Relative Permeability

In multiphase flow, relative permeability can be described as the ratio between the effective permeability of each phase to absolute permeability. There are two methods to measure the relative permeability either steady-state and unsteady state. The steady-state method takes more time than the unsteady-state method, but the unsteady-state method is more complicated to solve mathematically (Honarpour et al., 1986). Both methods can be used, and it should give the same value.

𝑘_𝑟𝑖 =^{𝑘𝑒𝑓𝑓𝑖}

𝐾 , 𝑖 = 𝑤, 0 (4)

Where 𝐾 is absolute permeability, 𝑘_𝑟 is relative permeability, and 𝑘_{𝑒𝑓𝑓𝑖} is effective permeability. Index 𝑖 = 𝑤, 0 represent phase properties water and oil. This thesis focus on oil and water system in which relative permeability is usually plotted as a function of water saturation.

7 Figure 2 illustrates typical relative permeability curves. The value of relative permeability is between zero and one as a ratio. Water saturation value is also between zero and one as a fraction. When producing oil, the relative permeability of oil decreases and the relative permeability of water increases as the water saturation increases. The highest value of relative permeability of water is reached when the water saturation is equal to one minus residual oil saturation (Sor) as no more oil can be produced.

Figure 2. Relative permeability oil-water system curves.

8 2.4 Capillary End Effect

Hadley and Handy (1956) described capillary end effect as the holdup of the preferentially wetting phase at the outlet of the porous medium during fluid flow. Figure 3 illustrates the capillary end effect. Water is injected from x=∞ in the negative direction with a known rate 𝑢_𝑇. The blue area indicates the water saturation at zero capillary pressure. The blue line shows the saturation distribution along the core due to the capillary end effect. If there is no capillary end effect, then the blue line will be a straight line. 𝑆_𝑤^𝑚𝑎𝑥is the maximum saturation from the known injected rate (𝑢_𝑇), the value is depend on the injection rate.

Figure 3. Illustration of the capillary end effect.

The steady-state end effect is the focus of this thesis. The core flooding experiment is executed by continuously injecting brine to replace oil through the core. Steady-state is reached when oil is perfectly immobilised due to entrapment, and the rate of injected water is equal to the rate of produced water (Virnovsky et al., 1995; Andersen et al., 2017). Figure 4 shows the experimental procedure where brine is injected into the core with several steps of the rate under the steady condition, following the procedure from Virnovsky et al. (1995). The dashed line in illustrates the steady state condition where oil is immobilised, the saturation does not change with time, and the amount of water injection is equal to water production. Most oil is produced at the initial rate.

Figure 4. Illustration of core flooding experiment and a steady-state condition.

9 2.4.1 Mathematical model

The mathematical model in this thesis is based on Andersen et al. (2017) for one dimensional, two-phase immiscible flow of oil (𝑜) and water (𝑤), and incompressible fluids in a porous

Where 𝑢_𝑖 is darcy velocity, 𝐾 absolute permeability, 𝑃_𝑖 pressure, 𝑘_𝑟𝑖 relative permeability, 𝜇_𝑖 viscosity and 𝑆_𝑖saturation. Index 𝑖 = 𝑜, 𝑤 represent phase properties. In this case, gravity effect is neglected.

The pressures are related by the capillary pressure function 𝑃_𝑐 = 𝑃_𝑜− 𝑃_𝑤. 𝑢_𝑇 is the total flux from equation (5) for oil and water phase, and can be expressed as

𝑢_𝑇 = 𝑢_𝑤+ 𝑢_𝑜 = −𝐾𝜆_𝑤𝜕𝑃_𝑤

By using constraint 𝑆_𝑜+ 𝑆_𝑤 = 1, and adding the transport equation (6) implies that

𝜕𝑢_𝑇

𝜕𝑥 + 𝜙𝜕(𝑆_𝑤+ 𝑆_𝑤)

𝜕𝑡 = 0,

𝜕𝑢_𝑇

𝜕𝑥 = 0 (8)

𝑢_𝑇 is constant along the x-axis, however, it can be changed at any time along the x-axis. From equation (7), ^𝜕𝑃𝑤

Mass balance of water phase can be expressed as 𝜙𝜕𝑆_𝑤

𝜕𝑡 = −𝜕𝑢_𝑤

𝜕𝑥

(10)

10 By inserting equation (9), the mass balance of water phase (10) can be written as

𝜙𝜕𝑆_𝑤

Where 𝑓_𝑤 is the fractional flow function.

𝑓_𝑤 =𝜆_𝑤

𝜆_𝑇 = 𝜆_𝑤

𝜆_𝑤+ 𝜆_𝑜 (12)

In (Buckley & Leverett, 1942) equation, it is assumed that capillary pressure is neglected.

Therefore, equation (11) can be written as 𝜙𝜕𝑆_𝑤

𝜕𝑡 = −𝑢_𝑇 𝜕

𝜕𝑥(𝑓_𝑤) (13)

However, in thesis capillary pressure is investigated and equation (11) is used together with the initial and the boundary conditions.

2.4.1.1 Boundary conditions

Andersen et al. (2017) mentioned that the brine is assumed to be injected from 𝑥 = ∞ in the negative direction, i.e. 𝑢_𝑇 (< 0) is known and equal to 𝑢_𝑤, since only water flows in the direction towards the outlet at the inlet. The boundary condition can be seen from Figure 3.

𝑓_𝑤(𝑥 = ∞) = 1, 𝜕𝑃_𝑐

𝜕𝑥|

𝑥=∞

= 0 (14)

Zero capillary pressure is defined as the boundary at the outlet, and this defines the capillary effect.

𝑃_𝑐|_𝑥=0= 0 (15)

2.4.1.2 Steady-state

As explained by Andersen et al. (2017), there is no saturation and pressure changes with time in steady state condition:

𝜕_𝑡𝑆_𝑖 = 0, 𝜕_𝑡𝑃_𝑖 = 0 , (𝑖 = 𝑜, 𝑤) (16) By inserting steady-state equation (16) into the mass balance of water phase (11), then water saturation as a function of spatial coordinate (𝑆_𝑤 = 𝑆_𝑤(𝑥)) can be written as:

0 = 𝜕

𝜕𝑥(−𝑓_𝑤𝑢_𝑇− 𝐾𝑓_𝑤𝜆_𝑜𝜕𝑃_𝑐

𝜕𝑥) (17)

By doing the integration of equation (17) and the boundary conditions thus give:

11

By gathering variable 𝑢_𝑇 on the same side, equation (18) can be rewritten as:

𝑢_𝑇(1 − 𝑓_𝑤) = 𝐾𝑓_𝑤𝜆_𝑜𝜕𝑃_𝑐

𝜕𝑥, the saturation gradient along the core at steady-state can be written as:

𝜕𝑆_𝑤 along the core and 𝐿 is the length of the core, then

𝜕𝑆_𝑤

𝜕𝑥_𝐷 =𝜕𝑆_𝑤

𝜕𝑥 ⋅ 𝜕𝑥

𝜕𝑥_𝐷 =𝜕𝑆_𝑤

𝜕𝑥 ⋅ 𝐿 (21)

Capillary pressure is assumed following Leverett J-function scaling:

𝑃_𝑐 = 𝜎√𝜙

𝐾𝐽(𝑆_𝑤) (22)

Where 𝜎 is the interfacial tension between oil and water, and 𝐽(𝑆_𝑤) is Leverett J-function scaling. Interstitial Velocity (𝜈_𝑇) can be expressed as

12 strong. Equation (24) shows the similar correlation as scaling factor 𝐿𝑢_𝑇𝜇_𝑤 from Rapoport and Leas (1953). Capillary end effect can be minimized at long core, high rate, high viscosity of water, and low permeability.

2.4.2 Wettability on the capillary end effect

• From equation (20) that is built from the mathematical model by Andersen et al. (2017), capillary pressure along the spatial coordinate is affected by the fractional flow and the constant flux on steady-state condition. The result of a study from Abeysinghe et al. (2012) shows that at the water-wet condition, there is no rate dependency in water saturation distribution. At water-wet media, there is no difference between the value of 𝑃_𝑐 between saturation where 𝑃_𝑐 equal to zero and 𝑃_𝑐 at 1-𝑆_𝑜𝑟. Thus, at the water-wet system, it does not matter on saturation distribution how strong the capillary number is. While, at the mixed-wet condition, capillary end effect can be minimised by using the higher rate. From equation (20), the capillary end effect is significant when the low rate is applied.

• A study from Abeysinghe et al. (2012) mentioned that there is no holdup of oil during water injection into a water-wet core. Capillary pressure has positive and negative value for the water-wet system and oil-wet system. However, at the water-wet system, the negative part becomes minus infinite almost for the same saturation value where 𝑃_𝑐 equal to zero. At more oil-wet system, the negative part after 𝑃_𝑐 equal to zero varies along the saturation until 1-𝑆_𝑜𝑟. Spontaneous imbibition is displayed on the positive part of the curve, and force imbibition is displayed on the negative part of the curve. The difference in the wettability of the core can be seen from the value of water saturation after spontaneous imbibition. If the change of water saturation in spontaneous imbibition is small, the core is most likely oil-wet. On the other hand, if the change is significant, then the core is most likely water-wet.

13

3 Sendra Simulator

SENDRA is a core flooding simulator that is used to simulate and verify SCAL experiment (Lenormand et al., 2017). History matching is performed by using simulator SENDRA (version 2016.1). It covers the whole experiment process including the steady-state flow corresponding to the assumption in this experiment (constant rate).

3.1.1 Relative permeability curve fitting

In this thesis, Brooks and Corey (1964) equation is used for history matching.

𝑘_𝑟𝑤 = 𝑘_𝑟𝑤⁰ (𝑆_𝑤^∗)^𝑁𝑤 (25)

𝑘_𝑟𝑜 = 𝑘_𝑟𝑜⁰ (1 − 𝑆_𝑤^∗)^𝑁𝑜 (26) Where 𝑘_𝑟𝑤⁰ is the relative permeability of water at residual oil saturation (𝑆_𝑜𝑟), and 𝑘_𝑟𝑜⁰ is the relative permeability of oil at initial water saturation (𝑆_𝑤𝑖).

The shape of water and oil relative permeability curve is attained by the parameter 𝑁_𝑤 or 𝑁_𝑜. The curve that will be used in the simulation is constructed by changing these parameters and keeping the end point of saturation, 𝑆_𝑤𝑖 and 𝑆_𝑜𝑟.

The normalised water saturation can be calculated as:

𝑆_𝑤^∗ = 𝑆_𝑤− 𝑆_𝑤𝑖

1 − 𝑆_𝑤− 𝑆_𝑜𝑟 (27)

Where 𝑆_𝑤^∗ is the normalized water saturation, 𝑆_𝑤𝑖 is the irreducible water saturation water saturation, and 𝑆_𝑜𝑟 is the residual oil saturation.

3.1.2 Two-phase capillary Pressure correlation

In this thesis, Skjaeveland et al. (2000) equation are used for history matching. Skjaeveland et al. (2000) generated a capillary pressure equation for a mixed-wet reservoir that comprises the imbibition as can be seen in Figure 5.

14 Figure 5. Schematic of bounding curves, capillary pressure Pc as a function of water saturation

Sw: (b) imbibition. Redrawn from Skjaeveland et al. (2000).

The general expression for imbibition curve is expressed as 𝑃_𝑐𝑖 = 𝑐_𝑤

(𝑆_𝑤− 𝑆_𝑤𝑖 1 − 𝑆_𝑤𝑖 )^𝑎𝑤

− 𝑐_𝑜

(1 − 𝑆_𝑤 − 𝑆_𝑜𝑟

1 − 𝑆_𝑜𝑟 )^𝑎𝑜 (28) Where 𝑃_𝑐 is the capillary pressure [Pa], 𝑆_𝑤 is the water saturation [-], 𝑆_𝑤𝑖 is the initial water saturation [-], and 𝑆_𝑜𝑟 is the residual oil saturation [-]. The 𝑐_𝑤, 𝑎_𝑤, 𝑐_𝑜, and 𝑎_𝑜 are constants for imbibition curve from 𝑆_𝑤𝑖 to 𝑆_𝑜𝑟. The value of 𝑎_𝑤, and 𝑐_𝑤 are used to define the positive term of the curve. While the value of 𝑎_𝑜, 𝑐_𝑜 are used to define the negative term of the curve. At 𝑆_𝑤𝑖, the negative term has 𝑆_𝑜 = 1 − 𝑆_𝑤𝑖 and the value is not zero. It means that both terms are needed to make a capillary pressure equal to zero whether the saturation point at capillary pressure equal to zero is at 𝑆_𝑤𝑖 or any other saturation value.

15

4 Experimental Work

In this part, two types of the core material is used, Bentheimer and Berea. The cores are treated by using Quilon to change the wettability to be more oil-wet. Spontaneous imbibition is performed to get the water saturation at capillary pressure equal to zero. Next, water flooding is performed at a constant rate with rate increases in steps. Differential pressure and cumulative oil production versus time are recorded to provide the capillary pressure and relative permeability curves of the core.

4.1 Experimental Material

4.1.1 Core

Four cores are used in this experiment with two type of core materials, Berea and Bentheimer core plug. Frantz Jr. et al. (1993) that Berea sandstone is believed to have deposited in a shallow marine environment. The core is drilled from an outcrop in Ohio, United States. In the core flood experiment, the upper Berea sandstone unit is the most commonly used. The rock has well-sorted and well-rounded sand grains (Churcher et al., 1991). Bentheimer sandstone is a type of shallow marine deposition. This core is taken from an outcrop of Netherland and German border. Bentheimer is an ideal rock for experimental study due to its lateral continuity and homogeneous block-scale (Peksa et al., 2015). The Bentheimer cores were drilled from the same large block.

The main component on both Bentheimer and Berea is quartz (Peksa et al., 2015; Frantz Jr. et al., 1993). Therefore, initially, the cores have more preference on water. Table 1 gives the necessary physical core data of Berea and Bentheimer cores.

Table 1. Core Properties.

Core Type

Parameter Berea Bentheimer_1 Bentheimer_2 Bentheimer_3

Length (mm) 89.24 89.89 90.03 90.03

Diameter (mm) 37.72 37.79 37.81 37.74

𝜙 (%) 20.89 22.89 21.79 21.82

𝐾 (Darcy) 0.63 3.20 3.00 3.10

Type Sandstone Sandstone Sandstone Sandstone

16 4.1.2 Fluid and Chemical

Table 2 shows the properties of fluids that are used in this experiment.

Table 2. Properties of fluids.

Quilon-L (3% Weight) in distilled water 0.999 -

n-Decane 0.73 0.92

4.1.2.1 Brine

Sodium chloride (NaCl) 1 M and 0.1 M are used during the experiment. The liquid density is measured by using a densitometer Anton Paar model DMA 4100 M shown in Figure 7 at room temperature 20^oC. Firstly, the tube is cleaned with acetone. Next, the density is calibrated by using distilled water. The densitometer is considered valid if the measurement of distilled water has the same value as in the literature. Then, the density of brine is recorded by a densitometer.

The measurement is done several times for accuracy. The viscosity of brine is determined based on the experiments by Kestin et al. (1981) at pressure 6.5 bar and room temperature 20^oC.

4.1.2.2 Quilon

In this thesis, Quilon is used to alter the wettability of the core. Quilon complexes are manufactured by Zaclon LLC. Quilon is a dark green solution, largely in isopropanol, of a chemically reactive compound in which a C14-1C18 fatty acid is coordinated with trivalent chromium. The fatty-acid group in the Quilon is hydrophobic (Quilon data sheet, 2018).

Therefore, after a reaction with the core, it is oriented away from the surface to give the core its oil-wet characteristic.

Initially, the wettability alteration by using Quilon was proposed by (Tiffin & Yellig, 1983).

He used Quilon-C to change the wettability of Berea from strongly water-wet to oil-wet, while Maini et al. (1986) used Quilon-S with some minor differences related to the grade of Quilon.

In this experiment, wettability alteration is based on a method which proposed by Abeysinghe et al. (2012). Initially, Quilon-H was used in the experiment with the assumption that the behaviour of H is similar to L. There is no study about the solubility of

Quilon-17 H in n-decane and distilled water. However, after performing the wettability alteration in this experiment. It is found that Quilon-H is not soluble in n-decane.

Therefore, in the end, Quilon-L is used to perform the wettability alteration on the core. Criollo (2011) observed that Quilon-L is soluble in both distilled water and n-decane. The differences between C, S, H and L is the grade. H and L are more concentrated than Quilon-C and S (Quilon data sheet, 2018). The more concentrated grade provide both economic and storage advantages. The fluid properties of Quilon-H and Quilon-L solution can be seen in Table 2.

4.1.2.3 Oil

N-decane is used as oil in the experiment. The chemical formula of decane is CH3(CH2)8CH3. From the physical appearance, the colour of n-decane is like water which is colourless with a density less than water ("PubChem Compound Database,"). Table 2 shows the properties of n-decane.

4.1.2.4 Sudan Blue GN

Distinguishing oil and brine inside the flow line is difficult. Therefore, for the Bentheimer_3 experiment, Sudan Blue GN is used for n-decane colouration. Sudan Blue GN is a blue-black powder that is soluble in n-decane but, not in water. It is manufactured by Waldeck GmbH &

Co. KG, Division Chroma. The chemical formula of Sudan Blue GN is C22H18N2O2. Sudan blue GN is not reactive with the rock.

4.2 Experimental equipment

4.2.1 Quizix QX20K pump

The pump that is used during the experiment is QX20K. This pump is manufactured by Chandler Engineering. There are two cylinders to store and to pump fluid which have a total volume of 7 mL. These two cylinders make it possible to inject fluid continuously, by changing the cylinder simultaneously. The maximum cylinder pressure rating of 20,000 psi, the minimum of the flow rate of 0.0001 mL/min and a maximum flow rate of 7.5 mL/minute ("Quizix QX Series", 2018).

18 Figure 6. Quizix QX20K pump.

4.2.2 Anton Paar 4100 density meter

The density meter is manufactured by Anton Paar. This density meter can show 4-digit density value. In this thesis, the density meter is used to measure the density of the brine.

Figure 7. Anton Paar 4100 Density Meter.

4.2.3 Separator

Usually, burette is used to collect and measure the volume of the effluent. However, the volume of the burette is limited, and brine is injected continuously. In this experiment, the volume of burette is only 25 mL, while the fluid production is larger than 25 mL per day at a higher rate.

Therefore, the separator was used in the force imbibition procedure where a higher rate was applied.

The separator that is used in this experiment makes use of the gravity effect where the fluid with lower density will stay on the top, and the fluid with higher density will stay in the bottom.

19 Figure 8 illustrates the schematic of the separator. The burette is placed upside down, and it is connected to a rubber line. Burette and the left section of the rubber line are initially filled with brine at the same fluid level, while the other rubber line is empty. The production fluid flow through the orange line from the core holder to the burette. When the fluid is produced, the oil will stay in the top part of the burette, while the brine will flow through the right section of rubber line to the effluent collector. By using this separator, oil and brine can be measured at the same time.

Figure 8. (Left) Pictures of the separator; (Right) Schematic of the separator.

4.3 Experimental Procedure

Figure 9 demonstrates general experiment steps that are conducted to the core. There are two types of treatment applied to the core depends on the wettability of the core. For mixed-wet core, saturation establishment is performed before wettability alteration. In the forced imbibition process, several values of rates are applied to retrieve the recovery performance and to estimate the negative value of capillary pressure to obtain residual saturation.

20 Figure 9. Schematic of experiment steps.

4.3.1 Core Preparation

In this stage, the dimensions of the core are measured by using a calliper to calculate the bulk volume of the core. The bulk volume of the core is calculated by using the formulation to measure cylinder (equation (29)) where D is the diameter of the core and L is the length of the core.

𝑉_{𝑏𝑢𝑙𝑘} = 𝜋 4×𝐷²

𝐿 (29)

Figure 10. Illustration of measuring the dimensions of the core.

21 4.3.2 Brine and oil preparation

4.3.2.1 Brine preparation

In this experiment, there are two solutions of sodium chloride with different concentration (1 M and 0.1 M). Firstly, 58.44 gram of NaCl (manufactured by VWR BDH Chemicals) is dissolved in distilled water to make one litre of Sodium Chloride 1 M. Then the brine is stirred overnight to form a homogeneous solution. Figure 11 illustrates the filtration setup. Stirred brine is filtered by using 0.22 µm filter paper and a filter to remove unwanted particle. Finally, brine is stored inside the plastic jar to prevent reaction with silica.

Figure 11. Filtration setup.

NaCl 0.1 M is made by adding distilled water to NaCl 1 M with proportion 90% distilled water and 10% NaCl 1 M.

4.3.2.2 Oil preparation and colouration

N-decane is used to saturate the core after initial water saturation establishment and to make Quilon in n-decane solution for changing the wettability. N-decane can be used without filtration process. The amount that of n-decane to saturate the core is 5 PV.

For the Bentheimer_3 experiment, Sudan Blue GN is added to n-decane. First, a tip of Sudan Blue GN is mixed to 20 mL of n-decane inside a small glass jar (Figure 12 (a)). It should be mixed instantly, but it is allowed to shake or even stir the jar with a magnet if needed. The syringe is used to collect and to add droplets to the main n-decane until the significant colour is seen (Figure 12 (b)). For additional, a tip of Sudan Blue GN can be added to distilled water to check the solubility with water. From Figure 12, it is proved that Sudan Blue GN is not soluble in distilled water.

22 Figure 12. Illustration of decane colouration. (a) a mixture between a tip of Sudan Blue GN and

n-decane in a small glass jar; (b) a final mixture between a tip of Sudan Blue GN and n-n-decane that is injected to Bentheimer_3; (c) a mixture between a tip of Sudan Blue GN and DI Water in a small

glass jar.

4.3.3 Initial Saturation Establishment

4.3.3.1 Saturating core with NaCl 1 M

Figure 13 represents a schematic of equipment used for core saturation. The core is placed inside the plastic container, and it is put inside a sealed setup. A vacuum pump is run to remove any air from the installation. The pressure in the setup should be lower than 0.6 mbar. Later, the vacuum pump is turned off, and the brine flows through a valve until the water column is higher than core height. When the core is fully saturated, it is taken out from the sealed setup, and the saturated core is weighed.

Figure 13. Schematic of equipment used for core saturation.

4.3.3.2 Porosity measurement

Porosity calculation is based on the weight difference between dry core and fully saturated core. The density of the brine is measured by a densitometer. The weight of the core is measured

In document Core Flooding for Analysis of Capillary End Effect and Multiphase Flow Properties (sider 19-0)