pressure–relative permeability–saturation relationships is due to the presence specification of functional relationships between capillary pressure, Pc, . goal is to illustrate the combined influence of various types of boundary conditions and. It is crucial to express the relationship between capillary pressure and . same purpose of the dynamic modeling of capillary transition zones. Capillary Pressure Petrophysical Reservoir Model. .. Highlights rock type differences. · Distinguishes potential water sands in a gross . estimate permeability from porosity/water saturation relationships. These calculations.
Because 4 to 24 hours are needed to reach equilibrium at each spin rate, most centrifuge data sets consist of eight or fewer spin rates. Ruth and Chen  recommend at least 15 spin rates for accurate evaluation of capillary pressure. The capillary pressure distribution Pc r at each spin-rate step depends on: At each spin rate, the capillary pressure at the inside face of the sample at ri is So, although the centrifuge method is faster than the porous-plate method, it is not as accurate.
Means for applying overburden stress also have been included in centrifuge designs. Baldwin and Spinler  and Spinler et al. Measurement of relative permeabilities There are many variations in methods for measuring relative permeabilities; only their general features will be discussed here.
Usually, a cylindrical porous sample is mounted in a holder similar to that shown in Fig. The cylindrical surfaces of the sample are sealed to prevent flow. The seal is accomplished in Fig. Fluids are injected and produced from the sample through ports at each end. Often, additional ports are added for pressure measurement. In addition to the sample holder, other apparatus are needed to: Inject and collect fluids Apply confining pressure Measure saturations, and so forth Some of these external features are shown in Fig.
Phase saturations can be estimated from: The change in mass of the rock sample The change of electrical conductivity The change of absorption of X-rays  or other radiation Acoustic methods  and CT scanning   are also used. To measure the change in mass, the rock sample is quickly removed from the assembly, weighed, then returned to the assembly.
Because this procedure could cause the saturation to change, in-situ techniques such as electrical conductivity or X-ray absorption have an advantage. Steady-state and unsteady-state methods for measuring relative permeabilities are discussed further in the subsections below.
Steady-state methods In steady-state methods, both phases oil and water, gas and oil, or gas and water are injected simultaneously at constant rates. Injection continues until a steady state is reached, as indicated by constant pressure drop and constant saturations. Four subcategories of the steady-state methods are introduced below. Multiple-core method In the multiple-core method, frequently called the Penn State method, a rock sample is sandwiched between two other rock samples to build a lengthened sample.
The upstream and downstream rock samples distribute flow of the multiple phases over the cross section of the rock and reduce the influence of capillary end effects  on the central rock sample. Pressure drop is measured across the central sample while two fluid phases are pumped through the sample at constant flow rates. In some early applications of this method, the saturations in the central rock sample were measured by quickly removing the sample and measuring its mass.
High-rate method In the high-rate method, two fluid phases are injected into a rock sample at high and constant flow rate. The actual magnitude of rate that is required for this method will depend on the length of the rock sample as well as its capillary pressure properties.
Measurement of capillary pressure and relative permeability -
The injection rate must be sufficient so that capillary end effects are negligible. Of the steady-state methods, the high-rate subcategory is used most frequently.
Stationary-liquid method In the stationary-liquid method, relative permeability of one highly mobile phase is measured in the presence of an essentially immobile second phase.
- Measurement of capillary pressure and relative permeability
- Journal of Petroleum Engineering
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Typically, the immobile phase is a liquid phase, while the mobile phase is usually gas. Because of the high mobility of gas, the liquid phase can be essentially immobile as long as the pressure gradient is small.
Uniform-capillary-pressure method In the uniform-capillary-pressure method, often called the Hassler  method, the capillary pressure between two flowing phases is kept uniform throughout a rock sample by keeping the pressure gradients in both phases equal. This is accomplished by incorporating porous plates or membranes at the entrance and exit faces of the porous sample not shown in Fig. The membranes allow passage of just one of the injected fluids, so the pressure drop in each flowing phase can be measured separately.
PEH:Relative Permeability and Capillary Pressure -
Although the Hassler method is rarely used, measurement methods with selective membranes are frequently encountered in the literature. Unsteady-state methods In unsteady-state methods, just one phase is injected at either a constant flow rate or a constant pressure drop.
Throughout the injection, the pressure drop and production of phases are measured. Three subcategories are described next. High-rate methods For measurements with high-rate unsteady-state methods, the injection rate must be sufficient so that capillary spreading effects and capillary end effects can be eliminated.
The injection and production data and the differential pressure data must be differentiated to obtain the relative permeabilities. These high-rate methods are used most frequently in the oil and gas industry.
They provide results for the least cost and with the least delay in time. The quality of the results has been questioned, but there is evidence in the literature that the methods can give results equivalent to those obtained with other methods. Low-rate methods The enormous increase in computing power and its availability in the last 20 years has facilitated measurement of relative permeabilities in low-rate unsteady-state tests.
These tests are preferred to high-rate tests for samples that have fines that become mobile at high rates.
The test equipment is identical to that used for high-rate methods, but numerical models are used for interpreting the production and pressure-drop data. The low-rate methods are not widely used. Centrifuge methods Relative permeabilities can be measured in centrifuge tests using the same apparatus as that described for measurements of capillary pressure Fig. Standard practice provides for measurement of the relative permeability of the lowest-mobility phase.
To obtain this relative permeability, the production of one phase as a function of drainage time must be measured. Then, differentiation of the data per the algorithm devised by Hagoort  gives the relative permeability.
Measurement of endpoint saturations Endpoint saturations are often valued more highly than capillary pressures and relative permeabilities for several reasons. First, the residual oil saturation for a waterflood defines the maximum amount of oil that can be recovered, so it is very useful for economics calculations. Irreducible water saturation is very useful for assessing the volume of oil in place in a reservoir. Furthermore, the endpoints can be measured more accurately than capillary pressure and relative permeability relationships.
As such, some discussion of methods for measuring endpoint saturation is included here. To measure residual oil saturations after a waterflood or gasflood, the apparatus of Figs. As a result, residual oil saturations are much less costly to measure than relative permeabilities and capillary pressures.
Still, care is needed to ensure proper wetting conditions for the measurements. On the other hand, most common aquifer materials such as quartz, carbonates, and sulfates are strongly water wet.
Wetting fluid water and nonwetting fluid mercury. It is the wettability of the reservoir rock that controls the distribution of oil and water and affects their movement through pore spaces. Understanding wettability in porous media is, by itself, a difficult problem.
PEH:Relative Permeability and Capillary Pressure
Controlling it to modify the behavior of reservoir rock presents a more complex problem. Numerous methodologies for studying, measuring, and altering the wettability of reservoir rocks are found in literature.
No satisfactory method exists for in situ measurement of wettability, and therefore it is necessary to estimate the wettability of reservoir rocks from laboratory measurements.
It is known that a porous material can be defined as water-wet, oil-wet, or mixed-wet. The degree to which a reservoir is one or another of these can be determined by considering the capillary pressure curve, or by characterizing it in terms of wettability indices.
Several experimental procedures have been proposed to assign quantitative wettability indexes to reservoir rock surfaces. The most recent of these proposals are those of Morrow [ 5 ], Graue et al. These indexes are designed to show a continuous variation from the preferential oil-wet to the preferential water-wet systems. They require measuring some property of the rock which is a function of surface wettability.
The quantities are measured on unaltered core material and compared with values obtained for known oil-wet and water-wet extremes on the same material. These methods are useful but are semiempirical in nature. They have the disadvantage that the measured quantities may be functions of other variables in addition to surface wettability.
Referring to Figure 2the Amott indices are defined as If the material is completely water-wet, then and. If the material is strongly oil-wet then and. For connected pathways of oil and water then both indices can be greater than zero. Capillary pressure diagram used to characterize wettability. This index is based on the ratio of the two areas representing forced imbibition in Figure 2: The range is from for a completely water-wet material to for a completely oil-wet material.
In general this index is not used very much. In this work, the wettability of the tested samples was not determined. Most common aquifer materials such as quartz, carbonates, and sulfates are strongly water-wet and since the tested samples are quartz and carbonates materials, it was therefore assumed that they are water-wet. When two immiscible fluids are in contact in the interstices of a porous medium, a discontinuity in pressure exists across the interface separating them.
The difference in pressure is called capillary pressure. The capillary pressure is dependent on the interfacial tension, pore size, and wetting angle. Capillary pressure curves directly determine the irreducible water saturation, residual oil saturation, and rock wettability and can be used to determine water oil contact point and approximate oil recovery.Relative Permeability, Petrophysics Lecture 5, Petroleum Reservoir Engineering free course
Figure 2 is a capillary pressure diagram showing how it can be used to characterize wettability and the capillary pressure is the difference in pressure as exemplified by Figure 3 where the porous medium can be described by a capillary tube where a clear interface exists between the immiscible fluids. Water flood performance is also significantly affected by the capillary pressure of the rock [ 9 ]. Fluid interface in a tapered capillary tube.
By definition, the capillary pressure is the nonwetting fluid pressure minus the wetting fluid pressure: The capillary pressure can be calculated by the Laplace equation: The capillary pressure equation can be expressed in terms of the surface and interfacial tension by where is interfacial tension between the two fluids and and are principle radii of curvature and is the contact angle.
Capillary pressure data are not only important for obtaining reservoir rock properties such as pore size distribution, permeability, and water saturation profile within the oil reservoir but also provide important information for water flooding designs and reservoir simulation studies. Capillary pressure is typically measured in the laboratory by mercury injection, porous plate, or centrifugation techniques. The porous plate method PP has been used for years in acquiring reliable capillary pressure data representative of reservoir rock fluid properties.
In recent years, the method is also found to be reliable and subject to less experimental errors and analysis when used for electrical resistivity RI measurements as well. A major problem has been the long time scales required for achieving reliable data.