Algorithm for fluid velocity field quantification from image sequences in complex geomaterials

Introduction
For reactive transport modelling in geosciences a velocity field v(xi,t) with i = 1, 2, or 3 is required. This velocity field can be a) obtained by definition, b) calculated on the basis of a given geometry, a set of partial differential equations and initial and boundary conditions or, c) obtained from observations:
By now, various tomographic methods have been applied to observe fluid flow also in dense geological material under realistic conditions. These are magnetic resonance imaging (MRI, e.g. (Greiner et al., 1997), neutron transmission tomography (e.g. (Pleinert and Degueldre, 1995), X-ray computed tomography (CT) (e.g. (Goldstein et al., 2007; Klise et al., 2008), electrical resistivity tomography (ERT), e.g. (Bowling et al., 2006; Gheith and Schwartz, 1998) and last but not least positron emission tomography (PET), e.g. (Khalili et al., 1998; Kulenkampff et al., 2008; Richter et al., 2005). Still, the extraction of quantitative velocity fields from observed concentrations fronts that pass through complex geological media is not a trivial task.
One option to solve this kind of problem is to inject a tracer pulse into a sample, record image sequences of the tracer's flow through the sample and generate local break through curves (BTC). Numerical simulations – e.g. on the basis of finite difference methods (e.g. (Yoon et al., 2008) – may then vary hydraulic conductivity, porosity and dispersivity values within appropriate ranges and evaluate the model fits of the data over different scales. The authors conclude that predicting water flow at fine scales (relative to permeability variations) is very challenging and that this may have large implications for modelling reactive transport, where reactant residence time and mixing can be greatly impacted by water flowpaths.

Methods
To overcome such problems that accompany the fitting of parameter values such as hydraulic conductivity, porosity or dispersivity in 3D numerical flow models, we designed, implemented and tested a new algorithm. It is conceptualized for application to real-world 3D PET image series of transport process observation in geological media that may be affected to some degree by noise, image artifacts and detection limits. Our algorithm does not need prior information about the internal geometry of the sample, but only the global flow rate and the geometric boundaries of the sample.
Still, the foundation of the algorithm is the continuity equation. Its validity serves as an optimization criterion to fit segments of flow paths to the images. In this way, the network of flow paths is recovered and the velocity can be computed using a robust and universal approach.
We model the flow path network inside a rock sample as a network of flow path segments. Each segment is a straight and typically a short part of a single flow path. For these segments, we assume that the fluid is incompressible and that there are no sinks and sources within a segment. These assumptions are typically true for water in a closed flow path.
As a first step, the algorithm identifies regions that show a significant increase in mass at some point in time (maxima in a BTC). At such regions nodes are placed, that are to be connected later with segments of the flow path network.
For each straight flow path segment, it is sufficient to use a 1D model, which greatly reduces computation time without sacrificing much accuracy, and makes the algorithm more robust against noise. For the algorithm, a segment is represented by a cylindrical tube that completely covers the flow path. Because there are no sinks and sources, the flow rate is constant when the tube covers exactly one flow path, but varies when it does not. Therefore, we can use the variation of the flow rate as an optimization criterion to decide where to place a tube, i.e. which of the aforementioned nodes to connect.
Finally, we can compute the velocity field from the flow rate and cross-sectional area of the tube.

Results & Discussion
For validating the algorithm we simulated a non-reactive tracer experiment in COMSOL Multiphysics® on a synthetic fracture network as a benchmark model for the algorithm. (Later, the image sequences obtained from the transport simulation are to be replaced by the PET image sequences). A velocity field (derived using the cubic law) was used to simulate transport of a conservative tracer. The resulting image sequence was provided to our new algorithm, which then computed the underlying velocity field.

Conclusion
Here we introduced our new algorithm (work in progress) that estimates velocity distributions from image sequences. It is robust against noise and static image artefacts, and requires no prior knowledge about the, possibly complex geometry of the sample. The current run-time for the example shown is well under ten seconds . These properties make the new algorithm universally suitable for tracer experiments for a wide range of applications.
The obtained velocity distributions (fig. 2, left) can directly be used for further reactive transport modelling.

References
Bowling, J.C., Zheng, C., Rodriguez, A.B. and Harry, D.L., 2006. Geophysical constraints on contaminant transport modeling in a heterogeneous fluvial aquifer. Journal of Contaminant Hydrology, 85(1–2): 72-88.
Gheith, H.M. and Schwartz, F.W., 1998. Electrical and visual monitoring of small scale three-dimensional experiments. Journal of Contaminant Hydrology, 34(3): 191-205.
Goldstein, L., Prasher, S.O. and Ghoshal, S., 2007. Three-dimensional visualization and quantification of non-aqueous phase liquid volumes in natural porous media using a medical X-ray Computed Tomography scanner. Journal of Contaminant Hydrology, 93(1–4): 96-110.
Greiner, A., Schreiber, W., Brix, G. and Kinzelbach, W., 1997. Magnetic resonance imaging of paramagnetic tracers in porous media: Quantification of flow and transport parameters. Water Resources Research, 33(6): 1461-1473.
Khalili, A., Basu, A.J. and Pietrzyk, U., 1998. Flow visualization in porous media via positron emission tomography. Physics of Fluids, 10: 1031-1033.
Klise, K.A., Tidwell, V.C. and McKenna, S.A., 2008. Comparison of laboratory-scale solute transport visualization experiments with numerical simulation using cross-bedded sandstone. Advances in Water Resources, 31(12): 1731-1741.
Kulenkampff, J., Richter, M., Gründig, M. and Seese, A., 2008. Observation of transport processes in soils and rocks with Positron Emission Tomography. Geophysical Research Abstracts, 9: 02754.
Pleinert, H. and Degueldre, C., 1995. Neutron radiographic measurement of porosity of crystalline rock samples: a feasibility study. Journal of Contaminant Hydrology, 19(1): 29-46.
Richter, M., Gründig, M., Zieger, K., Seese, A. and Sabri, O., 2005. Positron Emission Tomography for modelling of geochmical transport processes in clay. Radiochimica Acta, 93: 643-651.
Yoon, H., Zhang, C., Werth, C.J., Valocchi, A.J. and Webb, A.G., 2008. Numerical simulation of water flow in three dimensional heterogeneous porous media observed in a magnetic resonance imaging experiment. Water Resources Research, 44: W06405.