Large Scale Modeling of Flow in Heterogeneous Media


Principal Investigator

Mark Knackstedt

Department of Applied Mathematics,

Research School of Physical Sciences and Engineering

Understanding flow phenomenon at the field (or reservoir) scale is crucial to the economics of oil
and gas recovery processes, geothermal energy extraction and groundwater pollution abatement in underground reservoirs. In petroleum reservoir engineering for example, significant errors in petroleum reservoir performance prediction have been caused by inaccurate characterisation of the reservoir. This can prevent the full potential of a reservoir from being achieved and lead to significant amounts of oil and gas remaining unrecovered or recovered inefficiently. A second source of error is a lack of understanding of the pore-scale mechanisms responsible for mobilization and recovery of oil. We have considered the problem of how to accurately characterise geological heterogeneity, and the implication to reservoir flow properties. Results to date hint at the drastic effect incorporation of a realistic heterogeneity has on the problem of fluid displacement in natural porous media. The result implies many results for displacement processes will have to be reworked if they are to accurately model processes in natural porous media. This work involves extensive collaborations with the School of Petroleum Engineering,UNSW, and CSIRO Division of Petroleum Resources



Adrian Sheppard

Siewert-Jan Marrink

Department of Applied Mathsematics,

Research School of Physical Sciences and Engineering

Ji-youn Lee

School of Petroleum Engineering,

University of New South Wales

L. Paterson

CSIRO Petroleum Resources




w09 - PC


What are the results to date and the future of this work?

The bulk of our work in the last year has been based on simulating porosimetry measurements on porous rocks. We detail the results below.

An experimentally simple and theoretically accessible technique for describing basic pore scale properties is based on rate-controlled or constant-volume porosimetry. Here a porous rock is flooded by fixed volume increments of a nonwetting phase and the pressure is monitored as a function of volume. We have used network models to simulate rate-controlled

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mercury injection experiments on porous media displaying both uncorrelated and long-range correlated disorder. We have shown that the introduction of long-range correlations has a marked effect on the nature of the capillary pressure curve. Moreover we have shown that it is only possible to account for the behaviour of the experimental data for sedimentary rocks by including correlated heterogeneity. This result suggests that correlated heterogeneity, which is clearly evident in porous rock at reservoir and log scales, persists down to the pore scale. While the above result has provided compelling "qualitative" evidence for the presence of correlations in sedimentary rock at the pore scale, the "quantitative" comparison between experiment and simulation on correlated grids is poor. We have used correlated fields based on fractional Brownian motion (fBm), originally introduced by Hewett to describe heterogeneity on the reservoir scale. fBm, a model for describing "reservoir" heterogeneity, does not capture detailed heterogeneity of rock at the pore scale. This result highlights the need to quantify heterogeneity at the pore scale.

  • Our current goals in this area are therefore: Measure pore-scale morphology and heterogeneity for rock types typical of the major classes of hydrocarbon reservoirs.
  • Use these measurements to accurately identify the statistical character of the pore morphology and the spatial heterogeneity for each rock type. Determine the validity of reservoir scale models for heterogeneity at the pore scale.
  • Perform core-scale multi-phase displacement experiments and obtain experimental tomographic images of two- and three-phase distributions in the four rock types. The data will be compared to network model simulations in order to confirm the ability of network models and to effectively scale-up macroscopic flow properties such as capillary pressure from a fundamental description of pore-scale physics and heterogeneity.

This latter item will be the main focus of our supercomputer usage in the next year. This research will not only be a major scientific achievement, but would be of great value to the oil and gas production industries.

Future work will include increased liasing with the VizLab for assistance with 3D rendering and display.

The service units used in the previous period were also used for development of accurate heterogeneity maps for the correlated geophysical data. We then considered very simple two-phase flow simulations (based on invasion percolation and ordinary percolation) as a first step in the development of our three-phase flow simulator. Results to date have been encouraging and work was presented at the October Society of Petroleum Engineers Conference held in the U.S. Moreover, the work has highlighted some important inconsistencies in the generation of fields with long-range correlations by a stochastic process known as fractional Brownian motion (fBm). Finally, generation of percolation quantities on these correlated maps has shown that when correlations are introduced, new interpretations are required. Various quantities which converge on the uncorrelated networks no longer do so on correlated systems. For example, the percolation threshold itself is subject to interpretation based on the application. This contrasts

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with common work on uncorrelated lattices where these subtle details are insignificant.

The group was successful in obtaining funding in the 1998 ARC Large Research Grant round for a study of Large Scale Modeling of Flow in Heterogeneous Media.

What computational techniques are used?

We simulate invasion percolation, with defender fluid trapping using a novel method that does not need to search the entire grid at each invasion step. Rather the invading interface is stored as a binary tree, in which elements can be accessed in 0(log(N)) time. The majority of the computation time is then spent in accounting procedures that maintain these interface structures. This method has enabled us to investigate numerical grids orders of magnitude larger than those studied previously.

The speed up is such that the method is now memory rather than CPU time limited, so that a shared memory machine like the PC is ideal for our needs. The code is not vectorisable so the VPP is not suitable.


S. J. Marrink, L. Paterson, and M. Knackstedt, Definition and Scaling of Percolation Thresholds on Self-Affine Surfaces, Phys. Rev. E, in press.

L. Paterson, J.Y. Lee, and W.V. Pinczewski, Simulating Residual Saturation and Relative Permeability in Heterogeneous Formations, Society of Petroleum Engineers Formation Evaluation, in press.

C. Mattisson, T. J. Senden, and M. Knackstedt, Transport Properties in Fractured porous solids, Geophysical Research Letters, 24, 495-498 (1997).

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