Analytical solutions for one-phase seepage flows impeded by wellbore seals

Kacimov, A. R.; Marketz, Franz; Pervez, Tasneem

doi:10.1016/j.petrol.2008.12.005

Cited by 3 publications

(3 citation statements)

References 24 publications

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“…In this manner, seepage through elliptical and rectangular PRBs has been studied in detail; see [9,10,15,23]. In the constructal approach, the design starts not from a given shape but from a given criterion (see [24][25][26] for other hydrogeological/geotechnical/petroleum engineering applications): we specify d (or Q) and determine what is the best shape of a PRB accomplishing the mission of plume interception. The mathematical theory of constructal design has been developed in [27] for Darcian seepage flows under impermeable subsurface contours of concrete dams and in [28] for general physical fields described by holomorphic characteristic functions.…”

Section: Model and Constructal Design Of A Highly Conductive Prbmentioning

confidence: 99%

“…In [25,26] shaping the boundaries of flow domains gave nontrivial 1 As our results for an elliptical lens are based on [37], we shall try to keep the corresponding notations as much as possible. For the ellipsoid solution by Poisson we refer to the most common source, [49], and, correspondingly, we will try to preserve the notations from there.…”

Section: Elliptical Prbmentioning

confidence: 99%

“…This minimum solves problem A3. Again, using (26) and the FindMinuimum of [34, pp. 703-704] we compute this minimum.…”

Section: Elliptical Prbmentioning

confidence: 99%

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Constructal design of permeable reactive barriers: groundwater-hydraulics criteria

et al. 2011

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Unidirectional, steady-state, Darcian flow in a confined homogeneous aquifer is partially intercepted by a permeable reactive barrier (PRB), the shape of which is optimized with the following hydraulic criteria: seepage flow rate through a PRB (equivalent to the width and frontal area of the intercepted part of the plume in 2-D and 3-D cases, correspondingly) and travel time of a marked particle through the PRB interior along streamlines. The wetted perimeter, cross-sectional area and volume of the reactive material are selected as isoperimetric constraints. The PRB contour is modeled as either a constant head line (if the reactive material is much more permeable than the aquifer) or as a refraction boundary (if the reactive material has an arbitrary permeability), on which the hydraulic head and normal flux components in the barrier and aquifer are continuous. In the former case, the complex potential domain of the flow is a tetragon and a broad class of PRBs can be studied. In the latter case, analytical solutions are available for ellipses and ellipsoids (only these classes of shapes are considered in optimization). In the 2-D case and constant head PRB, a novel shape-control technique through the kernels of singular integrals is implemented: the Zhukovskii function is introduced; a Dirichlet boundary-value problem is solved for this function by setting the orientation (with respect to the incident flow direction) of the Darcian velocity vector on the PRB contour as a control function. Unlike similar controls for impermeable airfoils in aerodynamic design, the kernel has two 123 320 A. R. Kacimov et al. discontinuities, which reflect the flow topology near a hinge (stagnation) point and the PRB tip. The integral is evaluated for V-shaped and curve-shaped PRBs and parametric expressions for the contours are obtained resulting (for the latter case) in a "pointy banana" shape. In the class of a V-shaped PRB, it is proved that a straight-line barrier minimizes the perimeter if the plume width is fixed. In 2-and 3-D refracting PRBs, the Pilatovskii (ellipse) and Poisson (ellipsoid) solutions for the flow field inside and outside the PRB are used for obtaining explicit formulae for the magnitude of the velocity, which is uniform inside the PRB. Simple expressions for the longest travel time within the PRB and the discharge intercepted by it are obtained. The ellipse/ellipsoid axes ratio/ratios are used as control variables in optimization. Extrema are obtained and analyzed for different PRB-aquifer conductivity ratios and for varying angles between the incident velocity vector and the ellipse/ellipsoid axes.

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