Index for Characterizing Wettability of Reservoir Rocks Based on

Nov 18, 2013 - An index for characterizing wettability of reservoir rocks is presented using slope analysis of spontaneous imbibition recovery data. T...
0 downloads 0 Views 2MB Size
Download PDF
Article pubs.acs.org/EF

Index for Characterizing Wettability of Reservoir Rocks Based on Spontaneous Imbibition Recovery Data Abouzar Mirzaei-Paiaman,*,†,‡ Mohsen Masihi,† and Dag Chun Standnes§ †

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Post Office Box 11365-9465, Azadi Avenue, Tehran, Iran ‡ Research and Technology Department, National Iranian South Oil Company (NISOC), Ahvaz, Iran § Bergen, Norway ABSTRACT: An index for characterizing wettability of reservoir rocks is presented using slope analysis of spontaneous imbibition recovery data. The slope analysis is performed using the known exact analytical solution to infinite acting period of counter-current spontaneous imbibition. The proposed theoretically based wettability index offers some advantages over existing methods: (1) it is a better measure of the spontaneous imbibition potential of rock (because the magnitude is directly proportional to the imbibition rate); (2) there is no need for forced displacement data; (3) there is no need for waiting until the spontaneous imbibition process ceases completely; and (4) the data needed to run the new method are all easy to measure. Experimental data from the literature are used as examples on how to use the new method.

1. INTRODUCTION Wettability is a key parameter affecting the petrophysical properties of reservoir rocks and is defined as the preference of a solid to be in contact with one fluid rather than another. Several quantitative methods have been proposed to characterize the wettability of rocks. These methods describe wettability in terms of numbers for the purpose of comparison between different systems. The quantitative methods include contact angle measurements,1 Amott test,2 Amott−Harvey test,3,4 United States Bureau of Mines (USBM),5 combined Amott/ USBM,6 relative imbibition rate,7 total capillary pressure curves,8 relative pseudo-work of imbibition,9 modified Amott,10 nuclear magnetic resonance (NMR) relaxometry,11 and chemical adsorption12 methods. Among these methods, Amott and USBM methods have been widely used. Limitations, advantages, and disadvantages associated with these methods have been discussed by Anderson13 and Morrow.14 The aim of this paper is to present a new simple to use theoretically based wettability index using slope analysis of spontaneous imbibition recovery data. Sections 2 and 3 discuss the case in which the exact analytical solution to countercurrent spontaneous imbibition is used to quantify the cumulative non-wetting phase produced versus time during the process. From the resulting recovery expression and using a normalization technique, a wettability index is presented. In section 4, an approach for easier use of the new index is established, which applies to initially 100% crude-saturated Berea samples. In section 5, using two sets of spontaneous imbibition data from the literature, applicability of the proposed index to characterize wettability of rock samples with modified surface properties because of aging in crude oils is demonstrated. Finally, conclusions are drawn from this work.

equilibrium effects16−18) with only one face open to flow (i.e., countercurrent flow), eq 1 appears by combining Darcy’s law for wetting and non-wetting phases with the capillary pressure definition and the mass continuity equation for the wetting phase19

2. ANALYTICAL SOLUTION

Received: September 29, 2013 Revised: November 18, 2013 Published: November 18, 2013

ϕ

(1)

in which ϕ is the porosity, Sw is the saturation of the wetting phase, t is the time, x is the coordinate distance, and D(Sw) is the capillary diffusion function defined as19 D(Sw) = − f (Sw)k

k rnw dPc μnw dSw

(2)

where k is the absolute permeability, krnw is the relative permeability of the non-wetting phase, μnw is the dynamic viscosity of the non-wetting phase, and Pc is the capillary pressure. Moreover, f(Sw) is the fractional flow function and is defined as19 f (Sw) =

k rw μnw k rw μnw + k rnw μ w

(3)

in which krw and μw are the relative permeability and dynamic viscosity of the wetting phase, respectively. Schmid et al.20 notice that the McWhorter and Sunada21 solution to eq 1 is applicable to counter-current spontaneous imbibition without the use of the artificial boundary condition. Indeed, they show that the imposed boundary condition, as will be defined later in eq 6, is redundant for this process and the solution describes the standard situation found in the laboratory. Schmid and Geiger16,17 and MirzaeiPaiaman and Masihi22 use this solution to derive universal scaling equations for counter-current spontaneous imbibition. McWhorter and Sunada21 present the exact analytic solution by the method of McWhorter19 that makes use of a fractional flow function. The initial and boundary conditions used by McWhorter and Sunada21 are

For one-dimensional isothermal immiscible flow of two incompressible fluids in a porous medium (neglecting gravity effects15 as well as non© 2013 American Chemical Society

∂Sw ∂S ⎞ ∂ ⎛ ⎜D(Sw) w ⎟ = ∂t ∂x ⎝ ∂x ⎠

7360

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Sw(x , 0) = Swi

(4)

Sw(+∞ , t ) = Swi

(5)

u w (0, t ) = At −1/2

(6)

in which σ is the interfacial tension and θ is the contact angle. Afterward, inserting eq 13 into eq 12 gives Q = VP

Equation 4 is the initial condition stating that, at time zero, before imbibition starts spontaneously, the medium is at its initial wetting phase saturation Swi. The medium is assumed as a semi-infinite host having the saturation Swi at the far boundary x → ∞ for all times (eq 5). Equation 6 is the imposed inlet boundary condition equation used by McWhorter and Sunada21 to solve the problem. In this equation, A is a positive parameter that cannot be chosen freely and depends upon the characteristics of the fluid−rock system. For counter-current flow, A is defined as21 A=

ϕ 2

∫S

S w,BC wi

(Sw − Swi)D(Sw) dSw F(Sw)

F(Sw) = 1 −

∫S

t

(8)

The exact solution is given by the inverse formula

2A F ′(Sw)t 1/2 ϕ

(9)

t

u w dt = 2SAt 1/2

2k ϕμnwL2

Sw,BC

∫

(Swi − Sw )

k rw +

F(Sw )

μ k rnw μ w nw

dPc dSw

Equation 17 can then be written as Q =X VP

X=

dSw

∫ Swi

t 1/2 (18)

(Swi − Sw )

k rwk rnw cos(θ ) dJ μ k rw + k rnw w dSw μ nw

F(Sw )

dSw (19)

Investigation of the group of variables on right-hand side of eq 19 shows that this group is a function of the initial water saturation, pore structure, viscosity ratio μw/μnw, and wettability of the system. Specifically, krw and krnw are functions of the wettability and pore structure; J is a strong function of the pore structure; θ is a function of the wettability; and F is a function of the initial water saturation, J, krw, krnw, and fluid viscosities.

(12)

ϕ J(Sw ) k

kc ϕ μnwLc 2

2σ

in which

The capillary pressure Pc can be related to the Leverett dimensionless capillary pressure function J as23 Pc(Sw ) = σ cos(θ )

F(Sw )

dSw t 1/2 (17)

Swi

t 1/2

k rwk rnw cos(θ ) dJ μ k rw + k rnw w dSw μ nw

Sw,BC k rwk rnw

(Swi − Sw )

Swi

(11)

∫

kc ϕ μnwLc 2

2σ

Q = VP

Inserting eq 7 into eq 11 and using eqs 2 and 3 gives Q = VP

(16)

in which n is the total number of flow directions. Using the characteristic length and characteristic permeability definitions, eq 14 can be written as

(10)

Q 2A 1/2 = t VP ϕL

(15)

j=1

where S is the surface area perpendicular to the flow direction. Using the definition of the rock pore volume (i.e., VP = LSϕ, where L is the length of the porous medium), the ratio of the recovery to the rock pore volume can be written as

Sw,BC

Vma A s ∑i = 1 l ma,i ma, i

n

3. PROPOSED WETTABILITY INDEX In this section, the theoretical basis for the new wettability index is established. The reasons for developing this new wettability index are as previously mentioned, simplicity and foundation on a well-established theory. First, eq 6 is integrated to yield the cumulative volumetric recovery at any time Q as

∫0

(14)

kc = (∏ kj)1/ n

in which F′ is the derivate of F with respect to Sw. The procedure needed to use this solution can be found elsewhere.16,17,21,22With the exact analytical solution to the problem known, a suitable mathematically based index for characterizing wettability from spontaneous imbibition recovery data can be developed, which is the subject of the next section.

Q=S

dSw

where s is the number of the open faces to imbibition, Vma is the matrix bulk volume, Ama is the area of a surface area open to flow in a given flow direction, and lma is the distance from the open surface to the corresponding no-flow boundary.24 Furthermore, to account for the effects of directional permeability on the process, the permeability k may be substituted by the characteristic permeability kc as22

21

x(Sw , t ) =

μ nw

F(Sw )

1/2

Lc =

S w,BC (β − S w )D(β) dβ F(β) w

S w,BC (S w − S wi)D(S w ) dSw F(S w )

k rwk rnw cos(θ ) dJ μ k rw + k rnw w dSw

To account for the effects of different sample shape, size, and boundary conditions on counter-current spontaneous imbibition, the length L can be substituted by the characteristic length Lc as24

(7)

wi

∫

μnwL

(Swi − Sw )

Swi

in which Sw,BC is the wetting phase saturation at the open boundary. In this equation, F is a fractional flow function defined as21

∫S

Sw,BC

k ϕ 2

2σ

(13) 7361

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Table 1. Summary of Data for Counter-current Spontaneous Imbibition Experiments Performed Using Berea Samples with Zero Initial Water Saturationa Lc (m)

sample C3-1 C1-1 C3-9 C1-2 C3-6 C3-8 C3-7 C1-3 C1-16 C1-20 C1-25 C1-7 C1-9 C1-10 C1-8 C1-6 C1-4 C1-5 C1-22 C5-19 C1-26 C1-11 C1-12 C1-13 C1-15 C1-14 C1-21 C1-19 C1-17 C1-18 EV6-13 EV6-14 EV6-21 EV6-23 EV6-22 EV6-18 EV6-20 EV6-16 EV6-17 EV6-15 EV6-8 EV6-8a

0.013 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.078 0.077 0.077 0.074 0.072 0.076 0.075 0.078 0.075 0.077 0.013 0.013

k (m2) 6.35 6.23 7.05 6.45 6.67 6.27 6.45 6.27 6.99 7.12 7.15 6.51 6.54 6.88 6.97 6.55 6.61 6.80 6.97 8.57 6.78 6.87 6.25 5.56 7.04 7.00 8.20 6.99 7.72 6.74 1.13 1.27 1.07 1.32 1.09 1.40 1.33 1.37 1.28 1.07 1.51 1.39

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

−14

10 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−14 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13

ϕ (fraction)

σ (N/m)

μw (Pa s)

μnw (Pa s)

Xref

R2

0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.16 0.18 0.17 0.17 0.17 0.17 0.17 0.17 0.18 0.17 0.18 0.16 0.18 0.18 0.19 0.18 0.18 0.18 0.18 0.18 0.19 0.18 0.2 0.19

0.0489 0.0398 0.0376 0.0353 0.0344 0.0329 0.0321 0.0312 0.0306 0.0283 0.0277 0.0524 0.0412 0.0373 0.0359 0.0345 0.0331 0.0325 0.0308 0.0295 0.0286 0.0531 0.0417 0.0364 0.0346 0.0324 0.0319 0.0311 0.0296 0.0285 0.0505 0.0412 0.0343 0.0313 0.0289 0.0513 0.0417 0.0348 0.0321 0.0298 0.0528 0.0431

0.0011 0.0044 0.0087 0.0152 0.0218 0.0395 0.0594 0.0998 0.185 0.8269 1.6466 0.0011 0.0044 0.0087 0.0152 0.022 0.0396 0.0966 0.2284 0.5227 1.6466 0.0011 0.0044 0.0152 0.0354 0.0966 0.0999 0.2284 0.5227 1.6466 0.001 0.0041 0.0278 0.0977 0.4946 0.001 0.00410 0.0278 0.0977 0.4946 0.001 0.0049

0.004 0.0039 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.043 0.043 0.043 0.043 0.043 0.043 0.043 0.043 0.043 0.0039 0.0039 0.0039 0.0039 0.0039 0.0633 0.0633 0.0633 0.0633 0.0633 0.1731 0.1731

0.043 0.025 0.016 0.018 0.015 0.012 0.012 0.008 0.007 0.004 0.004 0.068 0.054 0.045 0.045 0.042 0.036 0.022 0.013 0.008 0.004 0.088 0.087 0.051 0.034 0.026 0.025 0.014 0.013 0.007 0.032 0.020 0.009 0.005 0.003 0.083 0.052 0.027 0.017 0.008 0.127 0.106

1 0.99 1 1 1 1 1 1 0.99 0.99 0.98 0.97 0.99 0.99 0.99 1 0.98 0.98 1 1 1 1 0.99 0.99 1 0.99 1 1 0.99 0.99 1 1 1 1 1 1 1 1 1 1 0.99 0.99

Samples denoted as “C” and “EV” are from Fischer and Morrow29 and Fischer et al.,30 respectively. The estimated values of Xref and the corresponding correlation coefficient R2 values are also included. a

On the basis of eq 18 plotting Q/VP versus (2σ(kc/ϕ)1/2/ μnwLc2)1/2t1/2 results in a straight line from which the slope X is a function of the mentioned parameters. Because the dependence of these parameters cannot be isolated, the term X cannot be described as a function of wettability only. The new wettability index Zw can, however, be defined as Zw =

X X ref

original rock being strongly water-wet by an appropriate cleaning method. To use the wettability index, Zw, a core with unknown wettability saturated with oil at its initial water saturation is immersed in a water reservoir and the oil recovery by spontaneous imbibition of water is monitored and recorded over time. Then, Q/VP data are plotted versus (2σ(kc/ϕ)1/2/ μnwLc2)1/2t1/2, and the slope of the resulting straight line X is obtained using simple regression analysis. The slope X is then normalized by Xref, which is the slope for a reference strongly water-wet system having the same pore structure, initial water saturation, and viscosity ratio as the original porous system to obtain Zw or the water wettability index using eq 20. It should be noted that the formulations presented in this study ignore

(20)

where Xref is a reference slope for a cleaned and strongly waterwet system with the same pore structure, initial water saturation, and viscosity ratio as the original sample. Because the pore structure of the reference and original samples should be the same, the recommended reference medium is the 7362

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Figure 1. Curves of recovery normalized by pore volume versus (2σ(kc/ϕ)1/2/μnwLc2)1/2t1/2 for strongly water-wet Berea samples with zero initial water saturation. For each experiment, the slope Xref can be obtained using a simple regression analysis. This figure is, for the purpose of compacting and improved representation, plotted on a horizontal log-scale axis. Furthermore, each experiment data series is shown by a continuous curve.

4. ALTERNATIVE METHOD FOR COMPUTATION OF XREF APPLICABLE TO INITIALLY 100% CRUDE-SATURATED BEREA SAMPLES To establish the proposed wettability index, the value of Xref is required. Such information can be established from individual experiments after cleaning the rock sample to strongly waterwet conditions followed by spontaneous imbibition tests. The initial water saturation and viscosity ratio should also be the same between the reference and original systems. For Berea samples, an alternative way to obtain the value of Xref when test data under strongly water-wet conditions are difficult to obtain can be estimated on the basis of available experimental data from the literature. A total of 42 published tests performed on strongly water-wet Berea samples with initial water saturation of zero, summarized in Table 1, are used to establish a correlation between Xref and the viscosity ratio. The samples denoted as “C” and “EV” are published by Fischer and Morrow29 and Fischer et al.,30 respectively. The reason for choosing these tests is that they cover a wide range of the viscosity ratio, which makes these data suitable for the purpose of this study. All rock samples used in the current study are, as a result of the lack of reported data, assumed to have equal permeability in each direction, implying the characteristic permeability22 as the same as the single-direction permeability. For each test, Q/VP data, obtained directly from the experiment, are plotted versus (2σ(kc/ϕ)1/2/μnwLc2)1/2t1/2, and from the straight portion of the resulting line, the slope Xref is obtained using simple regression analysis (Figure 1). In Figure 1, there may be some observations deviating much from the linearity, particularly at early times, which, besides the experimental reading errors, can be related to heterogeneities on the pore level (i.e., pore shape and pore-level roughness) and/or some geometrical effects that result in violation of one-dimensional flow assumption such that the spatial gradient of the capillary pressure is not linear (see the studies by Akin et al.31 and Fernø et al.32). Buoyancy effects are assumed negligible for the strongly water-wet smallsize rock samples. There is also some deviation from the linearity at very late times when the imbibition front hits the

the effect of gravity. Thus, the size of core samples used in the analysis should be such that the gravity does not play an important role. Ignoring gravity, however, seems to not be an issue because small size core samples are usually tested in the laboratory. The slope analysis of spontaneous imbibition recovery data is the center piece of the wettability index presented in this study. The use of slope analysis for evaluating wettability from spontaneous imbibition recovery data has been discussed before by Handy,25 Babadagli,26 Li and Horne,27 and Pordel Shahri et al.28 Making the assumption of piston-like displacement, Handy25 notices that a linear model applies when the volume of water imbibed is plotted against the square root of time for water imbibing into gas-bearing rock. From the resulting linear relation, Handy develops a method to derive an effective capillary pressure as a measure of the wettability properties of the rock. On the basis of literature works showing, in general, a linear relationship between the recovery by spontaneous imbibition and the square root of time, Babadagli26 and later Pordel Shahri et al.28 use the slope analysis to obtain wettability indicators for use in scaling processes. It can, however, easily be shown (not performed here) that the indices proposed by Babadagli26 and Pordel Shahri et al.28 are not sole functions of wettability but also depend upon other system parameters (the wettability indices that they establish rely on a general linear relationship). Li and Horne27 propose a method similar to Handy25 involving the computation of capillary pressure and global mobility (see their eqs 9 and 10 for related definitions) data from spontaneous water imbibition tests in oil/water/rock systems. They, however, make the assumption of piston-like displacement fronts, which is considered in general to be valid in some specific systems only. The distinguishing feature of the slope analysis presented here is its reliance on the exact analytical solution to the spontaneous imbibition problem, and no further assumptions rather than those needed for Darcy’s law are made. In this respect, the slope analysis presented in this study is the most complete form of its type introduced to date. 7363

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Figure 2. Plot of Xref versus viscosity ratio μw/μnw for all strongly water-wet experiments performed using Berea samples with zero initial water saturation.

Table 2. Summary of Data for Weakly Water-Wet Experiments Performed by Zhou et al.10 a sample Al:0hr Al:4hr Al:48hr Al:72hr Al:240hr Ak:0hr Ak:1hr Ak:4hr Ak:6hr Ak:12hr Ak:24hr Ak:72hr Ak:240hr a

k (m2)

Lc (m) 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013

3.60 3.55 3.65 3.00 3.64 3.65 3.63 3.60 3.36 3.43 3.74 3.67 3.55

× × × × × × × × × × × × ×

−13

10 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13 10−13

ϕ (fraction)

σ (N/m)

μw (Pa s)

μnw (Pa s)

Xref

0.217 0.215 0.214 0.225 0.217 0.223 0.220 0.226 0.220 0.225 0.226 0.223 0.218

0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242 0.0242

0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967 0.000967

0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925 0.03925

0.036 0.036 0.036 0.036 0.036 0.049 0.049 0.049 0.049 0.049 0.049 0.049 0.049

X 0.009 0.005 0.002 0.001 0.034 0.013 0.010 0.004 0.002 0.001 0.001

R2

Zw

1 1 0.99 0.99 0.92 0.98 0.98 0.99 0.99 1 0.99 0.99 0.98

1.00 0.50 0.37 0.24 0.17 1.00 0.83 0.52 0.45 0.29 0.20 0.14 0.14

The values of X, Xref, and the corresponding R2 values computed from regression analysis and computed values of Zw are also included.

It should be noticed that Xref should preferentially be determined by experimental measures. Use of the second and alternative method presented in this section (correlation) should only be considered when an experiment is not possible. The strongly water-wet experimental data used in this study to obtain the correlation between Xref and viscosity ratio are all obtained using initially 100% oil-saturated Berea sandstone. Such complete sets of experimental data covering a wide range of the viscosity ratio and initial water saturation are extremely rare in the literature for other rock types. Because of the possible differences in the imbibition rate related to (1) heterogeneities in the pore structure between different lithologies and (2) spontaneous imbibition behavior for different initial water saturations, application of the obtained correlation for lithologies deviating significantly from Berea sandstone and for non-zero initial water saturation conditions should be considered with caution and remains open for future works.

no-flow boundaries. Obviously, the analytical solution considered in this study is not valid anymore when the waterfront contacts the no-flow boundaries, and data points related to the nonlinear portions are therefore all excluded from the regression analysis. All relevant data and computed values of Xref as well as regression coefficients are listed in Table 1. Figure 1 is, for the purpose of compacting and improved representation, plotted on a horizontal log-scale axis, and each experiment data series is depicted by a continuous curve. All computed Xref values based on 42 strongly water-wet experiments on initially 100% oil-saturated Berea samples were plotted versus the viscosity ratio covering a wide range from 0.006 to 411.65 (Figure 2). It is seen that a curve with the following mathematical form can be fitted to data with a R2 coefficient of 0.925: ⎛ μ ⎞−0.363 X ref = 0.026⎜ w ⎟ ⎝ μnw ⎠

(21)

Therefore, for a given viscosity ratio, the value of Xref can be estimated simply by using eq 21 with acceptable accuracy when no access to experimentally determined Xref values is available. 7364

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Figure 3. Curves of recovery normalized by pore volume versus (2σ(kc/ϕ)1/2/μnwLc2)1/2t1/2 for the experiments performed by Zhou et al.10 by aging Berea sandstone samples for different time periods using Alaskan crude, Swi = 15%. The slopes X and Xref can be computed using a simple regression analysis from the resulting straight lines.

Figure 4. Curves of recovery normalized by pore volume versus (2σ(kc/ϕ)1/2/μnwLc2)1/2t1/2 for the experiments performed by Zhou et al.10 by aging Berea sandstone samples for different time periods using Alaskan crude, Swi = 20%. The slopes X and Xref can be computed using a simple regression analysis from the resulting straight lines.

5. EXAMPLES OF USING THE PROPOSED WETTABILITY INDEX

hence, conducted on rock samples having wettability shifted from initial strongly water-wet conditions and will therefore be referred to as weakly water-wet in the current paper. The two sets of data, with each one representing a case where all parameters affect the spontaneous imbibition process, except aging time/wettability, are approximately the same. These series are therefore very well-suited to investigate the impact of different wettability conditions on the spontaneous imbibition behavior. Such complete systematic sets of the experimental data, with all required information, are very rare in the literature. For all reported experiments, the curve for oil recovery normalized by pore volume is plotted versus (2σ(kc/ϕ)1/2/

Two sets of experimental data from the literature published by Zhou et al.10 are used to demonstrate the applicability of the new wettability index. The data are summarized in Table 2. The test series Al:0hr to Al:240hr (Swi = 15%) and Ak:0hr to Ak:240hr (Swi = 20%) were conducted by aging Berea samples saturated with Alaskan crude for different time periods. It is assumed that tests Al:0hr and Ak:0hr are strongly water-wet because of no aging. However, because Alaskan crude has been used as the non-wetting phase, a shift from strongly water-wet conditions is inevitable. The tests summarized in Table 2 were, 7365

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Figure 5. New and the modified Amott wettability indices to water versus aging time for the cores aged using Alaskan crude oil (see the study by Zhou et al.10), Swi = 15%.

Figure 6. New and the modified Amott wettability indices to water versus aging time for the cores aged using Alaskan crude oil (see the study by Zhou et al.10), Swi = 20%.

μnwLc2)1/2t1/2 (see Figures 3 and 4), and from the straight portion of the resulting line, the slope X is computed using regression analysis. It should be noted that, for the tests Al:0hr and Ak:0hr, the slope is Xref. As discussed before, there may be some deviations from the linearity, especially at early and late times. Data from these periods are excluded in the regression analysis. Table 2 contains estimated values for X and Xref with corresponding R2 values together with calculated wettability indices Zw. For the experiments Al:0hr to Al:240hr and Ak:0hr to Ak:240hr, the modified Amott water indices are reported by Zhou et al.10 Hence, in this study, a comparison between the proposed wettability index and the modified Amott index was made, depicted in Figures 5 and 6. It should be noticed that the modified Amott water index when positive and not very low corresponds closely to the normal Amott water index (see the study by Zhou et al.10).

Figures 3 and 4 show a systematic decrease in the imbibition rate as the aging time of the rock samples increases. Therefore, we expect the wettability indices to follow the same decreasing trend with an increase in the aging time. There is a general decrease in the modified Amott indices as the aging times become longer, as shown in Figures 5 and 6, but there is an exception where an increase in the aging time shows an increase in the wettability index (see Figure 6). The proposed wettability index, however, always decreases in line with the decreasing oil expulsion rate as the aging times increase. As discussed before, it should be noted that, because the reference tests Al:0hr and Ak:0hr may not be perfect strongly water-wet cases (because of use of Alaskan crude as the non-wetting phase), therefore, the actual Xref values (defined for strongly water-wet conditions) may be slightly greater than the regressed values of Xref. This causes the computed Zw values to be slightly greater than real values. The main difference 7366

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

Amott wettability index and the new proposed index shows large differences. (3) The new proposed wettability index seems like a more suitable measure for the rate of spontaneous imbibition than the traditional modified Amott water index, which also includes both spontaneous and forced displacement processes.

between the modified Amott water index and the proposed wettability index is that the former is a measure of the shift in the location of the wetting phase saturation where capillary pressure falls to zero as a fraction of the total wetting phase change because of spontaneous and forced imbibition. The magnitude of the modified Amott water index is hence determined by both changes in the spontaneous part of the wetting phase change as well as the forced part of the saturation change interval. Additionally, kinetics normally does not play any important role for the magnitude of the modified Amott index, except in the case where the spontaneous part of the wetting phase change takes a long time and a cutoff value needs to be defined for practical purposes. Hence, the magnitude of this wettability index can be interpreted as the “equilibrium” value of the wetting phase saturation associate with the processes of spontaneous and forced imbibition. This further implies that the shape of the imbibition front does not impact the magnitude of the modified Amott water index because recovery of oil after the waterfront hits the no-flow boundary is taken into account (gravity forces are assumed negligible as mentioned previously because of the small-size rock samples). Contrary, the proposed wettability index represents directly the rate increase of the wetting phase because of spontaneous imbibition normalized to the maximum rate (very strongly water-wet conditions). It is mainly impacted by the change of the capillary pressure curve because of wettability modifications before it approaches zero. The shape of the capillary pressure curve will together with the relative permeability curves determine the shape of the waterfront imbibing into the porous medium. This will, hence, have a strong impact on the magnitude of the proposed wettability index because only observations prior to the time when the waterfront hits the noflow boundary are included in the slope analysis. The relative large differences between the modified Amott water index and the new proposed wettability index should therefore be understood and interpreted with these differences in mind. In summary, the new proposed wettability index seems better suited to quantify the effect of wettability modifications on the oil expulsion rate because of reduction in capillary forces than the traditional modified Amott index, which is more suitable for quantifying the total saturation change because of both spontaneous and forced displacement processes. Aiming to characterize wettability rock samples in terms of spontaneous imbibition potential, it should be emphasized that the main advantages of the new wettability index is its ease of use and its reliance on a theoretically derived imbibition model. The proposed wettability index uses spontaneous imbibition recovery data only, and there is no need for any forced displacement data. Furthermore, there is no need for waiting until the spontaneous imbibition process ceases completely because only the data pertinent to the infinite acting period of the process are used. This could be advantageous because, in many cases, spontaneous imbibition tests may take months to complete and any challenge related to defining a relevant operationally time frame (e.g., cutoff time) for the imbibition tests is hence ruled out.

■

AUTHOR INFORMATION

Corresponding Author

*Telephone: +989168014851. E-mail: mirzaei1986@gmail. com. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Abouzar Mirzaei-Paiaman and Mohsen Masihi thank Sharif University of Technology for permission to publish this work. Abouzar Mirzaei-Paiaman appreciates very much the valuable comments from Hadi Saboorian-Jooybari, Ramin Roghanian, and Bagher Pourghasem of National Iranian South Oil Company (NISOC). Abouzar Mirzaei-Paiaman also thanks the Research and Technology Department of NISOC/National Iranian Oil Company (NIOC).

■

REFERENCES

(1) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; John Wiley and Sons, Inc.: New York, 1982; pp 332−368. (2) Amott, E. Trans. AIME 1959, 216, 156−162. (3) Boneau, D. F.; Clampitt, R. L. J. Pet. Technol. 1977, 501−506. (4) Trantham, J. C.; Clampitt, R. L. J. Pet. Technol. 1977, 491−500. (5) Donaldson, E. C.; Thomas, R. D.; Lorenz, P. B. SPE J. 1969, 13− 20. (6) Sharma, M. M.; Wunderlich, R. W. Proceedings of the 1985 SPE Annual Technical Conference and Exhibition; Las Vegas, NV, Sept 22− 25, 1985; SPE Paper 14302. (7) Denekas, M. O.; Mattax, C. C.; Davis, G. T. J. Pet. Technol. 1959, 330−333. (8) Longeron, D.; Hammervold, W.; Skjaeveland, S. M. Proceedings of the International Meeting on Petroleum Engineering; Bejing, China, Nov 14−17, 1995; SPE Paper 30006. (9) Ma, S.; Morrow, N. R.; Zhou, X.; Zhang, X. J. Can. Pet. Technol. 1999, 38 (13), 1−8. (10) Zhou, X.; Morrow, N. R.; Ma, S. SPE J. 2000, 5 (2), 199−207. (11) Fleury, M.; Deflandre, F. Magn. Reson. Imaging 2003, 21 (3−4), 385−387. (12) Strand, S.; Standnes, D. C.; Austad, T. J. Pet. Sci. Eng. 2006, 52, 187−197. (13) Anderson, W. J. Pet. Technol. 1986, 38 (11), 1246−1262. (14) Morrow, N. R. J. Pet. Technol. 1990, 42 (12), 1476−1484. (15) Mirzaei-Paiaman, A.; Masihi, M.; Standnes, D. C. Trans. Porous Media 2011, 89, 1−14. (16) Schmid, K. S.; Geiger, S. Water Resour. Res. 2012, 48, No. W03507. (17) Schmid, K. S.; Geiger, S. J. Pet. Sci. Eng. 2013, 101, 44−61. (18) Mirzaei-Paiaman, A.; Masihi, M.; Standnes, D. C. Energy Fuels 2011, 25 (7), 3053−3059. (19) McWhorter, D. B. Infiltration Affected by Flow of Air; Colorado State University: Fort Collins, CO, 1971; Hydrology Paper No. 49, p 43. (20) Schmid, K. S.; Geiger, S.; Sorbie, K. S. Water Resour. Res. 2011, 47 (2), No. W02550. (21) McWhorter, D. B.; Sunada, D. K. Water Resour. Res. 1990, 26 (3), 399−413. (22) Mirzaei-Paiaman, A.; Masihi, M. Energy Fuels 2013, 27, 4662− 4676. (23) Leverett, M. C. Trans. AIME 1941, 142, 159−172.

6. CONCLUSION The main conclusions drawn from this work can be summarized as follows: (1) Wettability of rock samples can be characterized quantitatively using a slope analysis of the spontaneous imbibition recovery data relying on the exact analytical solution. (2) A comparison between the modified 7367

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368

Energy & Fuels

Article

(24) Ma, S.; Zhang, X.; Morrow, N. R. Influence of fluid viscosity on mass transfer between rock matrix and fractures. Proceedings of the Annual Technical Meeting of the Canadian Society of Petroleum Engineers; Calgary, Alberta, Canada, June 7−9, 1995; DOI: 10.2118/ 95-94. (25) Handy, L. L. Pet. Trans. AIME 1960, 219, 75−80. (26) Babadagli, T. J. Pet. Sci. Eng. 1996, 14, 197−208. (27) Li, K.; Horne, R. N. SPE J. 2005, 10 (4), 458−465. (28) Pordel Shahri, M.; Jamialahmadi, M.; Shadizadeh, S. R. J. Pet. Sci. Eng. 2012, 82, 130−139. (29) Fischer, H.; Morrow, N. R. J. Pet. Sci. Eng. 2006, 52 (1−4), 35− 53. (30) Fischer, H.; Wo, S.; Morrow, N. R. SPE Reservoir Eval. Eng. 2008, 577−589. (31) Akin, S.; Schembre, J. M.; Bhat, S. K.; Kovscek, A. R. J. Pet. Sci. Eng. 2000, 25, 149−165. (32) Fernø, M. A.; Haugen, A.; Wickramathilaka, S.; Howard, J.; Graue, A.; Mason, G.; Morrow, N. R. J. Pet. Sci. Eng. 2013, 101, 1−11.

7368

dx.doi.org/10.1021/ef401953b | Energy Fuels 2013, 27, 7360−7368