TY - JOUR
AU - Ghaedi,, Mojtaba
AB - Abstract Spontaneous imbibition is the main oil production mechanism in the water invaded zone of a naturally fractured reservoir (NFR). Different scaling equations have been presented in the literature for upscaling of core scale imbibition recovery curves to field scale matrix blocks. Various scale dependent parameters such as gravity effects and boundary influences are required to be considered in the upscaling process. Fluid flow from matrix blocks to the fracture system is highly dependent on the permeability value in the horizontal and vertical directions. The purpose of this study is to include permeability anisotropy in the available scaling equations to improve the prediction of imbibition assisted oil production in NFRs. In this paper, a commercial reservoir simulator was used to obtain imbibition recovery curves for different scenarios. Then, the effect of permeability anisotropy on imbibition recovery curves was investigated, and the weakness of the existing scaling equations for anisotropic rocks was demonstrated. Consequently, an analytical shape factor was introduced that can better scale all the curves related to anisotropic matrix blocks. scaling, imbibition, shape factor, permeability anisotropy 1. Introduction A considerable percentage of oil and gas reserves are trapped in naturally fractured reservoirs (NFRs). In this kind of reservoirs, capillary and gravity are the main driving forces in oil and water flow between the matrix and the fracture (Prey and Lefebvre 1978). Although, in water invaded zones oil flow is mainly dominated by capillary forces, gravity forces could also be effective (Abbasi et al2016b). Several factors, such as the wetting phase and nonwetting phase viscosity (Mason et al2010), surface tension, porosity, permeability, size of matrix block, initial water saturation, fluid density and boundary conditions affect the imbibition process in NFRs (Mattax and Kyte 1962, Abbasi et al2016a). Different research studies have been performed on scaling counter-current imbibition recovery curves. Counter-current imbibition occurs when oil and water flow in opposite directions (Ruth et al2007). Rapoport (1955) presented a scaling equation for two-phase incompressible immiscible flow as follows: tD,R=kdpcdswuμwL, 1 where tD,R is the dimensionless time, k is the absolute permeability, u is the Darcy velocity, μw is the wetting phase viscosity, pc is the capillary pressure, L is the length of the matrix block and Sw is the saturation of wetting phase. Mattax and Kyte (1962) performed their experiments with sandstone samples and arrived at the following scaling equation: tD,MK=tkφσμw1L2, 2 where tD,MK is the dimensionless time, σ is the interfacial tension, φ is the porosity, and t is time. This equation does not include the effect of factors like nonwetting phase viscosity, permeability anisotropy, the shape of the sample, the effectiveness of the driving forces (i.e. gravity and imbibition) and boundary conditions. Kazemi et al (1992) modified the equation of Mattax and Kyte (1962) by substituting L with Ls which is a characteristic Kazemi–Gilman–Elsharkawy (KGE) length and is defined as: LS=1FS,KGE, 3 where FS,KGE is the shape factor. The shape factor, which includes the effects of size, shape and boundary conditions of the samples can be defined as: FS,KGE=1Vm∑Amdm, 4 where Vm is volume of the matrix, Am is area of a surface open to flow and dm is distance from the open surface to the center of block and summation is for all surfaces open to flow. Hence, the scaling equation is defined as follows: tD,KGE=tkφσμw1Ls2=tkφσFS,KGEμw, 5 where tD,KGE is the dimensionless time. Ma et al (1997) changed the definition of Kazemi et al (1992) and presented the shape factor as: FS,M=1Vm∑Amlm, 6 where FS,M is the shape factor and lm is the distance from an open surface to the no-flow boundary. Lc=1FS,M, 7 where Lc is the characteristic length. Hence, Ma et al (1997) introduced the following equation for the scaling equation of a counter-current flow: tD,M=tkφσμwμnw1Lc2, 8 where tD,M is the dimensionless time and μnw is the nonwetting phase viscosity. In the previously presented equations, matrix permeability is assumed to be the same in different directions. Permeability anisotropy, which is common in rock formations, is an important factor that influences oil production performance from matrix blocks. Permeability anisotropy is mainly a result of the orientation of both the mineral grains and the pores (Lewis 1988). Different researchers such as Clavaud et al2008, Ghous et al2005, Meyer and Krause 2006, Rasolofosaon and Zinszner 2002 have focused on various aspects of this factor. The ratio between the vertical and horizontal permeability values (kv/kh) is an indicator to represent the anisotropy (Widarsono et al2006). To solve the problem of scaling imbibition curves in case of permeability anisotropic rocks, Mirzaei-Paiaman and Masihi (2013) suggested the characteristic permeability is as follows: kc=∏j=1nkj1/n, 9 where n is the total number of flow directions. In anisotropic media, the k value in scaling equations (such as equation (8)) should be replaced by kcor its equivalent average absolute permeability (kavg). In this paper, the KGE shape factor is modified in such a way to consider the permeability anisotropy in scaling imbibition recovery curves. The effect of permeability anisotropic curves is also analyzed. To verify the applicability of the new shape factor, it was utilized to scale different anisotropic curves in defined scenarios. In the following section, by extension of the Kazemi et al (1992) method, a new shape factor including permeability anisotropy is developed. Then, numerical simulation models are used to analyze the new proposed shape factor. Finally, the ability of the new proposed shape factor is discussed and the results are analyzed. 2. Modeling 2.1. Deriving a new shape factor The mass conservation law for immiscible compressible two-phase flow can be defined as (Heinemann and Mittermeir 2012): div(ρu)p=-∂(φSpρp)∂t. 10 In this equation, the subscript ‘p’ refers to the type of existing phase (for instance, w and o stand for water and oil, respectively), ρ is fluid density, u is the phase velocity, φ is the porosity and sp is the phase saturation, where ∑p=w,oSp=1. 11 Based on the definition of Barenblatt et al (1960), the term of the interflow for a dual continuum system was introduced as follows: div(ρu)fp-qpmf=-∂(φSpρp)f∂t 12 div(ρu)mp+qpmf=-∂(φSpρp)m∂t, 13 where the subscripts m and f stand for matrix and fracture, respectively. Also, qpmf is the matrix-fracture fluid exchange. By assuming a pseudo-steady-state flow from the matrix block to the fracture the exchange term is introduced as follows: qpmf=FSkavgMP(φf-φm)p, 14 where kavg is the average permeability, φ is the potential and MP is the mass mobility, defined as: MP=ρPλP=ρPkrpμp, 15 where λP is the phase mobility, krp is the phase relative permeability and μp is the phase viscosity. By assuming all faces of the matrix to be open (a, b and c in figure 1), the total matrix-fracture exchange flow rate can be calculated using Darcy's law: Qpmf=2Qpx+2Qpy+2Qpz=2bca/2kx+abc/2×ky+acb/2kzMP(φf-φm)p 16 Qpmf=4abc1a2kx+1c2ky+1b2kzMP(φf-φm)p. 17 Qpmf can be exchanged to volumetric rate as below: qpmf=Qpmf/Vm, 18 where Vm=abc. By comparing equation (14) with equation (17) the following shape factor can be obtained: FS,newkavg=41a2kx+1c2ky+1b2kz 19 FS,new=41a2kx+1c2ky+1b2kz/kavg. 20 To implement the effect of permeability anisotropy in scaling of imbibition curves, equation (20) can be used in scaling of equations presented in the literature. Figure 1. View largeDownload slide Schematic of an idealized matrix block. Fluid exchanges between matrix and fracture on all six sides, as shown in equation (16). Figure 1. View largeDownload slide Schematic of an idealized matrix block. Fluid exchanges between matrix and fracture on all six sides, as shown in equation (16). 2.2. Numerical simulation Imbibition recovery curves can be obtained using core plugs or numerical simulation methods. Since core plugs are limited in size and properties, the numerical simulation approach is more applicable. A commercial reservoir simulator was used to generate the imbibition recovery curves. To reduce the sensitivity of the recovery curves to the number of grid cells, for each imbibition scenario, a grid sensitivity analysis was performed and the best matrix grid numbers were selected. To simulate the imbibition processes, the matrix is also assumed to be surrounded by a high permeability fracture model. Figure 2 shows a single matrix-fracture model. As this figure shows, all of the faces of the matrix are open to flow. All of the faces of matrix-fracture scenarios that are used in this paper are kept open to flow. This results in counter-current imbibition flow. Figure 2. View largeDownload slide Schematic of the static model (sliced) used in the simulation of imbibition flow between the matrix and fracture. Permeability values are 500 mD for the matrix and 20 D for the fracture. The red color cells show the fracture system and the blue color cells show the matrix system. Figure 2. View largeDownload slide Schematic of the static model (sliced) used in the simulation of imbibition flow between the matrix and fracture. Permeability values are 500 mD for the matrix and 20 D for the fracture. The red color cells show the fracture system and the blue color cells show the matrix system. Constant rock and fluid properties for different imbibition scenarios are listed in table 1. Also, the capillary pressure curve and relative permeability curves are shown in figures 3 and 4, respectively. Table 1. Rock and fluid properties used in the simulation. Parameter Value Unit Porosity 0.1 fraction Nonwetting phase viscosity 1.5 cP Wetting phase viscosity 1 cP Average capillary pressure 11.07 psi Maximum capillary pressure 27.02 psi Water density 60 lb ft-3 Initial water saturation 0.22 fraction Initial oil saturation 0.78 fraction Average relative water permeability 0.1 fraction Average relative oil permeability 0.30 fraction Parameter Value Unit Porosity 0.1 fraction Nonwetting phase viscosity 1.5 cP Wetting phase viscosity 1 cP Average capillary pressure 11.07 psi Maximum capillary pressure 27.02 psi Water density 60 lb ft-3 Initial water saturation 0.22 fraction Initial oil saturation 0.78 fraction Average relative water permeability 0.1 fraction Average relative oil permeability 0.30 fraction View Large Table 1. Rock and fluid properties used in the simulation. Parameter Value Unit Porosity 0.1 fraction Nonwetting phase viscosity 1.5 cP Wetting phase viscosity 1 cP Average capillary pressure 11.07 psi Maximum capillary pressure 27.02 psi Water density 60 lb ft-3 Initial water saturation 0.22 fraction Initial oil saturation 0.78 fraction Average relative water permeability 0.1 fraction Average relative oil permeability 0.30 fraction Parameter Value Unit Porosity 0.1 fraction Nonwetting phase viscosity 1.5 cP Wetting phase viscosity 1 cP Average capillary pressure 11.07 psi Maximum capillary pressure 27.02 psi Water density 60 lb ft-3 Initial water saturation 0.22 fraction Initial oil saturation 0.78 fraction Average relative water permeability 0.1 fraction Average relative oil permeability 0.30 fraction View Large Figure 3. View largeDownload slide Oil/water capillary pressure versus water saturation for a water-wet case. Figure 3. View largeDownload slide Oil/water capillary pressure versus water saturation for a water-wet case. Figure 4. View largeDownload slide Oil and water relative permeability versus water saturation for a water-wet case. Figure 4. View largeDownload slide Oil and water relative permeability versus water saturation for a water-wet case. In this study, capillary pressure versus wetting phase saturation are averaged using the following weighted arithmetic meaning method: pc,avg=∑i=1n1Swipci∑i=1n1Swi. 21 Also, to investigate the effect of using different permeability averaging methods, two different methods have been used. In the first averaging method, by assuming two flow directions in equation (9), the geometric average of matrix permeability in the x and y directions (kh=kxky) and vertical permeability values have been used: kavg=khkv. 22 In the second method, by assuming three flow directions in equation (9), both permeabilities in the x and y directions have been considered in averaging the equation: kavg=kxkykz3. 23 Interfacial tension is defined as follows: σ=pc2kφ. 24 2.3. Results and discussion With regards to the importance of considering anisotropic matrix permeability values in the imbibition process, in this section, different imbibition scenarios with a wide range of matrix permeability and matrix dimension values have been defined. Table 2 introduces the scenarios investigated in this study. In cases 27 and 28, gravity forces have been disabled (by considering negligible density differences) and only capillary forces are considered to influence the imbibition flow. Table 2. Important characteristics of different scenarios used to investigate the effect of permeability anisotropy on scaling of imbibition curves. Case number Matrix dimensions (ft3) Kx(mD) Ky(mD) Kz(mD) kavg = khkv (mD) kavg = kxkykz3 (mD) Oil density (lb ft-3) A1 30 × 30 × 50 400 400 4 40 86.18 48 A2 30 × 30 × 90 40 40 40 40 40 48 A3 30 × 30 × 90 400 400 4 40 86.18 48 A4 50 × 50 × 120 20 20 20 20 20 48 A5 50 × 50 × 50 40 40 40 40 40 48 A6 50 × 50 × 50 400 400 4 40 86.18 48 A7 50 × 50 × 50 30 30 30 30 30 48 A8 50 × 50 × 50 300 300 3 30 64.64 48 A9 30 × 30 × 50 40 40 40 40 40 48 A10 1 × 1 × 50 40 40 40 40 40 48 A11 1 × 1 × 50 4 4 400 40 18.57 48 A12 20 × 20 × 20 40 40 40 40 40 48 A13 20 × 20 × 20 400 400 4 40 86.18 48 A14 30 × 30 × 30 40 40 40 40 40 48 A15 30 × 30 × 30 400 400 4 40 86.18 48 A16 10 × 10 × 10 40 40 40 40 40 48 A17 10 × 10 × 10 400 400 4 40 86.18 48 A18 90 × 90 × 90 400 400 4 40 86.18 48 A19 1 × 1 × 10 40 40 40 40 40 48 A20 1 × 1 × 10 4 4 400 40 18.57 48 A21 1 × 1 × 20 4 4 400 40 18.57 48 A22 1 × 1 × 30 40 40 40 40 40 48 A23 1 × 1 × 30 4 4 400 40 18.57 48 A24 1 × 1 × 90 40 40 40 40 40 48 A25 1 × 1 × 90 4 4 400 40 18.57 48 A26 1 × 1 × 1 40 40 40 40 40 48 A27 50 × 50 × 50 40 40 40 40 40 60 A28 50 × 50 × 50 400 400 4 40 86.18 60 A29 30 × 30 × 10 40 40 40 40 40 48 A30 30 × 30 × 10 400 400 4 40 86.18 48 Case number Matrix dimensions (ft3) Kx(mD) Ky(mD) Kz(mD) kavg = khkv (mD) kavg = kxkykz3 (mD) Oil density (lb ft-3) A1 30 × 30 × 50 400 400 4 40 86.18 48 A2 30 × 30 × 90 40 40 40 40 40 48 A3 30 × 30 × 90 400 400 4 40 86.18 48 A4 50 × 50 × 120 20 20 20 20 20 48 A5 50 × 50 × 50 40 40 40 40 40 48 A6 50 × 50 × 50 400 400 4 40 86.18 48 A7 50 × 50 × 50 30 30 30 30 30 48 A8 50 × 50 × 50 300 300 3 30 64.64 48 A9 30 × 30 × 50 40 40 40 40 40 48 A10 1 × 1 × 50 40 40 40 40 40 48 A11 1 × 1 × 50 4 4 400 40 18.57 48 A12 20 × 20 × 20 40 40 40 40 40 48 A13 20 × 20 × 20 400 400 4 40 86.18 48 A14 30 × 30 × 30 40 40 40 40 40 48 A15 30 × 30 × 30 400 400 4 40 86.18 48 A16 10 × 10 × 10 40 40 40 40 40 48 A17 10 × 10 × 10 400 400 4 40 86.18 48 A18 90 × 90 × 90 400 400 4 40 86.18 48 A19 1 × 1 × 10 40 40 40 40 40 48 A20 1 × 1 × 10 4 4 400 40 18.57 48 A21 1 × 1 × 20 4 4 400 40 18.57 48 A22 1 × 1 × 30 40 40 40 40 40 48 A23 1 × 1 × 30 4 4 400 40 18.57 48 A24 1 × 1 × 90 40 40 40 40 40 48 A25 1 × 1 × 90 4 4 400 40 18.57 48 A26 1 × 1 × 1 40 40 40 40 40 48 A27 50 × 50 × 50 40 40 40 40 40 60 A28 50 × 50 × 50 400 400 4 40 86.18 60 A29 30 × 30 × 10 40 40 40 40 40 48 A30 30 × 30 × 10 400 400 4 40 86.18 48 View Large Table 2. Important characteristics of different scenarios used to investigate the effect of permeability anisotropy on scaling of imbibition curves. Case number Matrix dimensions (ft3) Kx(mD) Ky(mD) Kz(mD) kavg = khkv (mD) kavg = kxkykz3 (mD) Oil density (lb ft-3) A1 30 × 30 × 50 400 400 4 40 86.18 48 A2 30 × 30 × 90 40 40 40 40 40 48 A3 30 × 30 × 90 400 400 4 40 86.18 48 A4 50 × 50 × 120 20 20 20 20 20 48 A5 50 × 50 × 50 40 40 40 40 40 48 A6 50 × 50 × 50 400 400 4 40 86.18 48 A7 50 × 50 × 50 30 30 30 30 30 48 A8 50 × 50 × 50 300 300 3 30 64.64 48 A9 30 × 30 × 50 40 40 40 40 40 48 A10 1 × 1 × 50 40 40 40 40 40 48 A11 1 × 1 × 50 4 4 400 40 18.57 48 A12 20 × 20 × 20 40 40 40 40 40 48 A13 20 × 20 × 20 400 400 4 40 86.18 48 A14 30 × 30 × 30 40 40 40 40 40 48 A15 30 × 30 × 30 400 400 4 40 86.18 48 A16 10 × 10 × 10 40 40 40 40 40 48 A17 10 × 10 × 10 400 400 4 40 86.18 48 A18 90 × 90 × 90 400 400 4 40 86.18 48 A19 1 × 1 × 10 40 40 40 40 40 48 A20 1 × 1 × 10 4 4 400 40 18.57 48 A21 1 × 1 × 20 4 4 400 40 18.57 48 A22 1 × 1 × 30 40 40 40 40 40 48 A23 1 × 1 × 30 4 4 400 40 18.57 48 A24 1 × 1 × 90 40 40 40 40 40 48 A25 1 × 1 × 90 4 4 400 40 18.57 48 A26 1 × 1 × 1 40 40 40 40 40 48 A27 50 × 50 × 50 40 40 40 40 40 60 A28 50 × 50 × 50 400 400 4 40 86.18 60 A29 30 × 30 × 10 40 40 40 40 40 48 A30 30 × 30 × 10 400 400 4 40 86.18 48 Case number Matrix dimensions (ft3) Kx(mD) Ky(mD) Kz(mD) kavg = khkv (mD) kavg = kxkykz3 (mD) Oil density (lb ft-3) A1 30 × 30 × 50 400 400 4 40 86.18 48 A2 30 × 30 × 90 40 40 40 40 40 48 A3 30 × 30 × 90 400 400 4 40 86.18 48 A4 50 × 50 × 120 20 20 20 20 20 48 A5 50 × 50 × 50 40 40 40 40 40 48 A6 50 × 50 × 50 400 400 4 40 86.18 48 A7 50 × 50 × 50 30 30 30 30 30 48 A8 50 × 50 × 50 300 300 3 30 64.64 48 A9 30 × 30 × 50 40 40 40 40 40 48 A10 1 × 1 × 50 40 40 40 40 40 48 A11 1 × 1 × 50 4 4 400 40 18.57 48 A12 20 × 20 × 20 40 40 40 40 40 48 A13 20 × 20 × 20 400 400 4 40 86.18 48 A14 30 × 30 × 30 40 40 40 40 40 48 A15 30 × 30 × 30 400 400 4 40 86.18 48 A16 10 × 10 × 10 40 40 40 40 40 48 A17 10 × 10 × 10 400 400 4 40 86.18 48 A18 90 × 90 × 90 400 400 4 40 86.18 48 A19 1 × 1 × 10 40 40 40 40 40 48 A20 1 × 1 × 10 4 4 400 40 18.57 48 A21 1 × 1 × 20 4 4 400 40 18.57 48 A22 1 × 1 × 30 40 40 40 40 40 48 A23 1 × 1 × 30 4 4 400 40 18.57 48 A24 1 × 1 × 90 40 40 40 40 40 48 A25 1 × 1 × 90 4 4 400 40 18.57 48 A26 1 × 1 × 1 40 40 40 40 40 48 A27 50 × 50 × 50 40 40 40 40 40 60 A28 50 × 50 × 50 400 400 4 40 86.18 60 A29 30 × 30 × 10 40 40 40 40 40 48 A30 30 × 30 × 10 400 400 4 40 86.18 48 View Large In the above defined scenarios, some permeability values are not in the usual range of reported matrix permeability. Also, horizontal permeability (kh) is usually larger than vertical permeability (kv). But, in this work, we have intentionally neglected this fact to better analyze the generality of the proposed shape factor. Figure 5 shows the imbibition recovery curves for the different scenarios in table 2. The maximum horizontal distance between the curves are more than three orders of magnitude, which indicates that the cases in table 2 are within a wide range of properties. Figure 5. View largeDownload slide Imbibition recovery curves before scaling of the scenarios in table 2. The maximum horizontal distance between the curves is more than three orders of magnitude. Figure 5. View largeDownload slide Imbibition recovery curves before scaling of the scenarios in table 2. The maximum horizontal distance between the curves is more than three orders of magnitude. A scaling equation needs to be used to examine the ability of the new shape factor in scaling the imbibition curves of the anisotropic matrix blocks. In this work, the scaling equation presented by Ma et al (1997) has been used. Figure 6 shows the scaled imbibition curves when the KGE shape factor and equation (22) for averaging permeabilities have been used. This figure indicates that using average permeability instead of neglecting anisotropic permeability values may result in high error values in scaling the imbibition curves. Figure 7 shows the scaled imbibition recovery curves after using the new shape factor (equation (20)) and equation (22) as the permeability averaging method. It can be concluded that the new shape factor scales anisotropic imbibition curves successfully. Figure 6. View largeDownload slide The scaled imbibition recovery curves scaled using the KGE shape factor (equation (4)) and equation (22) as the permeability averaging method. The curves are scaled to three different classes of curves. Figure 6. View largeDownload slide The scaled imbibition recovery curves scaled using the KGE shape factor (equation (4)) and equation (22) as the permeability averaging method. The curves are scaled to three different classes of curves. Figure 7. View largeDownload slide The scaled imbibition recovery curves for the new presented shape factor (equation (20)) and equation (22) as permeability averaging method. Figure 7. View largeDownload slide The scaled imbibition recovery curves for the new presented shape factor (equation (20)) and equation (22) as permeability averaging method. Figure 8 illustrates the imbibition recovery curves scaled using the KGE shape factor and equation (23) as the permeability averaging method. Comparing the results of figures 6 and 8, it can be concluded that using equation (23) as the permeability averaging method gives better results in scaling of imbibition recovery curves. Finally, figure 9 shows the scaled imbibition curves after using the new shape factor and equation (23) as the permeability averaging method. Comparing the results of figures 7 and 9, one can realize that in contrast to the previous case, using equation (22) as the permeability averaging method results in more accurate results in comparison to equation (23) in the new shape factor. It is also obvious that permeability anisotropy is an effective parameter in the imbibition process in matrix-fracture systems of NFRs and should be considered in the scaling of imbibition curves. If considering the permeability anisotropy is not possible, selecting the best permeability averaging method is important. Here, considering three flow directions (equation (23)) resulted in better scaling than equation (22) where only two flow directions were considered. Obviously, it is because the number of horizontal flowing faces is twice the vertical ones. Therefore, the horizontal flow is more important. At field scale, permeability anisotropy may be more effective in comparison to lab scale experiments. Since the experimental imbibition tests are usually accomplished on core plugs, critical attention to coring direction is necessary. This is particularly important in one-end open imbibition tests, through which fluid flow is linear and the effect of permeability anisotropy is negligible. Upscaling these tests to more complex flow boundaries (e.g. all faces open) is challenging (Mason et al2009, Haugen et al2014). A comprehensive study on the challenges of considering permeability anisotropy in the upscaling of experimental scale imbibition tests to field scale imbibition processes is recommended for future works. Also, the effect of gravity force has not been investigated in this work. Many recent works such as that of Abbasi et al (2016b) investigated the effect of gravity force on the scaling of imbibition curves. The mutual effect of gravity forces and permeability anisotropy may be effective in the scaling of imbibition curves, particularly for large scale processes where the domination of capillary forces is reduced. Figure 8. View largeDownload slide The scaled imbibition recovery curves using the KGE shape factor (equation (4)) and equation (23) as the permeability averaging method. Compared with figure 6, it is clear that using the proper permeability averaging method is crucial. Figure 8. View largeDownload slide The scaled imbibition recovery curves using the KGE shape factor (equation (4)) and equation (23) as the permeability averaging method. Compared with figure 6, it is clear that using the proper permeability averaging method is crucial. Figure 9. View largeDownload slide Imbibition recovery curves using the newly presented shape factor (equation (22)) and equation (23) as the permeability averaging method. The scaling quality is improved significantly in comparison to using the KGE shape factor. Figure 9. View largeDownload slide Imbibition recovery curves using the newly presented shape factor (equation (22)) and equation (23) as the permeability averaging method. The scaling quality is improved significantly in comparison to using the KGE shape factor. In this work, to increase the effect of imbibition, it was assumed that the rock is fully water-wet which may not be true in most carbonate rock cases. Changing wettability from water-wet to oil-wet reduces the water imbibition ability of rock. The future study of the effect of permeability anisotropy on the imbibition process in other wettability states (e.g. oil-wet, mixed-wet) is recommended. 3. Conclusions In this paper, the effect of permeability anisotropy on the imbibition flow between the matrix and fracture systems has been investigated. The results indicate that permeability anisotropy is important. With regards to the importance of the scaling of imbibition recovery curves and the simulation of matrix-fracture flow systems, using an analytical method and by modifying the KGE shape factor, a new shape factor has been introduced which includes the permeability anisotropy of matrix blocks. The results show that scaling of anisotropic imbibition curves is not possible using the KGE shape factor. The new shape factor worked well and properly scaled all the curves considered. Also, an investigation on the permeability averaging methods has been conducted. It can be concluded that assuming three flow directions (equation (23)) was effective when the KGE shape factor was used. Also, assuming two flow directions (equation (22)) was successful when the new shape factor was used. References Abbasi J , Ghaedi M , Riazi M . , 2016a Discussion on similarity of recovery curves in scaling of imbibition process in fractured porous media , J. Nat. Gas Sci. 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TI - Improvements in scaling of counter-current imbibition recovery curves using a shape factor including permeability anisotropy
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2140/aa8b7d
DA - 2018-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/improvements-in-scaling-of-counter-current-imbibition-recovery-curves-4o5TfW20Dh
SP - 135
VL - 15
IS - 1
DP - DeepDyve
ER -