Single porosity-permeability

Since a permeability coefficient of 2 x 10 cm/sec was used in our theoretical calculation for case B, this is equivalent to a single porosity value for all calculations with Equation 6. For any given set of ethanol concentration at the boundary, the appropriate ethanol concentration distance profile was calculated from the experimentally determined profile for the pure ethanol/water system and taken as that of the system. For any given set of ethanol concentrations at the boundaries the appropriate portion of the experimentally determined ethanol concentration distance profile, was calculated from the experimentally determined profile for the pure ethanol/water system. For any given ethanol concentration distance profile, the corresponding solubility profile of 3-estradiol was obtained by interpolation of experimental solubility data (Figure 2). 

Figure 3 shows the confidence limits of the predicted bottom-hole flowing pressures using single-porosity estimate at 0.20 percent measurement error. The confidence interval is about 186 psi which is practically acceptable. The true pressures are all contained within the confidence interval also. As can be seen in Figure 2, the confidence regions for joint estimation of porosity and permeability at 0.20 percent measurement error indicate that even at the lowest confidence level of 95 percent, the confidence interval for porosity is very wide. The orientation and shape of the ellipses show that porosity is much less well determined than permeability. It seems, therefore, that porosity estimation is very sensitive to measurement error. Also, porosity estimates are not reliable when joint estimation of porosity and other parameter(s) is made or when there is a significant error in the matched performance data. 

The two most commonly measured rock properties are porosity, (f), and absolute (single-phase) permeability, k. The porosity (or voidage) is the fraction of the bulk volume of the porous sample that is occupied by pore or void spaces. For different types of porous media, the porosity can vary from nearly zero (e.g. certain volcanic rocks) to almost unity (e.g. insulators). The porosity can be measured using a variety of methods, which are described by Dullien (1979). 

The end modeling results are coupled immiscible flow equation systems, containing twice as many input parameters as the more rational single-porosity model would have two sets of relative permeability curves, two sets of capillaiy pressure curves, and so on. Consequently, such hopelessly ill-defined approaches, given the dearth of real-world data, not to mention errors likely to be found in laboratory measurement, may never see complete validation. Simpler flow models for periodic shales and fractures, such as those introduced in Chapter 5, shed greater physical insight. 

For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation ... 

If the amount of fluid within a fully saturated permeable medium is known as a function of position, the spatially resolved porosity distribution can be determined. If the medium is saturated with two fluids, and the signal from one can be distinguished, the fluid saturation can be determined. In this section, we will develop a method to determine the amount of a single observed fluid using MRI, and demonstrate the determination of porosity. In Section 4.1.4.3, we will demonstrate the determination of saturation distributions for use in estimating multiphase flow functions. 

All symbols are defined at the end of the paper. Equation 10 defines the pure water permeability constant A for the membrane which is a measure of its overall porosity eq 12 defines the solute transport parameter D /K6 for the membrane, which is also a measure of the average pore size on the membrane surface on a relative scale. The Important feature of the above set of equations is that neither any one equation in the set of equations 10 to 13, nor any part of this set of equations is adequate representation of reverse osmosis transport the latter is governed simultaneously by the entire set of eq 10 to 13. Further, under steady state operating conditions, a single set of experimental data on (PWP), (PR), and f enables one to calculate the quantities A, Xy 2> point... 

A liquid chromatographic column packed with 5.00 fim diameter solid support particles and having a porosity e = 0.400 is found to have a flow resistance parameter 750. Calculate the specific permeability K0. Assume that the flowrate/pressure drop relationship is identical to that of a bundle of identical parallel capillaries whose axes are spaced (in a square cross-sectional array) a distance of 5.00 /im from one another. What is the single capillary diameter of the hypothetical bundle ... 

Various direct and indirect methods are generally used to determine the permeability of a sedimentary basin. The direct methods include laboratory measurements on core samples wire-line formation tests, single-well tests and interference tests. The data from the different types of well test and interference test can be analysed and interpreted by well-established procedures (Da Prat, 1990 Earlougher, 1977 Kruseman et al., 1990 Matthews and Russel, 1967). The conventional, indirect methods are theoretical, semi-empirical and empirical procedures which are based on the relation between permeability, grainsize characteristics and porosity (e.g. the Kozeny-Carman method, Domenico and Schwartz, 1990 Van Baaren method. Van Baaren, 1979). The laboratory methods and the conventional indirect methods provide permeability values which are representative of only a very small portion of the subsurface (cm-scale). The single-well test and interference test provide information representative of a larger volume of the subsurface (m - km scale). 

Similar to studies on the porosity of capsule membranes using series of tracer molecules of different size, one may use molecules of similar size which differ in a single other parameter like polarity, shape, flexibility, etc., to yield additional information about the membrane structure. As all these observations are performed in the state of equilibrium distribution, there are no restrictions in terms of the overall duration of the measurement. Overall, systematic studies on the membrane permeability could elucidate a variety of details on the capsule structure and the possible release properties. 

Single-phase fluid flow in porous media is a well-studied case in the literature. It is important not only for its application, but the characterization of the porous medium itself is also dependent on the study of a single-phase flow. The parameters normally needed are porosity, areal porosity, tortuosity, and permeability. For flow of a constant viscosity Newtonian fluid in a rigid isotropic porous medium, the volume averaged equations can be reduced to the following the continuity equation,... 

A set of hypothetical data shown in Table I was assumed true for an undercompacted, stress-sensitive reservoir whose pressures were to be matched. The match period in all runs was 200 days, at which time a pseudo-steady state condition in the reservoir would have been attained. A single producing well was located at the center of the reservoir and was allowed to produce for 200 days. The drawdown data for the 200 days were then matched. Two sets of simulated drawdown data were used. One set was obtained assuming there was no measurement error in the data and another set was obtained with 0.20 percent measurement error. Reservoir permeability and porosity were separately and jointly estimated, and pressure prediction for an additional 60 days was obtained. The confidence limits of the point estimates and the predicted performance (pressure) data at 95 percent confidence level were then calculated. The results are presented as follows ... 

As shown in Table II, the point estimates for permeability are 19.996 md and 18.998 md from single-estimation of permeability and multi-estimation of permeability and porosity, respectively, when error-free drawdown pressure data were matched. The confidence interval from the single estimation is 0.003 md, less than one-tenth percent of the true value of 20 md, implying that the point estimate is very reliable. From the joint estimation, the confidence interval is 2.101 md, about 10.5 percent of the true value, implying much less reliable estimate than the single estimation. Table III shows the permeability point estimates from single and joint estimations when the matched performance data contained 0.20 percent measurement error. The confidence intervals indicate that more reliable estimate was obtained from the single estimation than from the joint estimation, but less reliable estimates were obtained in this case than from the error-free matched performance data case. 

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