Petroleum reservoir engineering practice pdf

 

    Petroleum reservoir engineering practice / Nnaemeka Ezekwe. p. cm. Includes bibliographical references and index. ISBN (hardcover: alk. Ezekwe N. Petroleum Reservoir Engineering Practice. Файл формата pdf; размером 10,20 МБ. Добавлен пользователем Silver 47 PVT and Phase Behaviour of Petroleum Reservoir Fluids. 48 Applied . The Practice of Reservoir Engineering has been written for those in the oil industry.

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    Petroleum Reservoir Engineering Practice Pdf

    Petroleum Reservoir Engineering Practice Nnaemeka Ezekwe. The complete, up-to-date, and practical guide to modern petroleum reservoir engineering: from basic to advanced topics * *Practices that work, presented صيغة الكتاب: pdf. The Complete, Up-to-Date, Practical Guide to Modern Petroleum Reservoir Engineering This is a complete, up-to-date guide to the practice of petroleum. 28 3 Basic Concepts and Definitions in Reservoir Engineering 31 Continuum .. In practice, it is necessary to repeat the stochastic simulation for different.

    Pearson Education, Inc. Nnaemeka Ezekwe covers topics ranging from basic to advanced, focuses on currently acceptable practices and modern techniques, and illuminates key concepts with realistic case histories drawn from decades of working on petroleum reservoirs worldwide. Ezekwe begins by discussing the sources and applications of basic rock and fluid properties data. Next, he shows how to predict PVT properties of reservoir fluids from correlations and equations of state, and presents core concepts and techniques of reservoir engineering. Using case histories, he illustrates practical diagnostic analysis of reservoir performance, covers essentials of transient well test analysis, and presents leading secondary and enhanced oil recovery methods.

    In Dr. During his career, Dr. Prior to his work with Sibneft, Dr. Diyashev was one of the key Schlumberger specialists to start the horizontal drilling project in Noyabrsk Western Siberia. He has authored 30 technical papers. Twice in his career Dr. He has 38 years' experience working for three global oil and gas companies.

    Petroleum Reservoir Engineering Practice - Nnaemeka Ezekwe - مكتبة علوم

    His specific contributions are in the areas of revitalizing old fields, remote and startup operations, petroleum economics, and introducing new reservoir management technologies internationally.

    His personal skills are in team development, specifically international cross cultural project teams of industry professionals. Kleinsteiber has over 24 years of petroleum engineering experience and has authored or co-authored papers dealing with production decline type curve analysis, CO2 flooding, and depletion of a rich gas condensate reservoir by nitrogen injection. Kleinsteiber has experience related to exploration well testing in the Mediterranean Ocean offshore Israel.

    He has also performed field development studies for coalbed methane reservoirs in the Bowen Basin of eastern Australia, and well test analyses for exploration wells in Hungary. Prior to joining MHA, he held various reservoir engineering positions with Amoco Production Company both in their Tulsa, Oklahoma research center and Denver regional production office.

    Kleinsteiber's last position with Amoco was Western Business Unit Technology Coordinator where he was an internal consultant to the business unit's engineering staff in the Rocky Mountain and Mid-Continent regions. Kleinsteiber and his colleagues at Amoco developed the initial plan of depletion for fields in Wyoming and Utah using compositional numerical simulation. His specific contributions were in the areas of fluid property characterization, well testing and simulation studies for various development options.

    Kleinsteiber also directs continued development of MHA's GAS3D reservoir simulator and software for production decline type curve analysis. He received a BS in petroleum engineering with highest honors from the University of Oklahoma in MHA Petroleum Consultants was incorporated in to provide a broad range of services from single-well valuations to fully integrated field studies.

    Their highly-trained professionals have assisted clients in maximizing the performance of reservoirs worldwide. The typical MHA instructor has over 30 years in the industry, and is professionally registered. Find more information at www. He has held various high-level technical and management positions. A ther- mostat controller raises the temperature of the core to a selected level, at which point the water within the core is vaporised and recovered.

    The vaporised fluids are first collected in the sample holder and then released vertically downwards through the hollow stem down- draft retort. They are subsequently condensed and measured in a calibrated receiving tube. The high temperature retort distillation method. Samples are usually destroyed in this test due to the high temperature and for this reason small-diameter samples or "plugs" small cores from the well core , are normally used.

    The calculation of the oil and water saturation is straightforward. The following parameter values are derived from the laboratory test: Chapter 3. The water and residual oil saturation are calculated as follows: The Dean-Stark Apparatus for Measuring Initial Fluids When the core to be analysed is weighed, the measurement includes the weight of rock grains, and the pore fluid. The sample is then placed in the tare apparatus to be sure that no sand grains are lost from the core sample during its analysis, which might otherwise lead to an erroneously high oil saturation!

    Toluene satisfies all of those requirements. When heat is applied to the solvent, it vaporises.

    The hot solvent vapour rises, surrounds the sample and moves up to the condensing tube, where it is cooled and condensed. The condensate collects into the calibrated tube until the fluid there reaches the spill point, where upon the solvent condensate drips back onto the sample containing the reservoir fluids. The solvent mixes with the oil in the sample and both are returned to the solvent flask below. At that point, the water vaporises, rises in the condensing tube, condenses therein and falls back into the calibrated tube.

    Because it is heavier than the solvent, it collects at the bottom of the tube, where its volume can be directly measured. When successive readings indicate no additional water recovery, the water volume is recorded for further calculations. After all the oil and water have been recovered from the sample, it is dried and weighed again. The difference between the original and final weights equals to the weight of the oil and water originally present in the sample. Because the water collected in the calibrated tube is distilled, with a density of 1.

    This information is subsequently combined with the estimated porosity of the clean, dry sample, the volumes of the oil and water can be converted into percent pore-space fraction saturation.

    The Dean-Stark apparatus. The samples in the process are not destroyed and can be further used in other mea- surements, i.

    The following parameters are derived from the laboratory test: Water and oil residual saturation is calculated according to Eqs. Basic Concepts and Definitions 38 in Reservoir Engineering 3. The migration and accumulation of petroleum in a reservoir leads to the replacement of the original pore water by gas and oil , even though the rock pores remain "water-wet" i.

    The density difference makes the gas accumulate at the top of the reservoir, and the oil directly below. Water underlies the petroleum, as an aquifer, but is continuously distributed throughout the reservoir as the wetting fluid. The distribution of fluid phases in a reservoir. Sw is the water saturation. The following fluid interfaces in the reservoir are important: Below the GOC, gas can be present only as a dissolved phase in oil.

    Below the OWC, oil is generally absent. In other words, FWL is the oil-water contact in the absence of the capillary forces associated with a porous medium, i. However, the term "oil-water contact" does not have a single, unique meaning in reservoir engineering considerations. The continuos distribution of water saturation in the reservoir zone see Sw in Fig. In most reservoirs, this is the level where So exceeds ca. This is the base of the reservoir, or the oil- col- umn level below which the capillary forces render oil completely arrested, or "imbibed", by the rock pores such that only thermal distribution can possibly remove the oil from the "dead-end" pores.

    Therefore, some engineers prefer to refer to this surface as the capillary oil displacement level or threshold pressure level. Needless to add, the distribution of these surfaces is of crucial importance when it comes to physical fluid dynamics and economical oil recovery considerations. The interfaces are usually determined on the basis of analysis and well drill-stem tests. The total pressure at any reservoir depth, due to the weight of the overlying fluid saturated rock column, is called the overburden pressure, pov.

    The total pressure at any depth is the sum of the overlaying fluid-column pressure p f and the overlaying grain- or matrix-column pressure pm , as sketched in Fig. Overburden pressure as the combined grain- and fluid column pressure. Basic Concepts and Definitions 40 in Reservoir Engineering Because the overburden pressure pov is constant at any particular depth D, then the differ- ential overburden pressure is zero, i. This means that any reduction of the fluid pressure, as it occurs during production, will lead to a corresponding increase in the grain pressure.

    Rock compressibility is therefore an important parameter to be considered when petroleum preferably oil production is estimated. The hydrostatic pressure is therefore identical to the water pressure, at any reservoir depth, as long as there is a continous phase contact in the water, all the way up to the sea surface. The reservoir pressure can then be corrected, relative the hydrostatic pressure, by using a constant C in the above pressure equations.

    The constant C accounts for the fact that the reservoir pressure is not in hydrostatic equilibrium, where the pressure in the reservoir is somewhat higher or lower than otherwise expected. In order to evaluate the pressure distribution in a reservoir, let us consider the reservoir which cross-section, as shown in Fig.

    Cross-section of a reservoir. Assuming normal pressure condition, we can evaluate the fluid-phase pressures at the dif- ferent reservoir "key" levels.

    Basic Concepts and Definitions 42 in Reservoir Engineering between water and oil and the two pressures are identical, i. Ideally there is no oil present in the zone between the FWL and the OWC, since the oil pressure is too low to allow the oil phase to enter the pore space the largest pore throats. Similar to the FWL, the definition of the GOC, is the level in the reservoir where the pres- sures in the oil and gas phases are identical. Often this pressure is referred to as thereservoir pressure.

    Different phase pressures are observed at the same elevation in the reservoir, as seen in Fig 3. The pressure difference between two coexisting phases is calledcapillary pressure and denoted Pc i j , where the subscripts i and j refer to oil-water, gas-oil or gas-water.

    Pressure distribution in a reservoir hypothetical exam- ple. The capillary pressure at the top of the reservoir, shown in Figs. The balance of these two forces result in an equilibrium distribution of phases within the reservoir prior to its devel- opment, as shown in Fig.

    Water pressure in a vertical cylindrical tube The water pressure at a depth D is found using Eq. Substituting the last results into Eq. The pressure variation in a reservoir is determined by the fluid densities alone when the gravitational coefficient is considered constant. Basic Concepts and Definitions 44 in Reservoir Engineering 3. Determine the porosity and lithology of a core sample, given the following data: Weight of dried core sample: A laboratory cylindrical cup contains cm3 water and weighs g.

    Carbonate sand limestone, CaCO3 is poured into the cup until the level of sand and water coincide. Calculate the bulk volume and porosity of this saturated porous medium knowing that the total weight of cup and its content water and limestone is g. How do you define the porosity?

    A glass cylinder has been filled with dolomite grains up to the cm3 mark. The mass of dolomite is g. Calculate the porosity of a sandstone core sample given the data from core analysis: Bulk volume of dried sample: A reservoir water pressure of bar is measured at a sub-sea depth of m. Formation water salinity will influence hydrostatic pressure estimation. It should also be noted that porosity is astatic parameter, defined locally as an average over the representative elementary volume of porous rock media considered.

    Genetically, the following types of porosity can be distinguished: Rock media having both fracture and intergranular pores are called double-porous or fracture- porous media. From the point of view of pores susceptibility to mechanical changes one should distinguish between consolidated and unconsolidated porous media. A consolidated medium means a rock whose grains have been sufficiently compacted and are held together by cementing material. An important characteristic of consolidated porous media is the ability to restore elastically, to a great extent, to their shape volume after the removal of the overburden pressure.

    Porosity is a statistical property dependent on the rock volume taken into consideration. If the volume selected is too small, the calculated porosity can deviate greatly from the "true" statistical average value [35]. Only a volume selected large enough a representative volume will result in a representative and correct statistical average see Fig. Porosity Domain of Domain of Bulk microscopic porous volume effects medium 1.

    Definition of a representative elementary volume for porosity measurements [35]. Therefore, several idealised models have been developed to approximate porous rock media and their varied characteristics. Idealised porous medium represented by a system of par- allel cylindrical pores pipes. Estimation of porosity accounting to this model, see Fig. It is rather obvious that rocks do not have pores like this and that this model gives a unre- alistically high porosity value.

    This model may though, be used in some situations where fluid flow under simplified conditions is modelled. Idealised porous medium represented by a regular sys- tem of cubic-packed spheres.

    The estimation of porosity according to this model, see Fig. Idealised porous medium represented by a regular sys- tem of orthorhombic-packed spheres. Idealised porous medium represented by regular system of rhombohedral-packed spheres. By drawing a graph with radii of the spheres plotted on the horizontal axis and heights equal to the corresponding frequencies of their appearance plotted on the vertical axis, one can obtain a histogram of distribution of particles spheres in sizes.

    Idealised porous medium represented by an irregular sys- tem of spheres with different radii. The different models, described above, may serve as a "mental image" or idealised con- cretization of a rather complex porous structure of porous rocks. The advantage of idealised models, in general and in particular in the case of porous media, is the opportunity they offer for simple quantification and representation of characteristic parameters.

    Since rock porosity has so many representations, it is important to maintain a representative image, though idealized, of the rock porosity, for further analysis and improved undersanding. Porous medium of irregular system of spheres A porous medium is blended with three types of sediment fractions: The three sediments are mixed in such proportions that the sand fills the pore volume of the fine pebbles and that the fine sand fills the pore volume of the sand.

    The volume of fine pebble gravel is equal to the bulk volume, i. Since the sand fills the pore volume of the pebble and the fine sand the pore volume of the sand, the following table is listed: Volume of sand: Volume of fine sand: Pore volume of fine sand: Porosity 4. The distribution may appear to be unimodal left or polymodal right. Such histograms may be constructed separately for the individual zones, or units, distinguished within the reservoir, and thus give a good basis for statistical estimates mean porosity values, standard deviations, etc.

    Unimodal and polymodal porosity distributions. Numerical simulation of fluid flow in porous media, related to laboratory tests on core sam- ples as well as full field production estimation, require a realistic picture of the rock porosity and its variation throughout the reservoir. This picture is not easily obtainable since porosity is measured locally in the well and porosity extrapolations introduce large uncertainty in the estimated average values.

    The grouping of porosity data according to the reservoir zones, depth profile or graphical co-ordination, may reveal spatial trends in the porosity variation , see Fig. The recognition of such trends is very important for the development of a bulk picture of the reservoir as a porous medium and representation of the reservoir porosity in mathematical simulation models reservoir characterisation, lateral correlation, numerical modelling, etc.

    This burial effect is illustrated by the two typical examples of sand and clay deposits in Fig. The same core-plug is a non-representative elementary volume for this type of rock. The porosity measurement in such rocks requires samples that are as large as can be obtained portions of full-diameter drilling cores. Examples of trends of porosity distribution in the depth profiles of two reservoir sandstone. Sediment compaction burial and porosity change.

    A full-diameter core sample usually has a diameter of 5 inches The full-diameter core technique does not differentiate between the actual types of porosity involved, but yields a singe porosity value that represents their effective combination. Several laboratory techniques used for porosity measurements, and the procedure is generally similar for full-diameter cores and core "plugs".

    Either the pore volume or the grain volume can be determined, depending upon the instrumentation and procedures used. Other gases, such as N2 and CO2 , might be good alternatives to Helium. N2 is also used, simply due to its availability. Successive measurements will increase the accuracy, due to effects of dehydration of the porous core sample.

    The core sample is first saturated with a wetting fluid and then weighed. The the sample is then submerged in the same fluid and its submerged weight is mea- sured. The bulk volume is the difference between the two weights divided by the density of the fluid. Such a holder is called the Hassler holder, or a hydrostatic load cell, see Fig.

    Hydrostatic load cell Hassler holder used for a direct measurement of pore volume. Helium or one of its substitutes is injected into the core plug through the end stem. The calculation of the pore volume Vp is as follows: If the core has been exposed to the open air for some time, some of the oil and water can evaporate and the saturation will be measured inaccurately.

    The volumes of the extracted oil, gas and water are added to obtain the pore volume and hence the core porosity. The core sample is divided into two parts. One part ca. The vaporised water and oil originally contained in the pores, move down and are subse- quently condensed and collected in a calibrated glassware, where their volumes are measured. The second part of the rock sample ca. Then the pressure of the mercury, pHg , is raised to 70 bar psi. At this pressure, the mercury enters the sample and compresses the gas, filling the pore space originally occupied with the gas.

    With an appropriate calculation, the volume of the mercury "imbibed" in the rock gives the gas volumeVg.

    The bulk volume and weight of the fresh sample allow the computation of the effective bulk density of the rock. This in turn is used to convert the weight of the first part of the sample, which was g to be retorted , into an equivalent bulk volume.

    The oil, water and gas volumes are each calculated as fractions of the bulk volume of the rock sample and the three values are added to yield the porosity value. The laboratory procedure provides the following information: From the second subsample, the fraction of the bulk volume occupied by gas i.

    Porosity and the sum of the fluid-volume factor then gives the porosity value: Use of pycnometer in matrix volume calculation. The pycnometer is a lab-tool occasionally used for measuring bulk- and pore vol- umes of core samples. A pycnometer is in principle a contained volume, a cell, where a defined amount of mercury can be injected or withdrawn.

    The sketch below illustrates the working principle of the pycnometer. Hg V0 Figure 4. Sketch of the pycnometer. In order to define the matrix volume, Vm of a core sample, the following mea- suring steps are carried out: The pycnometer cell is fully saturated with mercury.

    The pycnometer piston is withdrawn and a gas air volume of V0 is mea- sured. The core sample is placed in the cell, and the cell volume is sealed.

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    Mercury is injected into the cell and a new gas volume,V1 and gas pressure, p1 is measured. The mercury does not enter the pore system of the core sample, due to its high interfacial tension. Mercury, as laboratory fluid, has become less popular due to its toxic characteristics and is quite often replaced by other fluids.

    Finally, the matrix volume is found as follows: For characteristic pa- rameter estimation, like determination of the porosity, we will expect the uncertainty in the measured parameters to introduce an error in the estimate of the porosity found. Since the three parameters are dependent, i.

    Differentiation of Eq. Porosity and substitution in Eq. Every extra operation in the error propagation increases the final uncertainty.

    This method of porosity evaluation is not very accurate, but has the advantage of providing continous porosity data. Once these logs are obtained and converted into a porosity log, they can be calibrated using core-sample porisity data and serve as additional reliable source of porosity distribution evalu- ation.

    Porosity can be estimated from: The Formation resistivity factor is defined as the ratio of the resistivity of the porous sample saturated with an ionic solution Ro of the bulk resistivity of the same solution Rw , i. For more information regarding porosity evaluation using geophysical well logs, see refer- ence [7, 23, 37].

    Calculate the bulk volume of a preserved paraffin-coated core sample immersed in water, given the following data: Calculate the bulk volume of a dry core sample immersed in mercury pycnometer, given the following data: Calculate the effective porosity of a sandstone sample using the following data: The initial weight of the sample is The sample is the placed in aSoxhlet distillation apparatus, and 4. After drying the core sample, the weight is now The sample bulk volume, 95 cm3 is measured in a mercury pycnometer.

    Find the porosity, water saturation, oil saturation, gas saturation and lithology of the core sample. Another core sample is brought to the laboratory for compositional analysis, where 80 g of the sample is placed in a mercury pycnometer and the volume of gas found is 0.

    A piece of the same sample, weighing g is placed in a retorte, where the water and oil volume is measured to 2. Assume oil and water densities as in the exercise above and find the same characteristic parameters. Calculate the porosity of the sample described below: Is this effective porosity or the total porosity of the sample? What is the most probable lithology of the matrix material? A core, 2. It is saturated with oil and water, where the oil content is 1.

    If a formation is 2. Answer to questions: In general terms, the per- meability is a tensor, since the resistance towards fluid flow will vary, depending on the flow direction.

    In practical terms, however, permeability is often considered to be a scalar, even though this is only correct for isotropic porous media. If there were no interconnected pores, the rock would be impermeable, i. All factors affecting porosity will affect permeability and since rock permeability is difficult to measure in the reservoir, porosity correlated permeabilities are often used in extrapolating reservoir permeability between wells.

    Absolute permeability could be determined in the laboratory by using inert gas nitrogen is frequently used that fills the porous rock sample completely and limits the possibility of chemical interaction with the rock material to a minimum.

    Since the gas molecules will pen- etrate even the smallest pore-throats, all pore channels are included in the averaging process when permeability is measured. His results showed that the pressure drop across the filter is propor- tional to the water filtration velocity. In Henry Darcy proved that flow of water through sand filters, obeys the following relationship: Permeability where h is a difference in manometer levels, i.

    Experiments repeated after Darcy, have proved that if the manometric level, h, is kept constant, the same flow rate or flow velocity is measured, irrespective of the orientation of the sand filter see Fig. Orientation of the sand filter with respect to the direction of gravitation. The pressure difference across the sand filter in Fig.

    If the sand filter is made longer, a reduced flow velocity is expected and similarly if the water is replaced by a fluid of higher viscosity, a reduced flow velocity is expected. In these cases no flow is expected and static equilibrium is established, as observed in any reservoir where the fluid pressure increases with depth.

    In a historical context, the pressure potential has been associated with the energy potential energy pr. Ve- locity and flow rate are pr. At this point it is important to notice that the permeability, k, is introduced in Eqs.

    The permeability does pr. Permeability permeability is related to the transport capability of the porous medium, as often is the case in practical situations, the fact that this information about the porous medium is missing in Eq. The proportionality constant k, called permeability, describes not only the porous medium transport capability, as such, but represents all information about the porous medium etc. Linear horizontal core flow The minus sign "-" in the horizontal flow equation Eq.

    The pressures p1 , p2 and the positions x1 , x2 are labelled according to standard numbering and orientation. Horizontal flow in a core sample. The fluid velocity related to the cross-section area A is called the superficial i. This effect will increase the pore velocity even more, as illustrated in Fig.

    If, in addition, the porous medium contains a residual saturation of a non-flowing phase, e. Pore flow velocity in a porous medium. Permeability z is the elevation in the gravitational field and from Fig. In order to maintain a constant flow rate through the core sample, the pressure difference needs to be adjusted relative to the inclination angle dip angle.

    In a up-dip situation, as in Fig. In these tests, some important conditions have to be satisfied before permeabil- ity could be estimated from the measured data. These conditions are the following: Experimental determination of liquid permeability.

    Permeability is found by plotting the measured data as shown in Fig. The linear best fit through all experimental data-points will give a slope, from which the permeability can be calculated using Eq. The importance of linear representation of the measured data is the advantage of visual inspection, which may reveal non-linear effects in the data, e.

    It is not convenient to measure permeability of porous media in cm2 or in m2.

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    By convention the unit for the permeability is called theDarcy. The following definition of the Darcy has been accepted: Applying Darcy-units to Eq. Permeability The value 1 Darcy is defined in SI-units by substitution: It is important to remember that permeability is a tensor, which means that permeability might have different values in different directions.

    Vertical permeability i. In its turn, the horizontal permeability can be different in different directions. These permeability features should be taken into account while measuring permeability.

    Core sample liquid permeability. A cylindrical core sample is properly cleaned and all remains of hydrocarbons are removed from the pore space. The core is saturated with water and then flushed horizontally. Laboratory measurements, always contain uncertainty related to the technol- ogy used to obtain the lab-data.

    This uncertainty could be examined by plotting the data-pairs in an appropriate way, e. The advantage of data-plotting, compared to straight forward calculations, as in this example, is the opportunity to verify that the data used in the averaging process are "good" or representative.

    Because gas is a highly compressible substance, i. In a homogeneous porous rock, the mean flow rate is equal to the gas rate at the centre of the core sample. The Hassler core holder is commonly used for permeability measurements. It provides measurements of permeability in both vertical and horizontal directions. For permeability measurements in the vertical direction gas is injected through the core plug in the axial direction see Fig.

    The core plug is placed in an impermeable rubber sleeve protecting the gas flow at the outer-face of the core plug. Horizontal permeability measurements require a sealing of the top surfaces of the core with non-permeable rubber disks see Fig. The area of cylindrical surface at the inflow and outflow openings is covered with a screen and the sample is then placed into the core holder.

    Under high air pressure the rubber tubing is collapsed around the core. Low pressure air is introduced into the center of the holder and passes through the rubber boot and intersects with the screen, and then flows vertically through the screen.

    The air then flows through the full diameter sample along its full height and emerges on the opposite side, where the screen again allows free flow of the air to exit. The screen are selected to cover designated outer segments of the full diameter sample. In most cases the circumference of the core is divided into four equal quadrants. In this test the flow length is actually a function of the core diameter, and the cross-sectional area of flow is a function of the length and diameter of the core sample.

    It is common to furnish two horizontal permeability measurements on all full diameter samples. The second measurement is made at the right angles to the first.

    Core sample gas permeability. A gas permeability test has been carried out on a core sample, 1in in diameter and length. The core has been cleaned and dried and mounted in a Hassler core holder, of the type seen in Fig. The gas is injected and the pressure, p1 measured, at one end of the core sam- ple, while the gas rate, q2 is measured at the other end, at atmospheric pressure, i.

    The gas permeability could be estimated using Eq. The gas permeability k is found as a function of the mean core pressure. The following data is given: Gas permeability plotted as the reciprocal of mean pres- sure.

    Note that the gas permeability is pressure dependent. As the mean pressure in the core sample increases it is expected that the gas permeability will approach the absolute liquid permeability, since at such high pressure the gas itself, will start to behave as a liquid.

    This asymptotic limit is not reached unless the pressure, e. In order to adjust for the occurance of turbulence, the horizontal flow equation can be expanded by adding a term particularly describing the turbulent flow situation. For this purpose the Fanning Eq. According to the Fanning equation one may assume that pressure drop across a pore chan- nel is proportional to the square of the average gas velocity in the pore. Instead the gas rate is measured at the exhaust end, q0. In order to use Eq.

    Assuming there are three sets of data; set a, b and c. For each set there are three measure- ments; 1, 2 and 3, all together nine measurements. The three permeability values found form Fig.

    Sec- ondly, permeabilities are plotted as functions of the inverse average pressure, from where the absolute liquid permeability is found. Permeability 5. In the case of overburden pressure, exper- iments have shown that the permeability is even more dependent on the overburden pressure than the porosity.

    Permeability measurements are also sometimes strongly affected by the fluid, e. To avoid this effect, gases helium, nitrogen, carbon-dioxide and air are often used for permeability measurements. The use of gases introduce other problems, such as turbulent flow behaviour, increased uncertainty in gas rate measurements and at low pressure, theKlinkenberg effect. It follows from Eq. These facts should be considered when permeability from laboratory measurements is re- lated to reservoir permeability.

    This effect is known as the gas slippage effect or as the Klinkenberg effect, investigated by Klinkenberg in Klinkenberg found that the gas permeability of a core sample varied with both the type of gas used in the measurements and the average pressure p, in the core.

    At low gas pressure, in combination with small diameter pore channels, this condition is broken. At low p, gas molecules are often so far apart, that they slip through the pore channels almost without interactions no friction loss and hence, yield a increased flow velocity or flow rate.

    At higher pressures, the gas molecules are closer together and interact more strongly as molecules in a liquid. Experiments show that when gas permeability is plotted versus the reciprocal average pres- sure p, a straight line can be fitted through the data points.

    Extrapolation of this line to infinite mean pressure, i. Klinkenberg permeability determination. The parameter b depends on the type of gas used and reflects, to some extent, properties of the rock Fig. Corrections to measured gas permeability due to the Klinkenberg effect are normally mod- erate to small corrections, as seen for the table below. Non-corrected Klinkenberg corrected permeability, [mD] permeability, [mD] 1. In reservoirs, the pressure will be much higher and consequently the significance of the Klinkenberg effect of no importance.

    Onset of the Klinkenberg effect The onset of the Klinkenberg effect is considered in a system comprised of a bun- dle of identical capillary tubes.

    Irrespective of the fact that a bundle of cylindrical tubes is far from being a realistic model of a porous medium, one can estimate the permeability at which the Klinkenberg effect starts to become a significant effect. As an example helium gas might be chosen in the flow experiment.

    At higher pressures, lower mean free paths are observed, i. Since the Klinkenberg effect is said to become important when the mean free path of the gas and the size diameter of the pore channels are comparable, there is a maximum permeability limit, below which the Klinkenberg effect becomes active.

    Substituting the helium mean free path for the diameter of the tube radius in Eq. In an experiment where N2 or CO2 is used, the expected mean free paths are shorter and consequently the permeability limits are lower than in theHe case. For many gases, the mean free paths of their molecules at standard conditions room temperature and atmospheric pressure are in the range: Convert this equation to "Oil Field Units" where; k: The cylindrical pore model consists of cylindrical tubes stacked on top of each other.

    Use the laws of Darcy and Poiseuilles to estimate the lowest measurable permeability of a sandstone core sample, without detecting the Klinkenberg effect. The measurements are done under laboratory conditions, using N2. Calculate the air permeability, in two ways, for a cylindrical core sample where the following data is given. Verify that the two approaches used above give the same answer.

    Use the equation for gas rate at the effluent end qo and the equation for the average gas rate q. Length 3. An oil well is producing from a cylindrical reservoir with a drainage area of 20 acres. Calculate the well pressure, given the following data: Calculate the pressure in the reservoir at a distance 5 ft from the well.

    What is the pressure drop from the well to this position, in percentage of the total pressure difference in the reservoir? Linear flow in horizontal layers. Linear and horizontal flow through linear beds in series. Are the formulas above valid both for gas- and liquid flow?

    Radial and horizontal flow through cylindrical layers. An oil well has a intermediate zone with reduced reservoir permeability k1. Permeability What is the pressure at outside the damaged zone r1? Absolute permeability of a core sample is being measured by water flooding. The core sample is mounted in a transparent cylindrical tube, as shown in the figure below, and the air-water surface is monitored as function of time.

    The tube is placed in a vertical position and the water is assumed to flow through the whole core sample, evenly distributed over the surface. Calculate the absolute permeability of the sample when the air—water surface uses seconds to move 18 cm.

    Chapter 6 Wettability and Capillary Pressure 6. A central property, when giving an overall picture of the interfacial con- ditions, is the surface or interfacial tension or more correctly the surface or interfacial energy. This property is very sensitive to chemical changes at the interface. In this chapter, the interaction between wettability and surface tension is revealed.

    Some of the most commonly used techniques are reviewed. The equilib- rium bulk phases can be: Gases are basically miscible and thus, no interfacial tension is observed between gases.

    Any surface that is in the state of lateral tension, leads to the concept ofsurface tension. For curved interfaces, the definition is similar but slightly more complex. If two fluids, say water and oil is forming an interface, as seen in Fig.

    The molecules on or close to the interface may not move with the same degree of freedom and speed, due to the constraint put on them by the interface. Wettability and Capillary Pressure the molecules is mainly a function of temperature, the potential energy of molecules attached to the interface is greater than the potential energy of the bulk molecules. Molecular motion in bulk and close to the oil-water in- terface. Generally speaking, a molecule at a surface is in a state of higher potential energy than a bulk molecule, due to anisotropy and intermolecular interactions.

    This means that energy is required to move a molecule from the interior to the surface of a phase, i. Since a proportionality exist between surface area and potential energy of the system of molecules and since equilibrium is reached at minimum potential energy actually minimum Gibbs energy , the surface area of a system is always minimised.

    The unit of surface tension is therefore, the unit of energy pr. Note, that what is called surface or interface tension is in fact surface or interface energy and quite often it is more advantageous to use the energy perspective than it is to deal with tension and forces.

    The stronger the intermolecular attractions in the liquid, the greater is the work needed to bring bulk molecules to the surface, i. In Table 6. The term wettability can be defined as "the tendency of one fluid to spread or to adhere to a solid surface in the presence of other immiscible fluids" [29].

    The evaluation of reservoir wettability can be made through measurements of interfacial tensions, i. Note that wettability itself is a microscopic characteristic, that has to be measured by using micro-scale laboratory investigation techniques. The qualitative recog- nition of preferred spread is called a wettability preference, and the fluid which spreads more is said to be the wetting phase fluid.

    Contact angles are measured, by convention, through the fluid whose wettability is studied or through the fluid which is wetting the solid surface. A ta- ble of typical fluid pairs of interest in reservoir engineering is shown in the Table 6.

    Table 6. Fluid pair wettability under reservoir and laboratory con- ditions. Pure quartz sandstone or calcite surfaces are likely to be wetted preferentially by water. The presence of certain authigenic clays, particularly chamosite, may promote oil wet character. Wettability and Capillary Pressure can be illustrated by the following rule of thumb presented in Table 6.

    Wettability preference expressed by contact angle. Example of wetting preference. The three interfacial tension are not independent parameters, and in order to reveal the relationship between them a "gedanken" experiment is carried out on a droplet of water, surrounded by oil, placed in a contact with a water-wet reservoir rock, as seen in Fig.

    Geometry of the water droplet in oil, placed in a contact with a water-wet reservoir rock. The following definitions will be used: The water droplet is assumed to be in equilibrium with the surrounding medium.

    A small deformation of the surface area, will deform the droplet slightly and force the droplet to expand on the solid surface. The deformation is described by the equilibrium equation, expressing the change in energy due to the change in area. Wettability and Capillary Pressure 6. By convention, the Pc term is positive for unconfined immiscible fluid pairs, where Pc is defined as the pressure difference between the non- wetting and the wetting phase. Pressure difference across a curved spherical interface.

    Using an example with an oil drop floating in water where the density of oil and water are assumed similar, as seen in Fig. If the droplet is small, one may assume the interfacial tension to be far more important than the gravitational force acting on the droplet and thus, since the surface area is minimised, the droplet takes the form of a perfect sphere. A small perturbation, i. Taking typical values of a pore radius and an interfacial tension of oil and water, the capillary pressure can be obtained by the following evaluation: Normally, a curved surface is characterised by two radii of curvature;R1 and R2 , as seen in Fig 6.

    Curved surface and radii of curvature. With a pressure difference across the interface in the two phases, the interface will show a net curvature with the larger pressure on the concave side. Surface tension and surface energy The process of displacing water through a porous medium is comparable to the formation of droplets of sizes equal to the capillary pore throats.

    The energy in question, is the energy needed to increase the initial water sur- face A, of the initial volume V to N number of droplets with area, Ad and volume ,Vd.

    Wettability and Capillary Pressure. Formation of interface as function of the sign of the in- terfacial tension in pairs of immicsible unconfined fluids. The surface against the second type of molecules is min- imised and in the case of small droplets, spherical interfaces are formed.

    In these cases no preference with respect to mixing of the two fluids is observed. However, diffusion will lead to mixing of the two fluids. We may observe a chemical reaction where the final state is stable in time. An example of such a process is the hydrophilic ability of pure ethanol to mix with air more or less instantaneously. Using the Eq. Idealised model of a pore channel filled with two immis- cible fluids forming a curved interface between them. Oil - water displacement in a capillary tube Displacement processes in porous media are very often a competition between viscous- and capillary forces.

    In this example, the process by which oil displaces water in a cylindrical tube is considered in analogy with the production of oil from a water-wet reservoir where oil is forced through capillary pores which initially contained water.

    Consider a dynamical situation, as sketch in Fig. Cross-section view of a cylindrical "pore channel". In a situation where the two forces are assumed to be equally important, i.

    This means that the capillary forces alone decide which pore channels are going to be swept and which are not, in the reservoir. Labo- ratory studies have shown that the value of the Capillary number is directly related to the ultimate recovery of oil, where an increase in the Capillary number im- plies an increase in the oil recovery. Wettability of oil-water-solid system. The task is to define if possible a correlation between capillary pressurePc and the param- eters, mentioned above, being responsible for the numeric variation of the capillary pressure in the experiments.

    Wettability and Capillary Pressure M — mass, L — length, and T — time, which gives the following dimensions for all the parameters in Eq. This means that dimension of the capillary pressure can be defined through dimensions of those parameters, i.

    Using notation 6. The capillary pressure does also represent the response of interfacial tensions and rock wettability. Generally, is the capillary pressure char- acteristic of the reservoir heterogeneity. To reveal the relation between the capillary pressure and the microscopic heterogeneity of the reservoir, the following example is considered: Let Vi be the pore volume of a single capillary with radiusri and V be the total pore volume of the porous medium considered, i.

    Rock Water Oil Figure 6.

    Illustration of imbibition process in an idealised model of porous medium. Assume that at the starting point, the pressure in the oil phase po , is high enough to protect the water invading the pore channels. Then gradually decreasing the outlet pressure pressure in the oil phase , the water will invade the pores. Wettability and Capillary Pressure or, substituting Eq.

    Since the pore size distribution varies between the different layers in a reservoir, it is ex- pected that the capillary pressure curve shape will also vary from layer to layer. This phe- nomenon is frequently observed in laboratory tests, by using a mercury injection technique, on core samples taken from different elevations in the same well. Pore size distribution and capillary pressure curve In this example, it will be shown how data from a mercury drainage experiment could be used to produce a capillary pressure curve and how these data could be used further, to define the pore size distribution for the core sample tested.

    First, the core sample is properly cleaned, dried and placed in vacuum for some time, before it is sealed in a mercury pycnometer. The experimental data is as follows: The minus sign "-" is added due to convenience. When Pc Sg is known, then D r can be calculated, using the following steps. For a certain gas saturation Sg , the corresponding capillary pressure Pc Sg is calculated. Capillary pressure curve and corresponding pore size distribution [8].

    This is as a result of the non-wetting phase, nor- mally hydrocarbons, entering pore space initially occupied by the wetting fluid, normally water, during migration of hydrocarbons from a source rock region into a reservoir trap.

    A pressure differential is required for the non-wetting phase to displace wetting phase and this is equivalent to a minimum threshold capillary pressure and is dependent on pore size. The physical significance of threshold pressure in an oil reservoir may be appreciated by the analogy with a capillary rise of water in different vertical glass tubes suspended in an open tray of water, as seen in Fig. The FWL is a property of the reservoir system, while an oil-water contact observed in a particular well will depend on the threshold pressure of the rock type present in the vicinity of the well.

    Capillary water elevation in cylindrical tubes as func- tion of tube radii. The relation between height above the free water level and the capillary pressure is derived from consideration of the gravity-capillary pressure force equilibrium. Each unit can have its own capillary pressure characteristic and the static saturation distribution in the reservoir will be a superposition of all units, as seen in Fig.

    Observed water-oil contacts and their relationship with free water level FWL in a layered reservoir with a common aquifer [8]. Equilibrium in a capillary tube In this example, the relation between elevation of water above FWL in reservoirs is coupled to the pore dimension pore radius.

    Equilibrium in a vertical water wet capillary tube, as shown in Fig. The change in surface energy, caused by oil displacing water in a small fraction of the capillary tube see close up in Fig.

    Perturbation around equilibrium in a water-wet capil- lary tube. If small perturbations close to equilibrium are considered, the surface- and the potential energy changes is expected to be equal, i. This means that the saturation of initial water present in the reservoir above a certain height h is localised in those pores having a radii less than r.

    This dual characteristic of the capillary pressure gives the condition for the coexistence of oil and water in porous rock. Wettability and Capillary Pressure The migration of hydrocarbons into an initially water filled reservoir rock and subsequent equilibrium vertical saturation distribution is modelled in the laboratory by a non-wetting phase displacing a wetting phase drainage capillary pressure test.

    Air and brine are frequently used as the pseudo reservoir fluids, and the displacement is affected by increasing air pressure in a series of discrete steps in water saturated core plugs sitting on a semi-permeable porous diaphragm. The apparatus layout is shown in Fig. Gas-liquid drainage capillary pressure measurement.

    Portion of liquid in saturated core is displaced at a par- ticular pressure level by either gas or liquid. Liquid sat- uration measured after equilibrium saturation has been reached. Repetition for several successive pressure lev- els [8].

    In laboratory tests the final irreducible wetting phase saturation value is often beyond the breakdown pressure of the porous plate and is sometimes obtained by centrifuge spinning at a rotational force equivalent to about psi The pore size distribution in a given rock type is usually determined by a mercury injection test. Although this test is destructive, in the sense that the sample cannot be used again, it has the advantage that high pressures can be attained, where mercury, the non-wetting phase with respect to air, can be forced into very small pores.

    Height of water - oil transition zone. A laboratory air-brine capillary pressure of 1. The air-brine interfacial tension is 0. When two or more fluids flow through a porous medium simultaneously, the phase pressures pi , generally speaking, are not identical. The difference between the phase pressures of two coexisting phases is defined as the capillary pressure.

    The capillary pressure is inversely pro- portional to a generalised interfacial curvature, which is usually dominated by the smallest local curvature radius of the interface, as illustrated in Fig. Local curvature of the interface of two coexisting liq- uids. Wettability and Capillary Pressure withdrawn. The forcing of a non-wetting phase into a pore non-wetting saturation increasing is a drainage process. The reverse wetting saturation increasing is animbibition process. We imagine the pores have an exit for the wetting fluid somewhere on the right.

    Beginning at zero non-wetting saturation, injection up to the saturation shown incondition 1 in Fig.

    At static conditions, the pressure difference between the exit and entrance of the assemblage is the capillary pressure at that saturation. When the wetting fluid is introduced into the pore from the right, the non-wetting fluid disconnects leaving a trapped or non-flowing glob in the largest pore condition 2. At the static condition 2, the entrance - exit pressure difference is zero since both pressures are being measured in the same wetting phase. The distribution of a non-wetting phase at various satu- rations.

    Going from condition 2 to condition 3 is a second drainage process, that results in even higher non-wetting saturation, a higher capillary pressure, and a higher trapped non-wetting phase saturation after imbibition condition.

    At the highest capillary pressure condition 5 , all pores of the subtracted volume contain the non- wetting phase, and a post - imbibition trapped saturation is maximum. The capillary pressure curve going from the largest non-wetting phase saturation to the largest trapped non-wetting phase saturation is the imbibition curve condition 6.

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