Pores pressure differences

Figure 10.2 Mobilisation criterion for an NAPL droplet trapped in a pore. Pressure difference pi-pt must exceed the difference of capillary pressure p c — pc. 70w = interfacial tension NAPL/water, n, = radius of pore body, rn = radius of the pore neck.

The effect of decreased normal stress on a shallow imbedded crack makes it more likely to mp-ture to the surface above a critical depth as shown by Rudnicki and Wu (1995). The Coulomb friction condition is introduced such that slip occurs when Icr l = kapparent coefficient of friction, including effects of pore-pressure differences in the fault zone. The right-hand side of integral equation above is modified such that... 

Finally, it is worth remembering the sequence of events which occur during hydrocarbon accumulation. Initially, the pores in the structure are filled with water. As oil migrates into the structure, it displaces water downwards, and starts with the larger pore throats where lower pressures are required to curve the oil-water interface sufficiently for oil to enter the pore throats. As the process of accumulation continues the pressure difference between the oil and water phases increases above the free water level because of the density difference between the two fluids. As this happens the narrower pore throats begin to fill with oil and the smallest pore throats are the last to be filled. 

To prevent this flow, the pressure on the hotter side must be larger than the pressure on the colder side. The required pressure difference depends on the nature of the gas, its mean pressure and absolute temperature, the relation between its density and the pore size, and the temperature difference. However, it does not depend on the thickness of the plate. 

Besides shear-induced phase transitions, Uquid-gas equilibria in confined phases have been extensively studied in recent years, both experimentally [149-155] and theoretically [156-163]. For example, using a volumetric technique, Thommes et al. [149,150] have measured the excess coverage T of SF in controlled pore glasses (CPG) as a function of T along subcritical isochoric paths in bulk SF. The experimental apparatus, fully described in Ref. 149, consists of a reference cell filled with pure SF and a sorption cell containing the adsorbent in thermodynamic equilibrium with bulk SF gas at a given initial temperature T,- of the fluid in both cells. The pressure P in the reference cell and the pressure difference AP between sorption and reference cell are measured. The density of (pure) SF at T, is calculated from P via an equation of state. 

Other attempts to develop a pore pressure evaluation method from different drilling equations were made by Combs, by Bourgoyne and Young and by Bellotti and Giacca [101]. The three models follow the same general approach a drilling equation is developed assuming that all variables are independent. The ROP is then normalized to eliminate the effect of each variable but the pore pressure. These equations attempt to take more variables into account. 

Drilling break. A relatively sudden increase in the instantaneous drilling rate is called the drilling break. The drilling break may occur due to a decrease in the difference between the borehole pressure and formation pore pressure. When a drilling break is observed, the pumps should be stopped and the well watched for flow at the mud line. If the well does not flow, it means that the overbalance is not lost or simply that a softer formation has been encountered. 

Measures to reduce radon concentrations have been studied in an old house in which the radon decay-product concentration initially exceeded 0.3 Working Level (WL). Some of the measures were only partially successful. Installation of a concrete floor, designed to prevent ingress of radon in soil gas, reduced the radon decay-product concentration below 0.1 WL, but radon continued to enter the house through pores in an internal wall of primitive construction that descended to the foundations. Radon flow was driven by the small pressure difference between indoor air and soil gas. An under-floor suction system effected a satisfactory remedy and maintained the concentration of radon decay products below 0.03 WL. 

Compare Knudsen flows through a truncated cone of radii d0 and d1 with that through a uniform pore of the same length and volume and pressure difference. 

The linear relation between the rate of flow and the pressure difference leads one to suppose that the flow was streamline, as discussed in Volume 1, Chapter 3. This would be expected because the Reynolds number for the flow through the pore spaces in a granular material is low, since both the velocity of the fluid and the width of the channels are normally small. The resistance to flow then arises mainly from viscous drag. Equation 4.1 can then be expressed as ... 

Ruth et alS4-7) have made measurements on the flow in a filter cake and have concluded that the resistance is somewhat greater than that indicated by equation 7.1. It was assumed that part of the pore space is rendered ineffective for the flow of filtrate because of the adsorption of ions on the surface of the particles. This is not borne out by Grace18 or by Hoffing and Lockhart(9) who determined the relation between flowrate and pressure difference, both by means of permeability tests on a fixed bed and by filtration tests using suspensions of quartz and diatomaceous earth. 

Experimental work on the flow of the liquid under streamline conditions 0) has shown that the flowrate is directly proportional to the pressure difference. It is the resistance of the cloth plus initial layers of deposited particles that is important since the latter, not only form the true medium, but also tend to block the pores of the cloth thus increasing... 

One issue wifh fhis mefhod is fhaf, for larger pores shielded by smaller ones, the corresponding pressure necessary for the liquid to intrude them corresponds to the entry pressure for fhe smaller pores thus, the volume for the larger pores is incorrectly attributed to smaller ones. In addition, the assumption that the contact angle of fhe nonwetting liquid is the same on all solid surfaces is nof completely correct because diffusion layers with different treatments have pores with different wetting properties. For example, two pores of fhe same size may have different PTFE content and the entry pressure necessary for fhe liquid to penetrate them will be different for each pore, thus affecting the overall results and the calculated capillary pressures [196]. 

Nguyen et al. [205] designed a volume displacement technique that was used to measure the capillary pressures for both hydrophobic and hydrophilic materials. One requirement for this method is that the sample material must have enough pore volume to be able to measure the respective displaced volume. Basically, while the sample is filled wifh water and then drained, the volume of water displaced is recorded. In order for the water to be drained from fhe material, it is vital to keep the liquid pressure higher than the gas pressure (i.e., pressure difference is key). Once the sample is saturated, the liquid pressure can be reduced slightly in order for the water to drain. From these tests, plots of capillary pressure versus water saturation corresponding to both imbibitions and drainages can be determined. A similar method was presented by Koido, Furusawa, and Moriyama [206], except they studied only the liquid water imbibition with different diffusion layers. 

The Young-Laplace equation gives the equilibrium pressure difference (mechanical equilibrium) at the menisci between liquid water in membrane pores and vapor in the adjacent phase ... 

This method in principle can also be used to estimate the average pore size of the membrane when there are no defects in the membrane. First, gas flows through the membrane in the dry state. Typically, the gas flow rate is a linear function of the applied pressure difference. Then the membrane is saturated with the test liquid and the gas is forced to flow through the wet... 

According to the Washburn equation (10.23) a capillary of sufficiently small radius will require more than one atmosphere of pressure differential in order for a nonwetting liquid to enter the capillary. In fact, a capillary with a radius of 18 A (18 x 10 ° m) would require nearly 60 000 pounds per square inch of pressure before mercury would enter-so great is the capillary depression. The method of mercury porosimetry requires evacuation of the sample and subsequent pressurization to force mercury into the pores. Since the pressure difference across the mercury interface is then the applied pressure, equation (10.23) reduces to... 

Adsorption hysteresis is often associated with porous solids, so we must examine porosity for an understanding of the origin of this effect. As a first approximation, we may imagine a pore to be a cylindrical capillary of radius r. As just noted, r will be very small. The surface of any liquid condensed in this capillary will be described by a radius of curvature related to r. According to the Laplace equation (Equation (6.29)), the pressure difference across a curved interface increases as the radius of curvature decreases. This means that vapor will condense... 

FIG. 12.6 Electroosmotic flow through a pore. If the fluid flow occurs as a result of applied pressure difference along the length of the pore, the resulting potential difference is known as the streaming potential. (Adapted with permission from Probstein 1994.)... 

The rates of filtration of microbes (particles) at a constant pressure difference decrease with time due to an accumulation of filtered microbes on the surface and inside the pores of the membrane. Hence, in order to maintain a constant filtration rate, the pressure difference across the membrane should be increased with time due to the increasing filtration resistance. Such data as are required for practical operation can be obtained with fluids containing microbes with the use of real filter units. 

Synthesis of MCM-41 with Additives. The hydrothermal crystallization procedure as described earlier [10] was modified by adding additional salts like tetraalkylammonium (TAA+) bromide or alkali bromides to the synthesis gel [11]. Sodium silicate solution ( 14% NaOH, 27% Si02) was used as the silicon source. Cetyltrimethylammonium (CTA) bromide was used as the surfactant (Cl6). Other surfactants like octadecylltrimethylammonium (ODA) bromide (C,8), myristyltrimethylammonium (MTA) bromide (C,4) were also used to get MCM-41 structures with different pore diameter. Different tetralkylammonium or alkali halide salts were dissolved in little water and added to the gel before addition of the silica source. The final gel mixture was stirred for 2 h at room temperature and then transferred into polypropylene bottles and statically heated at 100°C for 4 days under autogeneous pressure. The final solid material obtained was washed with plenty of water, dried and calcined (heating rate l°C/min) at 560°C for 6 h. 

Equation 1 can be used to determine the pore diameter of an MCM-41 sample which exhibits capillary condensation at a certain relative pressure, or to determine the capillary condensation pressure for an MCM-41 sample of a certain pore diameter. To construct model adsorption isotherms for MCM-41, one also needs a description of the monolayer-multilayer formation on the pore walls. This description can be based on the experimental finding that the statistical film thickness in MCM-41 pores of different sizes (especially above 3 nm) is relatively constant for pressures sufficiently lower from those of the capillaiy condensation and can be adequately approximated by the t-curve for a suitable reference silica [29-31], for instance that reported in Ref. 35. In these studies [29-31], the statistical film thickness in MCM-41 pores, tMcM-4i, was calculated according to the following equation [29] ... 

As already discussed, DFT can be used to predict the capillary condensation and capillary evaporation pressures for pores with homogeneous surface and well-defined geometry. To generate model adsorption isotherms for heterogeneous pores, it is convenient to employ hybrid models based on both DFT data for homogeneous pores and experimental data for flat heterogeneous surfaces [6-9]. Such model adsorption isotherms can be used to calculate PSDs in mesopore [6-9] and micropore [9] ranges. This approach is particularly useful for pores of diameter below 2-3 nm (micropores and narrow mesopores), where an assumption about the common t-curve for pores of different sizes is less accurate, which in turn makes the methods based on such an assumption (even properly calibrated ones) less reliable [18],... 

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