Relative permeability/saturation

Bau and Torrance [128] use a different relative permeability-saturation relation and arrive at a slightly different relation. Jones et al. [129] use a similar treatment and find a relationship for qc, that gives values lower than those predicted by Eq. 9.130 by a factor of approximately 2. It should be noted that these predictions of qCI are estimations and that the effects of wettability, solid matrix structure (all of these studies consider spherical particles only), and surface tension (all of which influence the phase distributions) are included only through the... 

Current work on BMTl was reported. From the results, one can see that the dual relative permeability - saturation model used in the desaturation example might be useful for larger scale tests, such as the BMTl application, so that desaturation near the heater can be reproduced correctly by the TH model. 

Wettability Effects on Relative Permeability/Saturation Relationships... 

Mafiiematical models of DNAPL subsurface movement require capillary pressure/saturation and relative permeability/saturation constitutive relationships for DNAPL/water/soil systems. Wettability alterations may have a considerable effect on ciqiillary pressure/saturation relationships as discussed in the previous section. They may also have an intact on relative permeability/saturation relationships yet this has received less attention in file literature in large part due to file experimental difficulties associated with measuring these relationships. 

Figure 8. Two-dimensional infiltration photo and simulation result of PCE saturation at elapsed time = 60 minutes using Brooks-Corey/Burdine capillary pressure/relative permeability/saturation constitutive relationships.

Assuming that the solvent relative-permeability/saturation relationship is the same as the oil relative-permeability/saturation relationship,... 

The above experiment was conducted for a single fluid only. In hydrocarbon reservoirs there is always connate water present, and commonly two fluids are competing for the same pore space (e.g. water and oil in water drive). The permeability of one of the fluids is then described by its relative permeability (k ), which is a function of the saturation of the fluid. Relative permeabilities are measured in the laboratory on reservoir rock samples using reservoir fluids. The following diagram shows an example of a relative permeability curve for oil and water. For example, at a given water saturation (SJ, the permeability... 

The relative permeability to phase i, kri(s ), is taken to be a function of fluid saturation si which is the fraction of the pore space occupied by phase i it is supposed that any associated spatial variations are largely taken into account through the permeability. For two-phase flow, fluid saturations are related by... 

In this section, we describe our experimental and analysis methods to determine spatially dependent porosity and saturation distributions, permeability functions and saturation-dependent multiphase flow properties the relative permeability and capillary pressure functions. 

Relative permeability and capillary pressure functions, collectively called multiphase flow functions, are required to describe the flow of two or more fluid phases through permeable media. These functions primarily depend on fluid saturation, although they also depend on the direction of saturation change, and in the case of relative permeabilities, the capillary number (or ratio of capillary forces to viscous forces). Dynamic experiments are used to determine these properties [32]. 

Conventionally, the sample is initially saturated with one fluid phase, perhaps including the other phase at the irreducible saturation. The second fluid phase is injected at a constant flow rate. The pressure drop and cumulative production are measured. A relatively high flow velocity is used to try to negate capillary pressure effects, so as to simplify the associated estimation problem. However, as relative permeability functions depend on capillary number, these functions should be determined under the conditions characteristic of reservoir or aquifer conditions [33]. Under these conditions, capillary pressure effects are important, and should be included within the mathematical model of the experiment used to obtain property estimates. 

While capillary pressure can be determined independently through experiments implementing a series of equilibrium states, this can be very time consuming, particularly if the entire capillary pressure function is to be reconciled. Furthermore, as there can be difficulties in re-establishing identical states of initial saturation, it is most desirable to determine capillary pressure and relative permeability functions simultaneously, from the same experiment. 

We present a general approach for estimating relative permeability and capillary pressure functions from displacement experiments. The accuracy with which these functions are estimated will depend on the information content of the measurements, and hence on the experimental design. We determine measures of the accuracy with which the functions are estimated, and use these measures to evaluate different experimental designs. In addition to data measured during conventional displacement experiments, we show that the use of multiple injection rates and saturation distributions measured with MRI can substantially increase the accuracy of estimates of multiphase flow functions. 

The solution of these dynamic nonlinear differential equations is considerably more complex than the previous systems considered. In particular, stable solution methods are based on physically realistic multiphase flow functions that have the following properties relative permeability functions are non-negative, monotoni-cally increasing with their respective saturation, and are zero at vanishing saturations, and capillary pressure is monotonically increasing with respect to the saturation of the non-wetting phase. It is necessary that any iterative scheme for estimating the multiphase flow functions retain these characteristics at each step. 

More recently well treatments that do not interrupt normal water injection operations have increased in frequency. Addition of surfactant to the injection water (144,146) can displace the oil remaining near the production well. The lower oil saturation results in an increase in the water relative permeability (145). Consequently a greater water injection rate may be maintained at a given injection pressure or a lower injection pressure. Thus smaller and cheaper injection pumps may be used to maintain a given injection rate. While the concentration of surfactant in the injection water is relatively high, the total amount of surfactant used is not great since it is necessary only to displace the oil from a 6-10 foot radius around the injection well. 

Migration of free-phase NAPLs in the subsurface is governed by numerous properties including density, viscosity, surface tension, interfacial tension, immisci-bility, capillary pressure, wettability, saturation, residual saturation, relative permeability, solubility, and volatilization. The two most important factors that control their flow behavior are density and viscosity. 

Relative permeability is the reduction of mobility between more than one fluid flowing through a porous media, and is the ratio of the effective permeability of a fluid at a fixed saturation to the intrinsic permeability. Relative permeability varies from zero to 1 and can be represented as a function of saturation (Figure 5.8). Neither water nor oil is effectively mobile until the ST is in the range of 20 to 30% or 5 to 10%, respectively, and, even then, the relative permeability of the lesser component is approximately 2%. Oil accumulation below this range is for all practical purposes immobile (and thus not recoverable). Where the curves cross (i.e., at an Sm of 56% and 1 - Sm of 44%), the relative permeability is the same for both fluids. With increasing saturation, water flows more easily relative to oil. As 1 - SI0 approaches 10%, the oil becomes immobile, allowing only water to flow. 

Graphs of relative permeability are generally similar in pattern to that shown in Figure 5.10. As shown, some residual water remains in the pore spaces, but water does not begin to flow until its water saturation reaches 20% or greater. Water at the low saturation is interstitial or pore water, which preferentially wets the material and fills the finer pores. As water saturation increases from 5 to 20%, hydrocarbon saturation decreases from 95 to 80% where, to this point, the formation permits only hydrocarbon to flow, not water. Where the curves cross (at a saturation... 

If the values for all the factors are known and expressed in common units, this equation will provide a reasonable estimate of flow from recovery wells. The difficulty of actual application of this equation is the determination of the water-oil saturation in the aquifer matrix (thus, the relative permeability). The equation assumes a steady-state flow setting. In a dynamic situation where the oil reserves are being depleted and water-oil mixtures are variable, it is almost impossible (from an economic point of view) to use these equations precisely. However, inclusion of some assumptions based on results of the initial site investigation can often be of assistance in initial spacing of recovery wells and estimating recovery rates. 

Baildown tests have been used for decades during the initial or preliminary phases of LNAPL recovery system design to determine adequate locations for recovery wells and to evaluate recovery rates. Baildown tests involve the rapid removal of fluids from a well with subsequent monitoring of fluid levels, both the LNAPL-water (or oil-water) interface and LNAPL-air (or oil-air) interface, in the well with time. Hydrocarbon saturation is typically less than 1, and commonly below 0.5, due to the presence of other phases in the formation (i.e., air and water). Since the relative permeability decreases as hydrocarbon saturation decreases, the effective conductivity and mobility of the LNAPL is much less than that of water, regardless of the effects induced by increased viscosity and decreased density of the LNAPL. 

Relative permeability is defined as the ratio between the permeability for a phase at a given saturation level to the total (or single-phase) permeability of the studied material. This parameter is important when the two-phase flow inside a diffusion layer is investigated. Darcy s law (Equation 4.4) can be extended to two-phase flow in porous media [213] ... 

Nguyen et al. [205] used a technique in which a constant mass flow rate of water-saturated air was forced through a water-saturated sample. It was explained that the shear force of the gas flow dragged water out of the sample. In addition, the saturated air was needed in order to prevent water loss from the sample by evaporation. Once a steady state was achieved, the pressure difference between the inlet and outlet of the apparatus was recorded. After the tests were completed, the sample was weighed to obtain its water content. Thus, the relative permeability was calculated from the pressure drop, the water content in the sample, and the mass flow rate [205]. 

The overall gain of the multiphase mixture model approach above is that the two-phase flow is still considered, but the simulations have only to solve pseudo-one-phase equations. Problems can arise if the equations are not averaged correctly. Also, the pseudo-one-phase treatment may not allow for pore-size distribution and mixed wettability effects to be considered. Furthermore, the multiphase mixture model predicts much lower saturations than those of Natarajan and Nguyen - and Weber and Newman even though the limiting current densities are comparable. However, without good experimental data on relative permeabilities and the like, one cannot say which approach is more valid. 

ORR rate constant as defined by eq 61, 1/s ORR rate constant in Figure 11, cm/s thermal conductivity of phase k, J/cm K relative hydraulic permeability saturated hydraulic permeability, cm electrokinetic permeability, cm catalyst layer thickness, cm parameter in the polarization equation (eq 20) loading of platinum, g/cm molecular weight of species i, g/mol symbol for the chemical formula of species i in phase k having charge Zi... 

The relative permeability is difficult to quantify. It usually is estimated on the basis of laboratory experiments, as a function of relative saturation. In three-phase NAPL-water-air systems, each relative permeability is dependent on the relative saturation of each of the phases. Modeling based on empirical considerations can be employed. For example. Blunt (2000) discusses a model to estimate three-phase relative permeability, based on saturation-weighted interpolation of two-phase relative permeabilities. The model accounts for the trapping of the NAPL and air (gas)... 

The primary drainage and steady-state flow simulations, described in the earlier section and typically devised in the petroleum/reservoir engineering applications,53 54 58,60 were deployed to evaluate the capillary pressure and relative permeability relations as functions of liquid water saturation, respectively. 

The steady-state flow numerical experiment was primarily designed to evaluate the phasic relative permeability relations. The numerical experiment is devised within the two-phase lattice Boltzmann modeling framework for the reconstructed CL microstructure, generated using the stochastic reconstruction technique described earlier. Briefly, in the steady-state flow experiment two immiscible fluids are allowed to flow simultaneously until equilibrium is attained and the corresponding saturations, fluid flow rates and pressure gradients can be directly measured and correlated using Darcy s law, defined below. 

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