AbstractThe complex physics of multiphase flow in porous mediaare usually modeled at the field scale using Darcy-typeformulations. The key descriptors of such models are therelative permeabilities to each of the flowing phases. It iswell known that, whenever the fluid saturations undergoa cyclic process, relative permeabilities display hysteresiseffects.

In this paper we investigate hysteresis in the relative per-meability of the hydrocarbon phase in a two-phase system.We propose a new model of trapping and waterflood rela-tive permeability, which is applicable for the entire rangeof rock wettability conditions. The proposed formulationovercomes some of the limitations of existing trapping andrelative permeability models. The new model is validatedby means of pore-network simulation of primary drainageand waterflooding. We study the dependence of trapped(residual) hydrocarbon saturation and waterflood relativepermeability on several fluid/rock properties, most notablythe wettability and the initial water saturation.

The relevance of relative permeability hysteresis is thenevaluated for modeling geological CO2 sequestration pro-cesses. Here we concentrate on CO2 injection in salineaquifers. In this setting, the CO2 is the nonwetting phase,and trapping of the CO2 is an essential mechanism afterthe injection phase, during the lateral and upward migra-tion of the CO2 plume. We demonstrate the importance ofaccounting for CO2 trapping in the relative permeabilitymodel for predicting the distribution and mobility of CO2

in the formation. We conclude that a proper treatment ofthe nonwetting phase trapping leads to a higher estimateof the amount of CO2 that it is safe to inject.

IntroductionHysteresis refers to irreversibility or path dependence. Inmultiphase flow, it manifests itself through the dependenceof relative permeabilities and capillary pressures on thesaturation path and saturation history. From the pointof view of pore-scale processes, hysteresis has at least twosources: (1) contact angle hysteresis; and (2) trapping ofthe nonwetting phase.

The first step in characterizing relative permeability hys-teresis is the ability to capture the amount of oil that istrapped during any displacement sequence. Indeed, a trap-ping model is the crux of any hysteresis model: it deter-mines the endpoint saturation of the hydrocarbon relativepermeability curve during waterflooding.

Extensive experimental and theoretical work has focusedon the mechanisms that control trapping during multi-phase flow in porous media.1–3 Of particular interest tous is the influence of wettability on the residual hydro-carbon saturation. Early experiments in uniformly wet-ted systems suggested that waterflood efficiency decreaseswith increasing oil-wet characteristics.4,5 These experi-ments were performed on cores whose wettability was al-tered artificially, and the results need to be interpretedcarefully for two reasons: (1) reservoirs do not have uni-form wettability, and the fraction of oil-wet pores is a func-tion of the topology of the porous medium and initial watersaturation,6 and (2) the core-flood experiments were notperformed for a long enough time, and not enough pore-volumes were injected to drain the remaining oil films toachieve ultimate residual oil saturation. Other core-floodexperiments, in which many pore volumes were injected,the observed trapped/residual saturation did not followa monotonic trend as a function of wettability, and wasactually lowest for intermediate-wet to oil-wet rocks.7–9

Jadhunandan and Morrow10 performed a comprehensiveexperimental study of the effects of wettability on water-flood recovery, showing that maximum oil recovery wasachieved at intermediate-wet conditions.

An empirical trapping model typically relates thetrapped (residual) hydrocarbon saturation to the maxi-mum hydrocarbon saturation, that is, the hydrocarbonsaturation at flow reversal. In the context of waterflooding,a trapping model defines the ultimate residual oil satura-tion as a function of the initial water saturation. The mostwidely used trapping model is due to Land.11 It is a single-

Copyright 2005, Society of Petroleum Engineers This paper was prepared for presentation at the 2005 SPE Annual Technical Conference and Exhibition held in Dallas, Texas, U.S.A., 9 – 12 October 2005. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

SPE 96448

Relative Permeability Hysteresis: Trapping Models and Application to Geological CO2 Sequestration E.J. Spiteri, SPE, and R. Juanes, SPE, Stanford U.; M.J. Blunt, SPE, Imperial College; and F.M. Orr Jr., SPE, Stanford. U.

2 SPE 96448

parameter model, and constitutes the basis for a numberof relative permeability hysteresis models. Other trap-ping models are those of Jerauld12 and Carlson.13 Thesemodels are suitable for their specific applications but, aswe show in this paper, they have limited applicability tointermediate-wet and oil-wet media.

Land11 pioneered the definition of a “flowing satura-tion”, and proposed to estimate the imbibition relative per-meability at a given actual saturation as the drainage rel-ative permeability evaluated at a modeled flowing satura-tion. Land’s imbibition model11 gives accurate predictionsfor water-wet media,14 but fails to capture essential trendswhen the porous medium is weakly or strongly wetting tooil. The two-phase hysteresis models that are typicallyused in reservoir simulators are those by Carlson13 andKillough.15 A three-phase hysteresis model that accountsfor essential physics during cyclic flooding was proposedby Larsen and Skauge.16 These models have been eval-uated in terms of their ability to reproduce experimentaldata17,18 and their impact in reservoir simulation of water-alternating-gas injection.18,19 Other models are those byLenhard and Parker,20 Jerauld12 and Blunt.21 More re-cently, hysteresis models have been proposed specificallyfor porous media of mixed wettability.22–24

All of the hysteresis models above require a boundingdrainage curve and either: (1) a waterflood curve as input;or (2) a calculated waterflood curve using Land’s model.The task of experimentally determining the bounding wa-terflood curves from core samples is arduous, and the de-velopment of an empirical model that is applicable to nonwater-wet media is desirable. In this paper, we introducea relative permeability hysteresis model that does not re-quire a bounding waterflood curve, and whose parametersmay be correlated to rock properties such as wettabilityand pore structure.

Since it is difficult to probe the full range of relativepermeability hysteresis for different wettabilities experi-mentally, we use a numerical tool –pore-scale modeling–to predict the trends in residual saturation and relativepermeability. As we discuss later, pore-scale modeling iscurrently able to predict recoveries and relative permeabil-ities for media of different wettability reliably.25–30 We willuse these predictions as a starting point to explore the be-havior beyond the range probed experimentally.

In summary, this paper contains two main results:

1. A new model of trapping and waterflood relativepermeability, which is able to capture the behaviorpredicted by pore-network simulations for the entirerange of wettability conditions.

2. A demonstration of the importance of nonwetting-phase trapping and relative permeability hysteresis forthe assessment of geological CO2 sequestration pro-cesses.

In the next section we present a summary of the equa-tions of the proposed model. We follow with an overview ofpore-scale modeling of trapping and relative permeabilityhysteresis. We then present a new model of trapping andwaterflood relative permeability. As an application of theimportance of trapping and hysteresis we perform three-dimensional simulations of realistic scenarios of CO2 se-

questration in heterogeneous saline aquifers. In the finalsection of this paper we gather the main conclusions andanticipate ongoing and future work.

Summary of model equationsIn this section, for clarity, we present the trapping andhysteresis model. A detailed description of the tools usedand a physical interpretation of the results is provided inthe subsequent sections.

A trapping model attempts to capture the residual(trapped) oil saturation Sot after waterflooding, given aninitial oil saturation Soi achieved during primary drainageof oil into a water-filled medium. Therefore, the trappingmodel may be viewed as a curve on a diagram of initial oilsaturation versus residual oil saturation (initial–residual orIR curve). The development of a new trapping model ismotivated by the fact that, in contrast with most com-mon trapping models,11,12 initial–residual curves obtainedfrom pore-network simulations are not monotonic whenthe medium is not strongly water-wet. This non-monotonictrend has been confirmed experimentally.10

The simplest functional form that allows us to reproducea non-monotonic behavior is the quadratic expression:

Sot = αSoi − βS2oi. (1)

Parameters α and β correspond to the initial slope and thecurvature of this curve, respectively. They must satisfy thefollowing requirements:

0 ≤ α ≤ 1, β ≥ 0. (2)

The model contains two parameters, α and β, which mustbe calibrated for each rock–fluid system. Therefore, theywill depend on the pore-space morphology and the wetta-bility characteristics of the medium. Typical dependenceof these two parameters on the wettability of the mediumis discussed and explained later in the paper.

Inspired by Land’s model,11 we propose to calculate thewaterflood relative permeability ki

ro at the actual oil sat-uration So as the drainage relative permeability kd

ro at adifferent flowing saturation:

kiro(So) = kd

ro(Sof ). (3)

The flowing saturation is evaluated by means of the fol-lowing expression:

Sof =1

2β

[

(α − 1)

+√

(α − 1)2 + 4β[So − Sot + γ(So − Sot)(So − Soi)]

]

,

(4)

where Sot is the trapped oil saturation given by Eq. (1),and γ is a parameter related to the presence of film flow.It is dependent on the wettability characteristics of themedium, and is typically negative for strongly water-wetmedia and positive for weakly and strongly oil-wet media.This behavior will be discussed in detail later.

SPE 96448 3

Pore-scale modeling of trapping and hysteresisIn pore-network modeling the pore space is described bya network of pores connected by throats with an idealizedgeometry. A set of physically-based rules describe the con-figuration of the fluids within each pore and throat, as wellas the mechanisms for the displacement of one fluid by an-other. This approach was pioneered by Fatt31 and has re-ceived increasing attention over the past decade. Blunt32

and Blunt et al.33 provide a detailed description of thefundamentals and applications of pore-network modeling,together with an extensive literature review. One of thesuccessful application areas of pore-network models is theprediction of multiphase flow properties, such as capillarypressure and relative permeability. This success hinges onthe following:

1. The ability to reproduce the essential geometric fea-tures of the pore space of real rocks. A realis-tic three-dimensional pore-space characterization maybe obtained in a variety of ways: assembly of two-dimensional sections to form a three-dimensional im-age;34 direct X-ray microtomography of the three-dimensional pore space;35 stochastic 3D modelingwith statistics inferred from two-dimensional thin sec-tions;36 process-based reconstruction in which graindeposition, compaction and cementation are mod-eled.37,38

2. The ability to capture wettability effects. Most pore-network models used today introduce wettability ef-fects based on the pore-level scenario of wetting pro-posed by Kovscek et al.6 Their model mimics thesaturation change typical of a hydrocarbon reservoir.The medium is initially filled with water, and the rocksurfaces are water-wet. During oil migration, the oilinvades the pore space, altering the wettability of thesolid surface in contact with the oil. In this fashion,the network displays mixed wettability: a fraction ofa pore or throat may be oil-wet, while the corners andcrevices not in contact with the oil remain water-wet.

The combination of realistic pore geometry/topology andcorrect characterization of displacement and trappingmechanisms has allowed pore-network models to pre-dict hysteretic capillary pressure and relative permeabilitycurves under a wide range of wettability characteristics.

Experimental measurements of hysteretic relative per-meability for mixed-wet and oil-wet media are scarce.10,39

Given the success of pore-network models to reproduce ex-perimental data,26–28 in this work we have adopted the useof pore-network modeling as a way to investigate the fullspectrum of wettability conditions. Pore-network simula-tions results are taken as “data” to develop and validateempirical trapping and hysteresis models.

Description of the pore-network simulations. Weused the two-phase flow pore-network simulator developedby Valvatne and Blunt.28 The model has similarities withother network models.29,40,41 A full description of themodel is given in the references above and will be omit-ted here.

We used a three-dimensional pore-network of a Bereasandstone developed by Bakke and Øren.42 The model is

a cube of 3×3×3 mm3 containing 12,349 pores and 26,146throats. The absolute permeability of the rock is 2287 mDand the net porosity is 0.183, where 0.0583 is clay-boundor micro-porosity.

We investigated sequences of two consecutive displace-ments: primary drainage (oil invasion) and waterflooding.During primary oil drainage the network, which is initiallyfilled with water, is assumed to be strongly water-wet witha receding contact angle θr = 0◦. As the oil invades thelargest pores first in piston-like displacement, the water re-cedes and is squeezed to the crevices and pore throats untila very high capillary pressure or a target oil saturation isreached. At this point, the surface of the rock in con-tact with oil will undergo wettability alteration, while thecorners and elements that still contain only water remainstrongly water-wet. Wettability alteration is accounted forby changing the contact angle. In principle, one couldchange the advancing contact angle θa and the recedingcontact angle θr independently. In this work, however, weused a correlation proposed by Morrow43 to link both theadvancing and receding contact angles with an intrinsic

contact angle θi. This relationship is shown in Fig. 1.

Fig. 1— Relationship between receding and advancing contactangles on a rough surface, as a function of intrinsic contactangle measured at rest on a smooth surface.43 [Figure fromValvatne and Blunt.28]

During waterflooding there are several physical mech-anisms by which the water can displace the oil in place.2

These mechanisms include piston-type displacement, coop-erative pore-body filling and snap-off. The predominanceof any given displacement mechanism is strongly depen-dent on the wettability (specified by the advancing contactangle). These displacement processes and their implemen-tation are described in detail in the literature.28,29,40,41

After individual displacement events the transport proper-ties are calculated. The equations for absolute permeabil-ity, relative permeability and other transport parameterscan also be found in the literature.28,40,41,44 Followingthis procedure, pore-network simulations have been shownto reproduce experimental capillary pressure and relativepermeability curves both in primary drainage and water-flooding40 and for a variety of wettability conditions.26–28

We investigated the full range of wetting conditions after

4 SPE 96448

wettability alteration, by choosing average intrinsic con-tact angles between 20◦ and 160◦. Due to pore-scale in-homogeneities in the rock minerals and surface roughness,assigning a uniform contact angle throughout is unrealis-tic. Therefore, we assigned contact angles throughout thenetwork randomly within ±20◦ of the average value.

In this work we assume that the contact angle distri-bution in originally oil-filled pores is independent of Soi.In reality, a higher Soi represents a larger initial capillarypressure that may cause protective water films to collapsein oil-filled pores, causing a more significant wettabilityalteration than for low Soi.

6 However, this simple char-acterization of wettability has been shown to be sufficientto predict trends in recovery seen experimentally.28 Alsowe consider systems where there is a relatively uniformdistribution of wettability – we do not consider mixed-wetmedia where initially oil-filled pores may be water-wet oroil-wet with two distinct distributions of contact angle inthe same rock.

For each contact angle distribution we performed a seriesof displacement pairs (oil invasion and waterflooding), witha different target oil saturation Soi ranging from a verysmall value (almost no oil invasion) to the maximum valuepossible (connate water conditions). Next, we present re-sults from an extensive set of simulations for different con-tact angles and different target oil saturations.

Pore-network predictions of trapping. The use of apore-network simulator allows us to quantify the effectsof wettability and initial oil saturation on the trapping ofoil during waterflooding. The main results of the pore-network simulations are compiled in Fig. 2, where we plotthe initial–residual (IR) curves for different intrinsic con-tact angles. For a given curve (i.e. for a specific valueof the intrinsic contact angle after wettability alteration),each point denotes the trapped oil saturation Sot that cor-responds to a particular initial oil saturation Soi. The ini-tial oil saturation is dictated by the point at which primarydrainage ceases (and waterflood starts), and the trappedoil saturation is the value of unrecoverable (or residual) oilafter waterflooding.

The most noteworthy characteristic of the IR curvesshown in Fig. 2 is that they do not display a monotonicallyincreasing behavior for mixed-wet and oil-wet media. Thismeans that, for oil-wet media, higher oil saturation afterthe initial oil invasion may lead to lower residual oil satura-tion after waterflooding. This counterintuitive behavior isanalyzed and explained below, in terms of pore-scale fluidconfigurations and displacement mechanisms.

For water-wet media (small contact angles) trappingduring waterflooding is controlled by snap-off. As the ini-tial oil saturation is increased, oil is pushed into smallerand smaller pores. During waterflooding, water fills thesmallest pores first and snaps off more and more oil. Theamount of trapped oil increases monotonically with in-creasing initial oil saturation simply because there is moreoil to trap. As the contact angle increases there is acrossover from trapping by snap-off to trapping by bypass-ing, as the water tends to advance in a connected front withpiston-like advance on throats and cooperative pore-filling.We begin to see a non-monotonic behavior because, as the

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

ot)

θ = 20o

θ = 60o

θ = 90o

θ = 100o

θ = 110o

θ = 160o

Non−monoticity oftrapping relation

Fig. 2— Trapped oil saturation Sot as a function of the initialoil saturation (Soi) for different intrinsic contact angles (θ).

initial water saturation increases, we have more water fill-ing from the pores and throats in connected patches fromthroats that are initially water-filled. When these patchesjoin up, there can be trapping, as oil becomes strandedbetween these clusters. Low initial water saturation (i.e.high initial oil saturation) implies that there are few wa-ter clusters and little chance for bypassing. Trapping in-creases as the number of clusters increases and then de-clines again as there is less oil to trap in the first place.For intrinsic contact angles above 90◦ (advancing contactangles above 120◦) we observe the same behavior, but witheven less trapping. The reason is the presence of oil lay-ers within the network. Water fills the largest pores andthroats in an invasion percolation-like process. Oil remainsconnected in layers sandwiched between water in the cen-ter of an element and water in the corners. These layerslead to very little trapping, although the oil relative per-meability is very low. These layers are stable until the twowater/oil interfaces meet. High initial water saturationmeans that water bulges out in the corners causing theseinterfaces to meet, trapping more oil than for low initialwater saturation. Again, for sufficiently large Swi, (lowSoi), there is less trapping simply because there is less oilto trap.

We should also mention that the extremely low trappedoil saturations for very high initial oil saturations are anartifact of the criterion used for ascertaining the stability ofoil layers.28 In the future we plan to incorporate a stabilitycriterion based on free energy balance that predicts thatoil layers become unstable before the two water/oil inter-faces touch each other.45 For practical purposes, however,such high initial oil saturations are never achieved duringmigration of oil into realistic reservoirs—the initial watersaturation is typically much higher than 5%.

In conclusion, the trapping mechanisms that we haveindicated allow for a physical explanation of the nonmono-tonic behavior of the initial–residual curves. A completepicture of the trapping relation is given in Fig. 3 as a trap-

ping surface, that is, a surface that describes the residualoil saturation as a function of the initial oil saturation and

SPE 96448 5

the intrinsic contact angle after wettability alteration.

00.2

0.40.6

0.8

50

100

1500

0.2

0.4

0.6

Initial oil saturation (S oi)

Contact angle (θ)

Tra

pped

oil

satu

ratio

n (S

ot)

00.2

0.40.6

0.8

50100

150

0

0.1

0.2

0.3

0.4

0.5

0.6

Initial oil saturation (S oi)Contact angle (θ)

Tra

pped

oil

satu

ratio

n (S

ot)

Fig. 3— Two views of the trapping surface. Trapped oil sat-uration depending on the initial oil saturation (Soi) and rockwettability in terms of the intrinsic contact angle (θ).

The trapped oil saturation dictates the endpoint ofthe relative permeability waterflood curves. An impor-tant practical consequence of the nonmonotonic relation oftrapped vs. initial oil saturation for mixed- to oil-wet mediais that waterflood scanning curves will cross, as sketchedin Fig. 4.

Both of these features—nonmonotonic trapping relationand crossover of waterflood relative permeability curves—are not present in existing empirical models. This moti-vates the development of new empirical trapping and hys-teresis models that reproduce the observed behavior.

Pore-network predictions of waterflood relative

permeability. It is important to understand the trappingmechanisms that ultimately define the shape of the rela-tive permeability curves during waterflooding. The trap-ping model determines the endpoint residual saturationswhen the capillary pressure is lowered to an extremely low

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

(w))

So,max

Soi

Sot,max

kiro(w)

kdro(w)

Sot

Fig. 4— Waterflood relative permeability curves that crossdue to the nonmonotonic trapping relationship.

value. The trapping mechanisms at this point should notbe generalized for the entire waterflood process. Becausedifferent competing trapping mechanisms may dominateat different capillary pressure levels, this affects the shapeof the relative permeability curve.

In water-wet systems we have already mentioned thattrapping is primarily due to snap-off. After a certainpoint during waterflooding, the flowing oil phase becomestrapped due to the invading water phase. However, thismechanism does not operate right away. At the beginningof the waterflood process, the nonwetting phase relativepermeability is slightly higher than the drainage relativepermeability. This is seen in the two-phase experimentsperformed by Oak46 in water-wet Berea sandstone for bothwater/gas and water/oil systems (see Fig. 5).

During primary drainage, the oil preferentially fills largepores, leaving water residing in narrower throats and in thecorners of the pore space. At the end of primary drainage,many oil-filled pores have only a single connecting throatthat is also oil-filled. These dead-end pores may contain alarge saturation but do not contribute to the connectivityof the oil. During waterflooding, pore-filling is favored inpores that have many surrounding water-filled throats –essentially these dead-end pores.2 Thus initially there is acascade of pore-filling, where the oil saturation decreaseswith little decrease in oil relative permeability. This pro-cess competes with snap-off which traps oil and in con-trast leads to a large decrease in relative permeability. Forwater-wet media, the former process generally is more sig-nificant at high oil saturation, giving the typical hysteresispatterns seen in Fig. 5. The pore-network model is ableto reproduce this behavior with a quantitative agreementwith experiment,28 see Fig. 6.

In contrast, the trapping mechanisms that control theshape of the oil waterflood relative permeabilities in oil-wetmedia are very different from those of a water-wet rock. Atthe beginning of the waterflood, water percolates throughthe largest pores, leading to a significant reduction in theoil relative permeability. The oil, which remains connectedthrough oil films and layers, drains down to very low sat-

6 SPE 96448

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Gas saturation (Sg)

Gas

rel

ativ

e pe

rmea

bilit

y (k

rg)

Gas−Water experiment (200 md)

Drainageimbibition

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

Oil−Water experiment (1000 md)

DrainageImbibition

Fig. 5— Relative permeability curves of the nonwetting phasefor water/gas (top) and water/oil (bottom) systems fromOak’s46 experimental data in water-wet Berea sandstone.

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

Contact angle (0)

drainageimbibition

Fig. 6— Oil relative permeabilities generated from pore-network simulations in strongly water-wet media.

urations but at a low rate due to the small conductanceof these layers. The shape of these relative permeabilitycurves, as predicted by pore-network simulations, is shownin Fig. 7.

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

Contact angle (180)

drainageimbibition

Fig. 7— Oil relative permeabilities generated from pore-network simulations in strongly oil-wet media.

Development and validation of a new model ofrelative permeability hysteresis

In the previous section we used pore-network modeling tohighlight the following features of wettability effects on thewaterflood relative permeability:

1. The initial–residual (IR) curves are not monotonic formedia that are not strongly water-wet (Fig. 2).

2. For intermediate-wet and oil-wet media, the scanningcurves of oil relative permeability may cross (Fig. 4).

3. In strongly water-wet media, the trapped oil satura-tion is high but the waterflood relative permeabilitymay be higher than the drainage relative permeabilityat high oil saturations (Fig. 6).

4. In contrast, for strongly oil-wet media, the trappedoil saturation is low but the waterflood relative per-meability decreases sharply at high oil-saturations(Fig. 7).

Clearly, this markedly different behavior in water-wet andoil-wet media needs to be incorporated in the empiricalmodel. We start by describing a new trapping submodeland we follow with the proposed waterflood relative per-meability model.

The trapping model. We begin by reviewing some ofthe existing trapping models. These models were originallydesigned to account for gas trapping, but for consistencywe will treat them for oil trapping. We then formulate anew model and assess its performance for the full spectrumof wettability conditions.

SPE 96448 7

Land trapping model. The Land model11 is the mostwidely used empirical trapping model. Most relative per-meability models that incorporate hysteresis12,15,16,20–22

are based on this trapping model. It was developed topredict trapped gas saturation as a function of the initialgas saturation based on published experimental data fromwater-wet sandstone cores.47–49

In this model, the trapped nonwetting phase saturationis computed as:

Sot(Soi) =Soi

1 + CSoi, (5)

where Soi is the initial oil saturation, or the saturation atthe flow reversal, and C is the Land trapping coefficient.The Land coefficient is computed from the bounding oilinvasion and waterflood curves as follows:

C =1

Sot,max−

1

So,max, (6)

where So,max is the maximum oil saturation, and Sot,max isthe maximum trapped oil saturation, associated with thebounding waterflood curve. All these quantities are illus-trated in Fig. 8. The value of the Land trapping parame-ter is dependent on the type of rock and fluids.18

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

So,max

Soi

Sot,max

kiro

kdro

Sot

(Soi

)

Fig. 8— Parameters required in the evaluation and applicationof the Land trapping model.

Carlson trapping model. A simplified hysteresismodel proposed by Carlson13 implicitly defines a trappingmodel. The Carlson model requires the bounding drainageand waterflood curves. The trapped oil saturation is de-termined by shifting the bounding waterflood curve to in-tersect the intermediate initial oil saturation at the flowreversal. The idea behind Carlson’s interpretation is touse the model of the waterflood relative permeability scan-ning curves as being parallel to each other. This geometricextrapolation procedure is illustrated in Fig. 9.

The trapped nonwetting-phase saturation is computedas

Sot = Sot,max − ∆So, (7)

where ∆So is the shift in the waterflood scanning curvewith respect to the imbibition bounding curve (see Fig. 9).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

∆ So

Soi

So

Sof

Sot,max

kiro

kdro

Fig. 9— Geometric extrapolation of the oil relative permeabil-ity and trapped saturation during waterflooding, as proposedby Carlson.13

This model is adequate if the intermediate scanningcurves are almost parallel and there is little curvature inthe waterflood curve. The model is problematic when thesystem is oil wet. The large curvature of the bounding wa-terflood relative permeability curve at low saturations doesnot allow prediction of intermediate relative permeabilitycurves since any shifting will make the endpoint trappedgas saturation negative, a nonphysical value.

Jerauld trapping model. Jerauld’s trapping model12

is an extension of the Land trapping model that accountsfor the “plateau” observed in the initial–residual (IR)curves for mixed-wet rocks.50 The trapped nonwetting-phase saturation is given by:

Sot =Soi

1 + C S1+b/Coi

. (8)

In the original publication,12 the expression of the trappedsaturation Sot was given in terms of the residual satura-tion achieved when Soi = 1, Sor = 1/(C + 1). Jerauldintroduced a second tuning parameter b in addition to theLand coefficient. If this parameter is set to zero, Jerauld’smodel reduces to the Land trapping model. When this pa-rameter is equal to one, the trapping curve has a zero slopeat Soi = 1. Although Jerauld argued that the IR curvesshould not have a negative slope, his model allows for suchbehavior if b ≥ 1.

Although the fit of Jerauld’s model to the pore-networkdata was good for water-wet and intermediate-wet condi-tions (for which the model was designed), it was not assatisfactory for strongly oil-wet media. One of the reasonsis that Jerauld’s model assumes that the IR curve has aunit slope near the origin. This behavior does not conformto pore-network predictions (see the curves correspondingto θ = 110◦ and 160◦ in Fig. 2).

A new trapping model. We notice that the shapesof the trapping curves (Fig. 2) may be fit to a parabola.We establish the following simple quadratic relationship

8 SPE 96448

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 20

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 40

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 60

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 80

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 100

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 120

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 140

0 0.5 10

0.1

0.2

0.3

0.4

0.5

Initial oil saturation (Soi

)

Tra

pped

oil

satu

ratio

n (S

or)

Contact angle 160

Fig. 10— Performance of the new trapping model: initial–residual curves calibrated against pore-network simulation data.

between the trapped oil saturation Sot and the initial oilsaturation Soi:

Sot = αSoi − βS2oi. (9)

The parameters α and β correspond to the initial slopeand the curvature of this curve, respectively. These pa-rameters were tuned to minimize the least squared errorbetween the model prediction and the pore-network sim-ulation data. The optimization is constrained by the fol-lowing restrictions:

0 ≤ α ≤ 1, β ≥ 0. (10)

For an initial slope greater than 1, the trapping modelwould predict more trapped oil than what was originallypresent, which is not physically possible. The “optimal”parameters α and β are shown in Fig. 11 as functions ofthe intrinsic contact angle.

The performance of the optimization is illustrated inFig. 10. Notice that for water-wet media, the modeltends to slightly overestimate the trapped oil saturationwhen the initial oil saturation is high. This is becauseof the constraint in the optimization, and the inabilityof the model to achieve the desired curvature at the de-sired location. The important consideration is that the

50 100 1500

0.5

1

1.5

2

Contact angle (θ)

αβ

Fig. 11— Dependence of parameters α and β of the proposedquadratic trapping model on the intrinsic contact angle of ini-tially oil-filled pores for a network model of a Berea sandstone.

SPE 96448 9

trapping model reproduces the observed trapping behaviorfor all wettability conditions, even if it may slightly over-estimate the trapping of the bounding waterflood curve(Soi = So,max = 1 − Swc). We should also mention thatthe pore-network simulator is likely to underestimate thetrapping for the bounding curves due to an overly opti-mistic criterion for the stability of oil layers. If a new,free-energy based stability criterion is implemented,45 weexpect a better agreement between the trapping model andthe pore-network predictions.

When the parameters calculated from the optimizationare employed, the resulting trapping surface is shown inFig. 12. This surface should be compared with the oneobtained from pore-network simulations (Fig. 3).

00.2

0.40.6

0.8

50

100

1500

0.2

0.4

0.6

Initial oil saturation (S oi)

Contact angle (θ)

Tra

pped

oil

satu

ratio

n (S

ot)

00.2

0.40.6

0.8

50100

150

0

0.1

0.2

0.3

0.4

0.5

0.6

Initial oil saturation (S oi)

Contact angle (θ)

Tra

pped

oil

satu

ratio

n (S

ot)

Fig. 12— Trapping surface determined with the new trappingmodel.

The waterflood relative permeability model.

Most existing relative permeability hysteresis mod-els12,15,16,20,21 either require a bounding waterflood curveor model this curve according to Land’s11 waterflood rel-ative permeability model. The development of his modelis described below. The new relative permeability model

we propose is an extension of Land’s model to accountfor the different pore occupancies at different wettabilityconditions.

Land waterflood model. As a prelude to the devel-opment of the new waterflood relative permeability modelproposed in this work, we revisit the derivation of Land’srelative permeability model. The basis of Land’s formula-tion is to express the waterflood relative permeability ki

ro

at a given oil saturation (So) as being equal to the drainagepermeability kd

ro evaluated at a flowing oil saturation Sof

(see Fig. 13):

kiro(So) = kd

ro(Sof ). (11)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

(w))

Soi

So

kiro(w)

kdro(w)

Sot

(Soi

)

∆So

Sof

Fig. 13— Trapped and flowing saturations used to determinethe waterflood relative permeability from the drainage relativepermeability.

At any bulk saturation So, the flowing oil saturation Sof

and the trapped saturation ∆So are related by

∆So = So − Sof . (12)

Land makes the assumption that the trapped saturation∆So is the cumulative trapped saturation at a given pointin the waterflood process and that this quantity increasesas more of the flowing saturation becomes trapped. Heassumes that the maximum amount of cumulative trap-ping, equal to the trapped saturation determined by histrapping model (Equation (5)), occurs when the flowingsaturation becomes zero (So = Sot(Soi)). It is importantto note that in Land’s formulation it is necessary to obtainthe maximum trapped oil saturation Sot,max from a core-flood experiment in order to extract the appropriate Landtrapping coefficient C.

The intermediate trapped saturation ∆So is equal to thecumulative trapped saturation Sot minus the amount of oilthat is still flowing and will eventually be trapped:

∆So = Sot − Sot(Sof ), (13)

where

Sot ≡ Sot(Soi) =Soi

1 + CSoi(14)

10 SPE 96448

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 20o, C = 1.703

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 40o, C = 1.812

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 60o, C = 1.807

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 80o, C = 26.88

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 100o, C = 293.7

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)θ = 120o, C = 408.5

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 140o, C = 415.3

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 160o, C = 426

o

kdrg

(data)

kirg

(model)

kirg

(data)

Fig. 14— Comparison of Land’s trapping model for the bounding relative permeability curves with pore-network simulation data.

and

Sot(Sof ) =Sof

1 + CSof(15)

Substituting Equations (13)–(15) in Equation (12) one ob-tains

S2of − (So − Sot)Sof −

1

C(So − Sot) = 0. (16)

Solving this quadratic equation for Sof and taking the pos-itive root:

Sof =1

2

[

(So − Sot) +

√

(So − Sot)2 +4

C(So − Sot)

]

(17)Although Land’s assumptions are generally valid for

water-wet media, they do not hold for oil-wet media. Inwater-wet media, we noticed that the experimental andpore-network waterflood curves are sometimes above thedrainage curves for high oil saturations (Figures 5 and 6).Land’s model will generally underestimate the relative per-meabilities in this region under the assumption that thehydrocarbon phase will immediately be subjected to snap-off. In oil-wet media, the ultimate residual saturations areoften very low due to oil layer drainage. Initially, thereis a sharp decrease in the waterflood relative permeability(Fig. 7). However, at low oil saturations, oil layer drainageis the dominant mechanism, which leads to low oil resid-ual saturations achieved at very low oil relative permeabil-ities. In Fig. 14 we compare Land’s waterflood relativepermeability model to the pore-network simulation results.

The Land trapping model predicts the experimental datafairly well for water-wet media, but is unable to capture theconvex shape of the waterflood curve characteristic of oil-wet conditions. Indeed, for intrinsic contact angles greaterthan 80◦, Land’s model predicts reversible relative per-meability curves—waterflood relative permeability curvescoincide with the primary drainage curves.

A new waterflood model. Land’s waterflood modelhinges on the assumption that the trapped saturation in-creases monotonically during waterflooding. This assump-tion does not allow reproduction of the convex shape of theobserved relative permeability curves in intermediate-wetand oil-wet media. We modify Equation (13) as follows:

∆So = Sot(Soi) − Sot(Sof ) − γ(So − Sot(Soi))(So − Soi).(18)

The last term in this equation is designed to capturethe convexity of the waterflood curves in oil-wet media.It satisfies the following essential requirements: (1) theflowing saturation Sof equals zero when the bulk satura-tion reaches the ultimate trapped saturation Sot(Soi); and(2) the flowing saturation is equal to the bulk saturation(Sof = So) at the beginning of the waterflood (So = Soi)and no oil has yet been trapped. The parameter γ is anadditional parameter of the formulation, which should de-pend on rock type and wettability characteristics.

Substituting Equation (18) into Equation (12), we ob-tain:

Sof = So−Sot(Soi)+Sot(Sof )+γ(So−Sot(Soi))(So−Soi),(19)

SPE 96448 11

where the trapped saturation is given by the new trappingmodel:

Sot(Soi) = αSoi − βS2oi. (20)

We substitute Equation (20) in Equation (19) and solvefor Sof , to obtain a new model for the flowing oil satura-tion:

Sof =1

2β

[

(α − 1)

+√

(α − 1)2 + 4β[So − Sot + γ(So − Sot)(So − Soi)]

]

.

(21)

This expression of the flowing saturation is then used inEquation (11) to evaluate the waterflood relative perme-ability.

The tuning parameter γ can be obtained from fittingthe model to experimental bounding waterflood curves. Inthis investigation, we used bounding waterflood curves ob-tained from pore-network simulation. The dependence ofthe parameter γ on the intrinsic contact angle is shown inFig. 15. The trends in this relationship are the ones ex-pected. For water-wet media, the parameter γ is negative,indicating that the Land trapping model overestimates thetrapped saturation and subsequently underestimates therelative permeability. For oil-wet media this parametertakes positive values, which allows for the model water-flood curve to be below the one predicted by the Landmodel.

50 100 150−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Contact angle (θ)

γ

Fig. 15— Dependence of parameter γ of the proposed wa-terflood relative permeability model on the intrinsic contactangle of initially oil-filled pores for a network model of a Bereasandstone.

The performance of the combined trapping and water-flood models is shown in Fig. 16. Unlike Land’s trappingmodel, the new model provides a suitable fit to the bound-ing waterflood curves determined by pore-network simu-lations for all contact angles. The dark circle representsthe trapped saturation determined from the new trappingmodel. The trapped saturations predicted by the modeldo not always match the experimental end-points. This iswhy the model relative permeabilities do not fully agreewith the pore-network results, especially for saturationsclose to the ultimate residual saturation.

In Fig. 17 we compare the new waterflood model withpore-network simulated data for a set of intermediate scan-ning curves. Model predictions were obtained using thesame parameters determined from the trapping curves andthe bounding waterflood curves.

Wettability correlations. In this paper we have con-sistently used the intrinsic contact angle θi as a measureof wettability. This parameter is almost impossible to de-termine with any certainty in the laboratory since mostrocks are characterized by a large range of contact angles.Moreover, we have used a particular model43 that linksthe intrinsic contact angle with the receding and advanc-ing contact angles.

Ideally, one would correlate the trapping parameters αand β and the waterflood parameter γ with a measureof the overall wettability characteristics of the rock thatcan be determined in the lab. In fact, previous investiga-tions26,28 have shown that pore-network models are able toperform quantitative predictions of laboratory wettabilitymeasurements.

Common measures of wettability are the Amott wetta-bility indices,9 Iw and Io. A strongly water-wet medium isassociated with Iw = 1 and Io = 0, whilst values of Iw = 0and Io = 1 correspond to a strongly oil-wet medium. TheAmott–Harvey index Iwo is probably the most popularmeasure of wettability and is defined as:

Iwo = Iw − Io, (22)

which ranges between −1 and 1.These indices can be determined from two capillary pres-

sure curves corresponding to waterflood and subsequentoil flood, and can be computed directly from pore-networksimulations.26,28 The variability of the Amott oil and wa-ter indices and the Amott-Harvey wettability index withrespect to the intrinsic contact angle is given in Fig. 18

and Fig. 19, respectively. Although this was not pur-sued here, one could express the dependence of the trap-ping and waterflood relative permeability parameters withrespect to the Amott-Harvey wettability index directly,rather than the intrinsic contact angle.

Simulation of geological CO2 storageIt is well documented that carbon dioxide is a greenhousegas, and one of the main contributors to global warming.51

CO2 sequestration refers to the capture and long-term stor-age of anthropogenic CO2 in order to limit its emission tothe atmosphere.52 Injection into geological formations isone option to store CO2.

53,54 Different target formationshave been identified for this purpose, including depletedoil and gas reservoirs,55 unminable coal beds,56 and deepsaline aquifers.57,58

One of the major concerns in any sequestration projectis the potential leak of the CO2 into the atmosphere. Pos-sible causes of leaks are loss of integrity of the cap rockdue to overpressurization of the geological formation, andabandoned wells that may be present. When planning ge-ologic sequestration projects in saline aquifers or depletedhydrocarbon reservoirs, it is therefore essential to predictthe migration and distribution of the CO2 in the subsur-

12 SPE 96448

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 20o, γ = −1

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 40o, γ = −0.75

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 60o, γ = −0.1

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 80o, γ = 1.4

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 100o, γ = 1.65

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)θ = 120o, γ = 1.3

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 140o, γ = 1.3

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 160o, γ = 1.2

o

kdro

(data)

kiro

(model)

kiro

(data)

Fig. 16— Comparison of the new waterflood model with pore-network simulation data for the bounding relative permeability curves(initial saturation Soi = 0.76).

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 20o, γ = −1

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 40o, γ = −0.75

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 60o, γ = −0.1

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 80o, γ = 1.4

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 100o, γ = 1.65

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 120o, γ = 1.3

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 140o, γ = 1.3

0 0.5 10

0.2

0.4

0.6

0.8

1

Oil saturation (So)

Oil

rela

tive

perm

eabi

lity

(kro

)

θ = 160o, γ = 1.2

o

kdro

(data)

kiro

(model)

kiro

(data)

Fig. 17— Comparison of the new waterflood model with pore-network simulation data for scanning relative permeability curves(initial saturation Soi = 0.65).

SPE 96448 13

50 100 1500

0.2

0.4

0.6

0.8

1

Io

Iw

Contact angle (θ)

Wet

tability

index

Fig. 18— Amott oil and water wettability indices for all intrin-sic contact angles.

50 100 150−1

−0.5

0

0.5

1

Contact angle (θ)

Wet

tability

index

Fig. 19— Amott–Harvey wettability index for all intrinsic con-tact angles.

face structure so that injection can be maximized whilekeeping the risk of leakage at minimum.

Due to the density difference between the CO2 and thebrine, the low viscosity CO2 tends to migrate to the top ofthe geologic structure. This upward migration is some-times delayed or suppressed by low permeability layersthat impede the vertical flow of gas. There are severalmechanisms by which the CO2 can be sequestered. Theseinclude—in order of increasing time scales—dissolution inthe brine, gravity instabilities due to the larger density ofthe brine–CO2 liquid mixture, and geochemical binding tothe rock due to mineral precipitation.

In this paper we demonstrate that physical trappingmechanisms during the simultaneous flow of two fluidphases through the porous medium, such as snap-off,have a huge impact on the migration and distributionof CO2. Moreover, these trapping mechanisms act at amuch smaller time scale than other sequestration mecha-nisms such as aqueous dissolution and mineral precipita-tion. Therefore, relative permeability hysteresis becomesan order-one factor in the assessment of CO2 sequestrationprojects.

Fig. 20— Schematic of the trail of residual CO2 that is left be-hind due to snap-off as the plume migrates upwards [courtesyof Marc Hesse].

Basis for trapping in geologic CO2 sequestration.

Saline aquifers, predominantly water-wet, are prime candi-dates for geologic CO2 sequestration. In water-wet mediaand a capillary-dominated flow regime, snap-off is the dom-inant trapping mechanism. Trapping of the non-wettinggas phase occurs during waterflooding when the gas sat-uration is decreasing and the water saturation increasesas it invades the pore-space. During the injection of CO2

in the geologic formation, the gas saturation increases in adrainage-like process. Vertical flow paths are created as thegas phase migrates laterally away from the injection welland to the top of the aquifer due to buoyancy forces. Oncethe injection stops, the CO2 continues to migrate upwards.At the leading edge of the CO2 plume, gas continues todisplace water in a drainage process (increasing gas satu-ration), whilst at the trailing edge water displaces gas inan imbibition process (increasing water saturations). Thepresence of an imbibition saturation path leads to snap-off and, subsequently, trapping of the gas phase. A trailof residual, immobile CO2 is left behind the plume as itmigrates upwards (Fig. 20).

We perform a series of representative simulations to as-sess the impact of trapping and relative permeability hys-teresis on the migration and distribution of injected CO2

in a sequestration project.

Reservoir description. We carried out simulations ofCO2 injection in a synthetic but realistic model of ageologic formation. We selected the PUNQ-S3 model,which is a geometrically complex and heterogeneous three-dimensional geologic model originally designed as a testcase for production forecasting under uncertainty. Theoriginal PUNQ-S3 model is described in detail elsewhere,59

and the model data is publicly available for download.60

We modified the original model slightly to study hystere-sis and trapping effects in a CO2 injection scenario. Thegeometry of the model is characterized by a dome in thecenter. It is bounded by sealing faults on all sides andcontains five layers of fluvial sand and shale. The top ofthe reservoir is at a depth of 7677 ft. The average reser-voir thickness is 92 ft. The reservoir is discretized into19 × 28 × 5 grid blocks, of which 1761 blocks are active.

14 SPE 96448

The x and y dimension of each block is 590 ft. The averageporosity is 0.2, and the average horizontal permeability is100 md. The anisotropy ratio is about 3. A map of theabsolute permeability is shown in Fig. 21.

Fig. 21— Horizontal permeability distribution (md) of thePUNQ-S3 model.

The reservoir pore volume is approximately 109 ft3 withan initial reservoir pressure of 1020 psia at the top of thestructure. Our model has eight injection wells open to thebottom layer of the reservoir. The injection wells are rate-controlled and operate with a constraint in the maximumbottom hole pressure of 3000 psia. We assume that thefaults on all sides of the reservoir act as no-flow boundariesso the effects of regional groundwater flow through thereservoir are not included in this study.

The relative permeabilities of water and gas are takenfrom the Oak dataset46 for a water-wet Berea sandstoneand a gas–water system. The relative permeability curvesare plotted in Fig. 22. Because the medium is stronglywater-wet, hysteresis effects are significant in the gas rel-ative permeability only. We simulate hysteresis effects byproviding the simulator with the bounding drainage andwaterflood curves from the experimental data (Fig. 22).These data result in a Land trapping coefficient C ≈ 1.Intermediate scanning curves are computed using Kil-lough’s15 hysteresis model. The PVT properties of the gas(density, viscosity and formation volume factor) are repre-sentative of the properties of CO2 at reservoir conditions.

Setup of numerical simulations. The formation is ini-tially filled with brine. CO2 injection is simulated by con-trolling the volume of CO2 that is injected into the for-mation. A total of 0.15 pore volumes are injected into thebottom layer of the aquifer. The injection rate and volumeis the same for all eight injectors.

We simulated four different scenarios, summarized inTable 1. They were designed to assess the following fac-tors:

1. Hysteresis and trapping, by comparing the results ofCase 2 (in which hysteresis is modeled) and Case 1 (inwhich hysteresis is not modeled).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Gas saturation (Sg)

Gas

rel

ativ

e pe

rmea

bilit

y

Drainage kdrg

Imbibition kirg

Sgi

=0.69

Sgt

=0.4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Water saturation (Sw)

Wat

er r

elat

ive

perm

eaili

ty (

k rw) S

wc=0.31

Fig. 22— Relative permeability curves used in the CO2 sim-ulations, taken from Oak’s dataset46 for a water-wet Bereasandstone.

2. Injection rate, by comparing the results of Case 2(CO2 injection during one year) and Case 3 (injectionof the same volume over a period of ten years).

3. Injection of water, by comparing Case 4 (injection of0.05 pore volumes of water following CO2 injection)with Case 3, respectively (injection of CO2 alone).

For each case, we show results of the fluid distribution af-ter 50 years from the beginning of the injection phase. Weplot three-dimensional views of the water saturation Sw

(saturation of CO2 is simply Sg = 1 − Sw). Additionalinsight into the dynamic behavior of the system is gainedby plotting the evolution of the CO2 saturation at specificgridblocks. We chose three “observation” points: one nearthe top of the anticline structure (block 13, 18, 1), one ata slightly lower elevation (block 7, 21, 1) and a third oneat a lower elevation still (block 11, 11, 1). All of them arelocated at the top layer of the formation. The location ofthe injection wells and the observation gridblocks is shownin Fig. 23.

Effect of hysteresis and trapping. We begin by illus-trating the dramatic effect of relative permeability hystere-sis on the predictions of the fate of the injected CO2. We

SPE 96448 15

Table 1— Summary of CO2 sequestration simulations.Case no. Description Injection scheme Injection time

1 No hysteresis 0.15 PV gas 1 yr2 Hysteresis 0.15 PV gas 1 yr3 Hysteresis 0.15 PV gas 10 yr4 Hysteresis-water 0.15 PV gas, 0.05 PV water 10yr, 1yr

INJ-07

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2340 2350 2360 2370 2380 2390 2400 2410 2420

(7,21,1)

(11,11,1)

(13,18,1)

Fig. 23— Three-dimensional view of the depth (ft) of the topof the PUNQ-S3 formation. Also shown are the injection wellsand the “observation” grid blocks.

compare the results from Case 1 (no hysteresis) and Case 2(with hysteresis). In Case 1, the gas relative permeabilityis assumed to be reversible, and only the drainage curve isused. Both cases simulate injection of a total of 0.15 porevolumes of CO2 during one year, and the migration of theCO2 plume during the next 49 years.

In Fig. 24 we plot the distribution of water saturationpredicted by both models after 50 years from the beginningof injection. In Case 1, because the gas relative permeabil-ity is assumed to be reversible, the model does not predictany trapping of CO2. The CO2 plume migrates upwardsdue to buoyancy forces without leaving any residual sat-uration behind. After a sufficiently long time, the modelpredicts the formation of a gas cap of mobile CO2 at thetop of the formation. This scenario is unfavorable from asequestration standpoint: damage in the cap rock couldlead to fractures that might serve as conduits for leaks ofthe mobile CO2 to upper formations and, eventually, theatmosphere.

The predictions under Case 2 are entirely different. Af-ter the injection phase, the model predicts a trail of resid-ual, immobile CO2 during the migration of the plume. Asa result, while there is a net flow of CO2 in the verticaldirection, trapping prevents the injected CO2 from form-ing a gas cap. In fact, the simulations predicts that, after50 years or less, almost all the CO2 is trapped in the forma-tion. Accounting for hysteresis effects leads to a spread-outdistribution of trapped CO2, as opposed to a concentrateddistribution of mobile CO2. This scenario is in fact muchmore realistic and, importantly, much more favorable forthe effectiveness of CO2 sequestration: it minimizes therisk of leaks (the gas is immobile) and enhances other se-questration mechanisms such as dissolution into the brine

and geochemical binding (more interfacial area betweenthe CO2 and the initial pore water).

Fig. 25 shows the evolution of CO2 saturation with timeat the three different observation gridblocks in the reser-voir. For the block at the very top (13, 18, 1) we see theaccumulation of CO2 when hysteresis effects are ignored,reaching a saturation value close to 0.7 that correspondsto the connate water saturation. In contrast, very littleaccumulation (CO2 saturation of about 0.15) occurs whenhysteresis is accounted for. The evolution of the CO2 sat-uration in blocks at lower elevations (7, 21, 1 and 11, 11, 1)displays an interesting behavior. When hysteresis is ig-nored, the gas saturation first increases sharply—the lead-ing edge of the plume reaches the block during its migra-tion upwards—and then decreases to a very low value—theplume travels through the block without leaving any resid-ual CO2. On the other hand, simulations that account forhysteresis predict that the CO2 saturation decreases onlyto a finite, positive value. This is due to trapping duringthe imbibition process that occurs at the trailing edge ofthe plume, which results in residual CO2 being left behind.

Effect of injection rate. We now investigate the effectsof CO2 injection rate on the overall performance of thesequestration project. We do so by comparing Cases 2and 3, both of which account for hysteresis. We inject thesame amount of CO2 in both cases, but over a period of1 year in Case 2 and over a period of 10 years in Case 3.

The saturation distributions for both of these cases areshown in Fig. 26. Clearly, a slower injection rate leads tomore mobile CO2 reaching the top of the aquifer, whichhas an adverse effect for sequestration purposes. Theseresults can be explained as follows.

Higher injection rates lead to a more radial displacementpattern and higher gas pressures in the vicinity of wells.Physically, this higher pressure is responsible for the non-wetting gas to invade smaller pores, which have a highercapillary entry pressure. Snap-off occurs readily in smallerpores during imbibition, resulting in increased macroscopictrapping.

For lower injection rates, on the other hand, gravity af-fects the displacement pattern of water by CO2 early. Thelow-viscosity gas forms stable paths through the high per-meability regions of the porous medium. Microscopically,only the largest pores are invaded, which leads to reducedsnap-off during an eventual imbibition process. Moreover,the CO2 reaches the top of the formation before injectionceases. Since trapping only occurs during imbibition (wa-ter displacing CO2 at the trailing edge of the plume), thefraction of CO2 that has reached the top of the aquifercannot undergo imbibition and is therefore not subject totrapping.

The plots of CO2 saturation over time for these two cases

16 SPE 96448

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 24— Water saturation distributions after 50 years from the beginning of CO2 injection. Left: results from Case 1 (no hysteresis).Right: results from Case 2 (with hysteresis).

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

years

Sg

Block (13,18,1)

HysteresisNo hysteresis

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

years

Sg

Block (7,21,1)

HysteresisNo hysteresis

0 10 20 30 40 500

0.05

0.1

0.15

0.2

yearsS

g

Block (11,11,1)

HysteresisNo hysteresis

Fig. 25— Evolution of CO2 saturations at the three “observation” gridblocks for Case 1 (no hysteresis) and Case 2 (with hysteresis).

are shown in Fig. 27. As seen from the plots of grid-blocks 7, 21, 1 and 11, 11, 1, the system undergoes trappingof the CO2 in both cases. However, the case with lowinjection rate shows that CO2 reaches a saturation valueof 0.7 (connate water) at the top of the anticline (grid-block 13, 18, 1), whereas a value of just 0.15 is achieved inthe high-rate case.

Effect of subsequent water injection. Finally, we in-vestigate how the performance of the CO2 sequestrationproject is affected by the injection of a slug of water af-

ter injection of the CO2 into the aquifer. The motivationis to enhance the imbibition process that naturally occursat the trail of the nonwetting CO2 plume as it migratesupwards.

In Fig. 28 we compare the fluid distributions of Case 3and Case 4. In both cases we inject 0.15 pore volumesof CO2 for the first ten years. In Case 4 this is followedby injection of 0.05 pore volumes of water during a 1-yearperiod. These results confirm the expected response: sub-sequent water injection induces more trapping and reducessignificantly the amount of CO2 that accumulates at thetop of the aquifer.

The evolution of the CO2 saturation at the observationgridblocks (Fig. 29) offers additional insight into the be-havior of the displacement process. During the CO2 injec-tion period—the first ten years—the curves corresponding

to Case 3 and Case 4 are of course identical. Once wa-ter injection starts (in Case 4) the system experiences adrastic change in the flow dynamics. Water starts displac-ing the CO2 radially away from the wells. This can beseen as a forced imbibition process that leads to almostimmediate trapping of the CO2 phase in large portions ofthe reservoir. It explains why we see a sharp decrease inCO2 saturations at the observation gridblocks—the CO2

is being pushed elsewhere—down to some residual satura-tion. It is worth noting that this saturation value is lowerthan what would be predicted from Land’s trapping model,Sgt = Sgi/(1+CSgi), because part of the CO2 is being dis-placed through existing connected paths and is thereforenot trapped.

Summary. We offer the following brief summary of thefindings from the simulations of CO2 sequestration insaline aquifers:

1. Accounting for trapping and relative permeability hys-teresis of the nonwetting CO2 phase is essential in or-der to correctly characterize the migration and finaldistribution of the injected CO2. Trapping occurs asthe trailing edge of the plume ascends (and is replacedby water) after injection stops.

2. Trapping of the CO2 leads to more favorable scenariosfor sequestration purposes: a large fraction of the CO2

SPE 96448 17

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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INJ-02

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INJ-06

INJ-04

INJ-05

INJ-01

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 26— Water saturation distributions after 50 years from the beginning of CO2 injection. Left: results from Case 2 (injectionover 1 year). Right: results from Case 3 (injection over 10 years).

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

years

Sg

Block (13,18,1)

Hysteresis (1yr)Hysteresis (10yr)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

years

Sg

Block (7,21,1)

Hysteresis (1yr)Hysteresis (10yr)

0 10 20 30 40 500

0.05

0.1

0.15

0.2

yearsS

g

Block (11,11,1)

Hysteresis (1yr)Hysteresis (10yr)

Fig. 27— Evolution of CO2 saturations at the three “observation” gridblocks for Case 2 (injection over 1 year) and Case 3 (injectionover 10 years).

is trapped and immobile for practical purposes, andis more spread out throughout the aquifer, therebyincreasing the interfacial area for subsequent dissolu-tion.

3. High injection rates result in more effective seques-tration of the CO2. A shorter injection period leavesless time for the buoyant CO2 to reach the top of theformation, from which it is difficult to immobilize.

4. Injection of a water slugs after CO2 injection increasesthe effectiveness of the sequestration project. The in-jected water forces break-up of large connected CO2

plumes, enhancing trapping and immobilization of theCO2. A similar behavior would be observed if therewere a regional groundwater flow.

Conclusions and outlookWe have presented a new model of trapping and water-flood relative permeability. Development of the modelis motivated by the inability of existing models to cap-ture the trends observed for intermediate-wet and oil-wetmedia. Due to scarcity of reliable experimental data, wehave used pore-network simulation as a means to predictthe trends in trapping and relative permeability hystere-sis. The new model is able to capture two key features of

the observed behavior: (1) non-monotonicity of the initial–residual curves, which implies that waterflood relative per-meabilities cross; and (2) convexity of the waterflood rela-tive permeability curves for oil-wet media due to film flowof oil.

We have also investigated the impact of trapping andrelative permeability hysteresis in the context of CO2 se-questration projects in saline aquifers. We have providedconclusive evidence that hysteresis is an order-one factorin the prediction of the migration and final distribution ofthe CO2. Trapping of the CO2 occurs during the upwardsmigration of the CO2 plume, but only after injection hasstopped and the trailing edge of the plume is naturally be-ing displaced by water. This imbibition process leads totrapping of the CO2. A trail of residual CO2 is left be-hind as the plume migrates upwards. We have shown thathysteresis effects have a very favorable effect on the effec-tiveness of CO2 sequestration. From the point of view ofthe design of a CO2 project, we have shown that trappingof the CO2 can be enhanced by: (1) operating at high in-jection rates, and (2) alternatively injecting water and CO2

in the spirit of classical WAG for enhanced oil recovery.

18 SPE 96448

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 28— Water saturation distributions after 50 years from the beginning of CO2 injection. Left: results from Case 3 (no waterinjection). Right: results from Case 4 (water injection after CO2 injection).

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

years

Sg

Block (13,18,1)

Hysteresis (10yr)Hysteresis (10yr−WAG)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

years

Sg

Block (7,21,1)

Hysteresis (10yr)Hysteresis (10yr−WAG)

0 10 20 30 40 500

0.05

0.1

0.15

0.2

yearsS

g

Block (11,11,1)

Hysteresis (10yr)Hysteresis (10yr−WAG)

Fig. 29— Evolution of CO2 saturations at the three “observation” gridblocks for Case 3 (no water injection) and Case 4 (waterinjection after CO2 injection).

Nomenclature

Roman letters

b exponent parameter in Jerauld’s trapping modelC Land trapping coefficientIo Amott oil wettability indexIw Amott water wettability indexIwo Amott–Harvey wettability indexkrw water relative permeabilitykd

ro drainage oil relative permeabilityki

ro waterflood oil relative permeabilitySw water saturationSo oil saturationSo,max maximum oil saturationSof flowing oil saturationSoi initial oil saturationSot ultimate trapped oil saturation∆So intermediate trapped oil saturation

Greek letters

α initial slope of IR curve in new trapping modelβ curvature of the IR curve in new trapping modelγ film flow parameter in new waterflood modelθ, θi intrinsic contact angleθa advancing contact angleθr receding contact angle

AcknowledgementsEJS, RJ and FMO gratefully acknowledge financial sup-port from the industrial affiliates of the Stanford UniversityPetroleum Research Institute for Gas Injection (SUPRI-C)and support from the Global Climate and Energy Project(GCEP). MJB would like to thank the sponsors of theImperial College Consortium on Pore-Scale Modelling andthe NERC for funding.

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