Frontal Advance Theory for Flow in Heterogeneous Porous Media

Abstract

SPE Members Abstract This paper concerns the application of frontal advance theory to displacement processes in heterogeneous porous media. The assumptions under which a generalized frontal advance equation can be used to describe a flow process in a heterogeneous porous medium are examined, Material balance equations are derived based on these assumptions, and the theory is illustrated by application to the Dykstra-Parsons model of flow in noncommunicating layers. We show that the resulting formulation can be used to forecast performance and to solve the inverse problem: the determination of a permeability distribution from displacement data. This approach can also be used to determine pseudorelative permeability curves appropriate to a one-dimensional description of a layered system. We also consider the use of pseudorelative permeabilities in simulation studies to represent the effects of reservoir heterogeneities with length scales smaller than a grid block. The dependence of process performance and pseudorelative permeabilities on scale is described. Results show that frontal advance theory may be applied to flow in heterogeneous porous media for processes that exhibit linear characteristics. In such a process, a given saturation travels at a constant velocity through the porous medium. Displacement performance for the two-dimensional Dykstra-Parsons model is exactly duplicated using one-dimensional frontal advance theory for unit mobility ratio displacements. One-dimensional theory reproduces the qualitative performance of the two-dimensional model for nonunit mobility ratios, but exact quantitative agreement is not obtained, because the characteristics are nonlinear. Pseudorelative permeabilities determined for a system of noncommunicating layers apply to any scale as long as the layer structure does not change with system length. However, for processes that are not purely convective, process performance varies with the scale of the displacement. Thus, pseudorelative permeabilities obtained from flow processes that are not purely convective are only applicable to the scale at which the displacement data was measured. Introduction Experiments to determine the flow properties of a porous medium are inevitably interpreted in terms of some model representation of the flow. For example, if a miscible displacement is performed in a laboratory core, or in a field-scale experiment, the resulting effluent composition data may be interpreted by a convection-dispersion equation to determine a dispersion coefficient. That model, with the dispersion coefficient so determined, can be used to compute displacement performance for another miscible displacement in the same porous medium at the same scale, and it may or may not be useful for calculations at different scales. If the model used reflects faithfully the physical mechanisms that dominate the displacement, then calculations at different scales will be accurate. However, more than one model can be adjusted to match displacement data at a given scale. Thus, it may not be clear whether use of parameters so determined is appropriate for displacement in larger flow systems unless measurements have been made at several scales. Another example is use of the Buckley-Leverett equations to determine relative permeability from displacement data, In this case, it is assumed that the measured fractional flow can be interpreted as the result of variations of relative permeabilities with changes in phase saturations. There is, of course, no guarantee that the relative permeability description captures all the detail of the mechanisms that influence the transport of the phases. Thus, any model of a displacement is a compromise between a detailed representation of pertinent physical mechanisms and tractability of the calculation scheme, and availability of data required to support the model. Essentially all calculations of waterflood or other oil recovery process performance are based on the use of relative permeability functions to describe the local movement of phases based on their saturations. The functions used in a particular simulation are average representations of the effects of displacement mechanisms operating at smaller scales. For example, the analyses of Buckley and Leverett, Welge and Johnson, Bossler and Naumann are routinely used to determine relative permeability functions from measurements of the fractional flow of water leaving a core and the pressure drop across the core. These analyses assume that gravity and capillary pressure effects are negligible and that the core is one dimensional, even though most cores do not have strictly uniform permeability. P. 215^

Applying Fractional-Flow Theory Under the Loss of Miscibility

“…The WL method treats the SWAG injections as a combination of the miscible waves (gas/oil) and immiscible waves (water-oil/solvent) propagating according to the given initial and injection conditions. On the other hand, previous studies by Pande et al (1987), Sorbie et al (1994), and Li and Lake (1995) showed that a linear growth of the mixing zone with time will be observed if the simulation results are consistent with the fractional-flow solutions. Therefore, a similar behavior is expected for SWAG displacements when the WL solution is accurate.…”

“…Helfferich (1981) introduced the general theory of multicomponent and multiphase flow in permeable media by incorporating the theories of multi-component chromatography and fractional flow. Pande et al (1987) showed that the 1D fractional-flow solution of immiscible displacements can be used to qualitatively reproduce the displacement performance of 2D flow with noncommunicating layers.…”

Waterflooding Performance of Communicating Stratified Reservoirs With Log-Normal Permeability Distribution

“…It is therefore clear that extending the frontal advance theory in the sense of BL displacement to stratified systems is only valid for communicating systems with complete crossflow between layers. Pande et al 12 tried to extend this theory using the DP model ͑noncommunicating͒ but found agreement only for unit mobility ratio. At other mobility ratios, the velocities of the individual saturations were not constant with time as would be expected from Eq.…”

Tracer- and Pressure-Test Analysis for Characterization of Areally Heterogeneous Reservoirs

“…Such parameters can also be used with simple models to compute displacement performance. 27 It is obvious that when the scale of heterogeneities is comparable to system dimensions, the tracer flow response strongly depends on the exact arrangement of permeability in the flow field. Thus, even when average measures (variance and correlation length scale) of permeability variation are the same, tracer flow behavior (as well as the recovery behavior from an EOR process) might be completely different.…”