[1] We present a streamline-based formulation to model two processes: transport of a tracer undergoing rate-limited sorption and two-phase (water/oil or air/water) transport in fractured systems, using a dual-porosity approach. We show that these two processes can be simulated using mathematically equivalent formulations. In both cases the system conceptually has two components: a flowing fraction connected to stagnant regions with transfer between the two domains. Streamlines capture movement through the flowing fraction. Fluid transfer between flowing and stagnant regions enters as a source/sink term in the one-dimensional transport equations along a streamline. To model flow and transport in fractured systems, we develop a new formulation for the transfer function that matches experimental imbibition data. Then we illustrate the streamline approach with synthetic reservoir problems. We use a finely gridded (over one million grid blocks) threedimensional domain with a highly heterogeneous permeability field to study both fracture flow and tracer transport. We find breakthrough curves that are consistent with anomalous transport described by an exponent that characterizes the longtime tail of the transit time distribution. For fracture flow we demonstrate that the speed of fluid advance in the fractures is controlled by the imbibition rate. The run times for the simulations scale approximately linearly with system size, making the method appropriate for the simulation of large numerical models.

[1] We use fine-grid streamline simulation to demonstrate that anomalous (non-Gaussian) transport arises from purely advective movement through heterogeneous systems.The simulation model represents a channeled sandstone North Sea oil field and contains over one million grid blocks. We find the average location and spreading of the plume and the breakthrough curves. These features are consistent with anomalous transport described by an exponent b that characterizes the long-time tail of the transit time distribution. b depends on the degree of heterogeneity and whether the tracer was injected or originally uniformly distributed across the domain. Incorporating dispersion to account for sub-grid-block heterogeneity does not affect the results.

Summary We develop a physically motivated approach to modeling displacement processes in fractured reservoirs. To find matrix/fracture transfer functions in a dual-porosity model, we use analytical expressions for the average recovery as a function of time for gas gravity drainage and countercurrent imbibition. For capillary-controlled displacement, the recovery tends to its ultimate value with an approximately exponential decay (Barenblatt et al. 1990). When gravity dominates, the approach to ultimate recovery is slower and varies as a power law with time (Hagoort 1980). We apply transfer functions based on these expressions for core-scale recovery in field-scale simulation. To account for heterogeneity in wettability, matrix permeability, and fracture geometry within a single gridblock, we propose a multirate model (Ponting 2004). We allow the matrix to be composed of a series of separate domains in communication with different fracture sets with different rate constants in the transfer function. We use this methodology to simulate recovery in a Chinese oil field to assess the efficiency of different injection processes. We use a streamline-based formulation that elegantly allows the transfer between fracture and matrix to be accommodated as source terms in the 1D transport equations along streamlines that capture the flow in the fractures (Di Donato et al. 2003; Di Donato and Blunt 2004; Huang et al. 2004). This approach contrasts with the current Darcy-like formulation for fracture/matrix transfer based on a shape factor (Gilman and Kazemi 1983) that may not give the correct average behavior (Di Donato et al. 2003; Di Donato and Blunt 2004; Huang et al. 2004). Furthermore, we show that recovery is exceptionally sensitive to parameters that describe the physics of the displacement process, highlighting the need to make careful core-scale measurements of recovery. Introduction Di Donato et al.(2003) and Di Donato and Blunt (2004) proposed a dual-porosity streamline-based model for simulating flow in fractured reservoirs. Conceptually, the reservoir is composed of two domains: a flowing region with high permeability that represents the fracture network and a stagnant region with low permeability that represents the matrix (Barenblatt et al. 1960; Warren and Root 1963). The streamlines capture flow in the flowing regions, while transfer from fracture to matrix is accommodated as source/sink terms in the transport equations along streamlines. Di Donato et al. (2003) applied this methodology to study capillary-controlled transfer between fracture and matrix and demonstrated that using streamlines allowed multimillion-cell models to be run using standard computing resources. They showed that the run time could be orders of magnitude smaller than equivalent conventional grid-based simulation (Huang et al. 2004). This streamline approach has been applied by other authors (Al-Huthali and Datta-Gupta 2004) who have extended the method to include gravitational effects, gas displacement, and dual-permeability simulation, where there is also flow in the matrix. Thiele et al. (2004) have described a commercial implementation of a streamline dual-porosity model based on the work of Di Donato et al. that efficiently solves the 1D transport equations along streamlines.

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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 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,[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.