 2008] control large-scale behavior.

[3] In fractured sedimentary formations the matrix contribution to flow and transport can be much stronger and cannot be ignored or approximated. Here the most common approach is the opposite; the fracture contribution is approximated, while the matrix is modeled. One approach is to map the fracture permeability into the permeability of the background matrix [e.g., Jackson et al., 2000; Lee et al., 2001; Svensson, 2001; Botros et al., 2008; Roubinet et al., 2010a], conserving the overall flux through the formation. However, as the matrix blocks are much larger than the fractures, the fractures are smeared out to match the matrix geometry. In order to conserve the fracture transport velocities one needs to decrease the porosities in the respective blocks. Those matrix blocks are much larger than the fractures, but the fracture properties dominate its transport behavior and any fracture-matrix exchange is not well represented.

[4] Explicit particle tracking in a hybrid fracture-matrix system where particles move in both discrete fractures and the background matrix is complicated due to the different nature of matrix and fracture flow. Matrix flow is continuous in the entire domain (3-D), while fracture flow only takes place within the discrete fractures (2-D). A problem is that in many particle-tracking approaches the particles do not move continuously but jump from position to position. This leads to two apparent problems: first, particles transported in the matrix may jump across existing fractures instead of being transported within the fracture. Second, it is unclear how a particle, once moved into a fracture, can leave the fracture again. Also, it has to be determined what a deterministic particle does at any fracture intersection where flow is distributed to different paths. Recently, Roubinet et al. [2010b] developed a particle-tracking method that allows particles to jump from one fracture to another one depending on the homogeneous matrix diffusion parameters and the fracture proximity. However, advection within the matrix cannot be adequately represented.

[5] Our objective is to develop a particle-tracking method to model hybrid fracture-matrix transport where advection and diffusion within both the fractures and the matrix are accounted for explicitly.

2. Fracture and Matrix Geometry

[6] The new particle-tracking method has to account for the different nature of velocity fields in fractures and matrix. While the matrix is continuous in 3-D, the fractures are continuous within the 2-D fracture system but discontinuous within 3-D space. Thus, we superimpose both systems and enable exchange of particles between them. In order to obtain good transport simulations, the flow field should be as accurate as possible. In a formation with a complex fracture network and a strongly heterogeneous background matrix this is challenging. Both systems have to coexist within a domain and share nodes, where water and solutes are exchanged. Different methods exist to calculate combined fracture-matrix flow by modeling the fractures explicitly or as discontinuities between matrix cells [Martin et al., 2005]. We use here HydroGeoSphere (HGS) [Therrien and Sudicky, 1996; Therrien et al., 2004], which accounts for fractures explicitly. In our approach, the matrix requires an orthogonal grid, and the fractures are placed at the cell faces sharing the same nodes. HGS actually allows for inclined fractures, which still share the nodes of the matrix system but are able to cross the individual cells [Graf and Therrien, 2008]. This orthogonal form can lead to a steplike shape of fractures. This affects modeling in two ways: first, the path length within a fracture is larger for the model than for reality, and the overall geometry might not be matched. Finer discretization can improve the matching of the overall geometry, but the effect of increased path length is independent of discretization. The best way to account for the correct geometry would be to use some general finite element method that allows an arbitrary delineation of the fracture system and an explicit and heterogeneous description of the matrix. The method described in the following can be implemented on such a grid as well. Thus, any type of 2-D or 3-D fracture network could be used. On such a coupled system, flow is solved exchanging water between fractures and matrix at the common nodes. Matrix flow is solved based on Darcy's law and fracture flow based on Poiseuille flow considering a constant fracture aperture within a single fracture element.

3. Particle Tracking Within the Matrix

[7] The standard approach for particle tracking uses a constant time or space step. This is straight forward for calculations in (unfractured) porous media [Prickett et al., 1981; Kinzelbach, 1987; Delay et al., 2005; Salamon et al., 2006] but causes some complications for fractured media. A particle may cross from one side of a fracture to the other in a single jump. This means one has to check for each individual jump whether a fracture was crossed. To determine the point of intersection, time-consuming loops over all fractures and particles would be required. An alternative approach for porous media was proposed by Pollock [1988] for a finite difference scheme. From the fluxes at the cell faces and the initial position the outlet position from that cell was calculated using trilinear interpolating velocities. This approach was generalized for finite element cells by Cordes and Kinzelbach [1992] and Frind and Molson [2004]. The advantage of these approaches is that the end point of each individual jump is at the cell face of the matrix. No time-consuming search has to be performed whether a fracture was crossed during a space step. Only a simple check at this end point has to be performed. If a fracture exists at the cell face, the particle is passed on from the matrix system to the fracture.

[8] The Pollock method is based on a finite difference discretization that provides fluxes at the cell faces. If finite element methods are used, the fluxes at the center of an element (cell) are given. In order to use the Pollock method the cell face fluxes have to be consistently calculated. With arbitrarily shaped finite elements a continuous flux has to be determined for each pair of cells touching in a cell face [Frind and Molson, 2004]. Within this work we use a regular, orthogonal grid. This reduces our calculation to the harmonic mean of the respective cell fluxes. Next to fractures, the averaging was not performed, as there is no influence from the matrix cell opposite to the fracture. In this case the flux at the center of the cell was used to approximate the flux at the cell face.

[9] The Pollock method originally does not include diffusion or dispersion. Still, in sedimentary fractured aquifers, diffusion is known to be an important transport mechanism. The major difficulty of implementing diffusion into the Pollock method is that the particle has to be projected back to the cell face after the diffusive step. The concept of transport within the matrix is shown in Figure 1. First, the advective step xad is calculated according to Pollock [1988], which can be approximated by

display math(1)

with the matrix pore velocity vm and the advection time tad. Using tad, the three spatial components of the diffusive step, xd, are calculated:

display math(2)

where subscript n is the indicator for the spatial component, Dm is the molecular diffusion coefficient, and ξ is a random variable with mean zero between −1 and 1. Two diffusion components are moving along the fracture plane, and the third one is perpendicular to it. Thus, the new particle position does not lie on the cell face anymore. For the Pollock method, the particle has to be projected back to the cell face. This is done by adjusting the time, tad+d, the advection and diffusion time, or updated transport time:

display math(3)

Note, the diffusive step for a single particle can be both positive and negative, and therefore, the time can both decrease and increase with respect to tad. Finally, the diffusive components (2) and (3) have to be updated using the new time:

display math(4)

Remember, that time scales differently for advection and diffusion. For cases with very slow advection (tad very large), the diffusive steps become very large. Here the lateral steps inside the fracture plane might lead into the next fracture element, and the particle has to be projected back for these directions as well. Due to the different scaling of times this becomes complicated to implement. This is why we restrict ourselves to lateral diffusive jumps within a single fracture element. This makes our method valid for advection-dominated diffusive transport only. An advantage of not leaving a matrix cell during a time step is that there are no problems concerning the continuity of parameters like diffusion or porosity.

Figure 1.

Concept for transport (left) within the matrix and (right) within the fractures in two dimensions. For the matrix, first, the advective step is calculated (green). Then, the diffusive step perpendicular to the cell face is performed. This step is projected back while correcting the transport time. Finally, the lateral diffusive step is performed. For transport within the fracture, first, the advective time contribution is calculated. This time is then used to calculate the diffusive step perpendicular to the fracture face. Projecting this step back to the fracture wall renders the (real) washout time (tfad+fd), which is then used together with the fracture velocity to calculate the point where the particle leaves the fracture. Again, finally, the correction for diffusive step along the fracture is done.

[10] A note of caution has to be added about heterogeneous diffusion around the cell face. If the diffusion coefficient is different on both sides of the interface where the advective and diffusive spatial step takes place, the projection back to the cell face would lead to skewed Brownian motion. It has to be corrected for in order to represent diffusion correctly [Lejay, 2011]. As we use here the same homogeneous diffusion coefficients in both matrix and fractures, we do not have this problem. Further investigations are needed to quantify this effect. Particularly, for the case of a different diffusion coefficient in the matrix and the fractures, one would get a systematic bias for the washout times. For the opposite case when a particle moving within the matrix is stopped at a fracture, this effect is most likely small as the fracture aperture is small.

[11] The general exchange mechanism of a particle moving from the matrix to the fracture system is simple. Once a particle hits a fracture, it is moved within the fracture system.

4. Transport in the Fracture System

[12] Without any external force a particle would stay forever in the fracture as occurs in DFN (discrete fracture network) models. If additionally matrix diffusion is applied, particles reenter the fracture at the same point where they left it. So matrix diffusion is only a delay mechanism. If 2-D matrix diffusion is considered, additionally, a lateral diffusive component is introduced [Roubinet et al., 2012].

[13] If advection in the matrix is a process that cannot be ignored, two main processes control the residence time of a particle within a fracture: the advective flux perpendicular to the fracture plane and the diffusive jump perpendicular to the fracture plane (Figure 1).

[14] The position at which the particle is released again to the matrix is calculated using the fracture velocity, vf, and the fracture residence time, tf:

display math(5)

where xf is the position vector within the fracture, vf is given from the flow calculation, and tfad represents the time required to wash particles out of a fracture. The residence time tfad can be obtained from

display math(6)

where af is the fracture aperture and qmv is the matrix flux component vertical to the fracture element (cell face). The three diffusive steps are calculated analogously to the one in the matrix:

display math(7)

Analogous to matrix transport, the diffusive step perpendicular to the interface has to be projected back to the fracture wall by increasing or decreasing the transport time:

display math(8)

This gives us the updated transport time (tfad+fd), and using equation (5), an outlet position of the particle (xfad+fd) is calculated considering advection and perpendicular diffusion. Note, time can be decreased or increased depending on the sign of the diffusive jump. Finally, the final outlet position (xnew) is calculated. As for the matrix, the two remaining diffusive space steps are calculated using the updated washout time (tfad+fd).

[15] Special attention has to be given to the fracture aperture, af. The fracture aperture is not uniquely defined. Tsang [1992] and Zheng et al. [2008] distinguished between hydraulic aperture (ahyd) and mass balance (or transport) aperture (atransp). The hydraulic aperture is based on the concept of parallel plate voids using the cubic law and is known to model correctly hydraulic tests in fractured rocks. This aperture was also used in HGS for the calculation of the flow field used here. Tracer tests in fractured aquifers interpreted using ahyd show generally a breakthrough arrival time that is too small. This is why the mass balance or transport aperture was introduced. Both are equivalent apertures. The difference between them is that ahyd is the geometric mean of the true heterogeneous apertures, while atransp is the arithmetic mean [Tsang, 1992]. Both are averages over fracture depth and represent averaged quantities like a breakthrough curve. However, water (and therefore tracer) is known to move along preferential flow paths due to heterogeneity rather than homogeneously within the fracture plane. This means that the actual aperture (awashout) at a preferential flow path that is relevant for the washout is expected to be larger than proposed by the ahyd and atransp:

display math(9)

All apertures are equal for completely homogeneous parallel plate fractures and diverge with increasing fracture roughness. Their exact relation is not easily quantified, and more research will be needed. Porosity is set to 1, as we use here the concept of parallel plates. This might not be realistic for many field cases, as a fracture can be partly filled with sedimentary material. Such a decrease in porosity would also require an increase in fracture aperture to allow the same flux to pass through as the cubic law is not valid anymore. It is likely that the washout aperture has to be modeled stochastically in order to account for the variability in the size of these preferential paths. We here use the hydraulic aperture. This means our residence times within the fractures are likely to be underestimated. Generally, more research is needed to understand the heterogeneity of individual fractures and its consequences for the interplay between fractures and matrix.

4.1. Particle Behavior at Fracture Intersections

[16] An additional problem arises when a particle is moved within a fracture that is intersecting with another one. The particle can move along each fracture that has a flow component away from the intersection point of the particle path. A maximum of three possibilities exists in 3-D for a regular orthogonal grid, as a particle cannot leave a fracture at its edge of intersection. Different approaches to mixing at fracture intersections in 2-D models can be considered. Either no mixing takes place, and the particles follow their streamlines or mixing mechanisms like diffusion and heterogeneity are taken into account. Berkowitz et al. [1994] and Mourzenko et al. [2002] studied the particle behavior for homogeneous fractures and found that particles follow streamlines for high Péclet numbers, while complete mixing is obtained for low Péclet numbers. However, it was found that the different regimes have little influence on the overall particle behavior [Park et al., 2001]. Additional mixing effects due to the heterogeneity of fractures probably increase mixing at fracture intersections. Thus, we assume here complete mixing and weight the probability by the relevant fluxes in that fracture. This, together with the diffusive step, implies that our method is not deterministic anymore but becomes stochastic. Consequently, many particles have to be tracked to get meaningful results.

4.2. Implementation and Validation of the Method

[17] The presented method was implemented numerically in a 3-D particle tracker that uses HGS output files as input for velocities and fracture apertures. The method was validated for a small example of two homogeneous blocks (matrix cells) of different permeabilities and a fracture between them, which was finely discretized (Figure 2). The discretized fracture had to be enlarged with respect to its true aperture to ensure the correct fracture fluxes. The fracture hydraulic conductivity was calibrated against the flux through the fracture of the corresponding model with a fracture represented by the corresponding open aperture. A fracture porosity was used to adjust the fracture velocities. As the fracture width changed for our two examples, washout had to be scaled accordingly. Both the advective and the diffusive components of the washout were scaled by the ratio of true fracture aperture and the discretized fracture width.

Figure 2.

Validation of the method. (right) An explicit fracture with a small aperture. A particle is moved from the bottom to the fracture, then carried along the fracture, and finally released to the matrix again. (left) The equivalent of a discretized fracture. The fracture is enlarged for numerical reasons. This enlarged aperture means the washout has to take place over a larger distance. If this effect is accounted for, then the two models behave identically.

[18] The general particle behavior can be observed in Figure 3. A single fracture is crossing the domain from bottom to top and is parting into two fractures before hitting the upper boundary. Water flows from bottom to top. The two cases shown differ only in the ratio between matrix and fracture velocities. The weaker the matrix flux is, the longer the particles stay within the fracture, and the more the fracture system dominates.

Figure 3.

Particle behavior for a simple synthetic setup without diffusion. Flow is from bottom to top. (left) The particles are hardly affected by the fracture. Matrix flow is too strong, and particles are released from the fractures almost instantly. (right) All particles stay longer times inside the fracture. Here the matrix flow is small compared with the one in the fracture.

5. Application to the Effingen Member

[19] Finally, we apply the new method to a 2-D representation of a heterogeneous sedimentary formation, the Effingen member. The Effingen member of the Wildegg Formation (Oxfordian Malm stage strata of the Upper Jurassic) is currently under investigation for deep geological disposal of radioactive waste. In the area of interest, the Effingen member is 220 to 240 m thick at depths of 300–700 m below the surface and consists of interlayered calcareous marls to limestones. The sequence comprises two major hydrostratigraphical units: a more permeable internally fractured limestone, the Gerstenhuebelschichten (GH), and a less permeable clay-rich marl background (Ton-Mergel Ablagerungen (TMA)) [Roeser et al., 2008]. Three (sub)vertical fracture systems are crossing the whole formation, which are modeled stochastically (both orientation and aperture). The mean transmissivity of the fracture is T = 10−9 m/s. Here, for a better visualization, only the 16% of the fractures that have the highest permeability are shown [Lanyon, 2008]. Using some quasi-2-D (no flux in vertical direction) simulations, we want to investigate the general transport behavior in the two sedimentary units. A model domain of 2048 m × 2048 m with a discretization of 8 m is used resulting in 256 × 256 cells (Figure 4). Constant head boundaries assure a gradient of 0.005 and mean flow from North to South, and a line source of 34,700 particles evenly distributed at the inflow boundary allows characterization of the whole domain. One can observe the steplike shape of the fractures discussed above, but the discretization is fine enough to respect the general geometries. The matrix is assumed homogeneous with a hydraulic conductivity value representative for GH (K = 10−9 m/s). The diffusion coefficient in the matrix was set to an upscaled value of Dm = 3 × 10−10 m2/s [Churakov and Gimmi, 2011]. The same diffusion value was used for fractures. This leads to Péclet numbers of the range of 10–100 for the fractures. The hydraulic heads and the particle paths are shown in Figure 4. The particle paths are not much affected by the fractures. There is some minor draining effect that leads to the areas without particle traces. This observation is confirmed by the histogram of arrival times (Figure 4). Arrival times are normalized to the advective matrix travel time. The arrival is fairly homogeneous and around the time of advective transport. This means transport is clearly dominated by advection within the matrix. In Figure 5 the case for the lower permeable matrix is shown. Here the background hydraulic conductivity has a representative value for the TMA (K = 10−12 m/s). The fractures are identical to the previous example. Here the advection in matrix is expected to play a smaller role, as the fracture system transports water 20 times faster than the matrix (Table 1). The observed transport behavior is completely different. Here the particles are effectively drained by the fractures and moved over larger distances within the fracture system. Then, they are released to the matrix again and are transported along the fractures. Transport appears to be dominated by the fractures. Large parts of the matrix are not contributing to transport. However, if we look at the histogram of arrival times (Figure 5), we see that the arrival times are not faster than for pure matrix advection but slower. This is due to diffusion within the matrix. The relatively large diffusion (compared with advection in the matrix) has two effects. First, it leads to a faster ejection of particles from the fracture to the matrix. Remember, without diffusion and with small advection within the matrix the particles would almost stay forever in the fracture. Second, if a particle stays within the matrix, transport is diffusion dominated. Overall, we can say that for TMA the transport is diffusion dominated. However, the fracture system leads to some draining of particle paths that might be important for reactive transport simulations.

Figure 4.

Particle behavior for a simple 2-D representation of the Effingen member having a uniform background hydraulic conductivity representative for the more permeable GH (K = 10−9 m/s). For illustration purposes, only 247 particles are shown. Particles move downward and appear not to be affected by the fracture system. Also, the histogram of arrival times shows an almost uniform behavior close to the advection time with the matrix only. Advection is the main transport mechanism.

Figure 5.

Particle behavior for a simple 2-D representation of the Effingen member having a uniform background hydraulic conductivity representative for TMA (K=10−12 m/s). For illustration purposes, only 247 particles are shown. Particles move downward. They are drained by the fracture system and are moved in and around fractures. However, mostly, they are transported within the matrix. This is also shown in the histogram of arrival times. It shows that the particles are much slower than by advection in the matrix only. Diffusion in the matrix is the main transport mechanism.

Table 1. Flow Connectivity of the Fractured System Depending on the Hydraulic Conductivity of the Matrix for Fixed Conductivities in the Fracture System
Matrix K (m/s)Fracture and Matrix Q/Matrix Q
10−9 (GH)1.03
10−12 (TMA)19.47

6. Discussion and Conclusions

[20] We present a new method for particle tracking in fractured formations considering explicitly both fractures and matrix. No a priori assumptions on transport mechanisms in the matrix have to be made. Such an approach is of particular interest in fractured sedimentary formations where the advective flux within the matrix cannot be ignored with respect to the one in the fracture.

[21] The method is based on simplifying flow by forcing the fracture elements to lie on the matrix cell faces. This particular fracture geometry allows solution for particles traveling in the matrix using the method of Pollock. It determines the point where a particle leaves an individual matrix cell. Only at this point a check needs to be made whether a fracture exists or not. If a fracture exists, the particle is moved to the fracture system and is transported within the fracture planes. We use here an orthogonal matrix grid, but the method can easily extended to irregular grids. The time a particle stays within the fracture depends on the advective flux perpendicular to the matrix, the fracture aperture, as well as a diffusive component. One can say this (advective and diffusive) flux of the matrix is washing the particle out of the fracture. The stronger the background flux is, the shorter the time a particle stays in the fracture. Currently, this mechanism depends only on flow parameters (hydraulic aperture) and molecular diffusion. More research is needed to determine whether a transport-related aperture should be used and whether this should be done stochastically. Particle behavior at fracture intersections is implemented as a flux-weighted random choice. This together with the implementation of molecular diffusion makes the method stochastic.

[22] Applications to some simplified representations of a fractured sedimentary formation, the Effingen member, show that this method is very useful to understand transport behavior in such systems. Within the more permeable parts of the formation (GH) the fracture systems are effectively ignored due to a strong matrix flow. Transport is dominated by advection within the matrix. Within the less permeable example (TMA) the fractures are dominating the overall shape of transport. The path lines are concentrated inside and along fractures. Fractures appear to drain particles. However, even so, diffusion is strong enough to eject the particles soon back to the matrix, and the residence times within the fractures are never very long. As in the matrix diffusion dominates transport, overall transport is dominated by diffusion. The effect of diffusion might be weaker if a larger transport aperture is used.


[23] We gratefully acknowledge funding from Nagra (Swiss National Cooperative for the Disposal of Radioactive Waste), Wettingen, Switzerland.