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Background - Relationship of Time-of-Flight and Grid Block Properties

Streamline-based history matching has been a subject of intensive research over the last several years (see references below).  History matching using streamlines has been shown to be an attractive alternative to traditional history matching when streamline simulation is applicable.  All streamline-based history matching methods take advantage of the simple relationship between the time-of-flight (TOF) of the streamline, and the effective porosity and effective permeability along the streamline.  The TOF relationship was first derived by Pollock (1988) as,

 
Eq. 1
 
The TOF is the time required for a neutral particle to travel from source to sink along streamline path.  The path that the particle traverses traces the path of the streamline.  Since the total interstitial velocity is directly related to the effective permeability and porosity along the streamline, it follows that
 

Production Data Mismatch as a Time Shift

Eq. 2 above implies that if the effective porosity or permeability of the streamline is modified, the change in the TOF can be inferred directly.  For example, if the permeability were doubled, the time-of-flight (and by analogy the arrival time of fluids traveling down the streamline) would be halved.  We describe the ratio of TOFs as the correction factor c*. 

 

The old TOF is the time required to travel along a streamline derived using an old static model, while the new TOF is the time required to travel along the same streamline but with a new static model.  The correction factor is determined by analysis of the water cut mismatch between historical and simulated data at each well.  An example of how c* can be calculated is given in several references in the following section.  

Direct Modification of Grid Block Properties

This correction factor denoted above in Eq. 3, can be used to directly modify the underlying petrophysical properties (like permeability and/or porosity) of the reservoir model.  The concept is simple - if the water cut must be accelerated to breakthrough at 50 days (according to historical data), and the breakthrough in the simulation is at 100 days, then the permeability must be doubled (100/50 = 2) along the streamline.  Of course, in practice, the application is more complex.  But several authors have illustrated very simple yet effective approaches to history matching following this concept, including: 

  • Wang, Y. and Kovscek, A.R. (2000), "Streamline Approach for History Matching Production Data," SPE Journal (December 2000) 353.
  • Agarwal, B. and Blunt, M.J. (2003), "Streamline-Based Method with Full-Physics Forward Simulation for History-Matching Performance Data of a North Sea Field," SPE Journal (June 2003) 171.
  • Agarwal, B. and Blunt, M.J. (2003), "A Streamline-Based Method for Assisted History Matching Applied to an Arabian Gulf Field," SPE Journal (December 2004) 437.
  • Batycky, R.P., Seto, A., and Fenwick, D.H., "Assisted History Matching of a 1.4-Million-Cell Simulation Model for Judy Creek 'A' Pool Waterflood/HCMF Using a Streamline-Based Workflow". proceedings of the 2007 SPE ATCE, November 11-14 2007, Anaheim, CA.

Smart Paint Workflow

Streamsim has implemented this direct approach in its "Smart Paint" workflow, and is being applied in several companies on real fields.  An example of this approach is shown below.  The streamlines identify injector-producer pairs and the drainage zones over all time steps, shown in the figure below (left).  In addition, each streamline has a saturation and fractional flow profile going from injector (or aquifer) to the producers.  The streamlines in the figure below (left) are colored by water saturation (red is 100% water, purple is oil and irreducible Sw).  This information is processed by the smart paint algorithm, along with the historical mismatch in production (in this case water).  The result is a map of "corrections", called correction factors (CFACTORS) associated with particular grid blocks, shown in the figure below (right).  The algorithm selects the grid blocks to reduce the historical mismatch, yet at the same time minimize the modifications to the underlying grid.  
 

Streamline-Based Geologically-Consistent History Matching

The Streamsim/Stanford HM JIP has developed history matching tools that perform geologically consistent history matching. A method has been developed to modify the input trend of the grid properties using the correction factors calculated from Eq. 3. The trend is described as a locally varying mean (LVM) which is defined for each grid block over the entire reservoir. Bundles of streamlines are considered in order to capture large scale trends. The grid properties are reconstructed using geostatistics. The locally varying mean is adjusted iteratively to match water cut data using the streamline-based corrections.  Since we reconstruct our grid properties using geostatistics, we can account for well data, spatial correlation, and correlations between grid properties automatically, thus maintaining geological consistency in the reservoir model.

Another tool that has been developed uses the Probability Perturbation Method (PPM). The PPM provides a powerful methodology for modifying grid properties to match historical water production. With the PPM, the overall histogram, LVM trends, and facies proportions are maintainted, but the grid scale heterogeneity within the model is redistributed in order to match production data. The LVM updates and PPM can be combined in a useful manner. Through the iterative LVM updates, the approach accounts for variability in the large scale permeability trends in the reservoir. After the trend is determined, the fine scale variability of the given trend can be modified using the PPM. After PPM iterations, the LVM procedure can be repeated.

  • Caers, J., Krishnan, S., Wang, Y., and Kovscek, A.R. (2002),  "A Geostatistical Approach to Streamline-Based History Matching," SPE Journal (September 2002) 250.
  • Gross, H., Thiele, M.R., Alexa, M.J., Caers, J., and Kovscek, A.R. (2004), "Streamline-Based History Matching Using Geostatistical Constraints:  Application to a Giant, Mature Carbonate Reservoir," proceedings of the 2004 SPE ATCE, September 26-29, Houston, Texas.
  • Hoffman, B. T., and Caers, J.: "Geostatistical History Matching Using a Regional Probability Perturbation Method," proceedings of the 2003 SPE ATCE, October 5-8, Denver, Colorado.
  • Hoffman, B.T., and Caers, J.: "Regional probability perturbations for history matching," Journal of Petroleum Science and Engineering, 2005 46: 53-71.
  • Fenwick, D.H., Thiele, M.R., Agil, M., Hussain, A., Humam, F., and Caers, J.K., "Reconciling Prior Geologic Information with Production Data Using Streamlines:  Application to a Giant Middle-Eastern Oil Field". proceedings of the 2005 SPE ATCE, October 9-12 2005, Dallas, Texas.
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