A multiscale method for effective handling of wells (source/sink terms) in the simulation of multiphase flow and transport processes in heterogeneous porous media is developed. The approach extends the multiscale finite volume (MSFV) framework. Our multiscale well model allows for accurate reconstruction of the fine‐scale pressure and velocity fields in the vicinity of wells. Accurate and computationally efficient modeling of complex wells is a prerequisite for field applications, and the ability to model wells within the MSFV framework makes it possible to solve large‐scale heterogeneous problems of practical interest. Our approach consists of removal of the well singularity from the multiscale solution via a local change of variables and the computation of a smoothly varying background field instead. The well effects are computed using a separate basis function, which is superposed on the background solution to yield accurate representation of the flow field. The multiscale well treatment accounts for both types of well constraints: fixed pressure and fixed rate. The details of modeling wells with one or multiple completions (perforations) are also presented. The accuracy of the method is assessed using a variety of examples including highly heterogeneous permeability fields.

MSC codes

  1. 65N06
  2. 65N22
  3. 76S05
  4. 76T


  1. reservoir simulation
  2. multiscale finite volume
  3. well modeling
  4. well constraints
  5. permeability heterogeneity

Get full access to this article

View all available purchase options and get full access to this article.


J. E. Aarnes, On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation, Multiscale Model. Simul., 2 (2004), pp. 421–439.
T. Arbogast, Implementation of a locally conservative numerical subgrid upscaling scheme for two‐phase Darcy flow, Comput. Geosci., 6 (2002), pp. 453–481.
T. Arbogast and S. L. Bryant, A two‐scale numerical subgrid technique for waterflood simulations, SPEJ, 7 (2002), pp. 446–457.
K. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers, London, England, 1979.
G. F. Carey and S. S. Chow, Well singularities in reservoir simulation, SPE Res. Eng., 2 (1987), pp. 713–719.
A. Chen and T. Y. Hou, A mixed finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 72 (2003), pp. 541–576.
Z. Chen and X. Yue, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media, Multiscale Model. Simul., 1 (2003), pp. 260–303.
M. A. Christie and M. J. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPERE, 4 (2001), pp. 308–317.
G. Dagan, Flow and Transport in Porous Formations, Springer‐Verlag, New York, 1989.
C. Deutsch and A. G. Journel, GSLIB: Geostatistical Software Library and User’s Guide, 2nd ed., Oxford University Press, Reading, MA, 1998.
Y. Ding and L. Jeannin, New numerical schemes for near‐well modeling using flexible grids, SPEJ, 9 (2004), pp. 109–121.
L. J. Durlofsky, W. J. Milliken, and A. Bernath, Scale up in the near‐well region, SPEJ, 5 (2000), pp. 110–117.
T. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169–189.
P. Jenny, S. H. Lee, and H. A. Tchelepi, Multi‐scale finite‐volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187 (2003), pp. 47–67.
P. Jenny, S. H. Lee, and H. A. Tchelepi, Adaptive multiscale finite‐volume method for multi‐phase flow and transport in porous media, Multiscale Model. Simul., 3 (2005), pp. 50–64.
P. Jenny, S. H. Lee, and H. A. Tchelepi, An adaptive fully implicit multi‐scale finite‐volume algorithm for multi‐phase flow in porous media, J. Comput. Phys., to appear.
S. H. Lee, H. Tchelepi, P. Jenny, and L. J. DeChant, Implementation of a flux‐continuous finite difference method for stratigraphic, hexahedron grids, SPEJ, 7 (2002), pp. 267–277.
D. W. Peaceman, Interpretation of wellblock pressures in numerical reservoir simulation, SPEJ, (June 1978), pp. 183–94.
M. Slodicka, Mathematical treatment of point sources in a flow through porous media governed by darcy’s law, Transp. Porous Media, 28 (1997), pp. 51–67.
C. Wolfsteiner, L. J. Durlofsky, and K. Aziz, Calculation of well index for nonconventional wells on arbitrary grids, Comput. Geosci., 7 (2003), pp. 61–82.

Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 900 - 917
ISSN (online): 1540-3467


Submitted: 20 September 2005
Accepted: 30 May 2006
Published online: 10 October 2006

MSC codes

  1. 65N06
  2. 65N22
  3. 76S05
  4. 76T


  1. reservoir simulation
  2. multiscale finite volume
  3. well modeling
  4. well constraints
  5. permeability heterogeneity



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.