Summary
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.
Introduction
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
[Equation 1],
where where Γc 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. Γc 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,
[Equation 2]
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.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
28 February 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
7 January 2008
- Manuscript approved:
17 January 2008
- Version of record:
20 September 2008
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