Macrodispersion in sand-shale sequences is investigated by a series of numerical tracer tests. Hydraulic conductivity is modeled as a binary, spatially correlated random function. Realizations of the random conductivity field are simulated on a nodal grid discretizing the heterogeneous formation. Corresponding realizations of the random velocity field are obtained by solving the equation for saturated steady state flow. Particle tracking, with flux-weighted tracer injection and detection, is used to generate experimental residence time distributions (RTDs). Moments of the RTD are used to characterize longitudinal tracer spreading. Results show that macrodispersive transport in sand-shale sequences cannot be represented by a Fickian model. RTDs display a bimodal structure caused by the fast arrival of particles traveling along preferential sandstone channels and by the much slower arrival of particles following tortuous routes through sandstone and shale. The relative importance of channeling and tortuous flow transport mechanisms is determined by sand-shale conductivity contrast, shale volume fraction, and conductivity spatial correlation structure. Channeling is promoted by high conductivity contrasts, low shale fractions, and flow parallel to bedding in anisotropic media. Low contrasts, high shale fractions, and flow perpendicular to bedding act to break up channels and to enhance tracer spreading. 1. ß ß ß ß ß ß ß ß 20Xx 6Xx 6X ß 10Xx 10Xx 7X _ ß _ ß ß _ ß i I ß ß ß BINARY MODEL ß BIMODAL MODEL .Ol 1.00 •--, .10 .01 ß ß ß ß ß i I ß ß ß I I ß ß ©11 ß ß ß ß I I ß Kss/KsH: 10 4 ß Kss/KsH: 10 2 ß Kss/KsH: 101 0 .1'0 .2•0 .3'0 .4'0 .5'0 .60 VSH GSC

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A numerical approach is used to estimate effective permeability in sand-shale formations under steady state uniform flow conditions. Permeability is modeled as a binary, second-order stationary random function taking on two possible values Ks_• and K_• h in sandstone and shale, respectively. This model is realistic since experience with sandstone reservoirs has shown that randomly dispersed low-permeability shales are the single dominant heterogeneity affecting flow behavior. The cases of both spatially correlated and uncorrelated permeabilities are considered. For the case of spatially correlated permeability, an autocovariance model was fitted to data from the Assakao fluvial sandstone which outcrops in the Tassili region of the central Sahara. The turning bands method was. used to simulate the spatially correlated permeabilities of blocks discretizing the flow field. Effective permeability was found to depend on the shale volume fraction, the spatial covariance structure, and the dimensionality of the flow system.Existing analytical methods for estimating effective permeability in a two-phase medium are found to be inaccurate when compared to numerical results or unapplicable to stratified environments. In addition to providing a check of analytical work, the numerical approach is found to be a useful tool for exploring the effects of reservoir heterogeneity on flow behavior in a qualitative sense. of randomly varying physical properties. The averaging protess must consider implicitly the effects of heterogeneity on flow behavior at scales smaller than that at which the flow model is formulated. Remaining larger-scale heterogeneity is then described in a deterministic fashion, and its effects are considered explicitly by the flow model. In the case of analytica! flow models this usually involves determining globally uniform, equivalent physical properties over the whole flow field. Numerical flow models require the determination of locally uniform effective properties at the scale of finite-difference grid blocks [Neuman, 1982]. Permeability is perhaps the single most important physical property affecting subsurface flow, and it is not surprising that considerable effort has been devoted to the problem of calculating effective permeabilities in heterogeneous porous media Axness, 1983; Glezen and Lerche, 1985'I. Using a Monte Carlo approach, Warren and Price [1961] found that the geometric mean provided a good estimate of effective permeability for a wide range of univariate distribution models when permeabilities do not present any spatial correlation. Matheron [1967] used a perturbation approach to derive an expression for effective permeability in an unbounded statistically isotropic medium. For the particular case of two-dimensional flow and a lognormal permeability distribution, he was able to prove that the effective permeability is equal to the geometric mean. Using a self-consistent approach, Dugart [1979] proposed bounds and an estimate of effective permeability for isotropic formations. Following a pe...

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Steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical-empirical approach. The objective of the study is to determine the effective transmissivity of a square field with a constant-head external boundary and a constant-rate well at its center. Transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function. Block transmissivities are defined using an empirical scaling up of point support values within the field. The scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well. Block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field. The principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries. The geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale. Exact expressions are derived for the ensemble moments of block-averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known. The ensemble mean of block-averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation. This result contradicts earlier published work. Numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems.