RESUMO
It is common to determine the effective conductivity of heterogeneous media by assuming stationarity of the random local properties. This assumption is not obeyed in a boundary layer of a body of finite size. The effect of different types of boundaries is examined for a two-phase medium with spherical inclusions of given conductivity distributed randomly in a matrix of a different conductivity. Exact solutions are derived for the apparent conductivity and the boundary layer thickness. The interaction between the spheres and the boundaries is fully incorporated in the solutions using a spherical harmonics expansion and the method of images. As applications, the corrections for the effective conductivity are given for two cases of finite bodies: the Maxwell sphere and a cylinder of flow parallel to the axis.
RESUMO
Solute transport is investigated in a heterogeneous aquifer for combined natural-gradient and well flows. The heterogeneity is associated with the spatially varying hydraulic conductivity K(x, y, z), which is modelled as a log-normal stationary-random function. As such, the conductivity distribution is characterized by four parameters: the arithmetic mean K(A), the variance sigma(Y)(2) (Y=lnK), the horizontal integral scale I of the axisymmetric log-conductivity autocorrelation and the anisotropy ratio e=I(v)/I (I(v) is the vertical integral scale). The well fully penetrates an aquifer of constant thickness B and has given constant discharge QB, while the background aquifer flow is driven by an uniform mean head-gradient, - J. Therefore, for a medium of homogeneous conductivity K(A), the steady-state capture zone has a width 2L=Q/(K(A)|J|) far from the well (herein the term capture zone is used to refer both to the zone from which water is captured by a pumping well and the zone that captures fluid from an injecting well). The main aim is to determine the mean concentration as a function of time in fluid recovered by a pumping well or in a control volume of the aquifer that captures fluid from an injecting well. Relatively simple solutions to these complex problems are achieved by adopting a few assumptions: a thick aquifer B>>I(v) of large horizontal extent (so that boundary effects may be neglected), weak heterogeneity sigma(Y)(2)<1, a highly anisotropic formation e<0.2 and neglect of pore-scale dispersion. Transport is analyzed to the first-order in sigma(Y)(2) in terms of the travel time of particles moving from or towards the well along the steady streamlines within the capture zone. Travel-time mean and variance to any point are computed by two quadratures for an exponential log-conductivity two-point covariance. Spreading is reflected by the variance value, which increases with sigma(Y)(2) and I/L. For illustration, the procedure is applied to two particular cases. In the first one, a well continuously injects water at constant concentration. The mean concentration as function of time for different values of the controlling parameters sigma(Y)(2) and I/L is determined within control volumes surrounding the well or in piezometers. In the second case, a solute plume, initially occupying a finite volume Omega(0), is drawn towards a pumping well. The expected solute recovery by the well as a function of time is determined in terms of the previous controlling parameters as well as the location and extent of Omega(0). The methodology is tested against a full three-dimensional simulation of a multi-well forced-gradient flow field test ([Lemke, L., W.B. II, Abriola, L., Goovaerts, P., 2004. Matching solute breakthrough with deterministic and stochastic aquifer models. Ground Water 42], SGS simulations). Although the flow and transport conditions are more complex than the ones pertinent to capture zones in uniform background flow, it was found that after proper adaptation the methodology led to results for the breakthrough curve in good agreement with a full three-dimensional simulation of flow and transport.
