Computational materials scientist Clint Van Siclen, from the Department of Energy's Idaho National Engineering and Environmental Laboratory, has demonstrated a theoretical approach to modeling fluid transport in porous, variable material (such as subsurface soil and rock) that may one day dramatically simplify the development of computer simulation models. Van Siclen's refined approach, called the Walker Diffusion Method (WDM), is highly accurate and requires less computer memory and computing time to use than other approaches. His research was published in the February 2002 issue of Physical Review E.
To use an analogy, the slow, creeping flow of fluid in the subsurface is a great deal like the flow of electricity. Fluid flow can only occur if pore spaces in soil are somehow connected - enabling water to trickle along from one place to another. In the same way, electricity can only flow if conducting materials are connected, forming a circuit or path for electricity to follow. This similarity is reflected as well in the respective sets of mathematical equations describing the flow.
Van Siclen capitalizes on this similarity using the Walker Diffusion Method, and calculates how electricity 'diffuses' through a composite material. The WDM is based on the concept of a single random walker - theoretical construct (think of an ant, or a little person) that 'walks' through the material, randomly taking a step in this direction, a step in that direction, and so on. Left alone long enough, the walker will eventually explore all the potential paths available. By tracking these paths, Van Siclen is able to map out the fluid flow routes in a permeable material - including sharp twists and turns or the tiniest of crack lines.
Van Siclen maps out the movements of the walker by first digitizing the structure of the porous material. That digitized representation is thus a square or cubic array of pixels or voxels, each of which is open or closed, corresponding to pore space and impermeable rock, respectively. After that, it's just a matter of probabilities. If a pixel is open, there's a high probability that a walker will travel through that space. If the pixel is closed, then a walker won't be able to occupy that space. In a relatively short period of time, a walker can explore the accessible space using these simple, quickly computed probabilities. The calculations go even faster if Van Siclen uses several noninteracting walkers. These paths reveal the overall physical structure of the material.
More typically, the system of interest is an aquifer or oil reservoir, where different rock layers have different permeability. The digitized representation is constructed so as to be faithful to any well logging data, electrical resistivity measurements, etc. - is then comprised of pixels having those permeability values. Again, the walker explores that space and discovers the various flow paths through the system.
"It's absolutely remarkable that this imaginary walker can solve the system of transport equations simply by diffusing according to very simple probabilistic rules," says Van Siclen. "Those equations are inherent in the probabilities for movement of the walker from one pixel to another."
The WDM is very different from the conventional approach to calculating flow paths, called the finite difference method (FDM). Using the FDM, researchers take the digitized sample and construct a very large set of finite difference equations - equations that define the difference between the values of a function at two discrete points. Those equations have to be solved simultaneously, a task that strains the capabilities of all but the largest computers for realistically sized systems. In contrast, the WDM can be performed on a typical PC.
A unique benefit of the WDM is that it obtains the 'correlation length' for the material under study. This parameter is the size above which a specimen is uniform (homogeneous) with respect to the transport property of interest, such as fluid permeability, and below which it is variable (heterogeneous). The existence of the correlation length, which may be tens or hundreds of meters in the case of fractured bedrock, thus fundamentally limits the extent to which results from laboratory experiments are applicable to field sites.
So why care that a theoretical method for determining flow paths is fast and accurate? In many cases it is far cheaper and more efficient to do a calculation than to set up a field scale experiment. The foundation for any predictive model is the ability to mathematically describe the system and the behavior that occurs in the system. The WDM enables very large, or highly resolved, material systems to be studied. At present the WDM is used as a research tool to determine relationships between the geological structure of the subsurface and fluid flow phenomena. For example, it produces the correct dynamics of the displacement of one fluid by another, such as oil by brine in an oil reservoir, and groundwater by heavier-than-water contaminants in an aquifer.
Scientists and decision-makers alike need to understand where and how quickly fluids containing contaminants can move through subsurface soils. To gain that understanding, scientists develop models that describe the subsurface environment - including properties such as the size of pore spaces between grains of soil, pH, water saturation, electrical conductivity and others. Theoretical mathematical research such as Van Siclen's work aids in the development of accurate, reliable predictive models.
Other applications for the WDM range from the purely practical to the esoteric. For example, it has been used to predict the thermal properties of new composite materials proposed for use in nuclear reactors. And it has been used to obtain important results in percolation theory, an area of mathematical research concerned with relations between the transport properties and the structure and composition of disordered media.
This research is funded by the INEEL's Environmental Systems Research and Analysis program.
The INEEL is a science-based, applied engineering national laboratory dedicated to supporting the U.S. Department of Energy's missions in environment, energy, science and national security. The INEEL is operated for the DOE by Bechtel BWXT Idaho, LLC.
Technical contact: Please contact Clint Van Siclen through email only at cvs@inel.gov.
Media contact: Deborah Hill, (208) 526-4723, dahill@inel.gov
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Journal
Physical Review E