General Transfer Functions for Multiphase Flow in Fractured Reservoirs
- Huiyun Lu (Imperial College) | Ginevra Di Donato (BP) | Martin J. Blunt (Imperial College)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2008
- Document Type
- Journal Paper
- 289 - 297
- 2008. Society of Petroleum Engineers
- 5.2.1 Phase Behavior and PVT Measurements, 5.8.6 Naturally Fractured Reservoir, 1.2.3 Rock properties, 5.1.2 Faults and Fracture Characterisation, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 4.3.4 Scale, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation
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We propose a physically motivated formulation for the matrix/fracture transfer function in dual-porosity and dual-permeability reservoir simulation. The approach currently applied in commercial simulators (Barenblatt et al. 1960; Kazemi et al. 1976) uses a Darcy-like flux from matrix to fracture, assuming a quasisteady state between the two domains that does not correctly represent the average transfer rate in a dynamic displacement. On the basis of 1D analyses in the literature, we find expressions for the transfer rate accounting for both displacement and fluid expansion at early and late times. The resultant transfer function is a sum of two terms: a saturation-dependent term representing displacement and a pressure-dependent term to model fluid expansion. The transfer function is validated through comparison with 1D and 2D fine-grid simulations and is compared to predictions using the traditional Kazemi et al. (1976) formulation. Our method captures the dynamics of expansion and displacement more accurately.
The conventional macroscopic treatment of flow in fractured reservoirs assumes that there are two communicating domains: a flowing region containing connected fractures and high permeability matrix and a stagnant region of low-permeability matrix (Barenblatt et al. 1960; Warren and Root 1963). Conventionally, these are referred to as fracture and matrix, respectively. Transfer between fracture and matrix is mediated by gravitational and capillary forces. In a dual-porosity model, it is assumed that there is no viscous flow in the matrix; a dual-permeability model allows flow in both fracture and matrix. In a general compositional model (where black-oil and incompressible flow are special cases) we can write
where where Gc is a transfer term with units of mass per unit volume per unit time--it is a rate (units of inverse time) times a density (mass per unit volume). c is a component density (concentration) with units of mass of component per unit volume. The subscript p labels the phase, and c labels the component. Gc represents the transfer of component c from fracture to matrix. The subscript f refers to the flowing or fractured domain. The first term is accumulation, and the second term represents flow--this is the same as in standard (nonfractured) reservoir simulation. We can write a corresponding equation for the matrix, m,
where we have assumed a dual-porosity model (no flow in the matrix); for a dual-permeability model, a flow term is added to Eq. 2.
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Aronofsky, J.S., Masse, L., and Natanson, S.G. 1958. A Model for the Mechanism of OilRecovery from the Porous Matrix Due to Water Invasion in FracturedReservoirs. Trans., AIME, 213: 17-19. doi: 10.2118/932-G
Barenblatt, G.I., Zheltov, I.P., and Kochina, I.N. 1960. Basic concepts inthe theory of seepage of homogeneous liquids in fissured rocks. J. Appl.Math. Mech. 24: 1286-1303.
Barenblatt, G.I., Entov, V.M., and Ryzhik, V.M. 1990. Theory of FluidFlows Through Natural Rocks. Dordrecht, The Netherlands: Kluwer AcademicPublishers.
Behbahani, H. and Blunt, M.J. 2005. Analysis of Imbibition in Mixed-WetRocks Using Pore-Scale Modeling. SPEJ 10 (4): 466-474.SPE-90132-PA. doi: 10.2118/90132-PA.
Behbahani, H., Di Donato, G., and Blunt, M.J. 2006. Simulation ofcounter-current imbibition in water-wet fractured reservoirs. Journal ofPetroleum Science and Engineering 50 (1): 21-39.doi:10.1016/j.petrol.2005.08.001.
Chen, J., Miller, M.A., and Sepehrnoori, K. 1994. An Approach for Implementing DualPorosity Models in Existing Simulators. Paper SPE 28001 presented at theUniversity of Tulsa Centennial Petroleum Engineering Symposium, Tulsa, 29-31August. doi: 10.2118/28001-MS.
Civan, F. and Rasmussen, M.L. 2001. Asymptotic Analytical Solutions forImbibition Waterfloods in Fractured Reservoirs. SPEJ 6 (2):171-181. SPE-71312-PA. doi: 10.2118/71312-PA.
de Swaan, A. 1978. Theory ofWaterflooding in Fractured Reservoirs. SPEJ 18 (2): 117-122.SPE-5892-PA. doi: 10.2118/5892-PA.
Di Donato, G., Huang, W., and Blunt, M.J. 2003. Streamline-Based Dual PorositySimulation of Fractured Reservoirs. Paper SPE 84036 presented at the SPEAnnual Technical Conference and Exhibition, Denver, 5-8 October. doi:10.2118/84036-MS.
Di Donato, G. and Blunt, M.J. 2004. Streamline-based dual-porositysimulation of reactive transport and flow in fractured reservoirs. WaterResources Research 40 (W04203). doi:10.1029/2003WR002772.
Di Donato, G. Tavassoli, Z., and Blunt, M.J. 2006. Analytical and numericalanalysis of oil recovery by gravity drainage. Journal of PetroleumScience and Engineering 54 (1-2): 55-69.doi:10.1016/j.petrol.2006.08.002.
Di Donato, G., Lu, H., Tavassoli, Z., and Blunt, M.J. 2007. Multirate-Transfer Dual-PorosityModelling of Gravity Drainage and Imbibition. SPEJ 12 (1):77-88. SPE-93144-PA. doi: 10.2118/93144-PA.
Gilman J.R. and Kazemi, H. 1983. Improvements in Simulation ofNaturally Fractured Reservoirs. SPEJ 23 (4): 695-707.SPE-10511-PA. doi: 10.2118/10511-PA.
Hagoort, J. 1980. Oil Recoveryby Gravity Drainage. SPEJ 20 (3): 139-150. SPE-7424-PA. doi:10.2118/7424-PA.
Huang, W., Di Donato, G., and Blunt, M.J. 2004. Comparison ofstreamline-based and grid-based dual porosity simulation. Journal ofPetroleum Science and Engineering 43 (1-2): 129-137.doi:10.1016/j.petrol.2004.01.002.
Kazemi, H., Merrill, L.S. Jr., Porterfield, K.L., and Zeman, P.R. 1976. Numerical Simulation of Water-Oil Flowin Naturally Fractured Reservoirs. SPEJ 16 (6): 318-326.SPE-5719-PA. doi: 10.2118/5719-PA.
Lu, H., Di Donato, G., and Blunt, M.J. 2006. General Transfer Functions forMultiphase Flow. Paper SPE 102542 presented at the SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, USA, 24-27 September. doi:10.2118/102542-MS.
Lu, H. and Blunt, M.J. 2007. General Fracture/Matrix TransferFunctions for Mixed-Wet Systems. Paper SPE 107007 presented at theEuropec/EAGE Conference and Exhibition, London, 11-14 June. doi:10.2118/107007-MS.
Matthäi, S.K., Mezentsev, A., and Belayneh, M. 2007. Finite Element--Node-CenteredFinite-Volume Two-Phase-Flow Experiments With Fractured Rock Represented byUnstructured Hybrid-Element Meshes. SPEREE 10 (6): 740-756.SPE-93341-PA. doi: 10.2118/93341-PA.
Morrow, M.R. and Mason, G. 2001. Recovery of oil byspontaneous imbibition. Current Opinion in Colloid and InterfaceScience 6 (4): 321-337. doi:10.1016/S1359-0294(01)00100-5.
Pruess, K. and Narasimhan, T.N. 1985. A Practical Method for Modeling Fluidand Heat Flow in Fractured Porous Media. SPEJ 25 (1): 14-26.SPE-10509-PA. doi: 10.2118/10509-PA.
Quandalle, P. and Sabathier, J.C. 1989. Typical Features of a MultipurposeReservoir Simulator. SPERE 4 (4): 475-480. SPE-16007-PA. doi:10.2118/16007-PA.
Sarma, P. and Aziz, K. 2006. New Transfer Functions for Simulationof Naturally Fractured Reservoirs With Dual-Porosity Models. SPEJ11 (3): 328-340. SPE-90231-PA. doi: 10.2118/90231-PA.
Tavassoli, Z., Zimmerman, R.W., and Blunt, M.J. 2005. Analytic analysis for oilrecovery during counter-current imbibition in strongly water-wet systems.Transport in Porous Media 58 (1-2): 173-189.doi:10.1007/s11242-004-5474-4.
Terez, I.E. and Firoozabadi, A. 1999. Water Injection in Water-WetFractured Porous Media: Experiments and a New Model With ModifiedBuckley-Leverett Theory. SPEJ 4 (2): 135-141. SPE-56854-PA.doi: 10.2118/56854-PA.
Vermeulen, T. 1953. Theoryfor irreversible and constant-pattern solid diffusion. Ind. Eng.Chem. 45 (8): 1664-1670. doi:10.1021/ie50524a025.
Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally FracturedReservoirs. SPEJ 3 (3): 245-255; Trans., AIME,228. SPE-426-PA. doi: 10.2118/426-PA.
Wu, Y.-S. and Pruess, K. 1988. A Multiple-Porosity Method forSimulation of Naturally Fractured Petroleum Reservoirs. SPERE3 (1): 327-336. SPE-15129-PA. doi: 10.2118/15129-PA.
Zhang, X., Morrow, N.R., and Ma, S. 1996. Experimental Verification of aModified Scaling Group for Spontaneous Imbibition. SPERE 11(4): 280-285. SPE-30762-PA. doi: 10.2118/30762-PA.
Zhou, D., Jia, L., Kamath, J., and Kovscek, A.R. 2002. Scaling ofcounter-current imbibition processes in low-permeability porous media.Journal of Petroleum Science and Engineering 33 (1-3): 61-74.doi:10.1016/S0920-4105(01)00176-0.
Zimmerman, R.W. and Bodvarsson, G.S. 1989. Integral method solutionfor diffusion into a spherical block. Journal of Hydrology111 (1-4): 213-224. doi:10.1016/0022-1694(89)90261-8.
Zimmerman, R.W., Chen, G., Hadgu, T., and Bodvarsson, G.S. 1993. A numerical dual-porosity model withsemianalytical treatment of fracture/matrix flow. Water ResourcesResearch 29 (7): 2127-2137. doi:10.1029/93WR00749.