Flow Through a Porous Medium With Periodic Barriers or Fractures
- Gary R. Chirlin (Woodward-Clyde Consultants)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- June 1985
- Document Type
- Journal Paper
- 358 - 362
- 1985. Society of Petroleum Engineers
- 5.3.1 Flow in porous media
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Apparent permeability of a reservoir may be affected by local variations in permeability whose extent and location are not known in detail. One way to approximate this condition is to postulate a regular pattern of typically shaped and oriented inclusions.
An example is the presence of a large number of very thin, relatively impermeable shale streaks distributed throughout an otherwise homogeneous reservoir. If these barriers to flow generally are oriented alike, then at a scale that is large compared to the individual barriers, apparent permeability will be anisotropic, being higher parallel to the barriers than perpendicular to them. Discontinuous horizontal shale streaks can be responsible for reduced vertical oil drainage to the pumping well ; therefore, a reasonable prediction of vertical permeability in such settings can be of value.
This paper examines the special case of a periodic array of infinitely thin, impermeable barriers opposed to single-phase, two-dimensional (2D) flow. Results provide a quantitative picture of the importance of barrier geometry and arrangement to the apparent anisotropy of permeability. The problem setting, formulated previously, allows a closed-form analytical expression to be obtained for anisotropy as a function of barrier geometry. However, a previously published solutions is incorrect. The purpose of this paper is to present a correct solution and to discuss some of its consequences. Analytical results for this and other geometries, in two and three dimensions, have been developed previously. Numerical results for this problem are also available for both single-phase and two-phase (gas/oil) 1 flow.
By exchanging streamlines and equipotential lines, the problem solved here can represent a periodically fractured medium. Mathematically, the infinitely thin, infinite-conductivity fractures sustain no pressure change along their (finite) length, and conduct a finite amount of fluid. Practically, one might select such a model if it appears reasonable to ignore pressure changes along fractures when solving for the potential field in the matrix between the fractures.
For clarity, we will develop the solution in terms of the barriers problem, and will refer in retrospect to the fracture problem.
The Problem Setting
The medium geometry considered here is that of an unbounded, homogeneous, isotropic matrix of permeability k, into which a periodic array of barriers is introduced (Fig. 1). These zero-thickness barriers are of length 2(1 -alpha) w, are impermeable, are spaced a distance 2 alpha w apart end-to-end and h apart broadside, and are staggered so that the center of each barrier is opposite to the center of the two passages broadside to it. The model is 2D; hence the matrix and barriers implicitly extend indefinitely in the third dimension.
The spatially periodic medium is characterized geometrically by a group of pure translations that bring the patterns of the medium into self-coincidence. One can identify a set of (nonunique) lattice nodes, from each of which the view is identical, and connect these to form a unit cell. In the medium of Fig. 1, we select Unit Cell BB'P'P. (Obvious alternatives are cells CQQ'C', BB'Q'C', and PRR'B.) We need study the flow field only within a single unit cell; that within each unit cell will be identical. In fact, because of mirror symmetry across Cells CC' or AA' (Fig. 2), we consider only Quarter-Cell ABCD.
The cell geometry parameters h, w, and ct are indicated in Fig. 2. These parameters will be the independent variables controlling changes in apparent anisotropy.
Conservation of mass and Darcy's law combine to require that the flow potential phi satisfy
within the Quarter-Cell ABCD. The potential phi is set equal to the constant phi on Entrance AE to Quarter-Cell ABCD, and to the constant phi o on Exit CF (Fig. 2). Constant potential boundaries are selected for simplicity of analysis. They are, however, the only case in which all streamlines are parallel as they pass through the gap between barriers; this is a reasonable case to examine. The barriers themselves (Segments EB and DF, Fig. 2) are assumed to be impermeable. From the symmetry of the cell geometry and the constant potential boundary conditions, Segments AD and BC of Fig. 2 are also no-flow boundaries.
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