• choose a set of seed points
• trace streamlines from seed points backwards to injectors and forwards to producers
• map saturations and properties from the grid to the streamlines
• solve one-dimensional transport equation along streamlines
• map resulting saturation distributions back to grid.

Challenges

Choosing seed points. Seed points should preferably be chosen so that every cell is traversed by at least one streamline, this may be hard to achieve in low-flow regions without a seed point strategy designed to guarantee this.

Tracing streamlines. This can be hard for complex grid topologies and geometries, for instance on corner-point grids with faults. In particular, one must handle non-conforming grids, and velocity fields only defined by face fluxes.

Features

Our current streamline-based solvers are capable of:

• fast tracing of streamlines on simplicial, cartesian and corner-point grids (with faults)
• efficient and accurate solution of one-dimensional equations by front-tracking methods.

Selected papers

1. A. F. Rasmussen. Streamline tracing on irregular geometries. Proceedings of ECMOR XII, Oxford, UK, 6-9 September 2010.
2. R. Juanes and K.-A. Lie. Numerical modeling of multiphase first-contact miscible flows. Part 2. Front-tracking and streamline simulation. Transp. Porous Media, Vol. 72, No. 1/March, pp. 97-120, 2008. DOI: 10.1007/s11242-007-9139-y.
3. H. Hægland, H. K. Dahle, G. T. Eigestad, K.-A. Lie, and I. Aavatsmark. Improved streamlines and time-of-flight for streamline simulation on irregular grids. Adv. Water Resour. Vol. 30, Issue 4, pp. 1027-1045, 2007. DOI: 10.1016/j.advwatres.2006.09.00.
4. J. E. Aarnes, V. Kippe, and K.-A. Lie. Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour., Vol. 28, Issue 3, pp. 257-271, 2005.
5. H. M. Nilsen and K.-A. Lie. Front-tracking methods for use in streamline simulation of compressible flow. Paper SPE 119099 presented at 2009 SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2-4 February.

Streamline-based history matching

Generalized travel-time inversion (GTTI) is a very efficient method for history matching. The GTTI method is robust and computationally efficient. Unlike conventional ‘amplitude’
matching, which can be highly nonlinear, it has been shown that the travel-time inversion has quasilinear properties. As a result, the minimization proceeds rapidly even if the initial model is not
close to the solution. Second, travel-time sensitivities are typically distributed more uniform between wells compared to ‘amplitude’ sensitivities that tend to be localized near the wells. This prevents over-correction in the near-well regions. Finally, in practical field applications, production data are often characterized by multiple peaks. Under such conditions, the travel-time inversion can prevent the solution from converging to secondary peaks in the production response. 

As many othermethods for data integration, GTTI is based on the use of production response sensitivities. Using a streamline formulation, accurate approximations to such sensitivities can be computed analytically based on a single flow simulation. To this end, one uses time-of-flight sensitivties that describe the change in time-of-flight caused by small perturbations in reservoir properties such as porosity and permeability.  Production response sensitivites can then be obtained by simplifying the flow physics and assuming a monotone Buckley-Leverett displacement along each streamline.

Selected papers

1. V. R. Stenerud, K.-A. Lie, and V. Kippe. Generalized travel-time inversion on unstructured grids. J. Pet. Sci. Eng. Vol. 65, Issues 3-4, pp. 175-187, 2009. doi: 10.1016/j.petrol.2008.12.030
2. V. R. Stenerud, V. Kippe, K.-A. Lie, and A. Datta-Gupta. Adaptive multiscale streamline simulation and inversion for high-resolution geomodels. SPE J., Vol. 13, No. 1, pp. 99--111, March 2008. DOI: 10.2118/106228-PA