Transp Porous Med DOI 10.1007/s11242-014-0338-z
Quantifying the Representative Size in Porous Media Saeed Ovaysi · Mary F. Wheeler · Matthew Balhoff
Received: 6 December 2013 / Accepted: 14 May 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We present a new approach to quantify the representative quality of pore-scale samples of porous media. It is shown that the flow field uniformity serves as a reliable criterion to decide if the computed flow properties are representative at larger sample sizes. The proposed approach is computationally inexpensive and requires minimal effort to implement. We rely on the correlation matrix of flow field to quantify the representative quality of the computed flow properties at the pore scale. Using this approach, we have been able to study several pore-networks and a high-resolution image of a sandstone, and quickly answer if the computed flow properties from these pore-networks/images are representative of the actual media. Keywords Porous media · Pore-scale modeling · REV · Network modeling · Direct pore-scale modeling
1 Introduction Flow in porous media plays a critical role in variety of natural and industrial processes. Hydrocarbon production from underground reservoirs, CO2 sequestration in subsurface aquifers, and underground environmental remediation processes are only few examples where thorough understanding of the relevant pore-scale processes is vital. However, the typical size of the problems, particularly in earth science applications, makes it impractical to study the important pore-scale processes in the entire domain. Instead, small core samples of the medium are extracted and studied in detail. The constitu-
S. Ovaysi (B) · M. F. Wheeler Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station ACES, C0200, Austin, TX 78712, USA e-mail:
[email protected];
[email protected] M. Balhoff Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, 1 University Station, C0300, Austin, TX 78712, USA
123
S. Ovaysi et al.
tive properties measured from these samples are then used in larger scale simulations to predict the behavior of the system as a whole (Dagan 1989; Gelhar 1993; Neuman 1994). Furthermore, high-resolution imaging of these samples can shed light on the rock’s porespace topology (Spanne et al. 1994; Fredrich et al. 1995; Lindquist and Venkatarangan 1999; Wildenschild and Sheppard 2013). This information can be exploited to digitally reconstruct the porous medium (Dong and Blunt 2009; Prodanovic et al. 2006) which is then used for either direct pore-scale simulations (Ovaysi and Piri 2010, 2012; Mostaghimi et al. 2012; Li et al. 2005; Arns et al. 2002; Oren et al. 2007; Fredrich et al. 2006) or pore-scale network modeling (Oren et al. 1998; Patzek 2001; Valvatne et al. 2005; Piri and Blunt 2005). The latter involves an additional intermediate step where the reconstructed medium is converted to a network of pores and throats with simplified geometrical shapes. Both approaches are useful to help understand and predict the pore-scale physics of flow and transport through naturally occurring porous media (Ovaysi and Piri 2011, 2013, 2014; Bijeljic et al. 2013b; Raoof et al. 2010; Joekar-Niasar et al. 2008; Ramstad et al. 2012). However, due to the shortcomings of high-resolution imaging techniques, the acquired pore-scale samples measure only few millimeters in size. This, compared to the vast extent of the subsurface formations which can easily span few hundred kilometers, is many orders of magnitude smaller. Although in an actual field study, several core samples, each covering a different part of the domain, are collected, nonetheless, upscaling the calculated parameters from pore- to field-scale is still an open challenge (Klov et al. 2003; Rustad et al. 2008). Discrepancies in a given parameter at different scales pose a great challenge to upscaling. To deal with that, representative elementary volume (REV) is defined as the sample size beyond which variations in a given property become insignificant (Bear 1972). This definition implies that the property of interest is uniformly distributed throughout the samples in the REV range. Furthermore, variations in the depositional environment often create rock samples that are heterogeneous at multiple scales which yield multiple REV ranges depending on the scale of interest (Norris and Lewis 1991). Additionally, it has been shown that for a given sample, one could have different sizes for REV depending on the property of interest (Ovaysi 2010; Mostaghimi et al. 2012). Therefore, in upscaling a pore-scale property, the sample in the study must be within the REV range for that specific property. Considering the objective of such studies, the REV range of interest is normally few cubic millimeters for naturally occurring porous media. Obviously, such limitations do not apply to homogeneous porous media where variations in pore structure are negligible across lengths of several orders of magnitude. However, obtaining detailed information about pore-space topology of an REV size sample is not always within reach. Therefore, finite element mortars (Arbogast et al. 2007; Balhoff et al. 2008) have been developed to couple multiple pore-networks to generate an REV size sample for the specific pore-space topology being studied. Although the subject of REV has been studied in the past (Norris and Lewis 1991; Zhang et al. 2000; Mostaghimi et al. 2012; Al-Raoush and Papadopoulos 2010), the traditional approach of REV examination for fluid flow properties, namely permeability, is cumbersome and computationally demanding. In this paper, we present a new approach to calculate the permeability REV based on the flow pattern obtained from pore-scale simulations. We first discuss the mathematical approach to calculate permeability REV for pore-scale samples. Then, we apply this approach on several pore-networks to evaluate their representative quality including a study of whether/how mortar coupling can be useful in pore to field upscaling process.
123
Quantifying the Representative Size in Porous Media
2 REV Examination Noting that permeability REV is the smallest sample size where all pore-scale features of a given rock are uniformly present, an alternative way to answer the REV question would be to evaluate the uniformity of flow field in the sample. Uniform flow pattern in a porescale sample is a good indicator that the sample is large enough to include all the porescale features that give rise to different flow patterns. The constitutive properties obtained from simulations/experiments on this sample are then representative of what we should expect from much larger samples with the same pore-space topology. This greatly lowers the computational cost of the simulations. Correlation coefficient, defined by Eq. (1), provides a statistical tool to investigate the correlation of a set of paired parameters ( p1 , p2 ) in a given medium. n ( p1,i − p¯1 )( p2,i − p¯2 ) , (1) r p 1 , p 2 = i n n 2 2 ( p − p ¯ ) ( p − p ¯ ) 1,i 1 2,i 2 i i where n and p¯ denote number of observations and the mean value of p, respectively. r p1 p2 is a real number varying between −1 and 1. For a completely uncorrelated set of p1 and p2 , the correlation coefficient is equal to 0. To measure the uniformity of flow field in a given porous sample, we need to determine how the flow field in the sample is correlated in three-dimensional space. This can be accomplished by simply building a correlation matrix R whose elements are filled by the correlation coefficients of velocity vector u versus x, the coordinates vector, i.e., ⎤ ru x ,x ru x ,y ru x ,z = ⎣ ru y ,x ru y ,y ru y ,z ⎦, ru z ,x ru z ,y ru z ,z ⎡
Ru,x
(2)
where the elements are computed using Eq. (1). Knowing that a uniform flow field is uncorrelated in three-dimensional space, we can scale the closeness to REV by evaluating the Frobenius norm of Ru,x , i.e., ||Ru,x || F = 2 i j Ru,x i j . Since each element of Ru,x is a real number between -1 and 1, we have 0 ≤ ||Ru,x || F ≤ 3 in three-dimensional space. The smaller this value is, the closer the sample is to REV. In previous studies (Bijeljic et al. 2013a,b; Siena et al. 2014), this problem has been approached through using variograms for velocity only in the direction of flow. Using our proposed method, however, velocity vectors in all directions are taken into account to ascertain the flow field in uncorrelated in all directions. Additionally, information about the isotropy of the flow field in x, y, and z directions can be extracted by investigating the individual elements of Ru,x . 2.1 Pore-Network Modeling In a pore-network, pore-space topology of the natural medium is represented by a network of inter-connected pores and throats with simplified geometrical shapes. Steady-state singlephase flow in pore-networks is modeled by performing mass balance over individual pores, i.e., Nth,i Nth,i qi j = gi j (Pi − P j ) = 0, (3) j
j
123
S. Ovaysi et al.
where qi j denotes the flow rate from pore i to j, gi j is the hydraulic conductance of the throat connecting pores i and j, and P is pressure. Nth,i denotes the number of throats/pores connected to pore i. Applying Eq. (3) to all the individual pores of the network results in a nonsymmetric system of linear equations which can be solved efficiently using an appropriate Krylov solver. This equation, however, does not reveal the flow pattern inside the medium which is necessary to perform our statistical analysis. To do that, after the solution to Eq. (3) is obtained, we compute Qi = −
Nth,i 1 gi j (Pi − P j )(xi − x j ), di j
(4)
j
where di j = (xi − x j )2 + (yi − y j )2 + (z i − z j )2 . The flow rate vector for every pore i, i.e., Qi , can then be used in Eqs. (1) and (2) to compute the correlation matrix of flow rate, i.e., ||RQ,x || F in the pore-network. In what follows, for brevity, we refer to both ||RQ,x || F and ||Ru,x || F by ||R|| F . 2.2 Mortar Coupling of Pore-Networks Distribution of pores and throats in a realistic pore-network follows that of an actual pore-scale sample. Therefore, location of the boundary pores of one pore-network does not generally match those of another pore-network. Mortar coupling (Arbogast et al. 2007; Balhoff et al. 2008) is a domain decomposition method that overcomes this by placing a finite element mesh on the boundaries between adjacent networks. Mass conservation on these boundaries is then enforced, in a weak finite element sense, by Eq. (5) for any mortar space Γαβ connecting network spaces Ωα and Ωβ . N p,i Nc i
Nth, j
φl (x j )
j
g j,k (P j − Pk ) = 0, l = 1, ..., Nn
(5)
k
where Nn is the number of finite element nodes on the mesh representing Γαβ . Nc = 2 denotes the number of networks connected through Γαβ , and N p,i is the number of pores that lie on the connected boundary of Ωi . Continuous piecewise linear basis functions, i.e., φ(x), are used in this study. Moreover, pressure on the connected boundaries is defined by Pi =
Nn
λ j φ j (xi ),
(6)
j
where λ j is the nodal value at j. Equations (3), (5), and (6) present a system of linear equations whose solution resolves the pressure field in the entire domain.
3 Results and Discussions As discussed earlier, ||R|| F scales how close a given pore-scale sample is to REV. To evaluate the correctness of this statement, we study a pore-network from sandstone S4 (Dong and Blunt 2009). First, we follow the traditional approach to calculate the permeability REV for this sample. To do that, we incrementally shrink the sample size by removing the pores and throats that fall outside the domain. Absolute permeability of the network is then computed at each step and plotted in Fig. 1. As illustrated, variation of permeability K versus size
123
Quantifying the Representative Size in Porous Media 1.4
Permeability Correlation Coeff.
0.2
1.2 0.15
0.8 0.1 0.6 0.4
||R|| F
K (Darcy)
1
0.05
0.2 0
0
0.0005
0.001
0.0015
0.002
0.0025
0 0.003
Size (m) Fig. 1 Variations of permeability and correlation coefficient versus size for a network of sandstone S4
4
Permeability A Permeability B Correlation Coeff. A Correlation Coeff. B
3.5
0.25
0.2
2.5
0.15
||R|| F
K (Darcy)
3
2 0.1
1.5 1
0.05 0.5 0
0
0.0005
0.001
0.0015
0.002
0.0025
0 0.003
Size (m) Fig. 2 Variations of permeability and correlation coefficient versus size for a network of sandstone S1
becomes negligible at larger network sizes. In addition to that, we also plot how ||R|| F varies versus size for this network. Clearly, ||R|| F decreases sharply at about the permeability REV for this sample. However, relying on the slope of this curve will not always lead to correct judgments regarding the size of REV. To highlight that, consider the pore-network of a different sandstone i.e., S1 (Dong and Blunt 2009). In Fig. 2, we plot variations of permeability and ||R|| F versus size for this sample under two different cases. In case A, we shrink the pore-network toward one of its corners, whereas in case B, the sample is shrunk toward its center. Looking at the different styles of variation in both permeability and ||R|| F shown in Fig. 2,we deduce the following two points: (1) considering the heterogeneity of a given sample, the ||R|| F versus size curve might decrease sharply at any size regardless of REV, and therefore sharp decrease in ||R|| F is not unique to REV. (2) Depending on the starting point of shrinking/expanding the sample, the traditional approach can lead to different results making the judgment about REV difficult. As shown in Fig. 2, permeability variation
123
S. Ovaysi et al. Table 1 Correlation coefficients of several networks Name
Berea
A1
C1
C2
S1
S2
S3
S4
S5
S6
||R|| F
.0470
.0837
.0787
.1380
.1130
.1120
.1140
.0285
.1720
.1920
Name
S7
S8
S9
F42A
F42B
F42C
LV60A
LV60B
LV60C
||R|| F
.1340
.2280
.1720
.0688
.0707
.0979
.0989
.0777
.0850
Network files are taken from Dong and Blunt (2009) X
Z Y
Fig. 3 : Pa Flow field in a pore-network of S4 sandstone with ||R|| F = 0.0285. The sample is (2.688 mm)3 . The velocity vectors are sized and colored using the vector norm of velocity and pressure at each point, respectively
in case B attenuates at larger sizes which, based on the traditional approach, indicates REV. This behavior, however, is not observed in case A where permeability increases constantly. If we were to rely only on the traditional approach to calculate REV, we had to consider all the possible cases to shrink the original sample which is a tedious and demanding process. In light of the above findings, we propose the following criterion: Yes if ||R|| F < 0.05 Permeability REV = (7) No if ||R|| F > 0.05. The 0.05 threshold is obtained based on analyzing the results for variety of samples (see Table 1) acquired from Dong and Blunt (2009). Using this criterion, we only need to calculate ||R|| F once to reach a judgment regarding permeability REV for a given sample which can
123
Quantifying the Representative Size in Porous Media X
Z Y
Fig. 4 : Pa Flow field in a pore-network of S1 sandstone with ||R|| F = 0.113. The sample is (2.6049 mm)3 . The velocity vectors are sized and colored using the vector norm of velocity and pressure at each point, respectively 1.4 1.2
K (Darcy)
1 0.8 0.6 0.4 0.2 0 0
0.0005
0.001
0.0015
0.002
0.0025
Size (m) Fig. 5 Variation of permeability versus size for a Berea sandstone network
easily lead to a 100× reduction in processing time. It should be stressed that a simplified network of pores and throats is far from being a faithful representation of a naturally occurring rock sample. Therefore, the flow fields computed using pore-network modeling may not always match the realities of the actual rock samples. However, that is not our concern in this
123
S. Ovaysi et al. 0.7 0.6
K (Darcy)
0.5 0.4 0.3 0.2 0.1 0
0
0.0005
0.001
0.0015
0.002
0.0025
Size (m) Fig. 6 Variation of permeability versus size for C2 limestone network X
Z Y
Fig. 7 : Pa Flow field in the pore-network of Berea sandstone with ||R|| F = 0.047. The sample is (2.138 mm)3 . The velocity vectors are sized and colored using the vector norm of velocity and pressure at each point, respectively
study. Rather, regardless of the tool used to obtain the flow field information, Eq. (7) is used to make a quick judgment about permeability REV for a given rock sample. Although the 0.05 threshold is obtained through compiling a collection of pore-scale networks, we believe that this threshold would be equally valid if the flow fields were to be obtained using other
123
Quantifying the Representative Size in Porous Media X
Z Y
Fig. 8 : Pa Flow field in a pore-network of C2 limestone with ||R|| F = 0.138. The sample is (2.138 mm)3 . The velocity vectors are sized and colored using the vector norm of velocity and pressure at each point, respectively
tools such as direct pore-scale modeling. Clearly, the preference should be to use the most accurate flow field information which, using Eq. (7), would yield correct decision regarding the representative quality of the rock sample at hand. Applying Eq. (7) on the above porenetworks, we conclude that for sample S4 where ||R|| F = 0.0285 < 0.05, permeability REV is reached. Sample S1 , on the other hand, is not large enough to be considered permeability REV because ||R|| F = 0.113 > 0.05. These conclusions are corroborated by the flow patterns shown in Figs. 3 and 4 where uniform and non-uniform flow patterns are observed for samples S4 and S1 , respectively. To further clarify our proposed approach, we study two more pore-networks obtained for Berea sandstone and a carbonate codenamed C2 . The traditional approach of REV calculation yields Figs. 5 and 6 for Berea and C2 , respectively. We can conclude from Fig. 6 that the sample is definitely not large enough to be considered REV. However, the situation for the Berea sample is different as the permeability fluctuations dampen at larger sizes. Despite that, there is no certainty whether the same behavior can be seen if we were to shrink the sample toward a different point in the sample similar to what was observed for sample S1 above. Using Eq. (7), we compute ||R|| F = 0.047 < 0.05 for the Berea sample which clearly indicates that REV is reached for this sample. We also compute ||R|| F = 0.138 > 0.05 which, as evidenced by Fig. (6), rejects the representative quality of this sample. These REV judgments are corroborated by the flow patterns shown in Figs. 7 and 8 where uniform and non-uniform flow patterns are observed for the Berea and C2 samples, respectively.
123
S. Ovaysi et al. Z
Y X
Fig. 9 : Pa Flow pattern in a (2.280 mm)3 pore-scale network of a sandstone. The velocity vectors are sized and colored using the vector norm of velocity and pressure at each point, respectively
Due to the limitations of imaging techniques, it may not be possible to obtain a complete REV size image of the sample. What can be done in such circumstances is to couple the images from contiguous parts of a rock using mortar coupling as described in Sect. 2.2. As an example, consider a sandstone network whose flow field is illustrated in Fig. 9. We calculate ||R|| F = 0.195, 0.116, 0.0940 depending on whether the pressure gradient is applied along x, y, or z axis, respectively. These numbers indicate that first, the medium is anisotropic, and second, this pore-network is not large enough to be representative of the actual sandstone. In Fig. 10, we illustrate pressure distribution and flow pattern on the mortars used in this study to couple a system of 3 × 3 × 3 identical pore-networks of this sandstone. Clearly, the coupled system behaves more uniformly which is evidenced by very low values for correlation coefficient, i.e, ||R|| F = .0446, .0310, .0136 when pressure is applied along x, y, and z axes, repectively. However, the analysis we presented in the previous section cannot be valid here as we have copied the same pore-network, and a uniform flow pattern is expected. The correct way to examine the representative quality of this pore-network would have been to couple the pore-networks obtained from adjacent blocks of the original sample. However, we did not have access to this information and could not complete the analysis. Nevertheless, we have presented the proof of the concept for a computationally inexpensive method to examine the representative quality of a set of pore-networks belonging to adjacent blocks of small pore-scale samples. Another advantage of our proposed approach is when flow simulations are performed directly on the high-resolution images of porous media. In such circumstances, flow simulations are very expensive and generating an REV curve similar to, for instance, Fig. 5 would
123
Quantifying the Representative Size in Porous Media
(a) Z
Y X
(b)
Z
Y X
Fig. 10 : Pa a Pressure distribution and b flow pattern in a mortar coupled 3 × 3 × 3 configuration of the pore-network shown in Fig. 9. The velocity vectors are sized and colored using the vector norm of velocity and pressure at each point, respectively
123
S. Ovaysi et al.
Fig. 11 : Pa Flow pattern in a (0.5 mm)3 image with ||R|| F = 0.3 of a Bentheimer sandstone. Only the most conductive flow channels are shown
take days using today’s computing resources. Given the limitations of high-resolution imaging facilities, it is likely that the acquired image is smaller than the size of REV. Therefore, rather than trying to calculate the representative size, normally we are interested to know if a given image is representative of the medium. Using ||R|| F , we only need to perform one flow simulation to answer this question. Shown in Fig. 11 is flow field calculated using modified moving particle semi-implicit (MMPS) method (Ovaysi and Piri 2010) in a (0.5 mm)3 image of Bentheimer sandstone with 1.7 µm resolution. We conclude that this image cannot be representative of the actual rock based on ||R|| F = 0.3 which is much larger than the standard we set earlier, i.e., ||R|| F < 0.05.
4 Conclusions In this paper, we used correlation coefficients to determine whether a given pore-scale sample is representative at larger scales. We studied a series of pore-networks and a high-resolution image to conclude that, in terms of fluid flow, REV size samples must have ||R|| F < 0.05. Using this approach, one can quickly decide if a given pore-scale sample is representative without spending time on the tedious computations required by the traditional approach. This becomes critical if one has to make quick decisions using direct pore-scale modeling techniques. In such circumstances, the computational cost of running a multitude of samples to construct the REV curve is often prohibiting.
123
Quantifying the Representative Size in Porous Media Acknowledgments This research was partly supported by the financial support from the Center for Frontiers of Subsurface Energy Security (CFSES) at The University of Texas at Austin under U.S. Department of Energy contract DE-SC0001114. Professor Martin J. Blunt from Imperial College and Hu Dong from iRock Technologies are gratefully thanked for sharing their network and micro-CT data with us.
References Al-Raoush, R., Papadopoulos, A.: Representative elementary volume analysis of porous media using X-ray computed tomography. Powder Technol. 200(1–2), 69–77 (2010) Arbogast, T., Pencheva, G., Wheeler, M.F., Yotov, I.: A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6(1), 319–346 (2007) Arns, C.H., Knackstedt, M.A., Pinczewski, W.V., Garboczi, E.J.: Computation of linear elastic properties from microtomographic images: methodology and agreement between theory and experiment. Geophysics 67, 1396–1405 (2002) Balhoff, M.T., Thomas, S.G., Wheeler, M.F.: Mortar coupling and upscaling of pore-scale models. Comput. Geosci. 12(1), 15–27 (2008) Bear, J.: Dynamics of Fluids in Porous Media. Courier Dover, New York (1972) Bijeljic, B., Mostaghimi, P., Blunt, M.J.: Insights into non-fickian solute transport in carbonates. Water Resour. Res. 49, 2714–2728 (2013a) Bijeljic, B., Raeini, A., Mostaghimi, P., Blunt, M.J.: Prediction of non-fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Phys. Rev. E 87(1), 013, 011 (2013b) Dagan, G.: Flow and Transport in Porous Formations. Springer, New York (1989) Dong, H., Blunt, M.J.: Pore-network extraction from micro-computerized-tomography images. Phys. Rev. E 80(3), 036, 307 (2009) Fredrich, J.T., Menendez, B., Wong, T.F.: Imaging the pore structure of geomaterials. Science 268(5208), 276–279 (1995) Fredrich, J.T., DiGiovanni, A.A., Noble, D.R.: Predicting macroscopic transport properties using microscopic image data. J. Geophys. Res. 11(B03), 201 (2006) Gelhar, L.W.: Stochastic Subsurface Hydrolog. Prentice Hall, Englewood Cliffs (1993) Joekar-Niasar, V., Hassanizadeh, S.M., Leijnse, A.: Insights into the relationships among capillary pressure, saturation, interfacial area and relative permeability using pore-scale network modeling. Transp. Porous Media 74, 201–219 (2008) Klov. T., Oren, P.E., Stensen, J.A., Lerdahl, T.R., Berge, L.I., Bakke, S., Boassen, T., Virnovsky, G.: Pore-tofield scale modeling of wag. SPE 84549 (2003) Li, H., Pan, C., Miller, C.T.: Pore-scale investigation of viscous coupling effects for two-phase flow in porous media. Phys. Rev. E 72, 026, 705 (2005) Lindquist, W.B., Venkatarangan, A.: Investigating 3d geometry of porous media from high resolution images. Phys. Chem. Earth 24, 639–644 (1999) Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on micro-ct images. Math. Geosci. 45, 103–125 (2012) Neuman, S.P.: Generalized scaling of permeabilities: validation and effect of support scale. Geophys. Res. Lett. 21, 349–352 (1994) Norris, R.J., Lewis, J.J.M.: The geological modeling of effective permeability in complex heterolithic facies. SPE 22692, 359–374 (1991) Oren, P.E., Bakke, S., Arntzen, O.J.: Extending predictive capabilities to network models. SPE J. 3(4), 324–336 (1998) Oren, P.E., Bakke, S., Held, R.: Direct pore-scale computation of material and transport properties for north sea reservoir rocks. Water Resour. Res. 43, W12S04 (2007) Ovaysi, S.: Direct pore-level modeling of fluids flow in porous media. Ph.D. thesis, University of Wyoming (2010) Ovaysi, S., Piri, M.: Direct pore-level modeling of incompressible fluid flow in porous media. J. Comput. Phys. 229(19), 7456–7476 (2010) Ovaysi, S., Piri, M.: Pore-scale modeling of dispersion in disordered porous media. J. Contam. Hydrol. 124(1– 4), 68–81 (2011) Ovaysi, S., Piri, M.: Multi-GPU acceleration of direct pore-scale modeling of fluid flow in natural porous media. Comput. Phys. Commun. 183(9), 1890–1898 (2012) Ovaysi, S., Piri, M.: Pore-scale dissolution of CO2 +SO2 in deep saline aquifers. Int. J. Greenhouse Gas Control 15, 119–133 (2013)
123
S. Ovaysi et al. Ovaysi, S., Piri, M.: Pore-space alteration induced by brine acidification in subsurface geologic formations. Water Resour. Res. 50, 1–13 (2014) Patzek, T.W.: Verification of a complete pore network simulator of drainage and imbibition. SPE J. 6(2), 144–156 (2001) Piri, M., Blunt, M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two- and threephase flow in porous media. i. model description. Phys. Rev. E 71, 026, 301 (2005) Prodanovic, M., Lindquist, W.B., Seright, R.S.: Porous structure and fluid partitioning in polyethylene cores from 3D X-ray microtomographic imaging. J. Colloid Interface Sci. 298(1), 282–297 (2006) Ramstad, T., Idowu, N., Nardi, C., Oren, P.E.: Relative permeability calculations from two-phase flow simulations directly on digital images of porous media. Transp. Porous Media 94(2), 487–504 (2012) Raoof, A., Hassanizadeh, S.M., Leijnse, A.: Upscaling transport of adsorbing solutes in porous media: porenetwork modeling. Vadose Zone J. 9, 624–636 (2010) Rustad, A.B., Theting, T.G., Held, R.J.: Pore scale estimation, up scaling and uncertainty modeling for multiphase properties. SPE 113005 (2008) Siena, M., Riva, M., Hyman, J.D., Winter, C.L., Guadagnini, A.: Relationship between pore size and velocity probability distributions in stochastically generated porous media. Phys. Rev. E 89(013), 018 (2014) Spanne, P., Thovert, J.F., Jacquin, C.J., Lindquist, W.B., Jones, K.W., Adler, P.M.: Synchrotron computed microtomography of porous media: topology and transports. Phys. Rev. Lett. 73(14), 2001–2004 (1994) Valvatne, P.H., Piri, M., Blunt, M.J.: Predictive pore scale modeling of single and multiphase flow. Transp. Porous Media 58(1–2), 23–41 (2005) Wildenschild, D., Sheppard, A.: X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour. 51, 217–246 (2013) Zhang, D., Zhang, R., Chen, S., Soll, W.E.: Pore scale study of flow in porous media: scale dependency, rev, and statistical rev. Geophys. Res. Lett. 27(8), 1195–1198 (2000)
123