Abstract
Near-well effects can have a strong impact on many subsurface flow processes. In oil production, because dissolved gas is released from the oil phase when the pressure falls below the bubble point, the detailed pressure field in the immediate vicinity of a production well strongly impacts gas (and thus oil) production. This effect is complicated by the interplay of fine-scale heterogeneity and two-phase flow physics, and can be difficult to capture in coarse-grid simulations. In this article, we develop and apply a new upscaling (coarse-graining) procedure to capture such near-well subgrid effects in coarse-scale flow simulation models. The method entails the use of preprocessing computations over near-well domains [referred to as local well models (LWM)] for the determination of upscaled single-phase and two-phase near-well parameters. These parameters are computed by minimizing the mismatch between fine and coarse-scale flows over the LWM. Minimization is accomplished using a gradient-based optimization procedure, with gradients calculated through solution of adjoint equations. The boundary conditions applied on the LWM can impact the upscaled parameters, but these boundary conditions depend on the global flow and are not, therefore, known a priori. In order to circumvent this difficulty, an adaptive local–global procedure is applied. This entails performing a global coarse-scale simulation with initial estimates for well-block parameters. The resulting pressure and saturation fields are then used to define local boundary conditions for the near-well computations. The overall procedure is applied to several example problems and is shown to provide results in close agreement with reference fine-scale computations. Significant improvement in accuracy over existing near-well upscaling treatments is demonstrated, particularly for a heavy oil case with oil viscosity of ~104 cp.
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References
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 Sim. 2(3), 421–439 (2004)
Auriault J.L., Geindreau C., Orgéas L.: Upscaling Forchheimer law. Transp. Porous Media 70(2), 213–229 (2007)
Aziz K., Settari A.: Petroleum Reservoir Simulation. Chapman & Hall, London (1979)
Brouwer D.R., Jansen J.D.: Dynamic optimization of waterflooding with smart wells using optimal control theory. Soc. Pet. Eng. J. 9(4), 391–402 (2004)
Cao, H.: Development of techniques for general purpose simulators. PhD thesis, Stanford University (2002)
Chen WH, Gavalas G.R., Seinfeld J.H., Wasserman M.L.: A new algorithm for automatic history matching. Soc. Pet. Eng. J. 14(6), 593–608 (1974)
Chen Y., Durlofsky L.J.: Adaptive local–global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media 62(2), 157–185 (2006)
Chen Y., Wu X.H.: Upscaled modeling of well singularity for simulating flow in heterogeneous formations. Comput. Geosci. 12(1), 29–45 (2008)
Chen, Y., Yan, L.: Local–global two-phase upscaling of flow and transport in heterogeneous formations. Comput. Geosci. (To appear) (2009)
Christie M.A., Blunt M.J.: Tenth SPE comparative solution project: A comparison of upscaling techniques. Soc. Pet. Eng. Reserv. Eval. Eng. 4(4), 308–317 (2001)
Deutsch C.V., Journel A.G.: GSLIB: Geostatistical Software Library and User’s Guide, 2nd edn. Oxford University Press, Oxford (1998)
Ding, Y.: Scaling-up in the vicinity of wells in heterogeneous field. In: Proceedings of the SPE Reservoir Simulation Symposium, SPE paper 29137, San Antonio, Texas, USA (1995)
Durlofsky L.J.: Coarse scale models of two-phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques. Comput. Geosci. 2(2), 73–92 (1998)
Durlofsky L.J., Milliken W.J., Bernath A.: Scaleup in the near-well region. Soc. Pet. Eng. J. 5(1), 110–117 (2000)
Efendiev Y., Ginting V., Hou T., Ewing R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220(1), 155–174 (2006)
Efendiev Y.R., Durlofsky L.J.: A generalized convection–diffusion model for subgrid transport in porous media. Multiscale Model Sim. 1(3), 504–526 (2003)
Emanuel A.S., Cook G.W.: Pseudo-relative permeability for well modeling. Soc. Pet. Eng. J. 14(1), 7–9 (1974)
Gerritsen M., Lambers J.: Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations. Comput. Geosci. 12(2), 193–218 (2008)
Holden L., Nielsen B.F.: Global upscaling of permeability in heterogeneous reservoirs: the output least squares (ols) method. Transp. Porous Media 40(2), 115–143 (2000)
Hui, M.: Upscaling of multiphase flow parameters for modeling near-well and miscible displacements. PhD thesis, Stanford University (2005)
Hui M., Durlofsky L.J.: Accurate coarse modeling of well-driven, high-mobility-ratio displacements in heterogeneous reservoirs. J. Pet. Sci. Eng. 49, 37–56 (2005)
Jenny P., Lunati I.: Modeling complex wells with the multi-scale finite-volume method. J. Comput. Phys. 228(3), 687–702 (2009)
Jiang, Y. A flexible computational framework for effcient integrated simulation of advanced wells and unstructured reservoir models. PhD thesis, Stanford University (2007)
Juanes R., Dub F.: A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source term. Comput. Geosci. 12(3), 351–367 (2008)
Killough, J.E.: Ninth SPE comparative solution project: A reexamination of black-oil simulation. In: Proceedings of the SPE Reservoir Simulation Symposium, SPE paper 29110, San Antonio, Texas, USA (1995)
Krogstad S., Durlofsky L.J.: Multiscale mixed-finite-element modeling of coupled wellbore/near-well flow. Soc. Pet. Eng. J. 14(1), 78–87 (2009)
Li R., Reynolds A.C., Oliver D.S.: History matching of three-phase flow production data. Soc. Pet. Eng. J. 8(4), 328–340 (2003)
Mascarenhas O., Durlofsky L.J.: Coarse scale simulation of horizontal wells in heterogeneous reservoirs. J. Pet. Sci. Eng. 25, 135–147 (2000)
Muggeridge A.H., Cuypers M., Bacquet C., Barker J.W.: Scale-up of well performance for reservoir flow simulation. Pet. Geosci. 8(2), 133–139 (2002)
Nakashima, T.: Near-well upscaling for two and three-phase flows (in preparation). PhD thesis, Stanford University (2009)
Peaceman D.W.: Interpretation of well-block pressures in numerical reservoir simulation. Soc. Pet. Eng. J. 18(3), 183–194 (1978)
Ramirez W.F.: Application of Optimal Control Theory to Enhanced Oil Recovery. Elsevier, Amsterdam (1987)
Sarma, P.: Efficient closed-loop optimal control of petroleum reservoirs under uncertainty. PhD thesis, Stanford University (2006)
Schittkowski, K.: NLPQLP: a Fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search—users guide, version 2.2. Tech. rep., Department of Computer Science, University of Bayreuth (2006)
Wen X., Chen Y., Durlofsky L.J.: Efficient 3D implementation of local–global upscaling for reservoir simulation. Soc. Pet. Eng. J. 11(4), 443–453 (2006)
Wolfsteiner C., Lee S.H., Tchelepi H.A.: Well modeling in the multiscale finite volume method for subsurface flow simulation. Multiscale Model Sim. 5(3), 900–917 (2006)
Woods E.G., Khurana A.K.: Pseudofunctions for water coning in a three-dimensional reservoir simulator. Soc. Pet. Eng. J. 17(4), 251–262 (1977)
Zhang P., Pickup G., Christie M.: A new practical method for upscaling in highly heterogeneous reservoir models. Soc. Pet. Eng. J. 13(1), 68–76 (2008)
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Nakashima, T., Durlofsky, L.J. Accurate Representation of Near-well Effects in Coarse-Scale Models of Primary Oil Production. Transp Porous Med 83, 741–770 (2010). https://doi.org/10.1007/s11242-009-9479-x
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DOI: https://doi.org/10.1007/s11242-009-9479-x