Derivation and relative performance of strings of line elements for modeling (un)confined and semi-confined flow

In the analytic element method, strings of line-sinks may be used to model streams and strings of line-doublets may be used to model impermeable walls or boundaries of inhomogeneities. The resulting solutions are analytic, but the boundary conditions are met approximately. Equations for line elements may be derived in two ways: through integration of point elements (the integral solution) and through application of separation of variables in elliptical coordinates (the elliptical solution). Using both approaches, two sets of line elements are presented for four flow problems: line-sinks and line-doublets in (un)confined flow, and line-sinks and line-doublets in semi-confined flow. Elliptical line elements have the advantage that they do not need a far-field expansion for accurate evaluation far away from the element. The derivation of elliptical line elements is new and applicable to both (un)confined flow and semi-confined flow; only the resulting expressions for elliptical line elements for semi-confined flow have not been found in the current groundwater literature. Existing solutions for elliptical line elements for (un)confined flow were intended for the modeling of isolated features. Four examples are presented, one for each flow problem, to demonstrate that strings of elliptical line elements may be used to obtain accurate solutions; elliptical line-doublets for semi-confined flow can only be strung together in combination with two integral line-doublets. For a zigzag canal in (un)confined flow, a string of elliptical line-sinks performed better than a string of integral line-sinks of the same order when the smallest angle between two adjacent segments is less than 130°. Elliptical line-doublets performed better than integral line-doublets for a square inhomogeneity in a uniform, confined flow field; the difference was smaller for an octagonal inhomogeneity. For semi-confined flow, the difference between the integral and elliptical line-sinks was small when modeling a zigzag canal.

[1]  Otto D Strack,et al.  Theory and applications of the Analytic Element Method , 2003 .

[2]  M. Bakker,et al.  Two‐dimensional steady unsaturated flow through embedded elliptical layers , 2004 .

[3]  S. P. Neuman,et al.  Laplace-transform analytic element solution of transient flow in porous media , 2003 .

[4]  C. Detournay.  A Cauchy integral element for power-type singularities , 1992 .

[5]  D. Spencer,et al.  Field theory handbook: Including coordinate systems, differential equations, and their solutions , 1988 .

[6]  Otto D Strack,et al.  Groundwater mechanics , 1989 .

[7]  Randall J Hunt,et al.  Ground Water Modeling Applications Using the Analytic Element Method , 2006, Ground water.

[8]  Karl Bandilla,et al.  Steady two‐dimensional groundwater flow through many elliptical inhomogeneities , 2004 .

[9]  Fayez A. Alhargan,et al.  Algorithm 804: subroutines for the computation of Mathieu functions of integer orders , 2000, TOMS.

[10]  Mark Bakker.  An analytic element approach for modeling polygonal inhomogeneities in multi-aquifer systems , 2006 .

[11]  Fayez A. Alhargan,et al.  Algorithms for the computation of all Mathieu functions of integer orders , 2000, TOMS.

[12]  David R Steward,et al.  Analytic formulation of Cauchy integrals for boundaries with curvilinear geometry , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[13]  Otto D Strack,et al.  Area sinks in the analytic element method for transient groundwater flow , 1993 .

[14]  Mark Bakker.  Modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities , 2004 .

[15]  Henk M. Haitjema,et al.  Analytic Element Modeling of Groundwater Flow , 1995 .

[16]  Mark Bakker,et al.  Analytic elements for multiaquifer flow , 2003 .

[17]  Randal J. Barnes,et al.  High-order line elements in modeling two-dimensional groundwater flow , 1999 .