A dimensionally reduced Stokes–Darcy model for fluid flow in fractured porous media
Introduction
Fractures are heterogeneities in porous media whose transversal extension is much smaller than their lateral extensions. Therefore, fractures are often modelled as lower-dimensional inclusions in the adjoining porous medium which is typically called the rock matrix. Fractures can be of two different types: i) fractures with much lower permeability than the rock matrix and ii) highly permeable fractures. Fractures of the first type act as geological barriers while fractures of the second type act as channels. Such fractures can either be empty or contain filling material. When a fracture has lower permeability than the surrounding porous medium, the fluid tends to avoid the fracture, while in the case of a highly permeable fracture the fluid has a tendency to flow into the fracture and then along the fracture.
Different mathematical models have been formulated for the description of fluid flow through fractured porous media [1], [2], [3], [4]. These models can be divided into two classes: 1) modelling fractured media as continuum flow domains and 2) discrete fracture modelling. The first approach is based on the homogenization and upscaling techniques [5], [6], [7], [8], [9] to derive single- and double-continuum models and is applicable mainly for single-fluid-phase flows in periodically distributed fine fractures and vugs. The second class contains i) discrete fracture network models, where only fluid flow through the fractures is modelled [3], [4], and ii) models, where fluid flow in the fracture network is coupled to the flow in the porous medium [2], [10], [11], [12], [13], [14]. There is often no separation of length scales in fractured media which makes homogenization difficult if not impossible and results in poor quality of averaged models. Therefore, we will consider the discrete fracture modelling in this work. Since the fluid flow in the fracture significantly affects the flow pattern in the rock matrix, we will couple the flow in the fracture to the flow in the matrix, e.g. consider approach ii) in the second class of models. In this case, fractures and surrounding porous media are modelled as two different coupled flow systems [12], [15], typically two different porous medium systems [1], [2], [11], [13]. Therefore, two flow models together with the appropriate set of coupling conditions at the fracture-matrix interface are considered.
The fracture aperture is significantly smaller than the fracture length and the size of the fractured domain, therefore fractures are often modelled as lower-dimensional inclusions in porous media [1], [10], [13], [14], [16], [17], [18], [19], [20]. Such dimensionally reduced models will be of interest in this work. The fluid flow in the fracture and in the surrounding rock matrix is often described by the same model. Usually, this is the single-phase or multiphase Darcy’s law [2], [13], [16], [18], [19]. The Forchheimer equation, which is an extension of Darcy’s law, is often applied in the fracture when the flow rate is large enough and Darcy’s law is considered in the rock matrix [11], [21]. A reduced Brinkman–Darcy model is proposed in [12] to describe fluid flow in open channels and highly permeable fractures. The Brinkman equation [22] is considered in the fracture, Darcy’s law is applied in the surrounding porous media, and the coupling between the lower-dimensional model in the fracture and the full-dimensional models in the porous media relies on the simplification of the Beavers–Joseph–Saffman interface condition [23], [24], which assumes zero tangential velocity at the fracture-matrix interface. However, such an assumption is non-physical for many applications, e.g. for filtration problems. Therefore, we propose to use the Stokes–Darcy model with the original Beavers–Joseph and the Beavers–Joseph–Saffman coupling conditions instead [15], [25] and to develop the reduced coupled model, where the fractures are considered as narrow channels in porous media and modelled as lower-dimensional domains (Fig. 1, right). The dimensionally reduced model is derived by averaging the Stokes equations across the fracture and by coupling these equations of co-dimension one to the full-dimensional Darcy’s equations describing fluid flow in the surrounding porous media.
The paper is organised as follows. The coupled Stokes–Darcy model for fluid flow in fractured porous media is presented in Section 2 and the reduced model formulation is derived in Section 3. A weak formulation of the reduced coupled model is developed and the existence of a unique solution is proved in Section 4. The validation of the proposed model via numerical simulations is done in Section 5. Finally, the discussion is provided in Section 6.
Section snippets
Coupled Stokes–Darcy model
The flow system of interest contains two porous medium domains Ω1 and Ω2 separated by the fracture Ωf (Fig. 1, left). The fracture is filled with an incompressible fluid having also constant viscosity and the pore space in the rock matrix Ω1 ∪ Ω2 is fully saturated with the same fluid. The interfaces between the fracture and the rock matrix are considered to be simple, i.e. they cannot store and transport mass, momentum, and energy [26]. The thickness of
Geometry
We consider a bounded domain and assume that the fracture aperture d is small with respect to the length of the porous medium domain Ω. This assumption allows us to model the fracture as a complex interface γ (Fig. 1, right). The fracture domain can be described aswhere s ∈ γ, and the unit normal vector n on γ points from the porous medium domain Ω2 to domain Ω1 (Fig. 1, right).
Assuming that γ is smooth enough, we introduce a local coordinate system with the
Analysis of the reduced model
In this section, we will establish the existence and uniqueness of a weak solution of the coupled problem for the case of the Beavers–Joseph–Saffman interface condition (11). In this case, we set both in the reduced Eq. (19) and in the closure conditions (21) and (22).
Numerical simulations
We validate the proposed reduced model (3), (13), (16), (19)–(23) against the full-dimensional model (3)–(9), (11) numerically by considering two test examples. The computational domain is with the complex interface . We choose the physical parameters [Pa s], [m2], . The fracture thickness is [m].
We consider homogeneous Neumann boundary conditions for the porous medium problems on the left, top and bottom boundaries Γi, (Fig. 1
Discussion
In this manuscript, we proposed a new dimensionally reduced model which describes fluid flow in fractured porous media. Fractures are modelled as lower-dimensional inclusions in geological formations that are able to store and transport fluid. The Stokes equations are used in the fracture and Darcy’s law is applied for the flow in the surrounding porous media. Averaging the Stokes equations across the fracture, we obtained a dimensionally reduced model which allows to simulate flow in fractured
Acknowledgment
This work was supported by the German Research Foundation (DFG) project RY 126/2-2.
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