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Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. (English) Zbl 1160.76373
Summary: We describe a sequential fully implicit (SFI) multi-scale finite volume (MSFV) algorithm for nonlinear multi-phase flow and transport in heterogeneous porous media. The method extends the recently developed multiscale approach, which is based on an IMPES (IMplicit Pressure, Explicit Saturation) scheme [P. Jenny, S.H. Lee, H.A. Tchelepi, Multiscale Model. Simul. 3, No. 1, 50–64 (2004; Zbl 1160.76372)]. That previous method was tested extensively and with a series of difficult test cases, where it was clearly demonstrated that the multiscale results are in excellent agreement with reference fine-scale solutions and that the computational efficiency of the MSFV algorithm is much higher than that of standard reservoir simulators. However, the level of detail and range of property variability included in reservoir characterization models continues to grow. For such models, the explicit treatment of the transport problem (i.e. saturation equations) in the IMPES-based multiscale method imposes severe restrictions on the time step size, and that can become the major computational bottleneck. Here we show how this problem is resolved with our sequential fully implicit (SFI) MSFV algorithm. Simulations of large (million cells) and highly heterogeneous problems show that the results obtained with the implicit multi-scale method are in excellent agreement with reference fine-scale solutions. Moreover, we demonstrate the robustness of the coupling scheme for nonlinear flow and transport, and we show that the MSFV algorithm offers great gains in computational efficiency compared to standard reservoir simulation methods.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76T10 Liquid-gas two-phase flows, bubbly flows
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[1] Jenny, P.; Lee, S.H.; Tchelepi, H.A., Adaptive multiscale finite volume method for multi-phase flow and transport, Multiscale model. simul., 3, 50-64, (2005) · Zbl 1160.76372
[2] Durlofsky, L.J., Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water resour. res., 27, 699-708, (1991)
[3] Durlofsky, L.J.; Jones, R.C.; Milliken, W.J., A nonuniform coarsening approach for the scale up of displacement processes in heterogeneous porous media, Adv. water resour., 20, 335-347, (1997)
[4] Gautier, Y.; Blunt, M.J.; Christie, M.A., Nested gridding and streamline-based simulation for fast reservoir performance prediction, Comput. geosci., 3, 295-320, (1999) · Zbl 0968.76059
[5] Hou, T.; Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comp. phys., 134, 169-189, (1997) · Zbl 0880.73065
[6] Chen, Z.; Hou, T.Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. comput., 72, 541-576, (2003) · Zbl 1017.65088
[7] Arbogast, T., Implementation of a locally conservative numerical subgrid upscaling scheme for two phase Darcy flow, Comput. geosci., 6, 453-481, (2002) · Zbl 1094.76532
[8] Arbogast, T.; Bryant, S.L., A two-scale numerical subgrid technique for waterflood simulations, Soc. petrol. eng. J., 446-457, (2002)
[9] Aarnes, J.E., 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, 3, 421-439, (2004) · Zbl 1181.76125
[10] Jenny, P.; Lee, S.H.; Tchelepi, H.A., Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. comput. phys., 187-1, 47-67, (2003) · Zbl 1047.76538
[11] Hoeksema, R.J.; Kitanidis, P., Analysis of the spatial structure properties of selected aquifers, Water resour. res., 21, 4, 563-572, (1985)
[12] Dagan, G., Flow and transport in porous formations, (1989), Springer New York, NY
[13] Deutsch, C., Geostatistical reservoir modeling, (2002), Oxford University Press New York, NY
[14] A.G. Spillette, J.G. Hillestad, H.L. Stone, A high-stability sequential solution approach to reservoir simulation, SPE 4542, Presented at the 48th Annual Fall Meeting of SPE of AIME, Las Vegas, NV, September 30-October 3, 1973.
[15] Coats, K.H.; George, W.D.; Chu, C.; Marcum, B.E., Three-dimensional simulation of steamflooding, Soc. petrol. eng. J., 573-592, (1974)
[16] Watts, J.W., A compositional formulation of the pressure and saturation equations, Soc. petrol. eng. reservoir eng., 243-252, (1986)
[17] Aziz, K.; Settari, A., Petroleum reservoir simulation, (1979), Applied Science Publishers London, England
[18] Christie, M.A.; Blunt, M.J., Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE reserv. eval. eng., 4, 308-317, (2001)
[19] C. Wolfsteiner, S.H. Lee, H.A. Tchelepi, Well modeling in the multiscale finite volume method for subsurface flow simulation, Multiscale Model. Simul. (under review), September 2005. · Zbl 1205.76175
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