Abstract
This chapter gives a brief review of the basic equations needed to simulate single-phase and two-phase (oil–water) flow through porous media. It discusses the governing partial-differential equations, their physical interpretation (especially the diffusive nature of pressures and the convective behavior of saturations), spatial discretization with finite differences, and the treatment of wells. It contains well-known theory and is primarily meant to form a basis for the next chapter where the equations will be reformulated in terms of systems-and-control notation.
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Notes
- 1.
The Darcy velocity or the filtration velocity , is the superficial velocity that would occur if the entire cross section, and not just the pores, would be open to flow. This is as opposed to the interstitial velocity \(\tilde{\nu}\), which is defined as \(\tilde{\nu}\,=\,\nu/\phi \), and which is the true fluid velocity in the pore space. The Darcy velocity can also be interpreted as a volumetric flux , i.e. the amount of volume flowing through a unit of surface area per unit time.
- 2.
We use an arrow above a vector or matrix to indicate that it its components are representing quantities in physical space. For example \( {\vec{\mathbf{v}}} \) is a velocity vector with one, two or three components, depending on whether we use a one-, two-, or three-dimensional reservoir description. Note that the spatial-coordinate vector \( {\vec{\mathbf{x}}} \) is unrelated to the state vector x as used in Chaps. 2 and 3. The use of the same symbol for two different quantities is somewhat unfortunate, but results from conventions in different disciplines.
- 3.
Permeability has a dimension of length squared and is therefore expressed in SI units in m2. In reservoir engineering use is often made of Darcy units, which are defined as: 1 D = 9.869233 × 10−13 ≈ 10−12 m2.
- 4.
More complicated boundary conditions are possible, e.g. by specifying a relationship between p and ∂p/∂n, a so-called mixed boundary condition. Furthermore, different boundary conditions may be specified along different parts of the boundary.
- 5.
This two-dimensional grid-block numbering is introduced to obtain a systematic description of the transmissibilities in a two-dimensional reservoir model. In a numerical implementation, however, one normally uses a one-dimensional grid-block numbering as displayed in Fig. 1.3, and a connectivity table to list the pairs of adjacent grid blocks. See also Table 1.2 which illustrates the two different numbering systems as applied to Example 1.
- 6.
Absence of source terms corresponds to considering the (1-D) flow between an injector and a producer, in which case the flow is driven through the boundary conditions.
- 7.
The coefficients are still functions of saturation.
- 8.
\( f_w(S_w) \) is sometimes referred to as the flux function.
- 9.
- 10.
We may either consider the water mass or the oil mass. Moreover, because we assume the fluids to be incompressible, it is sufficient to consider a volume balance rather than a mass balance.
- 11.
- 12.
In the upstream oil industry standard conditions are usually defined as a pressure p sc = 100 kPa (14.7 psi) and a temperature T sc = 15 °C (60 °F), which can be considered as typical for atmospheric conditions in temperate climates. Oil at standard conditions is often referred to as stock tank oil.
- 13.
To stay in line with the notation used in the single-phase flow case, we should have used ϕ 0, ρ o,0 and ρ w,0 to indicate the pressure-independence of these parameters, but we have dropped the subscripts 0 to simplify the notation.
- 14.
Here the sub-matrices are displayed as vectors, which form, however, building blocks for matrices when the equations for multiple grid blocks are combined.
- 15.
As discussed before, we disregard the dependency of the transmissibility terms on pressure.
- 16.
We use a bold-face italics font to represent vectors with properties at the grid-blocks boundaries, whereas the conventional bold-face font is used to represent vectors with properties at the grid-blocks centers. In particular we use p and p for pressures, λ and λ for mobilities, \(\varvec{\nu}\) and v for Darcy velocities, and \(\tilde{\varvec \nu}\) and \({\tilde{\bf{v}}}\) for interstitial velocities.
- 17.
This is an example of an incidence matrix as used in the analysis of e.g. electrical or mechanical networks, to define the pattern of nodes and edges. Other names used in the literature are topology matrixor connectivity matrix . Note that we use the term connectivity in a slightly different sense; see p. 69.
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Jansen, J.D. (2013). Porous-Media Flow. In: A Systems Description of Flow Through Porous Media. SpringerBriefs in Earth Sciences. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00260-6_1
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