Petrophysics and rock physics modeling of diagenetically altered sandstone

Zakir Hossain1 and Yijie Zhou2

Abstract

We worked to establish relationships among porosity, permeability, resistivity, and elastic wave velocity ofdiagenetically altered sandstone. Many such relationships are documented in the literature; however, they donot consider diagenetic effects. Combining theoretical models with laboratory measured data, we derived math-ematical relationships for porosity permeability, porosity velocity, porosity resistivity, permeability velocity,velocity resistivity, and resistivity permeability in diagenetically altered sandstone. The effects of clay and ce-mentation were evaluated using introduced coefficients in these relationships. We found that clean sandstonecould be modeled with Kozeny’s relation; however, this relationship broke down for clay-bearing and diageneti-cally altered sandstone. Porosity is the first-order parameter that affects permeability, electrical, and elasticproperties; clay and cement cause secondary effects on these properties. Rock physics modeling results re-vealed that cementation had a greater effect on elastic properties than electrical properties and clay had a largereffect on electrical properties than elastic properties. The relationships we provided can greatly help to deter-mine permeability, resistivity, and velocity from porosity and to estimate permeability from resistivity and veloc-ity as well as to determine resistivity from velocity measurements.

IntroductionSandstone is one of the most significant groups of res-

ervoir rocks because approximately half of the knownhydrocarbon reserves are in sandstone. Sandstone hasbetter granular geometry than carbonate rocks; there-fore, interpretation of this type of rock can be consideredan easier task. Sandstone is often found to have under-gone diagenetic processes (Figure 1 a schematic exam-ple). However, diagenesis comprises a broad spectrumof physical, chemical, and biological postdepositionalprocess by which origin sedimentary assemblies, andtheir interstitial pore waters, react in an attempt to reachtextural and geochemical equilibrium with their environ-ment (Curtis, 1977). Diagenesis of sandstone, resulting infor example, quartz overgrowth (Figure 1c) or pore-filling clay minerals (Figure 1d), can greatly change theshape of intergranular pores and the correspondingphysical properties. Clay and cement in sandstone alsoincrease the complexity of rock physics analysis anddegrade the overall reservoir quality compared to cleansandstone. Therefore, we cannot ignore such diageneticeffects during our rock physics studies.

Literature reviewPorosity and permeability are two of the most impor-

tant parameters required for reservoir modeling (Zhou

and King 2011), well placement, and reservoir manage-ment (Zhou and King, 2013), etc. Both vary as a functionof pore geometry, grain packing, grain shape, pore-fill-ing solids, sorting, and any associated diagenetic facies.In general, permeability increases with an increasein porosity; however, such correlation is not quitestraightforward: We may have two rocks with the sameporosity but with completely different permeability. Tofacilitate rock physics study, it is common practice toderive a porosity-permeability approximated relation-ship. Kozeny (1927) introduces the relationship amongporosity (ϕ), permeability (k), and specific surface areaof bulk volume (S) by using the well-known Kozeny’sequation as

k ¼ cϕ3

S2 ; (1)

where c is Kozeny’s constant. This constant can be cal-culated from porosity as Mortensen et al. (1998), ifsquare tubes of the porosity model are assumed:

c ¼�4 cos

�13arccosð2ϕ − 1Þ þ 4

3π

�þ 4

�−1. (2)

The specific surface area with respect to pores (Sp)can be estimated as

1Formerly Rock Solid Images, Houston, Texas, USA; presently ION Geophysical, Houston, Texas, USA. E-mail: drzakirhossain2012@hotmail.com.2Rock Solid Images, Houston, Texas, USA. E-mail: yijie.zhou@rocksolidimages.com.First presented at the SEG 82nd Annual International Meeting. Manuscript received by the Editor 25 March 2014; revised manuscript received 29

August 2014; published online 5 December 2014. This paper appears in Interpretation, Vol. 3, No. 1 (February 2015); p. SA107–SA120, 15 FIGS., 1TABLE.

http://dx.doi.org/10.1190/INT-2014-0048.1. © 2014 Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved.

t

Special section: Geologic, geophysical, and petrophysical interpretation of core data and well logs

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Sp ¼ Sϕ. (3)

Several permeability models are found in the litera-ture (Table 1, Figure 2). Most permeability models arebasically derived from Kozeny’s equation, and few ofthem are empirical. Among them, the Kozeny-Carman(Carman, 1961) model is mostly used, rather than theKozeny equation.

Archie (1942) describes the relation between resis-tivity of the fully saturated rock Ro and resistivity ofthe pore fluid Rw as the formation factor, F :

F ¼ R0

RW. (4)

Archie (1942) also describes an empirical relationbetween formation factor, cementation factor m, andporosity in brine-saturated reservoir rock as

F ¼ 1ϕm . (5)

The cementation factor varies with porosity and poresize (Focke and Munn, 1987) and grain shape (Wyllieand Gregory, 1955), and for sandstone it varies with per-meability (Raiga-Clemenceau, 1977). Archie’s law issometimes written as

F ¼ 1ðϕ − ϕoÞm

; (6)

where ϕo is the percolation porosity below which thereare no conducting pathways and the rock conductivityis zero (Mavko and Nur, 1997). Archie’s law can be sim-plified as

F ¼ aϕm ; (7)

where a is an empirical constant close to one. A value ofless than one or larger than one results from trying to fitan Archie-like model to rocks that do not follow Archiebehavior (Mavko et al., 2009). Therefore, a corrects forclay and other conducting minerals. Note that Archie’ssecond law that relates saturation to porosity and rockresistivity is also widely used in industry. Becausesaturation is not within the scope of this work, we willfocus on equation 7 only.

For clay-bearing sandstone, the electric conductivityis also influenced by clay. Rock conductivity is not only

Figure 1. Geologic processes of progressive diagenesis insandstone. (a) Unconsolidated packing of quartz grains,(b) dense packing of quartz grains, (c) quartz overgrowth ce-mented sandstone, and (d) pore-filling cemented/clay-bearingsandstone.

Table 1. Different types of porosity permeability relationship and their corresponding equations and authors.

Model name Reference Equation

Bernabe Bernabe (1991) k ¼ ktubes þ kcracks ktubes ¼ w2fcracksϕ30

kcracks ¼ r2ð1−fcracksÞϕ20

Bloch Bloch (1991) k ¼ 10ð−4.67þ1.34ðsizeÞþ4.08∕ðsortÞþ3.42ðcontent∕100ÞÞ

Coates-Dumanoir Coats and Dumanoir (1974) k ¼ ð90;000m4 Þϕ2m

S2wi

Coates Clavier et al. (1984) k ¼ 4900ϕ4ð1−SwiÞ2S2wi

Fredrich Fredrich et al. (1993) k ¼ 1000d2ϕ3

450ð1−ϕÞ3

Modified Kozeny-Carman Mavko et al. (2009) k ¼ ðBdÞ2ðϕ−ϕcÞ3ð1þϕc−ϕÞ2

Owolabi Owolabi et al. (1994) k ¼ 307þ 26; 552ϕ2 − 34; 540ϕ2S2wiPanda-Lake Panda and Lake (1994) k ¼ sCSP3þð3CDP2þ1Þ

ðcþCDP2Þ2DPM3ϕ3

72τð1−ϕ2ÞPanda-Lake Kozeny-Carman Panda and Lake (1994) k ¼ ϕ3

2τsð1−ϕÞ2

Revil Revil and Cathles (1999) k ¼ 1000 ϕ4.5

S2wi

Timur Timur (1968) k ¼ 8581 ϕ4.4

S2wi

Tixier Tixier (1949) k ¼ 62; 500 ϕ6

S2wi

Wyllie Gregory Kozeny-Carman Wyllie and Gregory (1955) k ¼ d2ϕ3

36Cð1−ϕÞ2

Nomenclature: k, porosity; ktubes , tube permeability; kcracks, crack permeability; ϕ, porosity; w, width of crack; r, radius of tube; size, grain size (mm); sort, sortingcoefficient; content, rigid grain content; m, cementation exponent; Swi, irreducible water saturation; d, pore diameter; τ, flow tortuosity; S, specific surface area; d, porediameter; B, geometric factor; ϕc, percolation porosity; s, skewness of particle-size distribution (PSD); CDP, coefficient of variation of PSD; DPM, mean particle sizes; s,specific surface area; and C, Kozeny’s constant

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controlled by pore water salinity, but also by an excessof ions in a scattered double layer around clay particles(Waxman and Smits, 1968). The dual-water model (Cla-vier et al., 1984) divides the total water content into thebound clay water and the far water, away from the clay.The Indonesian model (Poupon and Leveaux, 1971) isoften the main alternative for the Archie model. Figure 3summarizes some relationships between the formationfactor and porosity, for different sandstone, accordingto Hacikoylu et al. (2006). We observe that shale con-tent decreases the formation factor: Clean sand hasthe largest formation factor, followed by average sand,and finally shaley sand has the smallest formation fac-tor. These differences are small when the porosity isclose to the critical porosity of sandstone as 0.4. Also,consolidated sandstone and unconsolidated sandstonehave a similar porosity and formation factor relation-ship, and unconsolidated sandstone has a slightlyhigher formation factor at low porosities and slightlylower formation factor at high porosities. Even thoughcementation of sandstone also has an ef-fect on resistivity (Hossain and Cohen,2012), few studies have been performedto describe the contact cement effect onsandstone resistivity.

Porosity is the primary mechanismaffecting acoustic velocity (Wyllie et al.,1956). Acoustic properties of rocks arealso a function of saturation (Wyllie etal., 1958), pressure (Wyllie et al.,1958), pore type (Nur and Simmons,1969), and pore fluid distribution (Mavkoand Mukerji, 1995). The acoustic velocityof rocks also depends on the diageneticproperties of those rocks (Hossain et al.,2011b).

Figure 4 shows the rock physicsrelations between seismic velocity andporosity based on modified Hashin-Shtrikman bounds.The soft-sand model is a modified Hashin-Shtrikmanlower bound, whereas the stiff-sand model is a modifiedHashin-Shtrikman upper bound (Dvorkin et al., 1994;Dvorkin and Nur, 1996). These models assume thatthe starting framework of sand is a dense random pack-ing of identical spherical grains with an initial criticalporosity. The soft-sand model assumes that the cementis deposited away from grain contacts, and thus it givesthe lower bound (Dvorkin and Nur, 1996; Dvorkin et al.,1994). In contrast, the stiff-sand model assumes that thecement is deposited close to grain contacts, and thus itgives the upper bound (Dvorkin et al., 1994; Dvorkinand Nur, 1996).

Electrical and elastic data often allow us to accu-rately quantify hydrocarbon saturation. A combinationof elastic and electrical properties could offer a power-ful means for solving the problem of hydrocarbonsaturation. Therefore, we are aiming to relate resistivityand velocity. Faust (1953) presents an empirical rela-tion between the measured resistivity of a water-

saturated formation and velocity as a function ofdepth Z:

VPðkm∕sÞ ¼ 2.2888

�ZðkmÞ Ro

Rw

�1∕6

. (8)

Normally, acoustic velocity depends on the com-pressibility of rock pore fluids. In contrast, resistivity inporous media with resistive minerals depends on poresand pore fluids. As the porosity increases, elastic veloc-ity decreases and resistivity decreases (hence, the for-mation factor also decreases). There are other factorsthat may influence this relationship. Elastic velocitiesare influenced by diagenetic processes in rocks (Hos-sain et al., 2011b). However, the effects of diageneticprocesses on resistivity are not yet well described. Toexplain the diagenetic effect, we model the resistivityand velocity relationship by combining the soft-sandmodel and stiff-sand model with Archie’s equation. Go-mez (2009) finds the following velocity and resistivity

Figure 2. Model curves represent different type of porosity-permeability rela-tionship. Their corresponding equations and authors are listed in Table 1.

Figure 3. Formation factor versus porosity relationships fordifferent types of sandstone (modified after Hacikoylu et al.,2006).

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transform from the stiff-sand model and Archie’s equa-tion:

F ¼ 2þ�0.500þ 0.177 · VPðkm∕sÞ0.6886þ 0.118 · VPðkm∕sÞ

�. (9)

Hacikoylu et al. (2006) find the following velocityand resistivity transform from the soft-sand model:

VPðkm∕sÞ ¼ F0.9þ 27 · F

. (10)

Electrical resistivity and elastic properties are usu-ally easier to measure in the laboratory and in situ thanpermeability. Defining a relationship between perme-ability and electrical resistivity would be useful for de-termining fluid flow in reservoir rocks from resistivitydata. Therefore, we aim to relate resistivity and per-meability. Resistivity is directly related to porosity,and in the same way a relationship may be expected be-tween porosity and permeability. Therefore, a relation-ship between resistivity and permeability could inprinciple be found, although no rigorous solution hasyet been found. Physically, these relationships maybe explained by combining Archie’s equation (Archie,1942) and Kozeny’s equation (Kozeny, 1927). Combin-ing equations 1, 3, and 7, we find the following relation-ship between formation factor and permeability:

k ¼ c

�aF

�1∕m 1

Sp2. (11)

Walsh and Brace (1984) derive the Kozeny per-meability relation using the formation factor:

k ¼ ϕ2

cF2S2 ¼ϕ3

cτ2S2 ; (12)

where c is a Kozeny constant, Based on Walsh andBrace (1984), c is equal to two for circular pores andthree for flat cracks. The electrical tortuosity is τ,and it is related to the porosity and formation factorthrough the relationship τ ¼ ϕF .

Fredrich et al. (1993) give a comparable relationship:

k ¼ 1bF

�ϕ

Sv

�2; (13)

where Sv is the specific surface area of pores and b isthe shape factor.

Worthington (1993) defines the relationship betweenthe formation factor and permeability as

k ¼�bF

�1∕c

; (14)

where b and c are positive empirical constants. It isclear that equations 13 and 14 are the alternative formof equation 12, where empirical constants b and c arerelated to the specific surface area, cementation factor,and Kozeny’s constant.

Raiga-Clemenceau (1977) simplify the relationshipbetween permeability (mD) and the cementation expo-nent as

m ¼ 1.28þ 2log kþ 2

. (15)

Porosity is the primary controlling factor for perme-ability and velocities. Therefore, a relationship may beexpected between elastic wave velocities and permeabil-ity of rocks. However, elastic velocity and permeabilitycan behave in different ways to the characterizationof rocks.

The objective of this study is to reexamine the rela-tionships among electrical properties, elastic wavevelocity, porosity, and permeability of diagenetically al-tered sandstone. In this study, we used laboratory mea-sured porosity, permeability, resistivity, and velocitydata from glauconite-bearing greensand from the NorthSea, Fontainebleau sandstone, and clay-bearing sand-stone from the UK. We analyze how porosity relates topermeability, resistivity, and velocity using laboratorymeasured and theoretical models. We derive porosity-permeability models for clean sandstone, clay-bearingsandstone and glauconite-bearing sandstone. We alsoexamine the relationship among resistivity and velocity,velocity and permeability, as well as resistivity and per-meability.

Data and methodWe used three laboratory measured data sets: glau-

conite-bearing greensand from the North Sea (Hossain,2011), Fontainebleau sandstone (Gomez, 2009), and

Figure 4. The theoretical relationship between porosity andvelocity demonstrated using the soft-sand model (lowercurve) and stiff-sand model (upper curve) and the intermedi-ate-stiff-sand model (curves between the bounds) (modifiedafter Hossain and MacGregor, 2014).

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clay-bearing sandstone (Han, 2010). Each data set con-tains measurements of porosity, permeability, resistiv-ity, velocity, and clay content. The glauconite-bearingPaleocene greensand is obtained from the Hermod andTy Formations. Samples from both formations are do-minated by quartz but also contain a large volume ofglauconite (20%–25%). A smaller amount of feldspar,mica, as well as pore-filling and pore-lining mineralsare also present in the studied greensand samples.Greensands from the Hermod Formation have higherpermeability compared to the Ty Formation. The maindifference between these two formations is that sam-ples from the Hermod Formation are only weaklycemented, whereas samples from the Ty Formationcontain berthierine or microcrystalline quartz cement(Hossain et al., 2011a). The Oligocene Fontainbleausandstone is from the Ile De France, France. The min-eralogy of these samples is 100% quartz with an averagegrain size of 250 μm (Gomez, 2009). The clay-bearingsandstone (Han, 2010) includes 1 Berea sandstone sam-ple, 3 sandstone samples from the Andrew Field of theNorth Sea, 10 samples from the Whitchester test site,with the other samples from different parts of the UK.The clay-bearing sandstone is dominated by quartz(40%–83%) but also a large volume of clay (0%–30%).Larger amounts of feldspar as well as other minerals(0%–60%) are also present in the studied sandstone sam-ples. The clay mineral includes illite, smectite, kaolinite,and chlorite.

Figure 5 describes the correlationamong pore-filling berthierine and poros-ity, permeability, formation factor, andvelocity, based on the 16 core plugs ofthe North Sea greensand. Pore-filling ber-thierine seems to have a good linear re-lationship with these properties, with theexception of permeability. Pore-fillingberthierine content increases with a de-crease in porosity and permeability, butan increase in formation factor andvelocities. However, these correlationsare scattered. Apparently, some othergeologic factors have to be consideredto acquire a better understanding of suchcorrelations.

Figure 6 shows the correlation amongclay content, porosity, permeability, for-mation factor, and velocity, based on thesample from Han (2010). Similar to pore-filling berthierine, if the clay content islower than 15%, then the clay content in-creases with a decrease in porosity andpermeability, but an increase in forma-tion factor and velocities. If clay contentis more than 15%, the porosity andpermeability increase, but the formationfactor and velocities decrease, with theincrease of clay content. Even thoughsuch correlations are scattered, the clay

effect of the Han (2010) data can be described by theMarion model (Marion, 1990).

Initially, we established porosity-permeability, poros-ity-resistivity, and porosity-velocity relationships forclean sandstone, glauconite sandstone, and clay-bearingsandstone. Then, we extended these relationships in thepermeability-resistivity, permeability-velocity, and veloc-ity-resistivity attributes spaces. We used three laboratorymeasured data sets, and the fundamental assumptionsare: Fontainebleau sandstone (Gomez, 2009) representsthe clean sandstone, sandstone from Han (2010) is usedas the clay-bearing sandstone, and glauconite-bearinggreensand (Hossain, 2011) represents the cementedsandstone. In this study, we described two diageneticprocesses: effect of clay and effect of cement. We de-scribed the clay effect as the difference between cleansandstone and clay-bearing sandstone as well as the dif-ference between cemented sandstone and clay-bearingsandstone. Similarly, we described the cement effectas the difference between clean sandstone and cementedsandstone as well as the difference between cementedsandstone and clay-bearing sandstone. The effect of claytype, amount of clay, and distribution of clay were notaddressed in this study. Similarly, we described the ef-fect of cement, though, the effect of cement distributionwas not addressed in this study.

Figure 5. (a) Relationship between pore-filling berthierine and helium porosity,(b) relationship between pore-filling berthierine and permeability, (c) relation-ship between pore-filling berthierine and formation factor, and (d) relationshipbetween pore-filling berthierine with sonic velocity for greensand (data were ob-tained from Hossain, 2011).

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Results and discussionModeling of porosity permeability

Figure 7 shows the crossplot of porosity and per-meability for clay-bearing sands (black points, Han[2010]; the clay contents are shown in Figure 6), green-sands (green points, Hossain, 2011), and clean sands

(blue points, Gomez, 2009). Different porosity andpermeability model curves (Table 1) are also plottedfor reference. We observe that some clean sands andclay-bearing sands with low clay content, porosity-permeability relation may be captured using referencecurves. However, these references curves do not ade-

quately describe the clay-bearing sandsand cemented greensand. For clay-bear-ing sands, due to the clay effect, thepermeability decreases, given certainporosities. Similarly, permeability alsodecreases due to the cementation ingreensand.

Figure 8 shows a porosity-permeabil-ity relationship study for the greensandin the North Sea. To demonstrate theeffect of cementation on permeability,we show the crossplot of porosity andpermeability of samples obtained fromthe Ty (triangular points) and Hermod(circle points) Formations obtainedfrom Hossain (2011). In the crossplot,reference lines, which represent Sp, inμm−1, are calculated by using equation 3.The samples from the Hermod Forma-tion are weakly cemented (Figure 8b)and have higher porosities and per-meability. The porosity-permeability re-lationship in this formation follows thetrends predicted by Kozeny’s equation.The samples from the Ty Formation arecemented (Figure 8c), their porosity-permeability relationship is scattered,and the data do not follow Kozeny’strend. Figure 8d shows the comparisonbetween laboratory measured perme-ability versus predicted permeability us-ing Kozeny’s equation using specific

surface area predicted from image analysis. The predic-tions are good for the samples from the Hermod Forma-tion where diagenetic features are minimal; however,we overestimated the permeability for the samples fromthe Ty Formation (Figure 8d). Overpredicted per-meability for diagenetic samples does not mean thatKozeny’s equation is inadequate for predicting the per-meability of cemented samples. Rather, we are unableto accurately predict the specific surface area of ce-mented samples from the image analysis. Alternatively,the specific surface area measured from BET (Bruna-uer-Emmett-Teller) analysis will be too high to predictlow permeability using this method (Hossain et al.,2011a). Even though the specific surface area is an im-portant parameter for permeability prediction, we canseldom measure this parameter accurately, especiallyfor diagenetic samples. To simplify this, we provide arelationship between specific surface area and porosity.

Figure 9 shows the diagenetic modification of primarycontrols on the porosity permeability, and specificsurface area relationships. Figure 9a shows the three

Figure 7. (a) Crossplot of porosity and permeability for threetypes of sandstone. The reference lines represent publishedporosity-permeability relations by different authors as in Ta-ble 1. Clay-bearing and cemented sandstone show differenttrends than models obtained from the published literatureas shown in Figure 2.

Figure 6. (a) Correlation between the amount of clay and helium porosity,(b) correlation between the amount of clay and permeability, (c) correlation be-tween the amount of clay and formation factor and (d) correlation between theamount of clay and sonic velocity for clay-bearing sandstone (Data obtainedfrom Han, 2010). Schematic Marion models (Marion, 1990) describe the effectof clay on porosity, permeability, formation factor, and sonic velocity. Solid linesrepresent upper model curves, whereas dashed lines represent lower modelcurves. Vertical arrows represent difference between solid and dashed lines as-sociated with compaction or pressure-related issues.

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porosity-permeability trends, which areobtained through the curve fitting to dif-ferent reservoir sands. Blue points areclean sandstone samples from Gomez(2009), black points are clay-bearingsandstone from Han (2010), and greenpoints are the greensand from the NorthSea (Hossain et al., 2011a). The blue ar-rows show the effect of the diageneticmodification. Increasing the clay contentwill force the porosity permeability tomove in the direction that the arrow indi-cates. Similarly, increasing the cementwill decrease the permeability, but poros-ity does not suffer a significant decrease.Figure 9b shows the model derived poros-ity-specific surface area of the bulkvolume and how this relates to the poros-ity-permeability trends (three lines fromFigure 9a) using Kozeny’s equation (Ko-zeny, 1927). We observe that for each typeof sand, the porosity and specific surfacearea of the bulk volume have an empiricallinear relationship in the log-log plot.Thus, porosity and specific surface areaof bulk volume have the following rela-tionship:

logðSÞ ¼ a logðϕÞ þ b; (16)

Figure 8. (a) Crossplot of porosity and permeability for greensand (modified after Hossain et al., 2011a). Samples from the Her-mod Formation have higher porosity and permeability, whereas the samples from Ty sand are more scattered. The model curvesrepresent equal specific surface of pores (Sp) as determined by using Kozeny’s equation, (b) BSE images of weaklycemented sandstone from the Hermod Formation, (c) Cemented sandstone from the Ty Formation. (d) Laboratory-measuredpermeability versus predicted permeability by using Kozeny’s equation from specific surface area predicted from image analysis.Samples from the Ty formation contains berthierine and/or microcrystalline quartz cements, resulting in a relatively poor per-meability prediction using Kozeny equation.

Figure 9. Diagenetic modification of primary textural controls on porosity-per-meability relationships: 1, model-derived porosity-permeability trend for puresandstone; 2, model-derived porosity-permeability trend for clay-bearing sand-stone; and 3, model-derived porosity-permeability trends for pore-filling cemen-tation and grain rimming. (b) Model-derived porosity-specific surface area ofbulk volume (S) relationships from the porosity permeability trends. (c) Amodel-derived porosity Kozeny’s constant relation.

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where a and b are the coefficients that need to be deter-mined according to the rock type. Combining equation 16with Kozeny’s equation, we have a new permeabilitymodel:

k ¼ cϕ3−2a

102b; (17)

where c can be estimated from porosity using the Morten-sen et al. (1998) method. We proposed a simple linear re-lationship between porosity and Kozeny’s constant,compared to the Mortensen et al. (1998) method. The pur-pose is to simplify this equation and to plug this linear re-lationship into the Kozeny equation for the permeabilityprediction by replacing the Kozeny’s constant. To simplifythe Mortensen method, we also derive the empirical rela-tion for porosity Kozeny’s constant in Figure 9c throughcurve fitting:

c ¼ 0.15ϕþ 0.18. (18)

For clean sandstone, a is −1.0 and b is −2.2; for clay-bearing sandstone, a is −2.5 and b is −2.3; for greensandwith pore-filling cementation, a is –5.5 and b is −3.0. Forthese correlations, intercept b is quite stable (2–3), de-spite the clay or cement content, but the slope a in-creases with the clay and cement content. Intercept bis obtained at higher porosity end members where thediagenetic effect is minimal. Therefore, we should ex-pect stable intercept values for clean sand, clay-bearingsand, as well as cemented sand. However, slope a shouldbe related to diagenetic effects. Slope a increases as afunction of the clay effect and the cement effect, whichdescribe that the specific surface area is increasing fromclean sandstone to clay-bearing sandstone as well asspecific surface area increases from clay-bearing sand-stone to cemented sandstone. Therefore, a small amountof quartz cementation on the surface of the quartz graincan significantly reduce the permeability of cementedsandstone (Hossain et al., 2011a). With the simplifiedporosity-permeability model provided, assuming we candetermine the coefficients a and b, given a certain rocktype and porosity, we can easily determine the perme-ability by using the modified Kozeny’s equation, whichis from the empirical regressions (equation 17).

Modeling of porosity resistivityFigure 10 shows the formation factor and porosity

relationships for different sandstone: clean sandstone,clay-bearing sandstone, and greensand with pore-fillingcement. Also, the model curves shown in Figure 3 areplotted here for reference. As we have seen in Figure 7,such model curves can be present in the clean sandtrend, but they cannot adequately represent the clay-bearing sands or cemented sandstone. For clay-bearingsands, due to the clay effect, the formation factor willincrease, given certain porosities.

Figure 11a shows the formation factor and porosityrelationships for different sandstone. The samplesare plotted together with different model curves, using

Archie’s equation (Archie, 1942). Wegenerated three model curves usingcementation exponents of 1.8, 2.3, and2.5. The larger cementation exponentwill produce a larger formation factor;therefore, we see that most of the cleansandstone falls in the range of the ce-mentation exponent from 1.8 to 2.3,whereas most of the clay-bearing sand-stone and cemented sandstone withpore-filling cement fall in the range ofcementation exponent from 2.3 to 2.5.Basically, when the cementation expo-nent is 1.8, Archie’s equation will givethe lower bound of the formation factor,which better represents the clean sand-stone trend, and when the cementation

Figure 10. Formation factor and porosity relationship forthree types of sandstone; model curves represent differenttypes of porosity formation factor relationship for differentsandstone as shown in Figure 3. Diagenesis and pore structureare primary textural controls on the porosity-resistivity rela-tionship. Clay-bearing and cemented sandstones show differ-ent trends than models obtained from the published literatureas shown in Figure 3.

Figure 11. Formation factor and porosity relationship for three types of sand-stone: (a) Model curves represent the Archie porosity permeability relationshipswith varying cementation exponent (m ¼ 1.8 for the lower curve,m ¼ 2.3 for themiddle curve, and m ¼ 2.5 for the upper curve). (b) SC modeling for a grain ARof 1 and a pore AR from 0.01 to 1.

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exponent is 2.5, Archie’s equation will give the upperbound of the formation factor, which better representsthe cemented sandstone trend. However, a significantnumber of clay-bearing samples fall in the same regionas clean sandstone, they are related with the distribu-tion of clay, the type of clay, the amount of clay, as wellas the diagenetic history of clay (Figure 6). Further-more, if the percentage of porosity is higher than theclay content, then pore-filling clays do not occupy allthe available porosity (Marion, 1990); in that case, theformation factor of those samples is dominated by thegeometry of the grain framework (Han, 2010). Alterna-tively, if the percentage of porosity is lower than theclay content, then porosity is filled by clay content(Marion, 1990); in that case, the formation factor ofthose samples is dominated by geometry of the claymineral assemblages (Han, 2010). Similarly, few clay-bearing samples are in the same region with cementedsandstone; they are related with cementation of quartzgrains of clay-bearing sandstone. Han (2010) describedthe presence of quartz cementation in his studiedsamples.

We modeled resistivity as a function of porosity us-ing the self-consistent (SC) approximations (Berryman,1995) (Figure 11b). In our effective-medium modeling,we assigned very high resistivity to the mineral phasequartz (1015 Ωm). The porosity and resistivity labora-tory data set may be modeled with pore aspect ratios(ARs) between 0.01 and 1. In the case of greensand,we found that we had to include spherical pores to ex-plain the lower resistivity, whereas, for clay-bearingsandstone as well as for clean sandstone, we found thatwe had to include penny-shape-like pores to explain thehigher resistivity.

Modeling of porosity velocityFigure 12a shows velocity and porosity relationships

for different sandstone. The samples are plotted to-gether with the rock physical modeling as describedin Figure 4. Most of the clean sandstone (blue points)follows the stiff-sand trend, but the clay-bearing sand-stone (black points) and greensand withpore-filling cement (green points) devi-ate from the stiff-sand trend due to thediagenetic effects. The blue arrows showthat increasing the clay content will de-crease the porosity and increase thevelocity. Similarly, increasing the cementwill also increase the velocity, but theporosity does not significantly decrease.

We modeled velocity as a function ofporosity using the SC approximations(Berryman, 1995) (Figure 12b). The po-rosity and velocity laboratory data setmay be modeled with pore ARs between0.01 and 1 and a grain AR of 1. However,the effect of clay and cement cannot beproperly captured using this model.

We simplified soft-sand and stiff-sand model curveswith empirical regressions. Here, we can see that thesoft-sand model usually provides the lower bound ofthe velocity, and it can be fit with a power equation as

VP ¼ 1.86ϕ−0.28. (19)

Gardner et al. (1974) describe a similar power lawrelationship for velocity and density (ρ ≈ 1.74V0.25

P ).The stiff-sand model provides the upper bound of the

velocity, and it can be fitted with a linear equation as

VP ¼ −8.23ϕþ 6.04. (20)

Han (1986) defines a similar relationship (VP ¼−8.06ϕþ 6.08) for laboratory-measured clean-water-saturated sandstone at 40 MPa. With the above twoequations, we can simplify the estimation of the boundsof the velocity, given a certain porosity. The coefficientswithin the equations may change with different rocks.Soft-sand and stiff-sand models can be applied to differ-ent rock types, as long as the elastic constants (moduli),critical porosity, and coordination number in the Hertz-Mindlin model are known. However, the purpose for thecurve fitting is to simplify such models so they caneasily combine with the permeability and resistivitymodels to derive a model between seismic velocityand permeability as well as seismic velocity and resis-tivity.

Relationship between permeability and velocityFigure 13 shows the permeability-velocity relation

for three types of sandstone. By increasing the velocity,the permeability will decrease. Here, the permeability ismodeled using Kozeny’s equation (Kozeny, 1927). Theporosity and specific surface area of the bulk volumehave a linear relationship in the log-log plot (Figure 9b).By changing the coefficients a and b, we can derive thespecific surface area of the bulk volume and the per-meability for certain types of sandstone using Kozeny’sequation. As discussed in Figure 9a, the model for clean

Figure 12. (a) Modeling of porosity and velocity relations using rock physicsdiagnostics as described in Figure 4. Diagenesis and pore structure are primarytextural controls on porosity-velocity relationships. The green curve is obtainedfrom the soft-sediment model for greensand. (b) Modeling of porosity and veloc-ity relations by SC approximation for grain AR of 1 and pore AR from 0.01 to 1.

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sandstone gives the upper bound on permeability andthat for cemented sandstone gives the lower boundon permeability with the same porosity. Velocity is mod-eled using the stiff-sand and soft-sand models as dis-cussed in Figure 12. Similarly, the velocity from thesoft-sand model gives the lower bound on the velocityand velocity from the stiff-sand model gives the upperbound on the velocity with the same porosity. Thelower bound can be estimated with a power equation,and the upper bound stiff-sand model can be estimatedwith a linear equation. Combing equations 17, 18, and20, we can have the mathematical expressions for thesefour modeled curves. For the first and second casesmodeled with the stiff-sand model, we have

k ¼ c

�6.04 − VPðϕÞ

8.23

�3−2a

10−2b. (21)

For the permeability model for pure sandstone (thefirst case), a is −1.0 and b is −2.2; for the permeabilitymodel with cemented sandstone (the second case), a is−5.5 and b is −3.0.

For the third and fourth cases modeled with the soft-sand model, we have

k ¼ c

�VPðϕÞ1.86

�0.28ð3−2aÞ

10−2b. (22)

As before, for the permeability model for pure sand-stone (the fourth case), a is −1.0005 and b is −2.2; forthe permeability model with cemented sandstone (thethird case), a is –5.5 and b is −3.0. Such simplified re-lations taken from different models can help us todetermine the bounds of the permeability and velocity.Because we can obtain both bounds for the permeabil-ity and velocity for each value of porosity, these boundswill become a rectangular region where all the per-meability-velocity points with the same porosity willfall in the crossplot. Inside the rectangular region, thecementation effects can move the data from left to right(soft-sand model to stiff-sand model) and also from bot-tom to top (changing the coefficient a and b for thepermeability model). We observe that for the cleansandstone (blue points), most data follow the upperbounds of the velocity-permeability trend.

Relationship between resistivity and permeabilityFigure 14 shows the formation factor-permeability

relation for three types of sandstone. With the increas-ing formation factor, the permeability will decrease.Figure 14 also shows different rock physics modelingfor the permeability and formation factor for threetypes of sandstone. The permeability is modeled usingthe Kozeny’s equation (Kozeny, 1927). The permeabilitymodel for clean sandstone provides the upper boundon permeability, whereas the permeability model for ce-mented sandstone provides the lower bound of per-meability for the same porosity. The formation factoris modeled using Archie’s equation as discussed in Fig-ure 11. When the cementation exponent m is equal to2.5, it will give the upper bound of the formation factor,and when m is equal 1.8, it will give the lower bound ofthe formation factor.

Similarly, combing equations 17 with Archie’s equa-tion, we can have the mathematical expressions forthese four modeled curves:

k ¼ cðFðϕÞÞmð3−2aÞ10−2b; (23)

where m is the cementation exponent. For clay-bearingsandstone,m is 2.5, and for pure sandstone,m is 1.8; forthe permeability model for pure sandstone a is −1.0 andb is −2.2; for the permeability model with cementedsandstone (the second case), a is −5.5 and b is −3.0.

For each porosity value, these bounds of permeabil-ity and formation factor will become a rectangular re-gion where all the permeability-formation factor pointswith same porosity will fall in the crossplot. Inside therectangular region, clay effects (increasing cementationexponent) move data from left to right and cementationeffects (changing the coefficient a, and b for permeabil-ity model) move data from bottom to top.

Relating electrical and acoustic propertiesFigure 15a shows modeling of velocity-formation fac-

tor relations by SC approximation for grain AR of 1, andpore AR from 0.01 to 1. It is clear evidence that this

Figure 13. Modeling of permeability-velocity relationship: 1,model is derived from stiff-sand model and permeability modelfor pure sandstone; 2, model is derived from the stiff-sandmodel and permeability model for cemented sandstone; 3,model is derived from soft-sand model and permeability modelfor cemented sandstone; and 4, model is derived from the soft-sand model and permeability model for pure sandstone. Givena porosity value, for example 0.2, the possible region of per-meability velocity should fall in a rectangular square (marked)in the crossplot. The permeability model for pure sandstonegives the upper bound of permeability, and the permeabilitymodel for cemented sandstone gives the lower bound of per-meability. Similarly, the stiff-sand model gives the upper boundof velocity and the soft-sandmodel gives the lower bound of thevelocity.

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theoretical model does not describe the velocity-forma-tion factor relationship. Comparing Figure 11b andFigure 12b, we observed that for electrical propertymodeling, we require one type of ARs and for elasticproperties modeling, we require another type of AR.Because we need different a type of AR for differentproperty modeling, this model does not describe thevelocity-formation factor relationship.

Figure 15b shows velocity-formation factor relationsfor three types of sandstone. With increasing velocity,the formation factor will also increase. As we discussedbefore, permeability and the formation factor haveupper bounds and lower bounds. For velocity, thesoft-sand model will produce the lower bound andthe stiff-sand model will produce the upper bound. Forthe formation factor, Archie’s equation with m ¼ 1.8will give the lower estimation, while the same equationwith m ¼ 2.5 will give the upper estimation for thesesandstones.

Combing equation 17 and 20 with Archie’s equation,we can derive the mathematical expressions for thesefour modeled curves. For the first and second caseswith the stiff-sand model, we have

F ¼�6.04 − VPðϕÞ

8.23

�−m

. (24)

For clay-bearing sandstone (the first model), m is2.5; for the pure sandstone (the second model),m is 1.8.

For the third and fourth cases using the soft-sandmodel, we have

F ¼�VPðϕÞ1.86

�−0.28m

. (25)

Figure 14. Modeling of formation factor-permeability rela-tionship: 1, model derived from the formation factor modelfor clay-bearing sandstone and permeability model for puresandstone; 2, model derived from the formation factor modelfor clay-bearing sandstone and permeability model for ce-mented sandstone; 3, model derived from the formation factormodel for pure sandstone and permeability model for ce-mented sandstone; 4, model derived from the formation factormodel for pure sandstone and permeability for pure sand-stone. As in Figure 13, given a certain porosity, such as 0.2,the possible region of formation factor-permeability shouldfall in a rectangular region (marked) in the crossplot. The per-meability model for pure sandstone gives the upper bound ofpermeability, and the permeability model for cemented sand-stone gives the lower bound of permeability. Similarly, as wediscussed in Figure 11, clay-bearing sandstone’s formationfactor can be derived using Archie’s equation, when m ¼2.5. This will give the upper bound of the formation factor.And pure sandstone’s formation factor can be derived usingArchie’s equation whenm ¼ 1.8, which gives the lower boundof the formation factor.

Figure 15. (a) Modeling of velocity-formation factor rela-tions by SC approximation for a grain AR of 1 and a poreAR from 0.01 to 1. (b) Modeling of the velocity-formation fac-tor relationship: (1) model derived from the stiff-sand modeland formation factor model for clay-bearing sandstone; (2)model derived from the stiff-sand model and formation factormodel for pure sandstone; (3) model-derived from soft-sand model and formation factor model for pure sandstone;and (4) model derived from the soft-sand model and formationfactor model for clay-bearing sandstone. As in Figure 13, givena certain porosity, for example, 0.1, the possible velocity-for-mation factors should fall in a rectangular region (marked) inthe crossplot. The stiff-sand model gives the upper bound ofvelocity, and the soft-sand model gives the lower bound ofvelocity. Similarly, as we discussed in Figure 11, the formationfactor of clay-bearing sandstone can be derived using Archie’sequation, whenm ¼ 2.5. This will give the upper bound on theformation factor. The formation factor of pure sandstone canbe derived using Archie’s equation whenm ¼ 1.8, which givesthe lower bound of the formation factor.

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For the clay-bearing sandstone (the fourth model),mis 2.5; for the pure sandstone (the third model),m is 1.8.

Similarly, for each porosity value, these bounds onvelocity and the formation factor will become a regularregion where all the velocity-formation factor points withsame porosity will fall in the crossplot. Inside the rectan-gular region, moving data from left to right indicates acementation effect (soft-sand model to stiff-sand model),while moving data from the bottom to the top indicatesa change in the clay effect (increasing cementation ex-ponent).

Figure 15b describes porosity as the first-orderparameter that affects electrical and elastic properties;clay and cement cause secondary effects. Rock physicsmodeling results show that cementation has a greatereffect on elastic properties than electrical propertiesand clay has a larger effect on electrical properties thanelastic properties.

Practical applicationsThe relationships provided in this study can greatly

help to determine permeability, resistivity, and velocityfrom porosity; to estimate permeability from resistivityand velocity; as well as to determine resistivity fromvelocity measurements. Special core calibration may benecessary for some of the models. These relationshipscan be applied at the well log scale as well as at theseismic scale. The following steps can be applied to usethese relationships at the well log scale:

1) Define the clean sand interval and clay-bearing sandinterval using petrophysical well log analysis.

2) Define the cemented sand interval using advancedrock physics diagnostics (Hossain and MacGregor,2014).

3) Apply relationships defined for clean sandstone tothe clean sand interval; for clay-bearing sandstoneto the clay-bearing sand interval and for cementedsandstone to the cemented sand interval.

Scale is an important issue when applying these rela-tionships at log scale and at seismic scale. Our definedrelationships are based on clay and cementation effects,which can be observed at the well log scale. Therefore,we assume that we can apply these relationships at logscale without resolving any upscale-related issues. How-ever, upscale-related issues must be taken into consider-ation to apply these relationships to seismic scale. Infact, some of these effects are only visible at pore scale;they are not visible at log scale and seismic scale. There-fore, it is recommended to use probability distributionsto represent the uncertainty produced by these effects(Grana et al., 2012; Dvorkin et al., 2014).

ConclusionsWe have derived simple empirical relations for

porosity, permeability, elastic, and electrical propertiesfor different types of sandstone combining heuristic andtheoretical models with laboratory-measured data.

Clean sandstone can be modeled with Kozeny’srelation; however, it breaks down for clay-bearingsandstone and diagenetically altered sandstone. Weproposed new empirical permeability model consider-ing the diagenetic effects of sandstone. The new modelcan be used to predict the permeability of differenttypes of sandstone, including clay-bearing sandstoneand diagenetically altered sandstone.

The soft-sand model and stiff-sand model can beused to describe diagenetically altered sandstone. Forporosity-velocity relations, we simplified soft-sand andstiff-sand models. Combining these relations with thenew permeability models, we defined permeability-velocity relations for clay-bearing sandstone and diage-netically altered sandstone.

We defined Archie’s relations for clean, clay-bearingsandstone and diagenetically altered sandstone. Com-bining these relations with the new permeability model,we defined formation factor-permeability relations forclay-bearing sandstone and diagenetically altered sand-stone. Combining these relations with simplified soft-sand and stiff-sand models, we defined formation fac-tor-velocity relations for clay-bearing sandstone anddiagenetically altered sandstone.

We found that porosity is the first-order parameterthat affects permeability, electrical, and elastic proper-ties; clay and cement cause secondary effects. The rockphysics modeling results show that cementation has agreater effect on elastic properties than electrical prop-erties and clay has a larger effect on electrical proper-ties than elastic properties.

AcknowledgmentsRSI is acknowledged for giving permission to publish

this work. M. Ellis, L. MacGregor, U. Strecker, and R.Ackermann are acknowledged for their valuable com-ments on this work. We would like to thank the editorC. Xu and the reviewers for their valuable time, com-ments, and suggestions that improved this paper.

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Zhou,Y.,andM.J.King,2011, Improvedupscaling for flowsim-ulation of tight gas reservoir models: Presented at SPEAnnual Technical Conference and Exhibition SPE 147355.

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Zakir Hossain received an M.S.(2007) in petroleum engineering anda Ph.D. (2011) in petrophysics and rockphysics from the Technical Universityof Denmark. During his Ph.D. study,he worked with rock physics modelingand integration of nuclear magneticresonance (NMR) study to rockphysics. In 2009, he worked as a visit-

ing research scholar at Stanford University as a part ofhis Ph.D. study under the supervision of T. Mukerji. Duringhis M.S. study, he worked with the relationship betweenstatic and dynamic properties of reservoir rocks. Hossainis a specialist in rock physics, petrophysics, rock mechanics,and amplitude variation with offset (AVO) analysis. Cur-rently, he is concentrating on the rock physics and petrophy-sics of unconventional and conventional reservoirs fromlab to log to seismic techniques. He has considerableregional experience with onshore United States, Gulf ofMexico, North Sea, Norwegian Sea, Barents Sea, Malaybasin, etc. Hossain has been working as a senior rock physi-cist at RSI since February 2012 to develop rock physicsworkflows including rock physics modeling, statisticalrock physics modeling, rock physics templates, rock me-chanics, NMR, 4D, AVO to petrophysics, and interpretationprojects for the company. His research interest includesrock physics modeling, AVO analysis, NMR studies to rockphysics, pore fluids effect on reservoir properties includingCO2, direct hydrocarbon indicator (DHI), 4D rock physics,pore pressure prediction (PPP), static versus dynamicproperties, and integration of rock mechanics to rockphysics.

Yijie Zhou received B.S. and M.S.degrees in geophysics fromPekingUni-versity and a Ph.D. in petroleum engi-neering from Texas A&M University.He is a petrophysicist with RSI, devel-oping petrophysical/rock physicalapplications and performing well loganalysis, rock physics modeling, andquantitative reservoir characterization.

He has an integrated background of reservoir engineeringand geoscience, and his research interests include reservoirsimulation and management, geologic models upscalingfor flow simulation, infill well placement, finite elementmodeling, geomechanics, and rock mechanics.

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