Computational issues and applications of line-elements to model subsurface flow governed by the modified Helmholtz equation

Horizontal drilling has become an appealing technology for water resource exploration or aquifer remediation in recent decades, due to decreasing operational cost and many technical advantages over vertical wells. However, many previous studies on flow into horizontal wells were based on the Uniform Flux Boundary Condition (UFBC), which does not reflect the physical processes of flow inside the well accurately. In this study, we investigated transient flow into a horizontal well in an anisotropic confined aquifer laterally bounded by two constant-head boundaries. Three types of boundary conditions were employed to treat the horizontal well, including UFBC, Uniform-Head Boundary Condition (UHBC), and Mixed-Type Boundary Condition (MTBC). The MTBC model considered both kinematic and frictional effects inside the horizontal well, in which the kinematic effect referred to the accelerational and fluid-inflow effects. A new solution of UFBC was derived by superimposing the point sink/source solutions along the axis of a horizontal well with a uniform flux distribution. New solutions of UHBC and MTBC were obtained by a hybrid analytical-numerical method, and an iterative method was proposed to determine the well discretization required for achieving sufficiently accurate results. This study showed that the differences among the UFBC, UHBC, and MTBC solutions were obvious near the well screen, decreased with distance from the well, and became negligible near the constant-head boundary. The relationship between the flow rate and the drawdown was nonlinear for the MTBC solution, while it was linear for the UFBC and UHBC solutions.

Section: Introductionmentioning

confidence: 99%

Intrawellbore kinematic and frictional losses in a horizontal well in a bounded confined aquifer

Wang

Zhan

2017

“…Fourier series have been used to develop integral representations for coefficients by Marin et al . [], and by Bakker and Kuhlman [] to solve the modified Helmholtz equation. Direct methods to solve without integration were developed for circle elements by Barnes and Janković [] as done here for slit elements.…”

Section: Discussionmentioning

confidence: 99%

Analysis of discontinuities across thin inhomogeneities, groundwater/surface water interactions in river networks, and circulation about slender bodies using slit elements in the Analytic Element Method

Steward

2015

Groundwater and surface water contain interfaces across which hydrologic functions are discontinuous. Thin elements with high hydraulic conductivity in a porous media focus groundwater, which flows through such inhomogeneities and causes an abrupt change in stream function across their interfaces, and elements with low conductivity retard flow with discontinuous head. Base flow interactions at the interface between groundwater and surface water transport water between these stores and generate a discontinuous normal component of flow. Thin objects in surface water with Kutta condition generate circulation by the discontinuous tangential component of flow across their interface. These discontinuities across hydrologic interfaces are quantified and visualized using the Analytic Element Method, where slit elements are formulated using the Joukowsky transformation with Laurent series and new influence functions to represent sinks and circulation, and methods are developed for these applications expressing discontinuities as Fourier series. The specific geometries illustrate solutions for a randomly generated heterogeneous porous media with nonintersecting inhomogeneities, for groundwater/surface water interaction in a synthetic river network, and for a slender body with geometry similar to the wings of the Wright Brothers. The mathematical details are reduced to series solutions and matrix multiplications, which are easily extensible to other geometries and applications.

“…The multiporosity conceptualization may be appropriate for flow in fractured rock in which hydraulic fracturing is being performed, but the simple homogeneous cylindrical geometry of the radial solution presented here cannot adequately represent pumping discrete fractures under typical field conditions (Gringarten, 1982;Clarkson, 2013). The governing equations presented here could be solved for discrete linear fractures using analytical solutions for single fractures (e.g., Gringarten et al, 1974) or using analytic element superposition-based combinations of line element solutions (Bakker and Kuhlman, 2011;Biryukov and Kuchuk, 2012). These approaches can better represent the geometry encountered during hydraulic fracturing.…”

Section: Approachmentioning

confidence: 99%

Multiporosity flow in fractured low‐permeability rocks

Kuhlman

Malama

Heath

2015

Self Cite

A multiporosity extension of classical double and triple-porosity fractured rock flow models for slightly compressible fluids is presented. The multiporosity model is an adaptation of the multirate solute transport model of Haggerty and Gorelick (1995) to viscous flow in fractured rock reservoirs. It is a generalization of both pseudo steady state and transient interporosity flow double-porosity models. The model includes a fracture continuum and an overlapping distribution of multiple rock matrix continua, whose fracture-matrix exchange coefficients are specified through a discrete probability mass function. Semianalytical cylindrically symmetric solutions to the multiporosity mathematical model are developed using the Laplace transform to illustrate its behavior. The multiporosity model presented here is conceptually simple, yet flexible enough to simulate common conceptualizations of double and triple-porosity flow. This combination of generality and simplicity makes the multiporosity model a good choice for flow modelling in lowpermeability fractured rocks.