PUBLICATIONS RESEARCH ARTICLE Key Points: Simulations of drainage/imbibition in water-wet pipe networks Approximately universal of the sense percolation theory in network simulations Resistivity index did not generally obey Archie s simple power law Correspondence to: M. Li, hytlxf@126.com Citation: Li, M., Y. B. Tang, Y. Bernabé, J. Z. Zhao, X. F. Li, X. Y. Bai, and L. H. Zhang (2015), Pore connectivity, electrical conductivity, and partial water saturation: Network simulations, J. Geophys. Res. Solid Earth, 120, 4055 4068, doi:10.1002/ 2014JB011799. Received 24 NOV 2014 Accepted 11 MAY 2015 Accepted article online 15 MAY 2015 Published online 5 JUN 2015 Pore connectivity, electrical conductivity, and partial water saturation: Network simulations M. Li 1,Y.B.Tang 1, Y. Bernabé 2, J. Z. Zhao 1,X.F.Li 3, X. Y. Bai 3, and L. H. Zhang 1 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China, 2 Earth, Atmospheric, and Planetary Sciences Department, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 3 School of Geoscience, China University of Petroleum, Beijing, China Abstract The electrical conductivity of brine-saturated rock is predominantly dependent on the geometry and topology of the pore space. When a resistive second phase (e.g., air in the vadose zone and oil/gas in hydrocarbon reservoirs) displaces the brine, the geometry and topology of the pore space occupied by the electrically conductive phase are changed. We investigated the effect of these changes on the electrical conductivity of rock partially saturated with brine. We simulated drainage and imbibition as invasion and bond percolation processes, respectively, in pipe networks assumed to be perfectly water-wet. The simulations included the formation of a water film in the pipes invaded by the nonwetting fluid. During simulated drainage/imbibition, we measured the changes in resistivity index as well as a number of relevant microstructural parameters describing the portion of the pore space saturated with water. Except Euler topological number, all quantities considered here showed a significant level of universality, i.e., insensitivity to the type of lattice used (simple cubic, body-centered cubic, or face-centered cubic). Hence, the coordination number of the pore network appears to be a more effective measure of connectivity than Euler number. In general, the simulated resistivity index did not obey Archie s simple power law. In log-log scale, the resistivity index curves displayed a substantial downward or upward curvature depending on the presence or absence of a water film. Our network simulations compared relatively well with experimental data sets, which were obtained using experimental conditions and procedures consistent with the simulations. Finally, we verified that the connectivity/heterogeneity model proposed by Bernabé et al. (2011) could be extended to the partial brine saturation case when water films were not present. 1. Introduction Knowledge of the petrophysical properties of rocks under a wide range of physical conditions is very important for the discovery, evaluation, and exploitation of oil and gas reservoirs. Electrical resistivity borehole measurements are among the most commonly employed well logs to determine oil and gas saturations in situ. Their interpretation is usually based on Archie s [1942] empirical relationship between the electrical resistivity of partially brine-saturated rocks and brine saturation S w : I ¼ ρ w ρ ¼ F w F ¼ S n w ; (1) 2015. American Geophysical Union. All Rights Reserved. where I is the resistivity index, ρ is the resistivity of the rock fully saturated with brine (Ω m), ρ w is the resistivity of the partially saturated rock, F = ρ/ρ br and F w = ρ w /ρ br are the corresponding formation factors, and ρ br the resistivity of the brine. The empirical exponent n was found to be near 2 in clean sands and sandstones [Archie, 1942]. However, disconnection of the electrically conductive brine phase may occur at a finite value of S w, leading to the divergence of I and the breakdown of Archie s power law relationship [Suman and Knight, 1997; Zhou et al., 1997; Tsakiroglou and Fleury, 1999; Ewing and Hunt, 2006; Toumelin and Torres-Verdin, 2008; Han et al., 2009]. Equation (1) may also fail when the rugosity of the pore walls facilitates formation of a water film in pores invaded by the nonwetting fluid [e.g., Davis et al., 1990; Robbins et al., 1991; Tuller and Or, 2001; Tsakiroglou and Fleury, 1999; Berkowitz and Hansen, 2001]. In this case, the resistivity index approaches a finite limit at very low water saturations [Wang and Sharma, 1988; Suman and Knight, 1997; Tsakiroglou and Fleury, 1999; Han et al., 2007, 2009; Taylor and Barker, 2002, 2006; Toumelin and Torres-Verdin, 2005, 2008; Yanici et al., 2013]. Moreover, factors such as saturation history (e.g., hysteresis of drainage/imbibition cycles), rock wettability, density contrast of the fluids, immiscible LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4055
displacement dynamics, and electrokinetics effects related to the presence of an electrical double-layer may also affect Archie s power law significantly [Knight, 1991; Suman and Knight, 1997; Zhou et al., 1997; Man and Jing, 2000, 2001; Bekri et al., 2003; Aggelopoulos et al., 2005; Yue and Tao, 2013; Revil et al., 2012, 2014]. In this paper, we follow the work of Bernabé et al. [2010, 2011], who used three-dimensional (3-D) network simulations to study the effect of pore connectivity and pore size variability on permeability k and inverse electrical formation factor 1/F. They found that k and 1/F can be modeled as power laws of (z z c ), where z denotes the average pore coordination number and z c = 1.5 at the percolation threshold. They derived the following expression: 1 F ¼ CðσÞ R 2 ½z z c Š γσ ð Þ ; (2) L where R denotes the hydraulic radius (twice the ratio of the volume to the surface area of the pores), L the average pore length, and the parameters C(σ) and γ(σ) are the functions of the pore size variability measure σ (the ratio of the standard deviation to the mean of the pore radius distribution). Since brine is usually the only electrically conductive phase in partially saturated rocks, the model embodied by equation (2) may remain applicable to partially brine-saturated rocks, provided that the geometrical and topological parameters are restricted to the portion of the pore space saturated with brine. As a consequence, it is important to discover how the coordination number z w, hydraulic radius R w, and heterogeneity measure σ w of the brine saturated pore space vary with S w during these processes such as drainage and imbibition. Here we attempted to resolve this question using a network simulation approach. The paper is organized as follows. The necessary background on drainage/imbibition phenomena and their modeling as percolation processes is briefly introduced in section 2. The numerical procedures are described in section 3, and the results are reported in section 4. The discussion in section 5 is focused on two main issues, comparison of the numerical simulations to experimental results reported in the literature and devising a close form model to reproduce the simulated data. Finally, section 6 summarizes our conclusions. 2. Background In water-wet rocks, partial saturation can be achieved by forced injection of a nonwetting fluid into the initially water saturated pore space, a process called drainage. Conversely, capillary forces during imbibition lead to the spontaneous penetration of water in pores originally saturated with a nonwetting fluid. Drainage and imbibition play an important role during extraction of hydrocarbons from natural reservoirs and have been extensively studied in the past. In particular, a number of authors have attempted to visualize the spatial distribution of the wetting and nonwetting fluids during drainage/imbibition of laboratory micromodels, i.e., networks of microchannels etched between two plates of suitable transparent material [Chatzis and Dullien, 1981; Lenormand et al., 1983; Lenormand and Zarcone, 1985; Lenormand, 1986, 1990; Wardlaw and Li, 1988; Avraam and Payatakes, 1995]. One important result of Lenormand and coauthors studies is that in the quasi-static limit (when capillary forces dominate viscous and inertial forces), drainage can be well described as invasion percolation, a process according to which pores are invaded by the nonwetting fluid in decreasing order of their radii, provided that (1) they are accessible (i.e., connected to the nonwetting fluid source by a continuous path of previously invaded pores) and (2) there is an escape route allowing the water contained in them to flow away. Failure of the second condition leads to trapping of water, which may partially explain the irreducible water saturation always observed in drainage experiments [Morrow, 1970; Zhou et al., 2000]. Water trapping is greatly reduced when a water film remains on the walls of the invaded pores. Experimental evidence supports the presence of a thin, pervasive, continuous water film in water-wet partially saturated porous media [Diederix, 1982; Wang and Sharma, 1988; Sharma et al., 1991; Knight and Dvorkin, 1992; Toumelin and Torres-Verdin, 2005; Knackstedt et al., 2007; Yanici et al., 2013]. It has been suggested that the existence of the water film and its effective thickness are linked to the presumably selfaffine, pore wall roughness [Diederix, 1982; Tsakiroglou and Fleury, 1999; Toumelin and Torres-Verdin, 2005]. Pendular rings around grain contacts may also contribute to water film conduction but are not sufficient by themselves to explain the resistivity index observations [Yanici et al., 2013]. Although the water film is bound to have a complex structure, it was satisfactorily modeled in many of the studies cited above as a LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4056
uniform layer. Despite a very low hydraulic conductivity, the water film allows fluid flow to occur and, given enough time, permits escape of volumes of water that would otherwise get trapped during drainage [Davis et al., 1990; Tuller and Or, 2001]. In the quasi-static limit, imbibition also behaves as a percolation process, whose type depends on the existence of water films [Lenormand et al., 1983; Lenormand, 1986, 1990]. In absence of a water film, imbibition behaves as invasion percolation, similar to drainage except that accessible pores are resaturated in increasing order of their radii. When a water film is present, on the other hand, pore accessibility is continuously preserved and imbibition in the quasi-static limit acts as a simple bond percolation process [Lenormand et al., 1983; Lenormand, 1986, 1990]. In both cases, trapping of the nonwetting fluid occurs in relatively large pores surrounded by smaller ones, a situation more likely to occur in rocks with a broader pore size distribution and/or lower pore connectivity [Wardlaw and Cassan, 1978; Chatzis et al., 1983]. Although the description above does not depend on the nature of the wetting and nonwetting fluids (drainage/imbibition processes obey the same physics in water- and oil-wet rocks), the associated changes in electrical conductivity are very different in practice because brine is, in general, the only conductive phase. In particular, the presence of a water film in water-wet rock greatly reduces electrical resistivity at low water saturations, whereas oil films in oil-wet rock have no electrical effects. Because percolation processes can be straightforwardly implemented in discrete networks, multiphase flow phenomena, including drainage/imbibition, have been frequently investigated in the past using pipe network simulations (see reviews by Berkowitz and Ewing [1998] and Blunt [2001]). Network simulations have also been used to identify the microstructural parameters affecting resistivity index curves [Wang and Sharma, 1988; Suman and Knight, 1997; Tsakiroglou and Fleury, 1999; Man and Jing, 2000, 2001; Knackstedt et al., 2007; Han et al., 2009]. The following findings are among the most important: the resistivity index I is rarely a simple power law function of S w (in other words, Archie s saturation exponent n generally varies with S w ); n is strongly affected by wettability, with a tendency to increase when the wetting conditions are changed from water-wet to oil-wet; n decreases with increasing pore connectivity and increasing pore size distribution width; in log-log plots, the resistivity index curves are curved in the downward direction when water films are included in the simulations, while the opposite phenomenon happens when they are not. In the present study, we attempted to quantify these statements using the parameterization of Bernabé et al. [2010, 2011], i.e., z w, R w and σ w. For the sake of simplicity, we restricted our simulations to networks of water-wet cylindrical pipes, with systematic formation of a water film in pipes invaded by the nonwetting fluid (we varied the water film thickness h, including values approaching zero as a way to treat the case where the water film is absent). We also limited the study to the case where capillary forces largely dominate the viscous, inertial, and gravity forces. Under these conditions, drainage was simulated as invasion percolation without trapping of water (except for the formation of the water film) and imbibition as bond percolation with trapping of the nonwetting fluid. 3. Numerical Procedures 3.1. Constructing the Network Realizations In order to construct the network realizations, we applied the procedures described in Bernabé et al. [2010, 2011] and briefly summarized below. We systematically repeated the simulations in three-dimensional simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) networks. This helps us ensure that our observations have a reasonable degree of universality (i.e., are independent on lattice type) and can therefore be more confidently assumed to apply to porous rocks. The linear dimensions of the networks were different for the different lattices, but always such that the network realizations contained about 7000 nodes (we generally used networks with an aspect ratio of 1, but, for specific applications, networks longer in one of the principal directions were also constructed). The SC, BCC, and FCC lattices have a bond coordination number z of 6, 8, and 12, respectively. Lower values of z were obtained by randomly removing pipes according to a probability q =1 p, where p is the occupancy probability. It is easy to see that p must be equal to z/z max (with z max the maximum possible value of z for the different lattices considered) and is therefore different for SC, BCC, and FCC. However, the values of the percolation threshold p c for SC, BCC, and FCC are such that they correspond to z c 1.5, where z c denotes the (nearly universal) critical value of z at the onset of disconnection. Pipes totally disconnected from the LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4057
spanning subnetwork do not participate in the motion of the fluid phases and are therefore undesirable. In order to avoid them, we only constructed network realizations with z greater than or equal to 4, for which the probability to find a totally disconnected pipe is effectively zero. We assigned the values of the pipe radii according to an uncorrelated stochastic process using log-uniform distributions with different σ (standard deviation normalized to the mean). This parameter provides a convenient measure of pore size variability, a measurable microstructural characteristic of rocks, and the only contribution to pore scale heterogeneity considered in this study. The distributions were designed to produce a value of the hydraulic radius, R =40μm, in all cases (the specific formulas for the various distributions used here are given in Bernabé et al. [2010]). Unlike some past studies (see review by Berkowitz and Ewing [1998]), we did not attach specific volumes to the nodes. In this approach, the nodes do not store fluid or dissipate energy but are used to track the propagation of nonwetting fluid in the network and to express the Kirchoff conservation relations when simulating electrical conduction. We considered periodic networks (i.e., the terminal bonds on each side face are implicitly connected to the corresponding nodes on the opposite face) and simulated electrical conduction using periodic boundary conditions [see Bernabé et al., 2010, 2011, and references therein]. The electrical conductance g i of a water saturated pipe of radius R i is πr 2 i g i ¼ σ br L þ Σ 2πR i S L ; (3) where σ br =1/ρ br is the brine conductivity (in Sm 1 )andσ S is the specific surface conductivity (in S). After invasion by the nonwetting fluid and formation of a thin residual water film, the pipe conductance is changed to 2πR i h 2πR i g i ¼ σ br þ Σ S L L ¼ ðσ br h þ Σ S Þ 2πR i L ; (4) where χ D h R i is the effective thickness of the water film and χ D is the Debye length (a measure of the electrical double layer thickness). For the sake of simplicity, we will consider h to be a constant independent on water saturation, although there have been suggestions that h decreases with increasing pressure of the nonwetting fluid and therefore with decreasing S w [Toumelin and Torres-Verdin, 2005]. The specific surface conductivity Σ S is a parameter describing the anomalous ionic conduction along a charged solid surface due to the electrical double layer and to possible, additional, nonionic conduction mechanisms [Revil and Glover, 1997, 1998]. The excess charge on the solid surface depends on the minerals in contact with the saturating fluid. It is often characterized using the cation exchange capacity (CEC in meq kg 1 ) and is affected by the salinity and ph of the aqueous solution. The CEC is relatively low (1 to 10 meq kg 1 ) for most silicates except phyllosilicates (in particular, clays) and zeolites (10 to 1000 meq kg 1 )[Allard et al., 1983]. In reservoir rocks, clays usually form a separate microporous phase that cannot be handled using the simple network simulation approach used here (see Tang et al. [2015] for an example of modified network simulation approach including a microporous phase). Here we will restrict the study to low CEC minerals and high salinity solutions, for which the Σ S terms in equations (3) and (4) can be neglected. For example, if we consider Σ S 9 10 9 S (porous glass [Watillon and de Backer, 1970]) and σ br > 1Sm 1 (the lower bound for the conductivity of high salinity solutions, see Revil et al. s [2014] Figure 20), the second term on the right-hand side of equations (3) is less than 10% of the first one for pipes with a radius greater than 0.18 μm (the smallest pipes in our simulations have a radius of 1.35 μm). Under this condition (Σ S Rσ br ), neglecting surface conduction generally produces a slight underestimation of the formation factor. In the end-member cases, S w 1 (near full saturation) and S w 0 (when only the water film remains), the apparent formation factors F* and F w * can be estimated as F * Fð1 2 S =Rσ br Þ and F1 w * F w ð1 S =hσ br Þ, respectively [Revil and Glover, 1997]. Since h R, neglecting surface conduction thus leads to a slight underestimation of the resistivity index I*=F w */F* at S w 0 or, equivalently, a slight overestimation of the thickness of the water film (this effect obviously increases when infinitesimal water film thicknesses are used, but this is not important since, in this case, I* becomes practically divergent when the set of water saturated pipes becomes disconnected). Note also that separating the water film and surface conduction contributions may be impossible in practice, unless conductivity measurements performed with a broad range of brine conductivities are available. LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4058
3.2. Simulating Drainage and Imbibition 3.2.1. Drainage/Invasion Percolation After constructing a network realization, we assigned a Boolean label to each pipe and node of the network, with the values 0 and 1 corresponding to saturation by water and the nonwetting fluid, respectively. Initially, all pipes and nodes are labeled 0 except the nodes situated on the side faces assigned to be in contact with the source of nonwetting fluid. Two injection methods were used. The 1-face method consisted in forcing the nonwetting fluid into the network through a single face perpendicular to the direction of measurement of the electrical conductivity. In the 3-face method, injection simultaneously occurred through three mutually perpendicular faces. It should be noted that because of the assumed network periodicity, the injection faces were in direct contact with the opposite faces in the same directions. The 1-face procedure is similar to the technique usually employed in experimental studies. With the 1-face procedure, we can also simulate jacketing of the samples (as in laboratory experiments) by preventing flow through the transverse side faces. The main drawback of the jacketed 1-face procedure is that it produces an anisotropic distribution of the nonwetting fluid with an undesirable gradient of saturation in the injection direction [e.g., Argaud et al., 1989]. Saturation gradients are greatly attenuated when the 3-face procedure is used. Invasion percolation is implemented by reading a list of the pipes, sorted in descending order of their radii, until an accessible pipe (i.e., connected to at least one node labeled (1) is found. The following actions are then taken: (a) setting to 1 the labels of the selected pipe and the nodes attached to it, (b) adding the volume of nonwetting fluid injected in the pipe (accounting for the presence of a water film) to the total volume in the network (initially set to zero) and calculating the corresponding water saturation S w, and finally (c) removing the pipe from the list and calculating the number fraction p w of remaining water saturated pipes. The volume fractions S w and number fractions p w are not equal except when σ =0 (exactly homogeneous networks). On one hand, p w is the theoretical order parameter of the percolation processes simulated here. It is unaffected by the pipe radius distribution and is linked to z w through p w = z w /z. On the other hand, S w is the experimental order parameter and, unlike p w, is systematically measured in drainage/imbibition experiments. We also made note of the pipes, if any, skipped during the list search because skipping indicates that these pipes are surrounded by smaller pipes and act as trapping sites of nonwetting fluid during imbibition. We then repeated the steps just described, restarting from the top of the list, as many times as needed until the list is totally exhausted. During this process, we temporarily paused the procedure at certain preset values of S w (0.95, 0.90, 0.85, and so on, down to 0.05) to measure the coordination number z w, hydraulic radius R w and heterogeneity measure σ w of the set of currently water saturated pipes. We also calculated the current value of the formation factor F w including the effect of the water film. 3.2.2. Imbibition/Bond Percolation Imbibition is simulated using a similar method. The differences are that we now use a pipe list sorted in ascending order of the radii, considering the pipes one by one without regarding for accessibility and resaturate them with water (with a change of their label back to 0) unless they were classified as trapping sites during drainage. Nodes are relabeled to 0 only if they are not connected to a single label 1 pipe. It is easy to see that according to these procedures, imbibition does not depend on the number of faces through which water reenters the network. Notice also that if several drainage/imbibition cycles are performed, all cycles after the second drainage will repeat identically. 4. Results 4.1. Geometry and Topology of the Set of Water Saturated Pipes In order to assess the effect of pore scale heterogeneity and pore connectivity, we simulated drainage and imbibition in SC, BCC, and FCC networks with the following characteristics, σ = 0.05, 0.30, 0.55, 0.80, and 1.05, and, z = 4, 6, 8, and 10. For each combination of the input parameters, we performed simulations on 16 realizations and, given the size of networks used (e.g., 25 25 25 for SC), the expected uncertainty of the ensemble averaged output parameters was very small, reaching about 1% only for the cases with highest heterogeneity and lowest connectivity. One important observation was that within the expected uncertainties, the simulated parameters R w, z w, σ w, and I were independent of the type of lattice used, implying a satisfactory level of universality for the properties observed. LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4059
Figure 1. Examples of 3-face FCC networks simulations with z = 6 and σ = 0.05, 0.30, 0.55, 0.80, and 1.05, as indicated by the color of the lines. The diamond symbols connected by solid lines show the simulated values of (a) R w /R during drainage, (b) R w /R during imbibition, (c) z w /z during drainage, (d) z w /z during imbibition, (e) σ w during drainage, and (f) σ w during imbibition. Additional curves (dotted lines) corresponding to z = 4, 8, and 10 for σ = 0.30 and 0.80 are also represented for comparison. In general, these lines nearly coincide with the main z = 6 lines. Numbers giving the values of z are included on the diagram when the corresponding lines can be identified clearly. The star symbols represent the points where imbibition stopped due to trapping of the nonwetting fluid. Following experimental practice, we represented the microstructural parameters, R w, z w, and σ w, as functions of S w (typical examples corresponding to the FCC networks with z = 6 are shown in Figure 1). We observed that R w, z w, and σ w generally decreased with decreasing S w, following curves strongly dependent on pore heterogeneity and relatively weakly on initial connectivity (note, however, that in the case of z w, the sensitivity to the variability measure σ is exclusively due to S w ). The curves were almost identical during LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4060
Figure 2. (a) Average values of simulated volume fraction S nw of trapped nonwetting fluidattheendofimbibitionfor z =4,6,8,and10,andσ = 0.05, 0.30, 0.55, 0.80, and 1.05, as indicated by the color of the symbols. The vertical bars represent the ranges of values obtained by running 3-face simulations in SC, BCC, and FCC networks. (b) Values of the water saturation at the onset of disconnection of the set of water saturated pipes during drainage. Essentially identical values were obtained during imbibition. The theoretical values, z c /z, for exactly homogeneous networks (σ = 0) are also plotted for comparison (grey squares). drainage and imbibition when σ was lower than 0.55 while moderate differences appeared at higher heterogeneity levels. To help visualize the effect of initial connectivity, Figure 1 also includes a few curves corresponding to z = 4, 8, and 10. Since we assumed that trapping of water cannot occur except in the water film, the water saturation decreased all the way to 0 during drainage in the simulations performed with an infinitesimally thin water film (e.g., examples in Figure 1), whereas an irreducible water saturation on the order of 1.5% was obtained with a water film of thickness h =0.3μm. During imbibition, S w increased back to a value lower than 1, corresponding to the trapping of nonwetting fluid. As illustrated in Figure 2a, the volume fraction of trapped nonwetting fluid strongly increased with increasing heterogeneity and decreasing initial connectivity, in agreement with experimental observations [Wardlaw and Cassan, 1978; Chatzis et al., 1983]. Interestingly, trapping of the nonwetting fluid displayed a slightly higher sensitivity to lattice type (lower universality) than the other quantities simulated (see the variability bars in Figure 2a). Notice also that when large nodal pores are included in network simulations, they all act as additional trapping sites for the nonwetting fluid (for example, see Lenormand s [1986] Figure 3c). Thus, the trapped volume fractions obtained here can be viewed as lower bounds. In all simulations z w eventually crossed the critical value z c = 1.5, in which case the set of pipes still saturated with water became disconnected except through the water film remaining in the invaded pipes. The values of S w at disconnection of the set of water saturated pipes greatly decreased with increasing σ and z (Figure 2b). Connectivity measures other than the pore coordination number have been proposed in the past. Recently, Herring et al. [2013] performed drainage/imbibition experiments in various porous materials while recording microtomography images of the phase distribution, from which they estimated the connectivity of the nonwetting fluid (air) bodies. For that purpose they used the normalized Euler number χ/χ 100, where χ = b 0 b 1 + b 2 and χ 100 corresponds to the (100 % air saturated) dry samples. The parameters b 0, b 1, and b 2 refer to the zeroth, first, and second Betti topological invariants, respectively, i.e., the number of distinct air bodies in the pore space, the number of handles (independent loops) of the air bodies, and the number of volumes of wetting fluid (water) enclosed inside the air bodies (in practice, b 2 = 0). Negative values of χ/χ 100 indicate that the air phase is fragmented in many distinct, relatively small air bodies. It is straightforward to determine χ/χ 100 in network simulations because the first Betti invariant of the clusters of pipes saturated with the nonwetting fluid is equal to N n N b + b 0, where N n and N b are the numbers of nodes and bonds forming these clusters. Similarly to z w, χ/χ 100 is controlled by p w and is therefore independent of the heterogeneity measure σ. Indeed, for a given value of z, the curves of χ/χ 100 versus z w precisely fell on top of each other, forming a single line independent on σ (Figure 3a). However, when plotted against S nw =1 S w, χ/χ 100 followed a different, monotonically increasing curve for each value of σ (Figure 3b). The drainage and imbibition curves closely overlapped at high S nw and separated LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4061
Figure 3. (a) Examples of curves of χ/χ 100 versus z w during drainage (solid lines) and imbibition (dotted lines). The lines correspond to a single pore scale heterogeneity value, σ = 0.05, but different lattice types and pore connectivities, as indicated by the color of the lines. Since χ/χ 100 is a quantity independent of σ, the other curves of χ/χ 100 versus z w corresponding to σ = 0.30, 0.55, 0.80, and 1.05 fall exactly on top the lines shown here and therefore cannot be represented individually on the same diagram. (b) Examples of curves of the normalized Euler number characterizing the set of pipes invaded with nonwetting fluid versus S nw. The 3-face SC simulations were performed during drainage (solid lines) and imbibition (dotted lines) for z = 6 and σ = 0.05, 0.30, 0.55, 0.80, and 1.05, as indicated by the color of the lines. Although not entirely shown here, the drainage lines are defined over the entire range of S nw while the imbibition lines stop at the points indicated by the star symbols due to trapping of the nonwetting fluid. Experimental data from Herring et al. [2013] (solid squares) were also included in the diagram for comparison. for sufficiently low values of S nw with the imbibition curves dropping below the drainage ones and stopping at the trapped nonwetting fluid saturation (Figure 3b). Herring et al. [2013] experimentally observed a somewhat similar behavior although some significant differences can be noted. For example, the imbibition curves were substantially above the drainage ones (Figure 3b). These discrepancies were probably caused by differences between the numerical and experimental procedures. For example, the Herring et al. s [2013] experiments involved forced imbibition at relatively high fluid velocities in vertically oriented samples using fluids with large density contrast, suggesting that the viscous and gravity forces were not negligible. Finally, the most important result of the simulations was the strong sensitivity of the curves of χ/χ 100 versus z w to lattice type (for example, see the difference between the SC and BCC curves with z = 6, blue and red lines, in Figure 3a), showing that the normalized Euler number is not universal and therefore less useful in practice than the pore coordination number. Figure 4. Examples of resistivity index curves during 3-face drainage simulations in FCC networks with z = 6 and σ = 0.05, 0.30, 0.55, 0.80 and 1.05, as indicated by the color of the lines. Two values of the water thickness are shown, (a) h = 0.3 μm and (b) h approaching 0. Additional curves (dotted lines) corresponding to z = 4, 8 and 10 for σ = 0.30 and 0.80 are also represented for comparison. LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4062
4.2. Resistivity Index The simulated resistivity index curves did not obey a simple Archie power law but showed significant curvature when plotted in log-log scale (see 3-face injection examples in Figure 4). The curvature was downward when the water film thickness was sufficiently large and I approached a finite value (on the order of 100 or less for h =0.3μm) at very low values of S w. In simulations performed with an infinitesimally thin water film, the curvature was upward at moderate to high water saturations, when the set of water saturated pipes remained connected. The resistivity index curves did eventually bend down for z w < z c since divergence of I cannot occur unless Figure 5. Comparison of resistivity index curves simulated h is exactly set to 0 (the bending down of the using the 1-face jacketed (dotted lines) and 3-face (solid resistivity index curve is not shown in Figure 4 lines) procedures. These examples correspond to drainage because including it requires a tremendous increase simulations in FCC networks with z =6andσ = 0.05, 0.55, of the vertical scale in the diagrams). The resistivity and 1.05, as indicated by the color of the lines. For S w near 1, the curves are steeper when the 1-face jacketed index curves were generally steeper with decreasing procedure was used. The increase in slope is larger for heterogeneity and initial connectivity. As before, there higher values of σ. The inset box shows slopes of 1 and 2 was very little difference between drainage and for comparison. imbibition except for the largest heterogeneity levels. At high water saturations (for S w greater than 80% or so), the resistivity index curves were approximately straight with relatively small Archie s saturationexponents when the 3-face procedure was used (between 1 and 1.5, exceptionally reaching 2 for the lowest z and σ). Larger values were obtained with the 1-face jacketed procedure but this steepening of the resistivity index curves only occurred for water saturations near 1 and was always followed by bending down of the curves with decreasing S w (Figure 5). This effect was likely caused by the temporary formation of a saturation gradient in the injection direction as has been observed experimentally [Argaud et al., 1989]. 5. Discussion The results of our network simulations generally agree with those of previous numerical studies [Wang and Sharma, 1988; Suman and Knight, 1997; Tsakiroglou and Fleury, 1999; Man and Jing, 2000, 2001; Knackstedt et al., 2007; Han et al., 2009 among others]. The most important observation was that the simulated resistivity index I did not obey Archie s power law relation over the entire range of water saturations. The simulated resistivity index curves were curved in the downward direction when a relatively thick water film was present and in the upward direction when the film thickness was infinitesimally small. In fact, if we set h exactly to zero, the network simulation method will yield a zero inverse formation factor 1/F w at the critical water saturation corresponding to disconnection of the set of water saturated pipes. Thus, our results support the widespread idea that presence of a water film prevents disconnection of the water phase and divergence of the resistivity index at some finite value of the water saturation. Yet many experimental resistivity index curves show neither divergence of I for a finite value of S w nor downward bending with I approaching a constant value for vanishing S w. One possible explanation is that the water film thickness h decreases with decreasing S w. When, in our simulations, all pipes have been invaded and only the water film remains, I must scale with 1/h (now assumed variable) and therefore with 1/S w (see also the fractal roughness model of Tsakiroglou and Fleury [1999]). However, it should be remembered that water films have complex structures and that two different effective thicknesses may be required to account for the film volume, on one hand, and for the film electrical conductance, on the other. In such a case, a power law I S -n w with n not equal to 1 may occur. Another important outcome of this work, generally not included in the studies cited above, was the detailed information obtained on the evolution of the spatial distribution of the wetting phase illustrated in Figures 1 and 2. We now wish to focus on two questions. How realistic are the numerical simulations in comparison to LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4063
experiments reported in the literature? Can we devise a quantitative model that could be applied to laboratory and perhaps even field studies? Figure 6. Comparison of experimental and simulated resistivity index curves. The diagram represents a superposition of the results of the drainage experiments of Diederix [1982] (smooth glass beads, black dots, and solid lines; textured glass beads, dark blue dots, and solid lines) and Argaud et al. [1989] (sandstone, red dots, and solid lines) and of the corresponding jacketed 1-face simulations (gray, light blue, and orange solid lines, respectively). The input parameters of the simulations (h/r, σ, andz) are given in the inset box. 5.1. Comparison With Experimental Data In order to select experimental studies suitable for comparison with our simulations, our first consideration was that they used experimental procedures ensuring clear predominance of the capillary forces over the viscous, inertial, and gravity forces. This condition seemed to be best satisfied when (a) variations in saturation were produced by stepwise changes in nonwetting fluid pressure, (b) the duration of each pressure step was sufficiently long to ensure near equilibrium of the phases inside the sample, and (c) the experimental system included a semipermeable membrane to insure correct estimation of the saturations. A second consideration was that the material tested had a small CEC, and that high salinity brines were used as wetting fluid in order to avoid surface conduction effects. We ended up with two papers reporting, among other things, drainage experiments in unconsolidated packs of smooth and textured glass beads [Diederix, 1982] and a sample of Cretaceous sandstone from offshore Gabon [Argaud et al., 1989]. Diederix s [1982] glass bead experiments were a part of an investigation of Rotliegend sandstone cores from a North Sea gas reservoir. The glass beads used in the experiments are expected to have a low CEC [Watillon and de Backer, 1970] and the saturating solution simulated the highly saline formation water of the reservoir. The glass beads experiments were, in fact, control tests, performed to demonstrate the effect of water films with different thicknesses. They were not analyzed quantitatively and almost no information, such as bead size, porosity, water film thickness, and so forth, was given. Unconsolidated granular media are expected to have high pore connectivity, i.e., z = 5 to 6, and to be less heterogeneous than consolidated rocks, i.e., σ =0.3 to 0.5 [Bernabé et al., 2011]. Hence, we tried to perform jacketed 1-face injection simulations matching the resistivity index curves in Diederix s [1982] Figure 9, starting with values of z and σ in the ranges mentioned above (arbitrary values of R and L could be used however, since I is, by design, independent on length scale). In this approach, the normalized thickness of the water film h/r was freely adjusted to fit I at the lowest water saturations. Although we used the jacketed 1-face injection procedure in networks with an aspect ratio of 2 in the injection direction, we could not properly fit the high water saturation portion of the index resistivity curves (Figure 6). The slope was always too small at S w near 1 and, in order to fit I at the low S w end, we had to decrease both z and σ to values at or even slightly below their presumably reasonable lower limits (Figure 6). In agreement with Diederix s conjecture that the thickness of the water film increases with grain surface roughness, h/r was about 2 times larger for the textured beads than the smooth ones. If we assume the size of Diederix s beads to be 100 μm (as frequently used in experimental studies), the pore radius in the bead packs had to be about 10 to 20 μm, giving h on the order of 0.04 to 0.09 μm for the smooth beads and 0.1 to 0.2 μm for the textured ones. For comparison, Toumelin and Torres-Verdin [2008] estimated the water film thickness in rocks to be on the order of 0.03 μmand0.3μm for smooth and rough grain surfaces, respectively. The core sample T1 in Argaud et al. s [1989] Figure 11 had the following characteristics, porosity of 0.219, permeability of 0.203 μm 2, grain size from 80 to 150 μm, CEC lower than 1 meq kg 1, and a very smooth grain surface. An aqueous solution with a conductivity of 3.7 S m 1 was used in the tests. Again based on the results by Bernabé et al. [2011], we expected this relatively high porosity sandstone to have σ slightly larger and z marginally lower than unconsolidated granular materials. We obtained a good fit using σ = 0.45 (a value within the accepted range), z = 4 (somewhat below the expected lower limit), and h/r = 0.0005 (Figure 5). Using Kozeny-Carman model and a tortuosity angle of 45 we estimated the pore radius to be on the order of 4 μm, corresponding to h = 0.002 μm, a value consistent with the very smooth grain surfaces described in LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4064
Table 1. Formulas for Calculating the Functions C(σ), C h (σ), γ(σ) andυ(σ) a ð0:32894þ0:23339σþ1:14230σ2þ CðσÞ ¼ 10 γ(σ) = 1.29030 + 0.045527σ + 0.82390σ 2 ð0:093043 0:048672σþ0:60447σ2Þ C h ðσþ ¼ 10 υ(σ) = 1.36090 0.20750σ + 0.36126σ 2 a These expressions were empirically established based on networks simulations as explained in Bernabé et al. [2011]. Argaud et al. [1989]. The experiments of Argaud et al. [1989] were performed using unusually low fluid velocities and long run durations and may have therefore involved mechanisms closely matching the ideal invasion and bond percolation processes simulated here. 5.2. Close Form Models The trial-and-error approach used above is not very convenient in practical applications. An accurate close form model would be much more useful. We tested the two following ideas. 1. When water films are absent or thin enough to have a negligible contribution to 1/F w, the model of Bernabé et al. [2011] expressed in equation (2) can be generalized as 1 ¼ Cðσ w Þ R 2 w ½z w z c Š γσw ð Þ ; (5) F w L where 1/F w = 0 for z w lower than z c and the functions C and γ are the same as those in equation (2) (empirical expressions for C and γ were determined by Bernabé et al. [2011] from network simulations; see Table 1). The resistivity index is then given by I ¼ CðσÞ R 2 ½z z c Š γσ ð Þ Cðσ w Þ R w ½z w z c Š γσw ð Þ; (6) where I diverges to infinity for z w lower than z c.when the water filmconductancecannotbeneglected, the conductance of each water saturated pipe of radius R i is produced by the in parallel arrangement of a water film of thickness h and a water cylinder of radius R i h. Assumingthatthisin parallel configuration can be generalized to the entire network for any value of S w,1/f w becomes 1 ¼ Cðσ w Þ R 2 w ½z w z c Š γσw ð Þ þ C h ðσþ Rh F w L L 2 ½ z z cš υσ ð Þ ; (7) where the first term of the right-hand side vanishes for z w < z c.inequation(7),c h and υ are functions of σ similar but not identical to C and γ. Following Bernabé et al. s [2011] method, we calculated the formation factor of networks where the conductive phase had the form of a pervasive water film of thickness h and empirically determined appropriate expressions for C h and υ (see Table 1). Since h is a small quantity, we simplified equation (7) by omitting h in R w h (first term on the right-hand side). The resistivity index is then given by CðσÞ½z z c Š γσ ð Þ I ¼ Cðσ w Þ½R w =RŠ 2 ½z w z c Š γσw ð Þ þ C h ðσþ½h=rš½z z c Š υσ ð Þ: (8) Notice that according to this model, I does not diverge but becomes constant in the interval of S w corresponding to 0 < z w < z c. 2. An alternative idea is to apply mixing theory. A partially saturated network can be considered as a mixture of two end-member media: (i) the fully water saturated network and (ii) a pervasive water film. Since we can associate an equal material volume to each pipe in the network, the volume fractions of each end-member media are equal to the number fractions, p w and q w =1 p w, of pipes saturated with water and the nonwetting fluid, respectively. The number fraction p w is analogous to the occupancy probability p discussed in section 3.1 and is therefore equal to z w /z. If we assume that geometric averaging is appropriate here, 1/F w and I can be written as 1 F w ¼!zw CðσÞ R 2 z ½z z c Š γσ ð Þ L C h ðσþ Rh 1 zw ½ z z cš υσ ð Þ z (9) L 2 " # and I ¼ CðσÞRz ½ z cš γσ ð Þ 1 zw z : (10) C h ðσþhz ½ z c Š υσ ð Þ LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4065
Figure 7. Comparison of simulations and close form models. The examples shown here are two drainage resistivity index curves (solid black and red lines corresponding to σ = 0.05 and 0.80, respectively) calculated using the 3-face injection procedure in SC networks with z = 6 and (a) with a water film present and (b) without water films. The results of equation (6) are represented by solid purple and pink lines for σ = 0.05 and 0.80, respectively, while those of equation (8) were indicated by the dotted light blue and orange lines. The colored double arrows highlight the areas of good agreement between simulations and models. Equation (10) requires knowledge of only z w and is therefore much simpler than equations (6) and (8). Arithmetic and harmonic averaging can also be used but they always produce large discrepancies with the simulation resistivity index curves. Note that if we considered a saturating solution with an insufficiently high salinity and could not neglect surface conduction, we would simply have to replace h in the equations (7) (10) by the sum h + Σ S /σ br. However, this simple modification of equations (7) (10) is not adequate to represent the effect of a microporous clay phase. We tested equations (6), (8), and (10) by comparing their results with corresponding simulated resistivity index curves (drainage and imbibition curves were both considered but only drainage examples are shown in Figure 7). Equations (6) and (8) (i.e., the generalization of Bernabé et al. s [2011] model) produced a good fit with the simulated curves in high water saturation intervals (from some S w value depending on z and σ up to S w = 1). The extension of these good fit intervals increased with decreasing pore connectivity and pore-scale heterogeneity (see examples in Figure 7a), even reaching all the way to the point where the set of water saturated pipes becomes disconnected (z w = z c ), when the water film thickness had an infinitesimally small value (Figure 7b). This suggests that equation (6) is a good model of the resistivity index when water films are absent and therefore when the connection of the water phase is the fundamental factor controlling the resistivity index. Equation (10) (i.e., geometric averaging) does not generally produce a good fit except when thick water films are present and in conditions of water saturations corresponding to z w < z c (Figure 7a). From these observations we can conclude that the generalized Bernabé et al. s [2011] model as expressed in equation (6) does account correctly for the effect on electrical conductivity of changes in mean pore size, pore connectivity, and pore scale heterogeneity of the portion of pore space saturated with water, but that our attempt to include the effect of a thick water film as an in parallel conductive layer in equation (8) is unsatisfactory, especially at relatively low water saturations. We speculate that as long as water films are not present, this model may remain applicable even if drainage/imbibition does not reduce to strict percolation processes. In other words, it may be useful for interpreting laboratory experiments, even those where viscous, inertial and gravity forces are not negligible with respect to the capillary forces. The main drawback of equation (6) is that in addition to a complete preliminary characterization of the material under consideration (i.e., measurements of z, R, and σ), it requires knowledge of z w, R w,andσ w in a wide range of saturations. However, the study by Herring et al. [2013] shows that the needed information may be accessible now in the laboratory by means of the newest high-resolution imaging techniques. When thick water films are present, the generalized Bernabé et al. s [2011] model can still be used at sufficiently high values of S w (note that, in this range, equations (6) and (8) give nearly identical results), but it has to be replaced by the geometric averaging model of LI ET AL. PORE CONNECTIVITY AND WATER SATURATION 4066