# Reliability of Relative Permeability Measurements for Heterogeneous Rocks Using Horizontal Core Flood Experiments

^{1}

^{2}

^{*}

## Abstract

**:**

_{cv}= kLp

_{c}*A/H

^{2}μ

_{CO2}q

_{t}) smaller than a critical value, flows are viscous dominated. Under these conditions, saturation depends only on the fractional flow as well as capillary heterogeneity, and is independent of flow rate, gravity, permeability, core length, and interfacial tension. Accurate whole-core effective relative permeability measurements can be obtained regardless of the orientation of the core and for a high degree of heterogeneity under a range of relevant and practical conditions. Importantly, the transition from the viscous to gravity/capillary dominated flow regimes occurs at much higher flow rates for heterogeneous rocks. For the capillary numbers larger than the critical value, saturation gradients develop along the length of the core and accurate relative permeability measurements are not obtained using traditional steady-state methods. However, if capillary pressure measurements at the end of the core are available or can be estimated from independently measured capillary pressure curves and the measured saturation at the inlet and outlet of the core, accurate effective relative permeability measurements can be obtained even when there is a small saturation gradient across the core.

## 1. Introduction

#### 1.1. Literature Review

_{2}[1]. However, compared with oil and gas reservoirs, where a century of experience exists regarding multiphase displacement processes, our understanding of the fate and transport of CO

_{2}and brine in saline aquifers is still limited. When CO

_{2}migrates through a saline aquifer, the interplay between viscous, capillary, and buoyancy forces, as well as structural heterogeneities, will determine how far and how fast the plume will move, how much CO

_{2}will dissolve, and how much will be immobilized by residual trapping [2,3]. Figure 1 illustrates the conceptual CO

_{2}migration in deep saline aquifers. Three physical forces dominate CO

_{2}flow behavior in different flow regimes. Characterizing different flow regimes by two transition points is important and very useful for upscaling. Scaling from one system (core scale) to another (field scale) is possible by using a dimensionless group to study the multiphase flow system. The relative permeability of CO

_{2}/brine systems is an essential element to determine CO

_{2}injectivity and migration, as well as to assess the safety of potential CO

_{2}sequestration sites. Multiphase flow parameters (relative permeability) are best understood in the viscous dominated regime.

_{2}/brine/rock system, such as the capillary pressure and the relative permeability curves. The latter, specifically the drainage relative permeability, is the main subject of the literature review below.

_{2}–brine relative permeability [16,23,24,25,26,27,28,29,30]. However, the reliability of such published relative permeability is directly affected by the quality of the measured relative permeability curves, as recently highlighted in reviews of published relative permeability measurements [31,32,33]. In particular, factors that could affect these measurements are (a) the core heterogeneity that may be responsible for flow rate dependency and incomplete fluid displacement; (b) capillary end effects that are not properly accounted for; and (c) gravity segregation that may occur when relatively long cores are used in a horizontal core-flooding system. In the following, the above mentioned issues are discussed in more detail.

_{2}/brine system [2,51,52,53], as the large density difference can lead to gravity override based on the high Bond number, and hence causes both horizontal and vertical saturation gradients. Although using vertical experiments to measure relative permeability can avoid gravity segregation [54], the use of a vertical arrangement would make the use of X-ray CT (Computed Tomography) scanning to observe fluid saturation quite challenging without purpose-designed equipment.

_{2}trapping capacity [58] and may cause flow rate dependency, high residual water saturation, and low end-point relative permeabilities observed from the CO

_{2}/brine core flood experiment [16,23,29,30,59,60,61]. It has been shown that including heterogeneity characteristics in numerical simulator grid blocks can improve the accuracy of simulation prediction and enable reliable relative permeability measurements [8,9,42,44].

- The large body of multiphase flow studies, particularly relative permeability in oil/water and gas/liquid systems, provides a good starting point for understanding CO
_{2}/brine systems; - As the fluid properties of the CO
_{2}/water system are very different from those of the oil/water system, and because of the fundamentally empirical nature of the relative permeability concept, studies are needed to establish similarities and differences between multiphase flow oil/water and CO_{2}/brine systems; - Potential and unresolved influences of flow rate, capillary number, and small-scale heterogeneity on relative permeability in CO
_{2}/brine systems need to be investigated; - The end effect is an important factor that could lead experimental error. If we want to investigate the flow rate dependence on relative permeability curves, the end effect must be carefully understood and compensated for;
- Based on recent studies of heterogeneity, the effect of heterogeneity may be the reason for the observed dependence of relative permeability on flow rate;
- As the experiments to measure the influence of heterogeneity on relative permeability are time consuming, numerical simulations can be used to simulate, understand, and interpret laboratory experiments of multiphase flow in typical reservoir rocks.

_{2}/brine core flood experiments, the issues described above are addressed in this paper. This work builds on two studies of the influence of flow rate, gravity, and capillarity on brine displacement efficiency in homogeneous porous medium [11] and heterogeneous porous media [60]. 3D numerical modeling and 2D theoretical analysis were perfomed to understand and predict the combined influences of viscous, gravity, and capillary forces in heterogeneous rocks over the range of conditions relevant to storage of CO

_{2}in deep underground geological formations (Figure 2). The proposed 2D semi-analytical technique predicts the brine displacement efficiency for 3D two-phase flow simulations very well when the Bond number ranges from 0.02 to 0.2 and the degree of heterogeneity σ

_{lnk}/ln(k

_{mean}) is smaller than 0.5. The system considered in this series of work is supercritical CO

_{2}/water and, most significantly, drainage displacements.

#### 1.2. Summary of the Previous Work

_{2}and five percent brine injected simultaneously into a simulated core at a wide range of flow rates (around 0.001 mL/min to 24 mL/min) were performed. Average CO

_{2}saturations for homogeneous and heterogeneous cores (only the high contrast model is shown here) over a wide range of flow rates are analyzed in terms of gravity number N

_{gv}(LHS of Figure 3) and capillary number N

_{cv}(RHS of Figure 3), respectively. It was shown that, when the effect of gravity is important for the multiphase flow system, we should use the gravity number N

_{gv}(Equation (1)) to analyze the saturation data, for example, for the homogeneous and mildly heterogeneous cores [11]. On the other hand, when the capillary heterogeneity is taken into account, the impact of gravity is much smaller and the capillary number N

_{cv}(Equation (2)) is a better dimensionless number to characterize our system [60]. The advantage of using appropriate dimensionless numbers can be easily seen from Figure 3. The balance of viscous, gravity, and capillary forces can be properly captured through these dimensionless numbers and flow regimes can be characterized by two critical numbers.

_{2}and brine; g is acceleration; q

_{t}is the total volumetric flow rate; k

_{eff}is the effective permeability of the core; ${\mathsf{\mu}}_{\mathrm{CO}2}$ is CO

_{2}viscosity; L is the core length; H is the core height; and A is the core area. p

_{c}* is the characteristic capillary pressure of the medium, chosen as a so-called displacement capillary pressure. The displacement capillary pressure is a capillary pressure value at the brine saturation S

_{w}equal to 1, and it is tangent to the major part of the capillary pressure data. The p

_{c}* value for our core is about 3000 Pa, shown in Figure 4.

#### 1.3. General Rule of Thumb for Reliable Relative Permeability Measurements

_{cv}is small enough (below the critical value, ${\mathrm{N}}_{\mathrm{cv},\mathrm{c}1}^{\mathrm{Hete}}$), viscous forces dominate and the gravity impact can be neglected in this regime even with horizontal core flooding.

_{BL}

^{Hete}is defined as the average CO

_{2}saturation of the heterogeneous core in the viscous-dominated regime, which is the Buckley–Leverett saturationtaking into account heterogeneity (shown in Figure 3); ${\mathrm{k}}_{\mathrm{rCO}2}\left({{\mathrm{S}}_{\mathrm{BL}}}^{\mathrm{Hete}}\right)$ is the CO

_{2}relative permeability evaluated at ${{\mathrm{S}}_{\mathrm{BL}}}^{\mathrm{Hete}}$; ${\mathrm{R}}_{\mathrm{l}}$ is the aspect ratio (L/H); and f

_{CO2}is the fractional flow of CO

_{2}. Because the critical capillary number ${\mathrm{N}}_{\mathrm{cv},\mathrm{c}1}^{\mathrm{Hete}}$ depends on the rock heterogeneity, the higher the degree of heterogeneity in the core, the smaller the capillary number N

_{cv}required reaching the viscous-dominated regime.

_{lnk’}< 2.5).

_{2}saturation as compared with that expected for a uniform core (Figure 3). Consequently, the effective relative permeability for the whole core is different from the intrinsic relative permeability of each individual voxel in the core. Saturations in this “viscous-dominated regime” vary spatially in response to the establishment of gravity-capillary equilibrium in the core, which is shown at the LHS of Figure 5.

## 2. Materials and Methods

_{c}), and relative permeability curves (k

_{r}). These are then used as input for simulations using TOUGH2/ECO2N [64,65] that mimic the core-flooding procedures for making steady-state relative permeability measurements. Outputs from the simulation are used as synthetic “data sets” for calculating the relative permeability of the core. The influence of flowrate, rock heterogeneity, core length, gravity, and interfacial tension on the accuracy of the calculated relative permeability curves are systematically studied by varying these parameters over a wide range of values. Based on the comparison between the input (intrinsic) and calculated (effective) relative permeability curves, we draw conclusions about the important sources of error for these calculations as well as the conditions over which accurate measurements can be obtained. Simulations are repeated at a number of fractional flows to construct the full relative permeability curve.

#### 2.1. Simulation

_{2}saturation distribution resulting from co-injection of 5% brine and 95% CO

_{2}is shown in Figure 7 [16,66]. The experiment is modeled by a three-dimensional roughly cylindrical core (Figure 8). A total of 31 slices are used in the flow direction, including 29 rock slices, an “inlet” slice at the upstream end of the core, and an “outlet” slice at the downstream end. All of the simulations are carried out by co-injecting known quantities of CO

_{2}and brine at a constant flow rate into the inlet end of the core (Figure 7). The laboratory conditions and core properties are selected to replicate the laboratory experiments. All the TOUGH2 simulations are conducted at 50 °C temperature and 12.4 MPa initial pore pressure.

_{lnk’}are shown in Table 1. The permeability of each grid element is assumed to be isotropic. These two heterogeneous models are compared to a homogeneous one to study the effect of heterogeneity on the multiphase flow system. Note that the capillary pressure and the relative permeability functions are the same in as the previous studies.

#### 2.2. Boundary Conditions

_{2}and brine are mixed and co-injected through a tube and enter into a diffuser plate to distribute evenly before entering into the upstream end of the core. To avoid dry-out, carbon dioxide and water are pre-equilibrated at a high pressure and temperature (in this case, 50 °C and 12.4 MPa) prior to starting the experiment. The amounts of CO

_{2}and brine that enter each pixel are controlled by the relative mobility of CO

_{2}and brine (Equation (8)) such that the total rate is equal to the injection rate of each phase (Equation (9)):

_{c}|

_{outlet}= 0.

_{c}/dx)|

_{outlet}= 0.

_{2}at a total injection rate of 2.6 mL/min flow rate is shown in Figure 9a [67]. Similar saturation distributions have been measured for other rocks as described by Krevor et al. (2012) [30]. A relatively uniform saturation profile is observed over the whole core; in particular, there is no large saturation gradient at the outlet, and different fractional flows of CO

_{2}have different values at the end. If the boundary condition with P

_{c}= 0 at the downstream end is imposed, a large saturation gradient occurs and every fractional flow of CO

_{2}have zero saturation at the outlet, which is not observed in the experiments (Figure 9b). The Dirichlet boundary condition with the added constraint that dP

_{c}/dx = 0 between the last slice in the core and the outlet provides a much better match to the data at the outlet (Figure 9c). Consequently, we use this boundary condition in the rest of the simulations. Specifications for the boundary conditions are listed in Table 2.

#### 2.3. Simulated Synthetic Relative Permeability Data Sets

_{2}and brine are run until the pressure drop and core-averaged saturation stabilize. All of the simulations were confirmed to run for long enough (more than 10 pore volumes injected) to reach steady-state. Important output parameters include grid-cell CO

_{2}saturations, CO

_{2}pressures, and capillary pressures. Briefly speaking, we only analyze the core-averaged saturation in the previous study, but now, the slice-averaged quantities along the length of the core such as saturation profiles (S

_{CO2}), pressure in the CO

_{2}phase (P

_{CO2}), and capillary pressure profiles (P

_{c}) are evaluated in this study. Figure 10 shows a typical simulation result, including the CO

_{2}saturation distribution, pressure drop across the core, and core-averaged CO

_{2}saturation.

_{CO2}= P

_{CO2,inlet}– P

_{CO2,outlet},

_{w}= P

_{w,inlet}– P

_{w,outlet}.

_{CO2}, for example, then the water pressure drop can be rewritten in terms of the two output parameters ΔP

_{CO2}and ΔP

_{c}:

_{w}= (P

_{CO2,inlet}− P

_{CO2,outlet}) − (P

_{c,inlet}− P

_{c,outlet})= ΔP

_{CO2}− ΔP

_{c}.

_{c,inlet}and P

_{c,outlet}do not refer to the capillary pressure in the endcap, but inside the rock, just downstream or upstream of the endcaps. When P

_{c}is the same in the first and last slice of the core, the pressure gradient drop across the core is the same in both phases. The pressure drops in each phase are used to calculate the corresponding relative permeability values based on the simplified Darcy’s equation, shown later in Equation (15).

## 3. Results

#### 3.1. Relative Permeability Calculated When ΔP_{w} = ΔP_{CO2}

_{CO2}is the used as the “proxy” for the measured variable ΔP in the experiments. In this case, the effective relative permeability can be rearranged and calculated from Equation (15) based on the assumption that ΔP

_{w}= ΔP

_{CO2}:

_{lnk’}= 0 and 0.96).

#### 3.1.1. Homogeneous Cores (σ_{lnk’} = 0)

_{2}saturation as a function of the distance from the inlet at a 95% fractional flow of CO

_{2}over a large range of flow rates (0.1 mL/min–6 mL/min). The saturation is uniform across the core at high flow rates where no saturation gradient exists, and hence there is no capillary pressure gradient along the core. Decreasing the flow rate below this regime leads to a saturation gradient in the flow direction. Based on the simulation results, the critical flow rate for the homogeneous core is around 0.3 mL/min, or equal to the flow velocity (u) of 0.25 m/day.

_{2}is higher near the top of the core. This in turn creates higher-than-average factional flow of CO

_{2}near the top of the core as the fluid moves away from the inlet boundary. The net effect is to cause a saturation gradient along the length of the core.

_{critical}~0.3 mL/min), which corresponds to the negligible saturation gradients observed in Figure 11. On the other hand, a roughly 15% saturation gradient along the flow direction results in a significant deviation of wetting phase relative permeability (0.1 mL/min). Equation (16) is no longer valid for the wetting phase as pressure drops for the two fluids are different once saturation gradients occur. Using the pressure gradient in the CO

_{2}phase overestimates the pressure drop in the water phase, leading to underestimation of the water-phase relative permeability.

#### 3.1.2. Heterogeneous Core (High Contrast Model, σ_{lnk’} = 0.96)

_{2}saturation along the length of the core between the homogeneous and heterogeneous cores at the same flow rates. The general trends observed in the homogeneous core can apply to the heterogeneous one. First, the slice-averaged saturation is relatively uniform in the high flowrate regime (q > 1.2 mL/min or u > 1 m/day). The source of saturation variation along the core is a combination of capillary, gravity, and heterogeneity effects. Given the constant capillary pressure curve and no gravity effects, we would get constant saturation even with the most heterogeneous core. Similar patterns can also be observed in the experiments [30,66]. Second, a large saturation gradient across the core starts to occur once the flow rate is below the limit for establishing the quasi-viscous-dominated regime. Comparing the homogeneous and the heterogeneous cores, it is clear that the capillary heterogeneity will enhance the flow rate dependency, decrease the average saturation, and increase the saturation gradient.

_{critical}), the effective relative permeability is independent of the flowrate, which demonstrates that the relatively uniform slice-averaged saturation results in the rate-independent effective relative permeability values (Figure 14). In addition, once large saturation gradients develop (q < q

_{critical}), the wetting phase relative permeability is reduced significantly as it is an effective property that incorporated the capillary heterogeneity effects [56,66,69,70]. In this case, the assumption of ΔP

_{w}= ΔP

_{CO2}would also contribute to this deviation. For the same flow rate, the heterogeneous core results in larger saturation gradients compared with the homogeneous core. In general, the rate-independent drainage effective relative permeability can be obtained even with the highly heterogeneous core once the flow rate is high enough to eliminate saturation gradients from one end of the core to the other. It is not required that saturation gradients are eliminated in the middle of the core, as these result from the capillary heterogeneity of the rock.

_{2}relative permeability is higher than the input value. This non-intrinsic effective CO

_{2}relative permeability occurs when the heterogeneities are aligned parallel to the direction of the flow field, a well-known phenomenon as described by Corey and Rathjens (1956) and Honarpour et al. (1994) [71,72]. Permeability distribution for the high contrast model indeed has some channels across the core diagonally (Table 1) leading to the effective relative permeability for the non-wetting phase of the heterogeneous core higher than for a homogeneous core.

#### 3.2. Relative Permeability Calculated by Using Corrected Pressure Drops

_{rw}(Figure 15). Once the true pressure drop of the wetting phase is known and used in the calculation, the wetting phase relative permeability is identical or close to the intrinsic values even with a 15% saturation gradient along the core (0.1 mL/min for the homogeneous core and 0.5 mL/min for the high contrast model).

_{c}) needs to be included in the calculation. This concept may apply to core flood experiments and give us a more reliable relative permeability. However, capillary pressure gradients in general are not measured in the experiment. It is possible to estimate capillary pressure gradients based on the average saturation values at the inlet and outlet slices of the core. Once saturations at the ends of the core are measured (e.g., using X-ray CT scanning), the corresponding capillary pressure values can be estimated from independently measured capillary pressure curves:

_{c,inlet}= P

_{c}(S

_{inlet}), P

_{c,outlet}= P

_{c}(S

_{outlet}).

_{c}= P

_{c}(S

_{inlet}) − P

_{c}(S

_{outlet}).

#### 3.3. Sensitivity Studies for Different Core Properties

#### 3.3.1. Effects of Heterogeneity

_{cv}ranging from 10 to 10

^{5}. Based on the previous results, it is reasonable to hypothesize that reliable relative permeability curves can be obtained as long as the saturation is relatively uniform. To validate this conclusion, we use the homogeneous core and the four heterogeneous models to obtain the relative permeability curves calculated based on the true pressure drops in each phase (Figure 17). The permeability heterogeneity of Random 2 and Random 3 models are generated based on a random log-normal distribution with a standard deviation of σ

_{lnk’}= 0.254 and 1.42, respectively. Kozeny–Carman (KC) and high contrast (HC) models are the other type of heterogeneity distribution generated using a porosity-based approach, which has already been described earlier. The injection flow rates are chosen to be higher than the minimum flow rates for these five cases to make sure the system reaches the viscous-dominated regime (Table 3). The higher degree of heterogeneity results in higher minimum rates.

_{rw}is almost identical when the heterogeneity factor σ

_{lnk’}< 1.42, which implies that the effective relative permeability to water k

_{rw}is not as sensitive as the k

_{rg}to the small-scale heterogeneity.

_{lnk’}are very close (0.254 and 0.275, respectively).

#### 3.3.2. Effects of Core Length (15.24–45.72 cm)

_{2}saturations as a function of capillary numbers for the three different core lengths (L, 2L, and 3L) are shown in the LHS of Figure 18. The aspect ratios R

_{l}are 3.14, 6.29, and 9.43, respectively. The starred points at the LHS represent 0.1 mL/min injection flow rate, while the RHS illustrates the corresponding relative permeability. The simulation results show that, even with up to 15% saturation gradient, we can still obtain the intrinsic relative permeability for different lengths of homogeneous cores.

#### 3.3.3. Effects of Interfacial Tension (7.49–67.41 mN/m)

_{2}/brine systems has been reported in the literature [28,33,39]. Simulations covering a wide range of interfacial tension (IFT) values, which are purely hypothetical, are used solely to explore the sensitivity of relative permeability measurements to IFT values.

_{cv}corresponds to smaller IFT values, as smaller IFT values reduce capillary pressure, and hence reduce displacement capillary pressure p

_{c}*.

#### 3.3.4. Effects of Gravity

^{2}).

_{cv}are smaller than the critical values. It is verified that the effect of gravity due to the density difference between two fluids and the long core is small in the viscous-dominated regime, as mentioned before. However, even without considering gravity in the simulation, flow rate dependency is observed in the heterogeneous cores.

_{c,min}< P

_{c}< P

_{c,max}). This could explain why the flow rate dependency is still observed in the heterogeneous cores even without considering gravity in the simulation.

## 4. Discussion

#### 4.1. Observations from the Numerical and Semi-Analytical Models

_{2}causes lower displacement efficiency and results in a vertical saturation gradient, which leads to the deviations of relative permeability values observed in Figure 12 and Figure 14. In this regime, gravity not only causes the inaccuracy of relative permeability values, but also results in large flow rate dependency. The average saturation over a wide range of injection rates for the homogeneous and two porosity-based permeability cores as well as four random log-normal distribution permeability cores are shown in Figure 21. The simulation results show that capillary heterogeneity will increase this flow rate dependency in the transition regime, and reduce the average saturation in the viscous dominated regime.

_{cv}to reach the viscous-dominated regime, and hence to obtain the reliable relative permeability data (Figure 21).

#### 4.2. Conditions for Reliable Effective Relative Permeability Measurements

- (i)
- If the core is known as relatively homogeneous (τ ~ 1 and S
_{BL}^{Hete}~ S_{BL})

_{gv}= N

_{cv}N

_{B}(Equations (1)–(3)), Equation (19) can also be presented as follows:

_{critical}(Equations (21) and (22)) in order to enter the viscous-dominated regime:

- (ii)
- If the core is very heterogeneous,

_{2}pressure drop across the core. Equation (31) implies that the first transition point occurs when ${\left({\mathsf{\Delta}\mathrm{P}}_{\mathrm{CO}2}\right)}_{\mathrm{critical}}^{\mathrm{Hete}}=\Delta \mathsf{\rho}\mathrm{gL}\mathsf{\tau}$ for the heterogeneous cores. It is very important as it provides the theoretical basis that the dimensionless parameter $\mathsf{\tau}$ could be quantified from the known properties and the experimental measurement.

#### 4.3. Permeability Heterogeneity Parameter τ

_{lnk’}:

_{lnk’}is the standard deviation of ln(k

_{i}/k

_{mean}). For the homogeneous core, σ

_{lnk’}= 0 and we should obtain τ = 1, which will be consistent with the previous work [60]. In this study, the function ${\mathrm{f}(\mathsf{\sigma}}_{\mathrm{lnk}\u2019})$ is assumed to only be dependent on σ

_{lnk’}, which requires detailed information on the rock (but could be estimated based on the lithology and stratigraphy). However, we can expect that weak (but highly correlated) heterogeneities have an equally strong (if not stronger) influence as strong (but randomly oriented) heterogeneity. Therefore, the permeability heterogeneity parameter τ should be a function of the strength of heterogeneity as well as the spatial correlation. Further investigation (by means of, e.g., variograms in each spatial direction or the correlation structure of the permeability) should be attempted to discuss how the length scale of these heterogeneities may affect the permeability heterogeneity parameter τ. In addition, sensitivity studies for different degrees of heterogeneity will be required to generalize the results to a wide range of conditions.

#### 4.4. Initial Guess of Critical Flow Rate

_{BL}, k

_{rCO2}(S

_{BL}). S

_{BL}is a constant saturation, derived from the Buckley–Leverett theory, which neglects gravity and capillary effects.

_{rCO2}(S

_{BL}) is unavailable before we perform the relative permeability measurements, we know k

_{r}

**(S**

_{CO2}_{BL}) is always smaller than or equal to 1; therefore, Equations (33) and (34) provides the upper bounds and the lower bounds of first critical number and injection flow rate for the homogeneous core, which is also useful for the design of core flood experiments.

_{l}. The more information we know about the core, the larger the N

_{cv}(Equation (19)) (and hence the smaller the injection flow rate (Equation (22)) that can be tolerated to obtain reliable relative permeability data. The upper bound q

_{c,max}calculated based on Equation (35) is about 3.52 mL/min, which is indeed larger than the ${\mathrm{q}}_{\mathrm{critical}}$ = 0.3 for homogeneous cores and 1.2 for the high contrast model (Table 4). Therefore, the upper bound of the critical flow rate ${\mathrm{q}}_{\mathrm{c},\mathrm{max}}$ for the homogeneous core would probably be a good initial guess of critical flow rates for the very heterogeneous core.

#### 4.5. Practical Application

- Conduct steady-state drainage core-flooding experiment with initial injection rate q
_{c,max}(Equation (35)). - $\Delta \mathsf{\rho}\mathrm{gL}$ can be calculated based on the design of experiment.
- Equation (25) would give the lower bound of ${\mathsf{\Delta}\mathrm{P}}_{\mathrm{CO}2}$=$\Delta \mathsf{\rho}\mathrm{gL}$.
- Change the initial rate up and down, and measure the corresponding core pressure drop of CO
_{2}(${\mathsf{\Delta}\mathrm{P}}_{\mathrm{CO}2}$) and average CO_{2}saturation for each injection rate. - ${\left({\mathsf{\Delta}\mathrm{P}}_{\mathrm{CO}2}\right)}_{\mathrm{critical}}^{\mathrm{Hete}}$ could be obtained once saturation becomes constant.
- The permeability heterogeneity parameter τ could be obtained based on Equation (31).

## 5. Conclusions

_{2}/brine systems. We now summarize this part of the work:

- Despite the presence of heterogeneity, it is possible to obtain the accurate effective relative permeability measurements for heterogeneous cores. The incomplete fluid displacement is primarily due to the heterogeneity and unfavorable mobility ratio, not gravity segregation, but with a sufficiently high flow rate, these effects can be overcome.
- The critical flowrate for making these accurate measurements was identified based on the properties of the core, and most notably on heterogeneity (Equations (22) and (28)). Increasing the flow rate results in minimizing the saturation gradient caused by the combined effects of capillary, viscous, and gravity forces; hence, the relative permeability approaches the maximum value asymptotically and stabilizes when the uniform saturation is achieved.
- The simulation results shown here indicate that the flow-rate dependent saturation occurs not only in the heterogeneous core, but also in homogeneous cores. In addition, we show that the capillary heterogeneity will increase the flow-rate dependency.

_{2}scenarios (particularly in the post-injection phase). In addition, as no chemical reactions were considered in the simulation or the theoretical analysis, the results drawn from here may not apply to carbonate systems, which have significant chemical reactions between the fluids and rock.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | cross-section area of the core [m^{2}] | f | fractional flow |

H | height of the core [m] | g | acceleration [m/s^{2}] |

L | length of the core [m] | k | average permeability [md] |

R_{l} | aspect ratio, L/H | k_{r} | relative permeability |

N_{B} | Bond number, ΔρgH/p_{c}* | q | volumetric flow rate [mL/min] |

N_{cv} | capillary number, ${\mathrm{k}}_{\mathrm{eff}}$_{n}Lp_{c}*/H^{2}μ_{g} u_{t} | p | pressure [Pa] |

N_{gv} | gravity number, Δρg${\mathrm{k}}_{\mathrm{eff}}$L/Hμ_{g} u_{t} | p_{c}* | characteristic capillary pressure [Pa] |

Δρ | density difference between CO_{2} and brine [kg/m^{3}] | u | Darcy velocity [m/s] |

μ | viscosity [cp] | θ | contact angle, 0° |

φ | porosity | τ | heterogeneity parameter |

σ | CO_{2}–brine interfacial tension [N/m] or standard deviation | ||

ΔP | pressure difference between the average inlet and the outlet slice values |

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**Figure 2.**3D numerical modeling and 2D theoretical analysis for homogeneous and heterogeneous models.

**Figure 3.**Average CO

_{2}saturation as a function of gravity number N

_{gv}(LHS (left hand side)) and capillary number N

_{cv}(RHS (right hand side)) for homogeneous and high contrast models, respectively, for different Bond numbers.

**Figure 5.**Capillary pressure and saturation distributions for the high contrast model at an extremely high flow rate (200 mL/min) and high flow rate (6 mL/min), respectively.

**Figure 7.**The experimental steady-state three-dimensional views of CO

_{2}saturation in the core for a given fractional flow of CO

_{2}at a given flow rate. The fluids were injected from right to left [16].

**Figure 9.**CO

_{2}saturation along the Berea Sandstone core for different fractional flows of CO

_{2}at a total injection flow rate 2.6 mL/min: (

**a**) experimental results [67]; (

**b**) high contrast model with boundary condition P

_{c}= 0; and (

**c**) high contrast model with boundary condition dP

_{c}/dx = 0.

**Figure 10.**CO

_{2}saturation distribution at steady state for 95% fractional flow of CO

_{2}at a total injection flow rate 1.2 mL/min.

**Figure 11.**Flow rate effect on CO

_{2}saturation along the homogeneous core at a 95% fractional flow of CO

_{2}with flow rates ranging from 0.1 mL/min to 6 mL/min.

**Figure 12.**Relative permeability calculated by the same pressure drop (ΔP

_{w}= ΔP

_{CO2}) for the homogeneous core with 430 md permeability at different flow rates.

**Figure 13.**The effect of heterogeneity on CO

_{2}saturation along the core at a fractional flow of 95% over a wide range of flow rates.

**Figure 14.**Relative permeability calculated by the same pressure drop (ΔP

_{w}= ΔP

_{CO2}) for the high contrast model (σ

_{lnk’}= 0.96) with 318 md permeability at different flow rates; (RHS) the same relative permeability curves at log scale.

**Figure 15.**Flow rate effect on relative permeability calculated by the true pressure drops (${\mathsf{\Delta}\mathrm{P}}_{\mathrm{w}}={\mathsf{\Delta}\mathrm{P}}_{\mathrm{CO}2}-{\mathsf{\Delta}\mathrm{P}}_{\mathrm{c}}$) for the homogeneous and heterogeneous core at various flow rates.

**Figure 16.**The experimental data and relative permeability calculated by the corrected pressure drops for high contrast models at various flow rates; (RHS) the same relative permeability curves at log scale.

**Figure 17.**Relative permeability calculated by the true pressure drops for five different heterogeneous cores in the viscous-dominated regimes: homogeneous and the Random 2 cores, Kozeny–Carman models, high contrast models, the Random 3 cores, and the input relative permeability curves.

**Figure 18.**(LHS) Brine displacement efficiencies for three different lengths of homogeneous core with capillary number ranging from 10 to 10

^{7}; (RHS) relative permeability calculated by the true pressure drops for homogeneous cores at 0.1 mL/min flow rates.

**Figure 19.**(LHS) Average CO

_{2}saturation as a function of capillary number N

_{cv}for the homogeneous and high contrast models with three different values of interfacial tensions; (RHS) interfacial tension effects on relative permeability calculated in the true pressure drops for heterogeneous core (high contrast model) at 6 mL/min flow rates.

**Figure 20.**Average CO

_{2}saturation as a function of capillary number N

_{cv}for the homogeneous, Kozeny–Carman (KC) (small heterogeneity), and high contrast (HC) (large heterogeneity) models with and without gravity (1G/0G).

**Figure 21.**Average CO

_{2}saturation as a function of capillary number N

_{cv}for homogeneous and different heterogeneous models.

σ_{lnk’} | Porosity | Permeability (md) | Capillary Pressure (Pa) | Input Relative Permeability | |
---|---|---|---|---|---|

Homogeneous Model | 0 | φ_{i} = φ_{mean}_{ } | k_{i} = k_{mean}_{ } | Measured P_{c} Curve | power-law functions |

Kozeny–Carman Model | 0.27 | φ_{i}_{ } | ${\mathrm{k}}_{\mathrm{i}}\propto $ φ_{i}^{3}/(1 − φ_{i})^{2}^{ } | ${\mathrm{P}}_{\mathrm{c},\mathrm{i}}\propto $$\sqrt{{\mathsf{\phi}}_{\mathrm{i}}/{\mathrm{k}}_{\mathrm{i}}}$ | power-law functions |

High Contrast Model | 0.96 | φ_{i}_{ } | ${\mathrm{k}}_{\mathrm{i}}\propto $ exp(64φ_{i}^{4}) | ${\mathrm{P}}_{\mathrm{c},\mathrm{i}}\propto $$\sqrt{{\mathsf{\phi}}_{\mathrm{i}}/{\mathrm{k}}_{\mathrm{i}}}$ | power-law functions |

Inlet Slice | Rock Slices (29 Slices) | Outlet Slice |
---|---|---|

φ_{mean}, | φ_{i}, | φ_{mean}, |

k_{mean}: Anisotropic | k_{i}: Isotropic | k_{mean}: Isotropic |

(k_{z} = k_{y} = 100k_{x}) | ${\mathrm{P}}_{\mathrm{c},\mathrm{i}}\propto \sqrt{{\mathsf{\phi}}_{\mathrm{i}}/{\mathrm{k}}_{\mathrm{i}}}$ | Dirichlet boundary condition |

P_{c} = P_{c,mean} | dP_{c}/dx = 0 |

**Table 3.**Summary of different flow rates for different heterogeneous cores: homogeneous, Random 2, Kozeny–Carman (KC), high contrast (HC), and Random 3 models.

Degree of Heterogeneity σ _{lnk’} | Injection Flow Rate q, mL/min | q_{critical}, mL/min | Regime | |
---|---|---|---|---|

Homo | 0 | 0.5 | 0.24 | Viscous-dominated |

Random 2 | 0.254 | 0.5 | 0.25 | Viscous-dominated |

KC | 0.275 | 1.2 | 0.37 | Viscous-dominated |

HC | 0.96 | 2.6 | 0.97 | Viscous-dominated |

Random 3 | 1.42 | 6 | 1.19 | Viscous-dominated |

S_{BL}/S _{BL}^{Hete} | k_{rCO2}(S_{BL})/k _{rCO2}(S_{BL}^{Hete}) | q_{critical} [mL/min] from the Simulation Results | |
---|---|---|---|

Homogeneous core | 0.324 | 0.0554 | around 0.3 |

High contrast model | 0.30 | 0.0483 | around 1.2 |

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Kuo, C.-W.; Benson, S.M.
Reliability of Relative Permeability Measurements for Heterogeneous Rocks Using Horizontal Core Flood Experiments. *Sustainability* **2021**, *13*, 2744.
https://doi.org/10.3390/su13052744

**AMA Style**

Kuo C-W, Benson SM.
Reliability of Relative Permeability Measurements for Heterogeneous Rocks Using Horizontal Core Flood Experiments. *Sustainability*. 2021; 13(5):2744.
https://doi.org/10.3390/su13052744

**Chicago/Turabian Style**

Kuo, Chia-Wei, and Sally M. Benson.
2021. "Reliability of Relative Permeability Measurements for Heterogeneous Rocks Using Horizontal Core Flood Experiments" *Sustainability* 13, no. 5: 2744.
https://doi.org/10.3390/su13052744