Reservoir Heterogeneity: Should It Be Modelled as Conformance or Dispersion?

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A hierarchy of models for simulating experimental results from a 3D heterogeneous porous medium

In this work we examine the dispersion of conservative tracers (bromide and fluorescein) in an experimentally-constructed three-dimensional dual-porosity porous medium. The medium is highly heterogeneous ($\sigma_Y^2=5.7$), and consists of spherical, low-hydraulic-conductivity inclusions embedded in a high-hydraulic-conductivity matrix. The bi-modal medium was saturated with tracers, and then flushed with tracer-free fluid while the effluent breakthrough curves were measured. The focus for this work is to examine a hierarchy of four models (in the absence of adjustable parameters) with decreasing complexity to assess their ability to accurately represent the measured breakthrough curves. The most information-rich model was (1) a direct numerical simulation of the system in which the geometry, boundary and initial conditions, and medium properties were fully independently characterized experimentally with high fidelity. The reduced models included; (2) a simplified numerical model identical to the fully-resolved direct numerical simulation (DNS) model, but using a domain that was one-tenth the size; (3) an upscaled mobile-immobile model that allowed for a time-dependent mass-transfer coefficient; and, (4) an upscaled mobile-immobile model that assumed a space-time constant mass-transfer coefficient. The results illustrated that all four models provided accurate representations of the experimental breakthrough curves as measured by global RMS error. The primary component of error induced in the upscaled models appeared to arise from the neglect of convection within the inclusions. Interestingly, these results suggested that the conventional convection-dispersion equation, when applied in a way that resolves the heterogeneities, yields models with high fidelity without requiring the imposition of a more complex non-Fickian model.Comment: 27 pages, 9 Figure

Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random Walk and Fractional Derivative Formulations

The basic conceptual picture and theoretical basis for development of transport equations in porous media are examined. The general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations, for single realizations and for ensemble averages of the domain. The application of these transport equations is focused on accounting for the appearance of non-Fickian (anomalous) transport behavior. The general ensemble-averaged transport equation is shown to be equivalent to a continuous time random walk (CTRW) and reduces to the conventional forms of the advection-dispersion equation (ADE) under highly restrictive conditions. Fractional derivative formulations of the transport equations, both temporal and spatial, emerge as special cases of the CTRW. In particular, the use in this context of L{\'e}vy flights is critically examined. In order to determine chemical transport in field-scale situations, the CTRW approach is generalized to non-stationary systems. We outline a practical numerical scheme, similar to those used with extended geological models, to account for the often important effects of unresolved heterogeneities.Comment: 14 pages, REVTeX4, accepted to Wat. Res. Res; reference adde

Investigation of local mixing and its influence on core scale mixing (dispersion)

textLocal displacement efficiency in miscible floods is significantly affected by mixing taking place in the medium. Laboratory experiments usually measure flow-averaged ("cup mixed") effluent concentration histories. The core-scale averaged mixing, termed as dispersion, is used to quantify mixing in flow through porous media. The dispersion coefficient has the contributions of convective spreading and diffusion lumped together. Despite decades of research there remain questions about the nature and origin of dispersion. The main objective of this research is to understand the basic physics of solute transport and mixing at the pore scale and to use this information to explain core-scale mixing behavior (dispersion). We use two different approaches to study the interaction between convective spreading and diffusion for a range of flow conditions and the influence of their interaction on dispersion. In the first approach, we perform a direct numerical simulation of pore scale solute transport (by solving the Navier Stokes and convection diffusion equations) in a surrogate pore space. The second approach tracks movement of solute particles through a network model that is physically representative of real granular material. The first approach is useful in direct visualization of mixing in pore space whereas the second approach helps quantify the effect of pore scale process on core scale mixing (dispersion). Mixing in porous media results from interaction between convective spreading and molecular diffusion. The converging-diverging flow around sand grains causes the solute front to stretch, split and rejoin. In this process the area of contact between regions of high and low solute concentrations increases by an order of magnitude. Diffusion tends to reduce local variations in solute concentration inside the pore body. If the fluid velocity is small, diffusion is able to homogenize the solute concentration inside each pore. On the other hand, in the limit of very large fluid velocity (or no diffusion) local mixing because of diffusion tends to zero and dispersion is entirely caused by convective spreading. Flow reversal provides insights about mixing mechanisms in flow through porous media. For purely convective transport, upon flow reversal solute particles retrace their path to the inlet. Convective spreading cancels and echo dispersion is zero. Diffusion, even though small in magnitude, causes local mixing and makes dispersion in porous media irreversible. Echo dispersion in porous media is far greater than diffusion and as large as forward (transmission) dispersion. In the second approach, we study dispersion in porous media by tracking movement of a swarm of solute particles through a physically representative network model. We developed deterministic rules to trace paths of solute particles through the network. These rules yield flow streamlines through the network comparable to those obtained from a full solution of Stokes' equation. In the absence of diffusion the paths of all solute particles are completely determined and reversible. We track the movement of solute particles on these paths to investigate dispersion caused by purely convective spreading at the pore scale. Then we superimpose diffusion and study its influence on dispersion. In this way we obtain for the first time an unequivocal assessment of the roles of convective spreading and diffusion in hydrodynamic dispersion through porous media. Alternative particle tracking algorithms that use a probabilistic choice of an out-flowing throat at a pore fail to quantify convective spreading accurately. For Fickian behavior of dispersion it is essential that all solute particles encounter a wide range of independent (and identically distributed) velocities. If plug flow occurs in the pore throats a solute particle can encounter a wide range of independent velocities because of velocity differences in pore throats and randomness of pore structure. Plug flow leads to a purely convective spreading that is asymptotically Fickian. Diffusion superimposed on plug flow acts independently of convective spreading causing dispersion to be simply the sum of convective spreading and diffusion. In plug flow hydrodynamic dispersion varies linearly with the pore-scale Peclet number. For a more realistic parabolic velocity profile in pore throats particles near the solid surface of the medium do not have independent velocities. Now purely convective spreading is non-Fickian. When diffusion is non-zero, solute particles can move away from the low velocity region near the solid surface into the main flow stream and subsequently dispersion again becomes asymptotically Fickian. Now dispersion is the result of an interaction between convection and diffusion and it results in a weak nonlinear dependence of dispersion on Peclet number. The dispersion coefficients predicted by particle tracking through the network are in excellent agreement with the literature experimental data. We conclude that the essential phenomena giving rise to hydrodynamic dispersion observed in porous media are (i) stream splitting of the solute front at every pore, thus causing independence of particle velocities purely by convection, (ii) a velocity gradient within throats and (iii) diffusion. Taylor's dispersion in a capillary tube accounts for only the second and third of these phenomena, yielding a quadratic dependence of dispersion on Peclet number. Plug flow in the bonds of a physically representative network accounts for the only the first and third phenomena, resulting in a linear dependence of dispersion upon Peclet number.Petroleum and Geosystems Engineerin

Influence of the disorder on tracer dispersion in a flow channel

Tracer dispersion is studied experimentally in periodic or disordered arrays of beads in a capillary tube. Dispersion is measured from light absorption variations near the outlet following a steplike injection of dye at the inlet. Visualizations using dye and pure glycerol are also performed in similar geometries. Taylor dispersion is dominant both in an empty tube and for a periodic array of beads: the dispersivity $l\_d$ increases with the P\'eclet number $Pe$ respectively as $Pe$ and $Pe^{0.82}$ and is larger by a factor of 8 in the second case. In a disordered packing of smaller beads (1/3 of the tube diameter) geometrical dispersion associated to the disorder of the flow field is dominant with a constant value of $l\_d$ reached at high P\'eclet numbers. The minimum dispersivity is slightly higher than in homogeneous nonconsolidated packings of small grains, likely due heterogeneities resulting from wall effects. In a disordered packing with the same beads as in the periodic configuration, $l\_d$ is up to 20 times lower than in the latter and varies as $Pe^\alpha$ with $\alpha = 0.5$ or $= 0.69$ (depending on the fluid viscosity). A simple model accounting for this latter result is suggested.Comment: available online at http://www.edpsciences.org/journal/index.cfm?edpsname=epjap&niv1=contents&niv2=archive

Upscaling of the acidizing process in heterogeneous porous media

Coupled fluid flow, reaction and transport in porous media has been the topic of research in various disciplines for the past few decades. Conventional approach assumes a homogeneous and isotropic porous media, and simplifies the nature of coupling between fluid and rock interactions. However, including the reality of the process, i.e. assuming heterogeneous and anisotropic porous media with fully coupled rock fluid interaction, can lead to more advanced understanding of the fundamental physics behind the problem and developing efficient industrial applications. In the oil and gas industry optimization of different well stimulation techniques such as matrix acidizing in order to enhance oil recovery requires an advanced understanding of fluid flow and also reaction in heterogeneous formations. This thesis is a contribution to development of more general governing equations describing the reactive flow and transport in heterogeneous formations.;The heterogeneity of the porous medium is introduced in the formulation through random permeability field that possess the characteristics of stationary stochastic process. The heterogeneity in permeability field affects the reservoir dynamics over a range of length and time scales by making pressure, concentration, diffusion and reaction coefficients stochastic random fields. Stochastic nature of these parameters helps us to be able to upscale the process while keeping the local information associated with heterogeneous nature of the porous media.;Conventional approaches to deal with this problem are homogenization and smoothing the heterogeneous properties of the formation using averaging based techniques such as up-gridding. However, these techniques do not carry the fundamental physics governing the process and cannot mimic the experimental observations such as acid front movement and instability of the reaction process. The local variations in rock and fluid properties are also ignored in these techniques which might lead to significant impacts in field scale application of acidizing as one of the major stimulation techniques.;In order to upscale the isothermal reaction process in a heterogeneous porous medium, according to the nature of the process, spectral-based small perturbation theory (Gelhar, 1993; Gelhar and Axness, 1983) is used among the various numerical and analytical upscaling techniques. The reaction is a nonlinear dissolution of an injected acid in a homogeneous liquid with constant density in a stationary mineral with constant porosity. In order to follow the acid front a moving coordinate is introduced. The upscaled governing equations are obtained with explicit macro-scale expressions for the coefficients and solved using time adaptive implicit finite difference technique. The results are compared with homogeneous models and sensitivity analysis of the upscaled equations is performed. Finally conclusions and results are discussed showing the importance of applying upscaling techniques to capture the impacts of heterogeneity on fluid dynamics