MsRSB: Multiscale Restriction-Smoothed Basis Method
The Multiscale Restricted-Smoothing Basis (MsRSB) method is the current state-of-the-art within multiscale methods, originally developed by SINTEF and later implemented in the commercial INTERSECT MS SFI simulator. MsRSB is very robust and versatile and can either be used as an approximate coarse-scale solver having mass-conservative subscale resolution, or as an iterative fine-scale solver that will provide mass-conservative solutions for any given tolerance. The performance of the method has been demonstrated on incompressible 2-phase flow, compressible black-oil models, as well as compositional models. It has also been demonstrated that the method can utilize combinations of multiple prolongation operators, e.g., corresponding to coarse grids with different resolutions, adapting to geological features, adapting to wells, or moving displacement fronts. At the moment, only parts of this functionality is available in the public version of the module.


Introduction: SPE10 layer

This example goes through the construction of a simple multiscale solver based on restriction-smoothed basis functions. We begin by defining the fine-scale problem and plotting the permeability field, which represents a number of layers sampled from the SPE10 model. Layers from 1 to 35 will result in smooth, lognormal permeabilities, whereas Layers 36 to 85 will result in channelized formations that are considered to be significantly more challenging for upscaling and multiscale methods.

We solve the flow problem using three methods: a fine-scale TPFA solver from the 'incomp' module, the multiscale finite-volume (MsFV) method from the 'msfv' module, and the MsRSB method implemented in this module.


Grid adaption

Extensive numerical experiments have shown that contemporary multiscale methods provide approximate solutions of good quality for highly heterogeneous media. However, cases with large permeability contrasts are generally challenging and it is not difficult to construct pathological test cases on which a particular method fails to produce accurate solutions. Kippe et al. (DOI: 10.1007/s10596-007-9074-6) proposed a simple and illuminating example consisting of a narrow high-permeable channel in a low-permeable background, where the channel is aligned with the diagonal direction.

This example corresponds to Example 4.3.3 in [1].


2-phase, Upper Ness

This example is a simple demonstration in how to use the multiscale solver to solve a 2-phase quarter five-spot problem defined with petrophysical properties sampled from the bottom layer of Model 2, SPE10. The fluid has equal viscosities and quadratic relative permeabilities. Since the total mobility does not change significantly, we neither update basis functions nor introduce extra iterations to reduce the fine-scale residual towared machine precision. As a result, the multiscale approximation will deviate somewhat from the fine-scale TPFA solution.


Inactive cells

A tough test case for any multiscale solver is a grid with many inactive cells that make coarse grids highly challenging.  We consider a quater-five spot simulation with lognormal permeabilities and a number of circular impermeable inclusions, which are modelled by setting the corresponding cells to be inactive. This example demonstrates the robustness of MsRSB when faced with a large number of inactive cells.

This example corresponds to Example 4.3.5 in [1].

Bed model

Pinchouts create non-neighboring connections and is one of the principal gridding challenges for real-life reservoir models. To exemplify, we consider a highly detailed 30x30x6 cm^3 model of realistic bedding structures . The bedding structure consists of six different rock types and is realized on a 30x30x333 corner-point grid. Almost 2/3 of the cells are pinched or inactive, giving a fine grid of approximate dimensions 30x30x100. The volumes of the cells, as well as the areas of the (vertical) faces, vary almost four orders of magnitudes. We partition the fine grid into 6x6x5 coarse blocks.

The model not only has a difficult geometry, but also has a large number of low-permeable shale layers pinched between the other high-permeable layers. Impermeable regions like this are known to pose monotonicity problems for the MsFV method. Here, we demonstrate that this is not the case for the MsRSB method.



This example is a modified version of Example 4.3.6 in [1], and demonstrates the use of the MsRSB method to a synthethic waterflood on the grid and petrophysical properties from a real field. The setup uses the reservoir geometry and petrophysical properties from the Norne field, which is an oil and gas field from the Norwegian Sea. Wells are placed somewhat arbitrarily throughout the model and we use a simplfied 2-phase fluid model. To partition the domain, we use METIS with edge-weights determined from the fine-scale transmissibilities. This way, the coarse grid will adapt to the heterogeneity by trying to form blocks that have as small permeability variation as possible.

For efficiency, the example uses MEX-accelerated computation of basis functions, which requires that MEX is set up and configured to work with an installed C++ compiler on your machine.


  1. O. Møyner and K.-A. Lie. A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids. J. Comput. Phys, Vol. 304, pp. 46-71, 2016. DOI: 10.1016/
  2. O. Møyner and K.-A. Lie. A multiscale restriction-smoothed basis method for compressible black-oil models. SPE Journal, 2016. DOI: 10.2118/173265-PA
  3. K.-A. Lie, O. Møyner, J. R. Natvig, A. Kozlova, K. Bratvedt, S. Watanable, and Z. Li. Successful application of multiscale methods in a real reservoir simulator environment. Comput. Geosci., Vol. 21, Issue 5-6, pp. 981-998, 2017. DOI: 10.1007/s10596-017-9627-2.
  4. O. Møyner and H.A. Tchelepi. A mass-conservative sequential implicit multiscale method for general compositional problems. SPE J., Vol. 23, Issue 06, pp. 2376-2393. DOI: 10.2118/182679-PA.
  5. Ø. S. Klemetsdal, O. Møyner, and K.-A. Lie. Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions. Comput. Geosci., Vol. 24, pp. 459-476, 2020. DOI: 10.1007/s10596-019-9827-z.
  6. K.-A. Lie, O. Møyner, and J. R. Natvig. Use of multiple multiscale operators to accelerate simulation of complex geomodels. SPE Journal, Vol. 22, Issue 6, pp. 1929-1945, 2017. DOI: 10.2118/182701-PA.
  7. S. T. Hilden, O. Møyner, K.-A. Lie, and K. Bao. Multiscale simulation of polymer flooding with shear effects. Transport in Porous Media, Volume 113, Issue 1, pp. 111-135, 2016. DOI: 10.1007/s11242-016-0682-2
  8. S. Shah, O. Møyner, M. Tene, K.-A. Lie, and H. Hajibeygi. The multiscale restriction smoothed basis method for fractured porous media (F-MsRSB). J. Comput. Phys., Vol. 318, pp. 36-57, 2016. DOI: 10.1016/

Published December 8, 2016