4. FLUID SATURATION AND CAILLARY RESSURE 4.1 Fluid Saturations We have seen that the viability of a reservoir depends upon three ritial parameters. The first two of these are the porosity of the reservoir rok, whih defines the total volume available for hydroarbon saturation, and the permeability, whih defines how easy it is to extrat any hydroarbons that are present. The final ritial parameter is the hydroarbon saturation, or how muh of the porosity is oupied by hydroarbons. This, and the related gas and water saturations are ontrolled by apillary pressure. The pore spae in a rok is oupied by fluids. In hydroarbon reservoirs these fluids are hydroarbon gasses, oil and an aqueous brine. We define the pore fration of eah of these as S g, S o and S w, respetively. Hene, S g + S o + S w = 1. The amount of eah of these fluids present at a given level in the reservoir depends upon gravity (buoyany) fores, whih tend to stratify the reservoir fluids aording to their density, external hydrodynami fores suh as flow from a remote aquifer, and interfaial fores that at between the various reservoir fluids and between the fluids and the rok matrix. The interfaial fores either take the form of (i) fores on the interfae between two fluids, or (ii) between the fluid and the solid matrix of the rok. Both effets rely upon differenes in the relative strength of inter-moleular fores between gasses, liquids and solids. Liquid-gas or liquid-vapour fores result from the differenes in moleular attration of eah of the gas and liquid moleules for moleules of the same fluid ompared to the moleular attration for moleules of the other fluid. The liquid-liquid fores result from the differenes in moleular attration of eah of the liquid moleules for moleules of the same liquid ompared to the moleular attration for moleules of the other liquid. The fluid-solid fores result from the preferene for fluid moleules to be attrated to the solid mineral surfae rather than to moleules of the same fluid. The interfaial fores give rise to what is known as a apillary pressure. Capillary pressure is the differene in fluid pressure aross an interfae between two fluids in a onfined volume. In the disussion of apillary pressure, we will first examine surfae energy and wettability, and then go on to derive a general expression for apillary pressure. 4.2 Fluid-Fluid Interations (Surfae Tension) The fundamental property of that state of matter that we all a fluid is that its moleules are free to move. In gasses the freedom is great, but in liquids the freedom is onstrained by relatively strong ohesive (attrative) fores that operate between the moleules. Imagine a drop of liquid suspended in a gas. In the interior of the liquid drop, a moleule is ompletely surrounded by other liquid moleules. On average it is equally attrated to the moleules surrounding it in every diretion. By ontrast, a moleule that oupies the surfae of the drop is strongly affeted by the attrative fores of Dr. aul Glover age 32
its fellow moleules inside the drop, but only weakly attrated to moleules outside the drop. This is beause there are muh fewer moleules in the vapour phase that surrounds the drop. Hene, the moleules on the surfae of the drop experiene net inward attration. Figure 4.1 illustrates this shematially. This attration ensures that the drop attains the shape with least surfae area in any given environment. For example, in zero gravity, the drop will float and will be perfetly spherial; the sphere being the geometry with the least surfae area per volume. Figure 4.1 Shemati illustration of surfae energy. To summarize; in the drop there are attrative fores between the moleules. The moleules inside the drop are surrounded by balaned attrative fores. The moleules at the surfae experiene a net inward fore, whih results in the drop ontrating to the smallest surfae area. The fluid pressure inside the drop is higher than that outside it, therefore the drop behaves as though it was ontained in a skin. If we have a drop of liquid suspended in a gas, and we want to inrease the surfae area of the drop, we must do work to move moleules from the interior of the drop to the surfae. This is equivalent to saying that the surfae possess free surfae energy. Dr. aul Glover age 33
F F F F 2 πr R δr F F F F Figure 4.2 Calulation of surfae energy and surfae tension. Take the surfae of any liquid droplet, and imagine that a irle is drawn on this surfae with radius R. The area of the irle is πr 2 (Fig. 4.2). To inrease the surfae area of this drop by one unit of surfae area, work must be done to move moleules from the interior of the drop to its new surfae. The surfae energy σ is equivalent to the work done W per unit surfae area inrease δa, i.e., σ = W/δA. But the work done W is the fore operating to make the hange F multiplied by the distane the fore moves, so in the ase of an imaginary irle on the surfae of the liquid drop whih inreases its diameter from R to R + δr, we an see that W = FδR and hene, σ = FδR /δa. When the droplet expands the surfae area of the droplet and the area of the irle also expand. If the hange in the radius of the irle is δr, the expanded irle has area π(r+δr) 2, and the hange in area has been δa = 2πRδR (Fig. 4.2). Hene, σ = FδR/ 2πRδR = F/ 2πR = F/L p, where L p is the perimeter of the original irle. Hene we an see that surfae energy σ has the dimensions of [fore]/[length], and is therefore a form of tension. Hene, the surfae energy of a fluid interfae is also alled the surfae tension of the fluid interfae. Note therefore, that the surfae tension is the fore operating perpendiular to a line on a fluid interfae divided by the length of that line. In our ase the fore was uniform around the irle, and the length of the irle (perimeter) was 2πR. We an write: σ = F L (4.1) where: σ = the surfae tension ( surfae energy) in dynes/m or N/m F = the fore operating on the surfae (dynes or N) L = the length of the line that the fore operates on perpendiularly (m or m). Dr. aul Glover age 34
The surfae tension of a fluid-fluid interfae is ommonly measured by using a loop of wire. If loop of wire is plaed upon a fluid-fluid interfae, it will be held in plae by surfae tension. Figure 4.3 shows the general arrangement. In this ase we will be assuming that the lower fluid has the greater fores of attration between its moleules. If we apply a small upward fore to the loop it tries to leave the surfae, but the surfae tension hold the loop down (Fig. 4.3). There are two fluid interfaes that ontribute to this, the outer irle and the inner irle. Both need to be overome to remove the loop from the interfae. The fore required (from Eq. (4.1)) is F = F inner + F outer = 2πσR inner + 2πσR outer = 4πσR mean. Hene; σ = 4 π F R mean (4.2) where, F = the fore required to lift the loop from the surfae interfae (dynes or N) σ = the surfae tension (dynes/m or N/m) = the mean radius of the loop (m or m). R mean F 2R inner 2R mean 2R outer Figure 4.3 Calulation of surfae energy and surfae tension. Table 4.1 shows the surfae tensions of some ommon fluid-fluid interfaes. Table 4.1 Surfae tensions of some ommon fluid interfaes. Interfae (N/m) Surfae Tension @ 20 o C (dynes/m) Water-Air 0.0726 72.6 Benzene-Air 0.0289 28.9 Cylohexane-Air 0.0253 25.3 n-hexane-air 0.0184 18.4 n-otane-air 0.0218 21.8 Water-Oil Approx. 0.0350 Approx. 35.0 Merury-Air 0.3680 368.0 Units: 1 N/m = 1000 dynes/m Dr. aul Glover age 35
Note that the liquid drop annot exist in isolation. The arguments put forward so far have onsidered a single fluid in a gas. In pratie both the fluid in the droplet and the fluid in whih the droplet exists ontribute to the interfaial effets. It is the relative attrations of the moleules of eah fluid for themselves and eah other that are important. In the ase of a liquid drop in a gas, the effet of the gas an usually be ignored. This is beause the ohesion fores operating between the moleules in the fluid are muh greater than those in the gas beause the fluid has a muh higher density (number of moleules per unit volume). The interfaial tension of an interfae is a fundamental property of that interfae. If a liquid has an interfae with a gas, then some moleules of the liquid will be found in a vapour phase in the gas above the interfae. These moleules redue the surfae tension. As temperature inreases the onentration of the vapour phase inreases, and the surfae tension dereases. At the ritial point there are as many moleules in the gas phase as in the liquid phase, the interfae effetively disappears, and the surfae tension beomes zero. 4.3 Fluid-Solid Interations (Wettability) We have seen that a fluid has a preferential attration to itself, and the relative strengths of suh ohesive fores result in surfae tension that develops on a fluid-fluid interfae. However, the moleules of a fluid may also have a preferential attration to solid interfaes. If two fluids oupy a solid surfae, the fluid whose moleules display the greatest attration for the atoms that ompose the solid will be the fluid that oupies most of the surfae, displaing the other fluid. The wettability of a surfae is the type of fluid whih is preferentially attrated to that surfae. Figure 4.4a shows the general ase of a single solid immersed in two immisible fluids at an unstable initial state. Now, imagine that the moleules of Fluid A are attrated to the solid moleules more than those from Fluid B are. Fluid A will then displae Fluid B on the surfae. However, the moleules of Fluid A and Fluid B are ontemporaneously also experiening attrations from their fellow moleules and, to a smaller extent, the moleules of the other fluid. Consequently, Fluid A annot displae all of Fluid B from the surfae unless the attration between the moleules on the surfae of the solid and those in Fluid A is very great. In the general ase a stable state will be enountered where Fluid A oupies the majority of the surfae. At this equilibrium state, the ontat angle θ haraterizes the balane between the various ohesive fores (Fig. 4.4b). (a) (b) Fluid B Fluid A Fluid B Contat Angle Fluid A Solid Solid Figure 4.4 Surfae wettability. Dr. aul Glover age 36
Clearly, the ontat angle an be used as a measure of wettability. Figure 4.5 and Table 4.2 show three possible senarios. Fluid B (gas) Fluid A (Water) Solid (glass) Fluid B (gas) Fluid A Solid (glass) Fluid B (gas) Fluid A (Merury) Solid (glass) Figure 4.5 Contat angles for different wettabilities. Table 4.2 Contat angles for different wettabilities. Contat Angle Desription relative to Fluid A, θ (degrees) 0 Extremely Fluid A wet 0-30 Signifiantly A wet 30-60 Moderately Fluid A wet 60-90 Weakly Fluid A wet 90 Neutrally wet 90-120 Weakly Fluid B wet 120-150 Moderately Fluid B wet 150-180 Signifiantly B wet 180 Extremely Fluid B wet Dr. aul Glover age 37
4.4 Capillary ressure 4.4.1 Derivation of the Capillary ressure Equation Take a spherial droplet of fluid, of radius R. Its surfae area is 4πR 2. If the sphere is inreased in size by inreasing its radius from R to R+δR, the inrease in surfae area δa is 4π(R 2 +rδr+δr 2 )-4πR 2, whih as δr 2 0 is δa=8πrδr. To inrease the radius of the sphere from R to R+δR requires work to be done moving moleules from the inside of the droplet to the surfae. As work done W is the produt of the surfae tension σ, and the hange in surfae area δa, we an say W = σ δa = 8π σ RδR. But the work done is also the fore operating to ause the hange multiplied by the distane the fore moves. By definition any pressure is equal to the fore F operating per unit area A, perpendiular to that area. Hene, F = A. The fore operating on the spherial surfae of the droplet is therefore equal to the nett pressure ( in out ) multiplied by the area of the spherial droplet 4πR 2, i.e., F = 4πR 2 ( in out ). Sine in > out, the net pressure ( in out ) is positive, and the net pressure and resulting fore is outward. The distane the fore moves is also outward, and its value is learly δr. Hene, the work done is W = F δr = 4πR 2 ( in out ) δr. Equating these relationships to eliminate W, gives ( in out ) = 2 σ R. (4.3) Note that the derivation equates the work done. This type of derivation does not rely on the interfae of the spherial droplet being stati, as it is based upon the onservation of work (energy). It is also independent of diretion beause work is a salar quantity. The apillary pressure is defined as the differene between the pressures in eah of the two fluids forming an interfae. Hene Eq. (4.3) is an equation for apillary pressure. The apillary pressure is proportional to the surfae tension. Note that the greater of the two pressures is developed in the fluid whih ontains the entre of urvature of the urved fluid interfae. The geometry of pore systems in roks is omplex, but an be approximated by a bundle of apillary tubes. If the pore throat is represented by a apillary tube suh as that shown in Fig. 4.6, we an note that the radius of the apillary tube R tube an be expressed in terms of the radius of urvature of the surfae of the fluid-fluid interfae R by R tube = R osθ, where θ is the ontat angle defined in the previous setion. Hene, we an write the equation θ ap 2 σ osθ = ( in out ) = R tube (4.4) Figure 4.6 Capillary pressure in a tube (pore throat). Dr. aul Glover age 38
Note that the pressure is proportional to the surfae tension and osθ, but inversely proportional to the radius of the tube. Hene, as the radius of the tube inreases, the apillary pressure dereases. The differene in pressure (the apillary pressure) auses the interfae to rise up the apillary tube until the weight of the suspended olumn of fluid balanes the apillary fore that is assoiated with the apillary pressure. Hene Eq. (4.4) an be used to desribe the rise of fluids in a apillary tube. 4.4.2 Capillary Fore The apillary fore ating on the periphery of the menisus is, aording to Eq. (4.1) F=σ L=2πσR tube. A proportion of this fore ats up the apillary tube F up = F osθ = 2πσR tube osθ. At equilibrium (i.e., the menisus is stati), the upward apillary fore is balaned by an equal and opposite fore due to the weight of the suspended olumn of fluids. The weight of the suspended olumn of fluids is π R 2 tube h g (ρ liquid -ρ vapour ), where h is the height of the fluid olumn, g is the aeleration due to gravity, and ρ liquid and ρ vapour are the densities of the liquid phase below the menisus and the vapour phase above the menisus, respetively. Hene, we an write 2πσR tube osθ = π R 2 tube h g (ρ liquid -ρ vapour ), and by rearrangement h = R tube g 2σ osθ = ( ρ ρ ) g ( ρ ρ ) liquid vapour liquid ap vapour (4.5) Hene, the height to whih the fluid rises depends inversely upon the tube radius (pore throat radius); the larger the tube (pore throats) the less the fluid rises. The apillary pressure in a simple system of glass tubes of different radii an be alulated from Eq. (4.5) by measuring the height of the apillary rise, while knowing g, and the densities of the liquid and the vapour. In pratie the density of the vapour is small at surfae onditions and an be ignored. It annot, however, be ignored at down-hole onditions, where its density may approah the density of the fluid. 4.4.3 Reservoir Senarios There are two senarios that are of partiular importane in a hydroarbon reservoir. These are the gas/water and oil/water systems. In these senarios, it should be noted that the free water level is defined as the level at whih ap = 0, i.e., the pressure in the two fluids forming the interfae is the same. Gas/Water System. A model of the apillary pressure in a gas/water system is shown in Fig. 4.7. The free water level is the height of the interfae when the radius of the apillary tube tends to infinity (i.e., the apillary pressure is zero and h=0). In Fig. 4.7, this is approximated by a wide bath of fluid. The interfae at the free water level may exist at any given absolute pressure FWL. Capillary fores exist inside the restrited apillary tubing that result in the rise of the water to a height h above the free water level. Dr. aul Glover age 39
The pressure in the gas phase above the menisus in the apillary tube is gas, where gas = FWL ρ gas g h. The pressure in the water phase below the menisus in the apillary tube is water, where water = FWL ρ water g h. Hene the apillary pressure is ap = gas water = (ρ water - ρ gas ) g h, whih is onsistent with Eq. (4.5). Using Eq. (4.4), the height to whih the water rises an be obtained by rearranging this equation to give h = g ap = ( ρ ρ ) R g ( ρ ρ ) water gas 2 σ osθ water gas (4.6) gas water θ h Gas FWL Water Figure 4.7 Capillary pressure in a gas/water system. Oil/Water System. A model of the apillary pressure in a oil/water system is shown in Fig. 4.8. The free water level is the height of the interfae when the radius of the apillary tube tends to infinity (i.e., the apillary pressure is zero and h=0) as before. This interfae exists at a given absolute pressure FWL. Capillary fores exist inside the restrited apillary tubing that result in the rise of the water to a height h above the free water level. Dr. aul Glover age 40
The pressure in the oil phase above the menisus in the apillary tube is oil, where oil = FWL ρ oil g h. The pressure in the water phase below the menisus in the apillary tube is water, where water = FWL ρ water g h. Hene the apillary pressure is ap = oil water = (ρ water - ρ oil ) g h, similarly to the last ase. Using Eq. (4.4), the height to whih the water rises an be obtained by rearranging this equation to give h = g ap ( ρ ρ ) R g ( ρ ρ ) water oil = 2 σ osθ water oil (4.7) oil water θ h Oil FWL Water Figure 4.8 Capillary pressure in a oil/water system. Knowledge of arameters. The apillary pressure depends most ritially upon the interfaial tension and the wetting angle. These parameters hange with pressure and temperature for any given fluid/fluid/solid system. Measurements an be made aurately in the laboratory in simplified systems Dr. aul Glover age 41
suh as the apillary tube, but are very diffiult to arry out in the rok itself. The result is that we use a range of values whih are generally onsidered to be standard, but should not be onsidered to be aurate. These are given in Table 4.3. Table 4.3 Standard interfaial tension and wetting angle values for ommon mixtures of reservoir fluids and solids. System Wetting Angle, q (degrees) os q Interfaial Tension at 20 o C, s (dynes/m [N/m]) Water/Gas/Mineral 0 1 72 [0.072] Water/Oil/Mineral 0 1 35 [0.035] Air/Merury/Mineral 0 1 368 [0.368] The merury/air. solid system is inluded beause it is used in merury porisimetry for obtaining porosity, apillary pressure, and pore size distributions. This tehnique will be desribed later. 4.5 Impliations of Capillary ressure in Reservoirs So far we have restrited ourselves to a simple single apillary tube. Real roks ontain an array of pores of different sizes onneted together by pore throats of differing size. Eah pore or pore throat size an be onsidered heuristially to be a portion of a apillary tube. Real roks may be ompletely water-wet, and indeed are ommonly so. They may, however be oil-wet or neutrally wet or have a mixed wettability, where some mineral grain surfaes are water-wet and some are oil-wet. For the purposes of this disussion we will onsider only water-wet roks. 4.5.1 Capillary Rise and Reservoir Fluid ressures Assume that the pores of the rok are water-wet and an be represented by apillaries of five sizes: very small, small, medium, large and very large. The apillary rises in eah of these pores will be as shown in Fig. 4.9. The apillary pressure at eah of these menisi is the differene between the pressure in Fluid A above the menisi (say gas) and the pressure in Fluid B below the menisi (say water), as defined earlier (Eq. 4.3). Fluid A Fluid A ressure Line Capillary ressures h Fluid B Fluid B ressure Line FWL ressure Figure 4.9 Capillary rise and apillary pressures. Dr. aul Glover age 42
The apillary pressures are shown in the graph in Fig. 4.9. We an see that Fluid A (gas) has inreasing pressure with depth due to the olumn of Fluid A above it, and this is desribed by the line labeled Fluid A ressure Line, whih has a gradient ρ FluidA g h. Fluid B (water) has inreasing pressure with depth due to the olumn of Fluid B above it, and this is desribed by the line labeled Fluid B ressure Line, whih has a gradient ρ FluidB g h. Fluid A sits above Fluid B beause it is less dense. Its lower density results in it having a steeper gradient. Where the two line interset, the pressures in both fluids are the same and the apillary pressure is therefore zero. This is the free water level. 4.5.2 Displaement ressure If we have a apillary tube or a rok that ontains 100% saturation of a non-wetting fluid (e.g. gas) and we introdue a wetting fluid (e.g. water) to one end, the apillary pressure will draw the wetting fluid into the tube or the pores of the rok spontaneously. If the pathway is down or horizontal this proess an ontinue as long as there is more tube or rok for the wetting fluid to fill. If the pathway is upward or vertial, the proess will ontinue until the apillary fore pulling the fluid into the tube or rok pores is balaned by the gravitational fore ating on the suspended olumn of fluid (Fig. 4.10). Tube Rok Gas Gas Water Water Spontaneous Imbibition of Water (Wetting Fluid) Figure 4.10 Capillary pressure spontaneously drawing up a wetting fluid.. Tube Rok Water Water Oil Oil Displaement ressure Fored Imbibition of Oil (Non- Wetting Fluid) Displaement ressure Figure 4.11 A fore is needed to overome the apillary pressure to displae a wetting fluid with a non-wetting fluid. Dr. aul Glover age 43
If we have a apillary tube or a rok that ontains 100% saturation of a wetting fluid (e.g. water) and we introdue a non-wetting fluid (e.g. oil) to one end, the non-wetting fluid will not be spontaneously drawn up into the tube or the pores of the rok. This is beause the wetting fluid is held inside the tube or the rok by the apillary fore. To introdue the non-wetting fluid into the tube or the rok, we must apply an external fore to overome the apillary fore holding the wetting-fluid in plae (Fig. 4.11). This is alled the displaement fore or displaement pressure, and is equal in magnitude, but opposite in sign (diretion) to the apillary fore and apillary pressure respetively. IMORTANT: In oil reservoirs, the reservoir rok initially ontains water and is water-wet. Oil migrates into the reservoir rok displaing the water. A displaement fore is required to overome the apillary fores in the water saturated water-wet reservoir rok. This fore is supplied by gravity operating upon the differential buoyany of the two fluids that results from their different densities. There is a level at whih oil annot replae water further beause the driving fore is insuffiient to overome the apillary fore. This does not our at the free water level, but at some height above the free water level defined by the displaement pressure. Thus, in reservoirs, the oil/water ontat (OWC) is above the free water level (FWL). Hene, if we have a number of apillary tubes of different radii arranged horizontally and ontaining a wetting fluid (e.g., water) (Fig. 4.12), and we wish to fore a non-wetting fluid into the tubes (e.g., oil), a lower displaement pressure will be required to replae the water in the larger tubes than to replae the water in the thinner tubes. One an use the analogy with roks, but remember that the pore spae in roks is omposed of pores of different sizes onneted by pore throats of varying size that are smaller than the pores. Hene the displaement fore or pressure required to fore a non-wetting fluid into a rok saturated with a wetting fluid will be ontrolled by the size of the pore openings. Oil Water Oil Oil Water Oil In Figure 4.12 Displaement pressure. An Example. Assume that a rok is omposed of five sizes of pore in equal number, and for the sake of simpliity, these pores are onneted by pore throats of the same size as the pores. The rok is ompletely saturated with a wetting fluid. Table 4.4 shows the size, number and volume of pores together with their respetive displaement pressures. Dr. aul Glover age 44
Table 4.4 ore size and displaement pressure example. ore Radius [mm] Individual ore Volume [mm 3 ] ore Number [-] Volume of ores [mm 3 ] Contribution to Total ore Volume [%] Displaement ressure [a 10 3 ] 1 3.1415 10000 31415 0.132 70 2.5 19.6349 10000 196349 0.825 28 5 78.5498 10000 785398 3.301 14 10 314.1593 10000 3141593 13.206 7 25 1963.4954 10000 19634954 82.535 2.8 The data given in Table 4.4 an be plotted as a displaement versus pressure plot (Fig. 4.13). In this rok the pressure required to begin to displae water by oil is the initial displaement pressure (2800 a in this ase). The level in the reservoir at whih this is the ase is alled the 100% water level whih should be the same as the OWC, and is above the FWL by a height related to the size of the displaement pressure and ontrolled by the largest pore openings in the rok. If the rok was omposed of 100% of pores of 25 miron size, this would be the level above whih there would be 100% oil saturation and below whih there would be 100% water saturation. However, the rok also ontains smaller pores, with higher displaement pressures. Thus there will be a partial water saturation above the 100% water level oupying the smaller pores, and this water saturation will redue and beome onfined to smaller and smaller sized pores as one progresses to higher levels above the 100% water level. Water will only be displaed from a given pore size if there is a suffiiently large fore to overome the apillary fore for that size of pore. Commonly, the fore driving the oil into the reservoir rok (due to differential buoyany) is insuffiient to overome the apillary fores assoiated with the smallest pores. Hene, the smallest pores in an oil zone remain saturated with water, and there is an irreduible water saturation (S wi ) in the reservoir. Displaement ressure (a x 10^3) 80 60 40 20 0 0.13% 0.82% 3.3% 13.2% 82.5% 0 25 50 75 100 Water Saturation (%) Figure 4.13 Displaement/pressure urves. Initial Displaement ressure Dr. aul Glover age 45
Note that one oil is in plae, the operation of temperature and pressure may geohemially alter the properties of the mineral surfaes suh that they beome oil-wet. This ours only in the presene of oil, so it an be imagined that a mature reservoir may have a mixed wettability, with water-wet small pores ontaining water and oil-wet larger pores ontaining oil. This has impliations when we want to move oil and water through the rok during prodution. Depth 2000 2100 2200 2300 FWL Water Gradient RFT Data Oil Gradient 2400 0 25 50 75 100 Fluid ressure (psi) Capillary ressure (Displaement ressure) gives height above FWL for a given water saturation, h= / ρ g Capillary ressure ( a x 10^ 3) 80 60 40 20 Capillary ressure Data 0 0 25 50 75 100 Water Saturation 2000 2100 2200 Water Saturation (%) Water Saturation versus Height Height in Reservoir (m) Initial Displaement ressure Height of the FWL 2300 FWL 2400 0 25 50 75 100 Water Saturation (%) 100% Water Level Figure 4.14 Calulation of water saturation in a reservoir. The urve shown in Fig. 4.13 is ontinuous for a reservoir rok beause the rok ontains a ontinuous pore size distribution. This urve is alled the apillary pressure urve. The apillary pressure urve desribes the pressure required to displae from the rok a wetting fluid at initially 100% saturation to a given saturation. The apillary pressure is related to the height above the FWL by the relationship Dr. aul Glover age 46
= ρ g h (4.8) So, if the apillary pressure urve is known, and the FWL is known, we an alulate the water saturation at any point in the reservoir as in Fig. 4.14. If this water saturation agrees with the water saturations obtained from wireline tools and ore in a given well, the wireline water saturation data an be used with onfidene as a preditor of water saturation in the unored setions of other wells in the same field. 4.6 The Measurement of Capillary ressure Curves for Roks There are many ways of measuring apillary pressure urves in roks. The most ommon are desribed below. A detailed desription of these will be given in ourses given later in the MS ourse. 4.6.1 orous late The sample is saturated ompletely with a wetting fluid (usually water), and its effetive porosity is measured. It is then plaed on a porous plate in a pressure vessel. Gas or oil is introdued into the vessel around the sample at a low pressure (Fig. 4.15). The porous plate is onstruted so that it will only let through water. The pressurized oil or gas displaes some of the water from the sample (throughout the whole sample) and the displaed water passes through the porous plate, whereupon it is olleted and its volume is measured. The vessel is left at the set pressure until no more water evolves, whih may take several days. The pressure of the gas or oil is then inreased, and more water evolves and is measured. The proedure is repeated usually about 7 times at inreasing pressures (e.g., 1, 2, 4, 8, 16, 32, 64 psi). A apillary urve an then be onstruted with pressure versus the water saturation in the sample (Fig. 4.16). Capillary Gas ressure Core lug 1 of 10 per Apparatus Brine Outlet orous late Filter aper and Filter Aid Low ressure Vessel Figure 4.15 orous plate apillary pressure measurement. Dr. aul Glover age 47
32 Ca pilla ry re s s ure (ps i) 16 8 4 0 0 25 50 75 100 Water Saturation (%) Figure 4.16 Capillary press urve from porous plate measurements. The maximum pressure is limited by the pressure at whih the porous plate begins to let the gas or oil pass through with the water. There is also a range of speialized dynami and semi-dynami versions of the porous plate tehnique that an be used on onfined ores together with diret measurement of sample fluid saturations using X-Ray, CT and gamma radiometri imaging tehniques. Advantages: Disadvantages: Can use atual reservoir fluids for the measurements. Aurate. Slow. Only provides a few data points on the apillary pressure urve. Dr. aul Glover age 48
4.6.2 Centrifuge The sample is saturated ompletely with a wetting fluid (usually water), and its effetive porosity is measured. It is plaed in a entrifuge and rotated at progressively higher speeds. The speed of rotation generates a entripetal fore that displaes the wetting fluid from the sample replaing it with air or oil (if an oil reservoir is attahed to the inside of the rotating sample). The volume of the evolved fluid is measured by olleting it in a graduated vial attahed to the outside of the spinning arrangement, and read using a strobosope. At slow rotation speeds, the fore is only suffiient to displae water from the largest pores. At higher speeds the fore is able to displae water from smaller and smaller pores in the sample. The pressures on the fluids an be alulated from an equation based upon the rotary motion of the entrifuge. This is plotted against the water saturation in the sample alulated from the measured volume of evolved fluids for eah spin speed (Fig. 4.17). 40 Ca pilla ry re s s ure (ps i) 30 20 10 0 0 25 50 75 100 Water Saturation (%) Figure 4.17 Capillary pressure urve from entrifuge measurements. Advantages: Disadvantages: Fast. Commonly only used with air as the displaing fluid. Only provides a few data points on the apillary pressure urve. As pressure is a funtion of distane from the spin entre, the fluid saturation aross the sample varies, and the measured values are taken as the mean value. Dr. aul Glover age 49
4.6.2 Merury orosimetry A dry sample of any shape, but of known weight, is plaed in a hamber whih is filled with merury. The merury is a non-wetting fluid and does not spontaneously enter the pores of the rok. This the bulk volume of the rok an be measured. The pressure on the merury is inreased in a step-wise fashion. At eah step, the merury enters smaller and smaller pores overoming the apillary pressure operating against the non-wetting fluid (Fig. 4.18). At eah stage the amount of merury intruded is measured. A graph an be onstruted of merury pressure at eah step against volume of merury intruded. This is the apillary pressure urve. This data an also be inverted to provide the pore size and grain size distributions of the rok, and the raw data also provides a rough estimate of the onneted porosity of the rok. ressure Oil In ressure Vessel Valve Closed Valve Closed Valve Opened Glass Holder Drop in Hg Level Drop in Hg Level ressure Oil Sample Dried Sample is Evauated Merury is added until mark while the system remains evauated. The amount of merury needed provides a measurement of bulk volume Atmospheri pressure is applied to the system allowing merury to enter the larger pores. Drop in Hg level gives volume of maropores. ressure is raised to 200 Ma allowing Hg to enter progressively smaller pores. The drop in the Hg level gives the volume of the pores saturated with Hg at eah pressure level. Figure 4.18 Merury porisimetry. Advantages: Disadvantages: Fast. rovides porosity of sample. rovides pore size distribution. rovides grain size distribution. Can be used on small, irregular samples, suh as uttings. Gives many data points for intrusion. Cannot be used with reservoir fluids. Data must be orreted to give reservoir values. Can only be used in imbibition yle. Dr. aul Glover age 50
4.7 Capillary ressure Data Corretion We require the apillary pressure urve for the reservoir fluids and the reservoir rok. Measurements, however, are generally made with other fluids. We must, therefore have a method for orreting the data to give the apillary pressure for the reservoir fluids. The orretions are simple and take aount of the different interfaial tensions and wetting angles for different fluid ombinations at 20 o C only. The measured apillary pressure for a merury/air/sample system (Hg/air/rok) is 2σ osθ 2 368 1 (Hg/air/rok) = = R R (4.9) The required apillary pressure for a oil/water/sample system (oil/water/rok) is 2σ osθ 2 35 1 (oil/water/rok) = = R R (4.10) In both ases the values for interfaial tension and wetting angle ome from Table 4.3. Sine the value of R is the same whatever the fluids are, we an equate and rearrange Eqs. (4.9) and (4.10) to give (Hg/air/rok) (oil/water/rok) 368 = = 10.5 ; 35 and (oil/water/rok) (Hg/air/rok) = 35 368 = 0.09524 (4.11) Similarly, (Hg/air/rok) (gas/water/rok) 368 = = 5.1; 72 and (gas/water/rok) (Hg/air/rok) = 72 368 = 0.1960 (4.12) and, (oil/water/rok) (gas/water/rok) = 35 72 = 0.4861 ; and (gas/water/rok) (oil/water/rok) = 72 = 2.057 35 (4.13) Hene, to onvert (Hg/air/rok) to (oil/water/rok) we should divide by 368 and multiply by 35 (i.e, multiply by 0.09524). Table 4.5 summarizes the onversions. Table 4.5 Capillary pressure onversions. From To Multiply by Merury/Air/Rok Oil/Water/Rok 0.09524 Merury/Air/Rok Gas/Water/Rok 0.19608 Oil/Water/Rok Merury/Air/Rok 10.5 Oil/Water/Rok Gas/Water/Rok 2.057 Gas/Water/Rok Merury/Air/Rok 5.1 Gas/Water/Rok Oil/Water/Rok 0.4861 Note: This is only valid for 20 o C Dr. aul Glover age 51
These onversions are valid for the values of interfaial tension and wetting angle measured at 20 o C and given in Table 4.3. However, these parameters hange with pressure and temperature, and are therefore different in the sub-surfae. Values of interfaial tension and wetting angle are very diffiult to obtain for sub-surfae onditions, and one might ommonly have to make do with those at 20 o C. However, sub-surfae values should always be used when available as the interfaial angle is partiularly sensitive to hanges in temperature, with values as low as 10 dynes/m for oil/water systems, and as low as 25 dynes/m for gas water systems. Many of the values in Table 4.5 are given to 4 signifiant figures auray. However, it is ommon for only two signifiant figures to be used (i.e., 10 for 10.5 and 5 for 5.1). This is often justified on the grounds that our knowledge of the values of interfaial tension and wetting angle is not aurate. The lak of aurate data introdues errors in any final alulations. However, noting that these inauraies our is no exuse for introduing further voluntary inauraies by making suh assumptions as using 10 instead of 10.5. The error introdued by this assumption an result in errors of millions of dollars on the bottom line. Now we an onvert merury derived apillary pressure data to the apillary pressure for the reservoir fluids, we an write Eq. (4.8) in terms of the merury apillary pressure. ( Hg / air / rok) = 10.5 ( oil / water / rok) = 10. 5 ρ g h (4.14) and ( Hg / air / rok) = 5.1 ( oil / water / rok) = 5. 1 ρ g h (4.15) These equations are very useful, beause they allow the height in the reservoir for a given saturation to be alulated from apillary urves measured by merury porisimetry, whih is the most ommon apillary pressure tehnique pratied in the industry. 4.8 Fluid Distributions in Reservoirs Capillary pressure urves an tell us muh about the variation of saturations aross a reservoir. The following example is taken from the personal notes of Theo. Grupping, and is a partiularly good example. In deltai-type deposition sands may be laid down in fan-like bodies. The urrent that is responsible for the deposition is fast in the proximal region and beomes progressively slower as one progresses distally or laterally. Hene, in the proximal region the grain size of the resulting roks is large (only large grains have time to fall from the fast urrent). Distally and laterally the grain size of the resulting roks beomes finer as smaller sand partiles have time to fall out of the slower urrent. Eventually, at the extreme edges of the fan, the urrent is so slow that very fine lay partiles are also deposited, beoming interspersed with the fine sand grains. The proximal large grained lean sands have large pores and large pore throats. Therefore, they have small initial displaement pressures, and their water an easily be displaed by migrating oil. The distal fine grained lean sands have small pores and onsequently small pore throats. They have higher initial displaement pressures, and their water annot be displaed by migrating oil as easily as for the oarser grained roks. Dr. aul Glover age 52
The distal fine grained shaly sands (ontaining interspersed lay) have the smallest pores and onsequently the smallest pore throats, and these pores and pore throats are bloked by the lay partiles. They have very high initial displaement pressures; the great majority of their water is immobile and annot be displaed by migrating oil. Figure 4.19 shows the results of 5 wells drilled progressively more distally in the same sub-marine fan. Well Loation Well A Well D Well B Well C Well E roximal Sediment Supply Distal Well Sample Data Capillary ressure Curves Shaliness C apillar y r essur e H eight A bove FWL 50% Water Level 100% Water Level 0 20 40 60 80 100 Water Saturation (%) Shaliness Figure 4.19 Capillary pressure urves and faies. Dr. aul Glover age 53
The first part shows the fan geometry and position of the wells (labeled A to E). The seond part shows the apillary pressure urves for eah of 5 samples taken from eah of the wells with the 100% water point labeled S 100 A to S 100 E, the initial displaement pressure labeled D I A to D I E, the 50% water point labeled as S 50 A to S 50 E, and the irreduible water saturation labeled S wi A to S wi E. The third part shows the height above the free water level as a funtion of faies and water saturation, with the same saturation labels as in part 2. Note the following: The initial displaement pressure beomes greater as the sample beomes more shaly (finer grained) as one moves distally. The irreduible water saturation beomes greater as the sample beomes more shaly (finer grained) as one moves distally. The apillary pressure urve for the proximal samples are quite flat from 100% water to about 60% water indiating that only low pressures are required to replae about the first 40% of water with oil (in the largest pores with the largest and most aessible pore throats). At any given apillary pressure, more water is retained as the sample beomes more shaly (finer grained) as one moves distally. The height above the free water level of 100% water saturation inreases as the sample beomes more shaly (finer grained) as one moves distally. The height above the free water level of 50% water saturation inreases as the sample beomes more shaly (finer grained) as one moves distally. So, the grain size ontrols the height of the 100% water saturation level. Now imagine four layers of sand that have different grain sizes, porosities and permeabilities, and that have beome folded (Fig. 4.20). Eah layer has its own apillary pressure harateristis assoiated with its grain size, and that apillary pressure is related to the height of the 100% water level above the free water level as shown. It an be seen that it is possible to drill through layers whih are 100% water zones existing between oil or gas zones from the same reservoir. In this ase Well C has two oil water ontats. It is the bottom one that is the true OWC. Beause multiple OWCs are onfusing, we tend to talk about oil down to (ODT) and gas down to (GDT) levels when there are multiple hydroarbon/water ontats. Well A Well B Well C Well D orosity % 15 25 10 30 ermeability md 40 190 5 200 Figure 4.20 Suspended water zones. FWL Dr. aul Glover age 54