1] This paper is a follow up on Bernabé et al.'s (2010) study of the effect of pore connectivity and pore size heterogeneity on permeability. In the permeability model initially proposed, pore connectivity was characterized by means of the average coordination number z, a parameter rarely included in experimental investigations of the transport properties and microstructure of porous rocks. Obviously, lack of information on z makes it difficult to apply the model. One way around this problem is to eliminate z from the model by introducing the resistivity formation factor, an approach previously used by Paterson (1983), Walsh and Brace (1984), and many others. Using the network simulation approach of Bernabé et al. (2010), we extended the model to include the electrical formation factor. The new joint permeability-formation factor model consists of three equations, the first two expressing the relation of permeability and formation factor to z and the last one, obtained by elimination of z, linking permeability and formation with each other. We satisfactorily tested the model by comparison with published experimental data on a variety of granular materials and rocks. Furthermore, we show that, although our model does not explicitly include porosity, it is consistent with Archie's law. Citation: Bernabé, Y., M. Zamora, M. Li, A. Maineult, and Y. B. Tang (2011), Pore connectivity, permeability, and electrical formation factor: A new model and comparison to experimental data,

[1] Laboratory experiments were conducted to determine the effective pressure law for permeability of tight sandstone rocks from the E-bei gas reservoir, China. The permeability k of five core samples was measured while cycling the confining pressure p c and fluid pressure p f . The permeability data were analyzed using the response-surface method, a statistical model-building approach yielding a representation of k in (p c , p f ) space that can be used to determine the effective pressure law, i.e., p eff = p c À kp f . The results show that the coefficient k of the effective pressure law for permeability varies with confining pressure and fluid pressure as well as with the loading or unloading cycles (i.e., hysteresis effect). Moreover, k took very small values in some of the samples, even possibly lower than the value of porosity, in contradiction with a well-accepted theoretical model. We also reanalyzed a previously published permeability data set on fissured crystalline rocks and found again that the k varies with p c but did not observe k values lower than 0.4, a value much larger than porosity. Analysis of the dependence of permeability on effective pressure suggests that the occurrence of low k values may be linked to the high-pressure sensitivity of E-bei sandstones.

Summary The paper adopts the view that the liquid droplets entrained in gas wells tend to be flat and deduces new formulas for the continuous removal of liquids from gas wells. The results calculated from the formulas are smaller than those of Turner et al. However, the predicted results are in accord with the practical production performance of China's gas wells with liquids. The paper also gives the simple forms of these formulas and shows the load-up and near load-up production performance as well as unloaded gas wells through the wellhead, producing performance figures. Introduction Gas produced from a reservoir will, in many cases, have liquidphase material with it, which can accumulate in the wellbore over time when transporting energy is low enough in a low-pressure reservoir. The liquids accumulated in the wellbore will cause additional hydrostatic pressure on the reservoir, resulting in a continued reduction of available transportation energy and affecting the production capacity. In some cases, it even causes gas wells to die. It is essential to investigate the cause for gas-well load-up and to determine the minimum gas flow velocity and rate to transport liquids to surface. Turner et al.1 compared two models - the continuous-film and the entrained-drop-movement models. He proved that the entrained- drop-movement model was more adequate for explaining gas-well load-up and used it to further investigate this. Assuming the liquid droplets were spherical, Turner et al.1 deduced the formulas used to calculate the minimum gas-flow velocity and rate to remove liquid droplets with +20% adjustment. The minimum gasflow velocity and rate are known as the terminal velocity and the critical rate. Turner et al.1 also suggested that, in most instances, wellhead conditions controlled the onset of liquid load-up and the gas/liquid ratio in the range of 1370 to 178 571 m3 /m3 and did not influence the terminal velocity and critical rate. There are many gas wells producing at rates less than the minimum flow rate formula, and these wells are still in a good production state in China. To obtain a relatively accurate critical producing rate, the engineers in China's gas fields adjusted the critical rate, reducing by two-thirds. Steve2 found that the unadjusted liquid-droplet model tended to offer a better match to the field data. The model still cannot obtain a suitable critical rate to explain the phenomenon of some gas wells that should be loaded up with his model but are not. This paper presents formulas for predicting the terminal velocity and critical rate after analyzing the shape of a liquid drop entrained in a high-velocity gas stream. With the present model, the calculated results are in conformity with the practical daily production record of gas wells. For easier application, this paper puts forward simple forms of the deduced formulas, analyzing different factors that affect the removal of liquids from gas wells. Steve3 analyzed the wellbore behavior of load-up, near load-up, and unloaded gas wells. This paper shows the production performance of these with wellhead production-rate figures through which engineers can have a better understanding of their effect on a gas well's production performance. Terminal-Velocity and Critical-Rate Theory Shape of Entrained Drop Movement. Hinze4 showed that liquid drops moving relative to a gas are subjected to forces that try to shatter the drops, while the surface tension of the liquid acts to hold the drop together. He determined that it was the antagonism of two pressures - velocity and surface tension. The ratio of these two pressures is the Weber number, NWe = v2 ?gd/s. If the Weber number exceeded a critical value, the liquid drop would be shattered. For free-falling drops, the value of the critical Weber number was on the order of 20 to 30. Turner et al.1 deduced the terminal-velocity and critical production-rate formulas with the larger Weber number value (30). These formulas were put forward without taking the deformation of liquid droplets into consideration. As a liquid drop is entrained in a high-velocity gas stream, a pressure difference exists between the fore and aft portions of the drop. The drop is deformed under the applied force, and its shape changes from spherical to that of a convex bean (called a flat shape here) with unequal sides (Fig. 1). Spherical liquid drops have a smaller efficient area (held by gas) and need a higher terminal velocity and critical rate to lift them to the surface. However, the flat ones have a more efficient area and are easier to be carried to the wellhead. As mentioned previously, the critical rate determined from Turner et al.1 is by far higher than that of the field data in China. This also suggests that entrained liquid droplets can be flat. Formulas. A rigorous determination of the terminal velocity of a deformed liquid drop presents many difficulties. Nevertheless, the velocity can be estimated under the hypothesis that the drop tends to be flat, as shown in Fig. 2. When the liquid drop remains motionless relative to the wellbore (i.e., the velocity of the liquid drop relative to gas is v and equals gas velocity vg), it is clear that vg is the terminal velocity vt. With the condition that the gravity of a liquid drop equals the buoyancy plus the drag force (see Fig. 2), we haveEquation 1 When the liquid drop changes from a spherical shape to a flat one, its projected area differs. The area s is determined in the Appendix. Substituting Eq. A-9 into Eq. 1, we get the falling velocity v of the drop relative to a high-velocity gas. Because the velocity equals the terminal velocity on the balance condition, we now have the following form.Equation 2