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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