Non-Darcy Flow

Flow in porous media is often governed by the Darcy's law, in which the flux is a linear function of pressure gradient. However, Darcy's law may not work correctly in some cases. Forchheimer [1901] empirically found the relationship between the gradient of the water potential ∇Φ = (∇P −ρg) [Pa $m^{-1}$] and average velocity to be:

$-\varDelta\varPhi= {\mu \over k.k_r}\bold u+\beta\rho|\bold u| \bold u$ (7-45)

where u [m/s] is the average velocity, ρ [kg m-3] is the density, k [m2] is the absolute permeability, $k_r$ [-] is the relative permeability, which is a function of saturation, and μ [Pa s] is the dynamic viscosity. The second term on the right-hand side of Eq. (7-45) accounts for inertial forces. On the pore scale, as velocity increases, flow tends to locally separate from the tortuous pore walls, leading to local vortices and countercurrent flow regimes. The coefficient β [m-1] is a function of effective permeability ( $k.k_r$ ), phase content (θ =φS , where φ is porosity and S is the saturation of the considered fluid phase), and tortuosity ( $\tau={L \over l}$ , where L is the macroscopic flow distance, and l is the length of the corresponding microscopic, tortuous flow path). This dependence indicates a transition from viscous-dominated to form-dominated flow as the Reynolds number ( $Re=\rho\mu\sqrt K/\mu$ , where K is a form factor) increases beyond a critical value. The parameter β has been referred to as the non-Darcy flow coefficient, the Forchheimer or inertial resistance coefficient, or the turbulence factor. Note, however, that deviations from Darcy’s law are observed at Reynolds numbers that are at least one order of magnitude smaller than those where actual turbulence occurs. We will refer to β as the non-Darcy flow coefficient, which can be written in a general, parameterized form as follows:

$\beta=A_1\cdot{1 \over(k \cdot k_r)^{A_2}} \cdot{1 \over \theta^{A_3}}\cdot{1 \over \tau^{A_4}}$ (7-46)

Thirteen correlation models and user specified model for the non-Darcy flow coefficient have been implemented in iTOUGH2 which are also used in TOUGH4 (see Table 16). Details can be found in Finsterle, 2016. Most of the original models were developed for single-phase gas flow or gas flow at immobile water saturation. They are generalized here to two-phase conditions by replacing absolute permeability k with effective permeability, $k_{eff}=k\cdot k_r(S)$ , and porosity φ with phase content, θ =φS . Moreover, it is assumed that the Forchheimer equation applies to both liquid and gas flow. (Note that the user can restrict its applicability to gas flow only). The appropriateness of these extensions has not been examined.

Table 16 Parameters for Non-Darcy Flow Coefficient β

Non-Darcy flow based on the Forchheimer equation is implemented in TOUGH4 according to user selection of the non-Darcy flow coefficient model (eq. 7-46 with different parameters provided as in Table 16). The non-Darcy flow coefficient β is evaluated for each gridblock in a connection, and then averaged according to the selected weighting scheme. The quadratic Forchheimer equation (Eq. 7-45) with given β is solved for the absolute value of the non-Darcy velocity u.

The simulation of non-Darcy flow can be invoked by providing a new data block following keyword FORCH in the main input file.