pore-level modeling of gas and condensate flow in two- and three-dimensional pore networks: pore...

Transport in Porous Media 53: 281–315, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

281

Pore-Level Modeling of Gas and Condensate Flowin Two- and Three-Dimensional Pore Networks:Pore Size Distribution Effects on the RelativePermeability of Gas and Condensate

CECILIA I. BUSTOS and PEDRO G. TOLEDO∗Chemical Engineering Department and Surface Analysis Laboratory (ASIF), University ofConcepción, PO Box 160-C, Correo 3, Concepción, Chile

(Received: 7 February 2002; in final form: 11 March 2003)

Abstract. We present a mechanistic model of retrograde condensation processes in two- and three-dimensional capillary tube networks under gravitational forces. Condensate filling-emptying cyclesin pore segments and gas connection–isolation cycles are included. With the pore-level distribution ofgas and condensate in hand, we determine their corresponding relative permeabilities. Details of porespace and displacement are subsumed in pore conductances. Solving for the pressure field in eachphase, we find a single effective conductance for each phase as a function of condensate saturation.Along with the effective conductance for the saturated network, the relative permeability for eachphase is calculated. Our model porous media are two- and three-dimensional regular networks ofpore segments with distributed size and square cross-section. With a Monte Carlo sampling wefind the optimum network size to avoid size effects and then we investigate the effect of networkdimensionality and pore size distribution on the relative permeabilities of gas and condensate.

Key words: gas–condensate, two-phase flow, pore-level modeling, network modeling, pore-sizedistribution, relative permeability, Monte Carlo simulation, capillary condensation.

Nomenclature

a pore cross-section.A cross-sectional area of a porous medium.b vector of constants.B adjustable parameter in Table I.C function in Equation (12).D adjustable parameter in Table I.f factor in Equation (12).g fluid conductance of a pore segment.g acceleration of gravity.g effective conductance.gabove conductance of a gas column above the condensate in a vertical pore.gb conductance of a condensate bridge in a vertical pore.

∗Author for correspondence: Tel.: +56-41-204534; Fax: +56-41-247491;e-mail: [email protected]

282 CECILIA I. BUSTOS AND PEDRO. G. TOLEDO

gbelow conductance of a gas column below the condensate in a vertical pore.gj network fluid conductance of a j phase.gw conductance of array of wedges in a vertical pore.gH(x, y) conductance of a horizontal pore connecting nodes (x, y) and (x − 1, y).gV(x, y) conductance of a vertical pore connecting nodes (x, y) and (x, y − 1).G matrix of conductances.hc condensate height in a vertical pore.hcc critical condensate height in a vertical pore.k intrinsic or absolute permeability.kj effective permeability of fluid j.krc relative permeability of a condensate phase.krg relative permeability of a gas phase.krj relative permeability of phase j.l pore segment length.labove length of a gas column above condensate in a vertical pore.lb length of a condensate bridge.lbelow length of a gas column below condensate in a vertical pore.lw length of condensate wedge.L,L2, L3 network dimension.n number of sides of the polygonal cross-section of a pore.nx number of nodes in the x-direction.ny number of nodes in the y-direction.p vector of pressures. p pressure drop across a pore.P(x, y) pressure at node (x, y).Pc capillary pressure.Pg pressure in gas phase.Pin − Pout pressure difference across a network.Pl pressure in condensate phase.Ps snap-off pressure. P/L pressure gradient in flow direction.( P/L)j pressure gradient on fluid j in the direction of the main flow.q volumetric flow of fluid through a single pore.Q volumetric flow rate.Qj total flow of phase j throughout a network.reff effective radius.rt inscribed radius in a pore.rt,max maximum pore radius in Table I.rt,min minimum pore radius in Table I.rv equivalent volume radius.rw radius of curvature of a longitudinal meniscus.Sc condensate saturation.V volume of total pore space.Vg volume of gas in a pore segment.Vl volume of condensate in pore corners.(x, y) position of a node inside a network.

Greek Symbolsα pore corner half-angle.β dimensionless resistance factor.γ interfacial tension.

PORE-LEVEL MODELING OF GAS AND CONDENSATE FLOW 283

θ contact angle.θA advancing contact angle.θR receding contact angle.µ mean pore radius in Table I.µf fluid viscosity.µg gas viscosity.µj viscosity of fluid j.µl condensate viscosity.ρ condensate density.σ standard deviation in Table I.ψ1, ψ2, ψ3 functions in Equation (12).

1. Introduction

A better understanding of phase distribution and flow in gas–condensate reservoirsis essential to delineate optimum exploitation strategies. The task is not easy con-sidering that new reservoirs are becoming increasingly complex, deeper and withhigher pressures and temperatures. When pressure, either in the wellbore or in thereservoir, drops below the dew point a new phase appears, a phenomenon knownas retrograde condensation. At first, the new condensate phase remains immobileblocking few gas paths; the gas effective permeability remains high. As the pressuredecreases, condensate dropout tends to occupy more and more gas paths, attainingthe so-called critical condensate saturation, at which the liquid phase becomes mo-bile for the first time, and the gas effective permeability decreases. As the pressuredepletes even more, two scenarios are possible. The condensation process endswhen the maximum liquid dropout is reached; in this scenario the gas effectivepermeability reaches a minimum non-zero value. Alternatively, the condensationprocess continues and eventually the gas phase reaches the percolation thresholdof sample-spanning paths of gas-filled pore-segments; then, in the absence of asignificant external pressure gradient, the gas flow is stopped and the gas effectivepermeability decreases to zero, how abruptly depending on pore structure, fluidproperties and operating conditions. The critical condensate saturation and therelative permeability of gas and condensate are thus essential parameters for theevaluation and new designs of recovery strategies.

Accurate measurements of critical condensate saturation and relative permeabi-lity of gas and condensate are difficult and reliable data are still scarce. Wangand Mohanty (1999), Blom et al. (2000), Jamiolahmady et al. (2000) and Li andFiroozabadi (2000) provide brief literature reviews of laboratory studies. Interpreta-tion of experimental data and unresolved issues can be summarized as follows.For high permeability core samples, measured critical condensate saturations invertical orientations are often lower than those measured in horizontal orienta-tions (Danesh et al., 1991; Morel et al., 1992; Ali et al., 1993; Henderson et al.,1993). The effect is less pronounced in low permeability cores. Mukenrud (1989)


however found less significant effects of gravity on critical condensate saturation.Danesh et al. (1991) found that critical condensate saturation increases with gas–condensate interfacial tension and decreases with connate water. Regarding gasrelative permeability, Gravier et al. (1986) and Chen et al. (1995) observed anabrupt decrease near critical gas saturation, the point where gas ceases to flow. Onthe contrary, Mukenrud (1989) and Haniff and Ali (1990) did not observe suchan abrupt decrease but instead a smooth, monotonic decrease. Henderson et al.(1993), Chen et al. (1995) and Ali et al. (1997) found that gas relative permeabilityincreases with flow rate. Finally, Henderson et al. (1993) observed that the relativepermeability of both gas and condensate increases with interfacial tension.

Few works are available in the literature on microscopic modeling of gas–condensate systems attempting to explain experimental, and in some cases conflict-ing, data. Mohammadi et al. (1990) used percolation theory to model retrogradecondensation and revaporization on Bethe trees as pore space models. Observationsof the condensation process on glass micromodels provided pore-level mecha-nisms. Their experiments showed pressure communication throughout the micro-model, although in cases through condensate films only. In their model, condensatein pore corners contributes to the overall condensate saturation but not to the con-densate conductance. The authors reported simulated gas and condensate relativepermeabilities and their sensitivity to various factors such as pore geometry, poresize distribution, pore connectivity, phase trapping and irreducible water. How-ever, the important effect of gravity was not studied. Fang et al. (1996) studiedthe pore-level physics of capillary condensation in small two-dimensional squarenetworks of cylindrical pore segments with distributed radii when gravity is anexternally imposed condition. These authors modified the continuous condensa-tion process to occur in discrete steps. At each step, they add a small amountof condensate to each pore segment with less-than-critical radii. Pore segmentswith larger-than-critical radii collect condensate dripping from adjoining pores.They calculated critical condensate saturation, defined as the liquid saturation atwhich liquid production from the pore network becomes sustainable, but not rel-ative permeabilities. According to these authors, critical condensate saturation is afunction of surface tension and contact angle hysteresis. Toledo and Firoozabadi(1998) used Fang et al. (1996) gas and condensate distributions on square net-works to calculate gas relative permeability with a renormalization technique. Theyfound that the gas permeability drops abruptly to zero at a condensate saturationhigh enough to block the gas flow completely. Wang and Mohanty (1999) de-veloped a mechanistic network model for the critical condensate saturation. Forthe pore space they used two representations, a modified sphere-pack and a cubicnetwork of pore bodies and pore throats. They critically examined phase trappingand connectivity in pore corners. Only capillary and gravitational effects wereconsidered. They showed that the critical saturation depends on pore geometry,water saturation and interfacial tension. Phase permeabilities were not reported. Liand Firoozabadi (2000) extended the works of Fang et al. (1996) and Toledo and


Firoozabadi (1998) to include viscous forces through an externally imposed pres-sure gradient. Two-dimensional square networks of cylindrical pore segments withdistributed radii represented pore space. These authors also allowed for condensateoverflowing in gravity-aligned pore segments. Their results showed that wettabilityaffects more significantly both critical condensate saturation and gas relative per-meability. Jamiolahmady et al. (2000) developed flow visualization experimentsto gain a better understanding of rate effects on gas–condensate relative perme-abilities. The observations were used to develop a single-pore mechanistic modeldescribing the coupled flow of gas and condensate. With the single-pore modelthese authors captured most features of the reported rate effects but concluded onthe need of incorporating multiple pore interactions.

In this work, we develop a mechanistic model of the retrograde condensationprocess in three-dimensional pore networks under gravitational effects. Observa-tions of the condensation process on glass micromodels presented in the literatureprovide most pore-level mechanisms we use. The model constitutes a Monte Carlosimulation of the condensation process. Pore segments are rectilinear and haveregular polygonal cross-sections. Condensate connectivity is always preserved al-though in cases through condensate wedges in pore corners only. Condensatewedges contribute to both condensate saturation and conductance. In ourmodel, condensate remains liquid at all pressures and thus revaporization isnot allowed. We report new gas and condensate relative permeabilities as a func-tion of condensate saturation for various system and simulation parameters andconditions.

We first describe the network model, that is, pore space model, pore-level sat-urations and conductances, and pore-network saturation and conductance. Thenwe report gas and condensate relative permeabilities and their sensitivitiesto simulation parameters, such as network size, dimensionality, and pore sizedistribution.

2. Network Model

Two- and three-dimensional cubic networks of pore segments represent porousmedia. Nodes at which the pore segments are connected act only as volumelessjunctions with infinite conductance. Pore segments are rectilinear with regularpolygonal cross-section. A polygonal section has n sides circumscribing a circleof radius rt. In this work we use pores with square cross-section. Pore segmentradius is randomly assigned according to probability density functions. On verticalpores we allow for contact angle hysteresis, which is characterized by advancing,θA, and receding, θR, contact angles; the advancing contact angle being alwaysgreater than the receding contact angle. Contact angle hysteresis arises when theliquid–vapor interface is unable to retrace its original path when it recedes on asolid surface. Pore segment length l is constant and chosen to accommodate thehighest stable condensate column in vertical pore segments without overflowing


Figure 1. Left: Stable condensate bridge at the center of a pore segment. Right: Condensatebridge at the bottom of a pore segment.

Figure 2. Gas–condensate distribution in a pore corner with semiapical angle α. Contact anglethrough the condensate phase is θ and radius of curvature of the longitudinal gas–condensatemeniscus is rw.

(see Figure 1). Condensate accumulation in pore corners allows for condensateconnectivity throughout the network, no matter how high the pressure is (seeFigure 2). Contact angle hysteresis in pore corners is not permitted in our model;contact angle there remains constant and numerically equal to θR. Menisci move-ment we considered quasi-static and capillarity controlled.


3. Pore-Level Physics

In this section we describe the capillary condensation model we use to determinethe distribution of gas and condensate in pore networks as the capillary pressuredecreases, when gravity is an externally imposed condition. Pore networks are opento flow at upper and lower boundaries. Periodic boundary conditions are imposedon vertical sides aligned with gravity.

3.1. CAPILLARY CONDENSATION IN VERTICAL PORES

Figure 3 shows a condensation sequence in a single vertical pore segment withsquare cross-section as capillary pressure decreases. Vertical pore segments arealigned with gravity. For each saturation level, Figure 3 shows the fluid distributionin longitudinal- and cross-sections through the condensate phase. Configurations(a), (b) and (c) show condensate accumulation in corners as capillary pressure de-creases. Configuration (d) shows the formation of a condensate bridge upon furthercapillary pressure reduction. From configuration (d) to (g) condensate accumulatesin the center of the pore at constant capillary pressure. The condensate column inconfiguration (g) reaches a critical height or weight and becomes unstable. Con-figurations (h), (i) and (j) show the condensate column flowing to the lower endof the vertical pore. The lower gas–condensate interface in configuration (j) getspinned at the lower end of the pore segment allowing for additional condensateaccumulation as configuration (j) shows. The advancing contact angle looses itsmeaning in configuration (j). Critical height of the condensate column is attainedonce the lower meniscus becomes flat. Configuration (d)–(i) has the same capillarypressure. Configurations (g), (h) and (i) have the same condensate column height.

Figure 3. Capillary condensation in a vertical pore segment with square cross-section ascapillary pressure decreases.


An infinitesimal addition of condensate to configuration (j) turns the configurationunstable and condensate begins to drip from the pore. As the pore becomes empty,condensate appears again in the same pore the configuration of which is set by theprevailing capillary pressure. The volume of condensate residing in the corners ofa pore segment of arbitrary polygonal cross-section, see Figure 2, is given by thegeneral formula

Vl = n r2wl

[sin(α + θ) cos(α + θ) + cos2(α + θ)

tgα− π

2+ α + θ

](1)

where Vl is the volume of condensate in the corners of a pore, n is the number ofsides of the polygonal cross-section of the pore, rw is the radius of curvature of thelongitudinal meniscus define as

rw = γ

Pc(2)

where γ is the interfacial tension and Pc is the capillary pressure. l is the porelength, α is the pore corner half-angle, which is 45◦ for square pores, and θ is thecontact angle. Pc is defined as Pg − Pl, where Pg is the pressure in the gas phaseand Pl in the condensate phase. Condensate saturation is calculated as Sc = Vl/V ,where V is the volume of the total pore space. The capillary pressure that producesthe bridge of condensate corresponds to the pressure at which the contact betweenthe gas phase and the pore walls is lost as configuration (c) in Figure 3 shows.This configuration is highly unstable; an infinitesimal addition of condensate toconfiguration (c) produces the snap-off of the gas–condensate interface and theformation of a bridge, configuration (d) of Figure 3. The snap-off pressure Ps isgiven by (see, for instance, Lenormand et al., 1983; Mohanty et al., 1987)

Ps = γ cos(α + θ)

rt cos α(3)

where rt is the radius of a circle circumscribed by the section of the pore. Thecritical height of the condensate column for which the condensate begins to flow tothe lower end of the pore segment, configuration (g) of Figure 3, can be calculatedfrom (Fang et al., 1996)

hc = 2γ

ρgrt(cos θR − cos θA) (4)

where hc is the condensate height, g is the acceleration of gravity and ρ is thecondensate density (see also Figure 1). Condensate saturation is easily calculatedfrom hc and the shape of the pore segment. The critical condensate height inside apore, configuration (j) of Figure 3, that produces dripping of condensate from thebottom can be calculated from (Fang et al., 1996)

hcc = 2γ

ρgrtcos θR (5)


where hcc is the critical condensate height. In this case a flat meniscus is assumedat the bottom end of the pore segment (see Figure 1). From a simulation view-point, condensate dropout in vertical pores occurs by a cycling process comprising:condensation until a capillary bridge is formed, additional condensation until thecondensate column becomes unstable and dripping begins, brief reopening of thepore to the flow of gas, and formation of a new condensate bridge at the prevailingcapillary pressure. This process repeats endlessly.

3.2. CAPILLARY CONDENSATION IN HORIZONTAL PORES

For horizontal pore segments, Figure 4 shows longitudinal and cross-sections ofcapillary condensate in pore segments of square cross-sections as capillary pres-sure decreases. Configurations (a), (b) and (c) show condensate accumulation incorners as capillary pressure decreases. Once the snap-off pressure is reached, fur-ther condensate dropout forms a bridge, as configuration (d) in Figure 4 shows. Thecondensation process continues until the condensate fills the pore segment com-pletely, configuration (h) in Figure 4. Further condensate dropout drips from theends of the pore segment. Condensate configurations in horizontal pore segmentsare not affected significantly by gravity. From a simulation viewpoint, condensatedropout in horizontal pore segments occurs in two steps: condensation until thecapillary bridge is formed, and further condensation until the condensate saturatesthe pore completely and condensate dripping begins. This last configuration isstable.

3.3. CAPILLARY CONDENSATION IN PORE NETWORKS

Our work here builds from the model of Fang et al. (1996) for the distribution ofgas and condensate as capillary pressure decreases. Here we add new features suchas condensate accumulation and flow in pore corners, gas–condensate interactions

Figure 4. Capillary condensation in a horizontal pore segment with square cross-section ascapillary pressure decreases.


in three-dimensional pore networks, and gas–condensate cycling in vertical poresegments.

The continuous condensation process is discretized in steps, and idea first in-troduced by Fang et al. (1996). A small amount of condensate is added to eachpore segment per step. Capillary equilibrium throughout the network is inforcedat the end of each step. As a consequence of adding condensate, while decreas-ing the capillary pressure, a pore may either increase its condensate inventory aswedges in pore corners if Pc > Ps or form a bridge of condensate, or increase itslength if a bridge already exists, if Pc �Ps. The condensation process continuesand the gas phase reaches the percolation threshold of sample-spanning paths ofgas-filled pore-segments. In the absence of a significant external pressure gradi-ent, the gas flow stops. Accordingly, the gas effective permeability becomes zero.Figures 5 and 6 summarize, respectively, the various stable two- and three-dimen-sional gas–condensate configurations considered in our simulations. Small bridges

Figure 5. Gas (white)–condensate (grey) configurations in a two-dimensional pore network.Figure (a.i)–(a.viii) depict all possible configurations represented by Figure (a). Abbreviatedconfigurations (b)–(i) can be expanded similarly.


Figure 6. Gas (white)–condensate (grey) configurations in a three-dimensional pore network.Abbreviated configurations (a)–(r) can be expanded as in Figure 5(a).

in horizontal pore segments in Figure 5 (2D configurations (a), (b), (d), (e), (g)and (h)) and Figure 6 (3D configurations (a)–(e), (g)–(k) and (m)–(q)) and smallbridges in vertical pores segments in Figure 5 (2D configurations (a)–(c)) andFigure 6 (3D configurations (a)–(f)), represent condensate only when capillarypressure is less than or equal to the snap-off pressure. For capillary pressuresgreater than the snap-off pressure, only wedges of condensate are possible, andthe bridge does not exist. So then, as an example, configuration (a) in Figure 5actually represents eight configurations, that is, configurations (a.i)–(a.viii). Thisconvention applies also to the other configurations in Figures 5 and 6. Conden-sate saturating horizontal pore segments in Figure 5 (2D configurations (b), (c),(e), (f), (h) and (i)) and Figure 6 (3D configurations (b)–(f), (h)–(l) and (n)–(r))correspond to condensate at capillary pressure less than or equal to the snap-offpressure. In a typical vertical pore segment, as condensate accumulates from eithercondensate dropout or condensate overflow from adjoining pores, the condensatecolumn reaches a critical length, at the most equal to the pore length, becomesunstable, and falls, reopening the pore segment to gas flow for an instant. Forcapillary pressures less than the snap-off pressure the gas and condensate in thesepores are cycling, that is, the pore segment repeatedly opens and closes to gasflow. In our model, vertical connectivity of the condensate network is thus throughwedges and smallest pores. Pore half-length condensate columns in vertical pores


Figure 7. Gas (white)–condensate (grey) configuration changes in a two-dimensional porenetwork.

Figure 8. Gas (white)–condensate (grey) configuration changes in a three-dimensional porenetwork.

in Figure 5 (2D configurations (d)–(f)) and Figure 6 (3D configurations (g)–(l))represent condensate at a capillary pressure less than or equal to the snap-offpressure for that pore. Two situations exist. For capillary pressures equal to thesnap-off pressure, the condensate accumulates in excess to stay as a bridge, be-coming unstable for the first time and dropping to the bottom of the pore. Forcapillary pressures less than the snap-off pressure the situation is the same as beforealthough occurring in later cycles. Critical-height condensate columns in verticalpores in Figure 5 (2D configurations (g)–(i)) and Figure 6 (3D configurations


Figure 9. Schematics of gas islands reconnection to the gas flow-carrying backbone.

(m)–(r)), nearly condensate-saturated pores, represent condensate at a capillarypressure less than or equal to the snap-off pressure for that pore. Again, two situ-ations exist. For capillary pressures equal to the snap-off pressure, the condensatereaches its critical height for the first time in this pore. The pore is at the vergeof becoming empty of condensate and open to gas flow for an instant. For capil-


lary pressures less than the snap-off pressure the situation is as before, althoughoccurring in later cycles.

3.4. CONDENSATE SATURATION EVOLUTION

Once a condensate column in a vertical pore segment reaches its critical height,condensate flows downwards and out of the pore. Three events occur, almost simul-taneously. Gas flows again through the pore, although for an instant, if the pore ispart of a percolating gas cluster. Condensate reappears in the pore in the form ofa small bridge, preventing gas flow through the pore. Condensate leaving the porecontributes to the saturation of adjoining pores if they are small enough to growcondensate (2D configurations (g) and (h) in Figure 7 and 3D configurations (m),(n), (o), (p) and (q) in Figure 8) or simply hyperdisperses (Toledo et al., 1995)through the condensate network, composed of wedges and full pores (2D config-uration (i) in Figure 7 and 3D configuration (r) in Figure 8) until it drips from thebottom of the network.

A complex flow mechanism may arise for the gas phase. A gas island maybe reconnected to the gas flow-carrying backbone if two or more pores holdingcondensate and acting as barriers to the gas flow get free of condensate in a givencycle. Then the gas island reconnects to the main gas flow contributing to its over-all conductance, although just for a short moment. The mechanism that is shownschematically in Figure 9 has also been included in our simulations. In Figure 9(a)and (b) once pores 1 and 2 becomes condensate free, a large gas island reconnectsto the gas percolating cluster. Additional gas is reconnected in Figure 9(b) and (c)once pore 3 becomes free of condensate.

One last mechanism we considered in our simulations occurs during the down-flow of an unstable condensate bridge. For the gas phase this implies advancementof a gas column above the condensate and displacement of a gas column below thecondensate. This event contributes to the overall gas permeability if the gas phaseconnects with the sample-spanning paths of gas-filled pore segments.

4. Gas and Condensate Relative Permeabilities

Details of pore space and pore occupancy by gas and condensate are subsumed,respectively, in pore gas and pore condensate conductances. First, we find pore-level gas and condensate conductances for the various configurations adopted bythe two phases. Then we calculate the conductance of the gas and condensatenetworks. Finally, we find the gas and condensate relative permeabilities.

4.1. PORE-LEVEL GAS CONDUCTANCE

Poiseuille’s law defines the fluid conductance g of each pore segment, that is,

q = g p (6)


where q is the volumetric flow of fluid through a single pore and p is the pressuredrop across the pore. Three different gas configurations need to be considered.

4.1.1. Gas at Pore Center and Condensate at Pore Corners

When gas occupies the center of a pore and condensate its corners, the pore-level gas conductance is approximated here by Poiseuille’s law for an effectivegas cylinder (Blunt, 1997), that is,

g = πr4eff

8µgl(7)

where µg is the gas viscosity and reff is given by

reff = 12 (rt + rv) (8)

where rv is an equivalent volume radius given by

π r2v l = Vg (9)

with Vg corresponding to the volume of gas in the pore segment.

4.1.2. Bridge of Condensate

In this case, there is no gas flow through the pore segment and the gas conductanceis set to zero.

4.1.3. Downflow of Condensate Displacing a Gas Column

In vertical pore segments, downflow of an unstable condensate bridge impliesadvancement of a gas column above the condensate, the conductance and lengthof which are gabove and labove, respectively, and displacement of a gas columnbelow the condensate, the conductance and length of which are gbelow and lbelow,respectively. Thus, the length of labove plus lbelow plus the length of the condensatebridge equals the length of the tube. We use Equations (7)–(9) for calculating thecorresponding gas conductances using the actual length of the gas columns. Thetotal gas conductance in the pore is given by the series arrays of the two individualgas conductances. For a sudden drop of the bridge, the total gas conductance of thepore is gbelow × gabove/(gbelow + gabove).

4.2. PORE-LEVEL CONDENSATE CONDUCTANCE

Three different condensate configurations need to be considered.

4.2.1. Condensate-Saturated Pore

For a fully condensate-saturated pore segment, the conductance of the condensateis calculated from Equation (7) by replacing µg by µl, the condensate viscosity,


and assuming an effective cylinder of condensate with length equal to that of thepore segment, and radius given by

reff = 1

2

(rt +

√a

π

)(10)

where a is the pore cross-section.

4.2.2. Gas at Pore Center and Condensate at Pore Corners

For condensate wedges, the conductance is calculated according to (Blunt, 1997)

g = Vlr2w

βµl2(11)

where Vl is given by Equation (1), rw is given by Equation (2), and β is a dimen-sionless resistance factor that is calculated from (Hughes and Blunt, 2000)

β = 12 sin2 α(1 − C)2(ψ1 − Cψ2)(ψ3 + fCψ2)2

(1 − sinα)2C2(ψ1 − Cψ2)3(12)

where

ψ1 = cos2(α + θ) + cos(α + θ) sin(α + θ) tgα (13)

ψ2 = 1 − θ

π/2 − α(14)

ψ3 = cos(α + θ)

cos α(15)

C =(π

2− α

)tgα (16)

The factor f varies from 0 to 1 depending on the boundary condition at the fluidinterface. We assume, as Hughes and Blunt (2000) did, a free boundary for whichf = 0.

4.2.3. Condensate Bridge and Condensate at Pore Corners

Condensate conductance in this case is calculated as a series array of condensatewedges and bridge conductances. Equation (11) gives the conductance of con-densate wedges of length lw. Equation (7) gives the conductance of a condensatebridge of length lb and effective radius given by Equation (8). The total condensateconductance of the pore is gw × gb/(gw + gb), where gw is the conductance of theresulting array of wedges and gb is the conductance of a bridge.


4.3. GAS AND CONDENSATE NETWORK CONDUCTANCES

For any given gas–condensate capillary pressure each phase develops its own flownetwork to which conductance can be assigned in much the same way as forsingle-phase flow. Several approaches are available for computing the macroscopicflow properties once the pore-level saturations are established. Notably mean fieldtheory (Kirkpatrick, 1973), real-space renormalization group (for applications toimmiscible flow see King, 1989 and King et al., 1993; for applications to gas–condensate systems see Toledo and Firoozabadi, 1998 and Li and Firoozabadi,2000) and direct solution of nodal material balances for each phase (see for in-stance Mogensen et al., 1999) have been used. The later technique is employedhere. A nodal material balance for each phase leads to a system of linear equations,Gp = b, where G is a matrix of conductances, p is a vector containing the unknownpressures, and b is a vector depending on the pore pressures at the upper and lowerboundaries and the conductances of the throats connected to these boundaries. Forexample, for a given fluid phase, in two dimensions, a typical material balance in anode leads to

gH(x, y)P (x − 1, y) + gV(x, y)P (x, y − 1) −− [gH(x, y) + gV(x, y) + gH(x + 1, y) + gV(x, y + 1)]P(x, y) ++ gH(x + 1, y)P (x + 1, y) + gV(x, y + 1)P (x, y + 1) = 0,

x = 1, . . . , nx; y = 1, . . . , ny (17)

where (x, y) represents the position of the node inside the network; gH(x, y) is theconductance of an horizontal pore segment connecting nodes (x, y) and (x−1, y);gV(x, y) is the conductance of a vertical pore segment connecting nodes (x, y)

and (x, y − 1); P(x, y) is the pressure at node (x, y) and nx and ny are thenumber of nodes in the x and y directions, respectively. gH(x, y) and gV(x, y)

are nil when a different fluid phase occupies the throats. To find the distribution ofnodal pressures in each flow network once an external pressure gradient is imposedwe use an iterative solution of the system of equations. For the gas phase, first apreliminary reduction is applied before solving the system. The method of burningof Herrmann et al. (1984) is used to determine the sample-spanning cluster andthen, the connected cluster or flow-carrying backbone. A pore is not part of thebackbone if for topological reasons it is unable to carry flow. After such pores areeliminated, the remaining system of equations is optimally stored and solved witha conjugate gradient method with symmetric successive overrelaxation (Mogensenet al., 1999). This method is part of the ITPACK routine libraries that are publiclyavailable at the website http://rene.ma.utexas.edu/CNA/ITPACK. The reductionused for the gas phase is not required for the condensate phase because it is con-nected throughout the network always. We assign fixed pressures to the upper andlower flow boundaries and use a uniformly increasing pressure as an initial guess.To avoid size effects, periodic boundary conditions are imposed on the networksby connecting nodes along the vertical edges of the network through additional


pore segments. The relaxation parameter is chosen as 1.87 by trial and error. Withthe nodal pressures of a given flow network in hand, the flow rate everywhere iscalculated and the network conductance computed from

gj = Qj

(Pin − Pout)(18)

where gj is the network fluid conductance of the j phase, Qj is the total flow ofphase j throughout the network and Pin − Pout is the pressure difference across thenetwork.

4.4. GAS AND CONDENSATE RELATIVE PERMEABILITIES

Darcy’s law describes the creeping flow of a single viscous fluid in a porousmedium,

Q = k

µ fA P

L(19)

where Q is the volumetric flow rate, µf is the fluid viscosity, A is the cross-sectionalarea of the porous medium, P/L is the pressure gradient in the direction of flow,and k is the intrinsic or absolute permeability. By neglecting any viscous couplingeffects and assuming that the simultaneous flow of two fluids in a porous mediumcan be described by Darcy’s law, it is customarily to write (Morel-Seytoux, 1969;Peaceman, 1977)

Qj = kj

µj

A

( P

L

)j

(20)

where Qj is the volumetric flow rate of fluid j, µj is the viscosity of fluid j,( P/L)j is the pressure gradient on fluid j in the direction of the main flow, andkj is the effective permeability of fluid j. The relative permeability krj of phase j iskj/k and is calculated here as

krj = Qj

Q(21)

or simply,

krj = gj

g(22)

Q and g are, respectively, the volumetric flow rate and the effective conductancewhen the phase flows through the pore network alone. In what follows krg is therelative permeability of the gas phase and krc is the relative permeability of thecondensate phase.


5. Monte Carlo Simulation

Here we present the sequence of steps and the various parameters used to simu-late retrograde condensation in three-dimensional networks of pore segments withdistributed radii.

(a) Specify fluid properties such as density and interfacial tension.(b) Specify fluid–solid properties such as advancing and receding contact angles.(c) Choose a lattice of specified dimensions, mean coordination number, and

boundary conditions.(d) Decorate the underlying lattice with independent pore features. We select rec-

tilinear pore segments with square cross-section and distributed size. Choosepore segment length constant and high enough to prevent condensate overflow-ing in vertical pore segments, that is, high enough to accommodate the higheststable condensate columns that are possible according to pore size.

(e) Calculate the capillary pressure that is required to form a condensate bridgein each pore segment of the network, that is, the so-called snap-off pressurefor each pore. Here the continuous condensate process occurs in discretizedsteps. Add a small amount of condensate per step in each allowable poresegment.

(f) At the beginning of the process, when capillary pressure is highest, every poresegment in the network is open to gas flow. Condensate is confined to porecorners in capillary equilibrium and distributed according to pore size and cap-illary pressure level. Condensate bridges appear first in the smallest availablepore segments. Once the capillary pressure equals the snap-off pressure of agiven pore, liquid dropout continues at constant capillary pressure until thecondensate column becomes unstable flowing out of the pore for the first time.This condensate contributes to the condensate inventory of adjoining pores ifpressure is below their snap-off pressures or simply hyperdisperses throughand out of the network.

(g) Allow pressure to decrease to the snap-off pressure of the next size pore. Thejust-emptied pore segment in Step (f), at capillary pressures lower than itssnap-off pressure, enters an endless condensate filling and emptying cycle. Ascapillary pressure decreases condensate dropout occurs in larger and largerpore segments, in increasing order of size.

(h) A gas-permeable pore segment is not allowed to retain condensate overflowsif unable to build a condensate bridge first.

(i) Calculate gas and condensate saturations for each capillary pressure, that is,for each snap-off pressure.

(j) For each capillary pressure, calculate pore-level gas and condensate con-ductances.

(k) From pore-level conductances, calculate the overall network conductance toboth gas and condensate as a function of condensate saturation. Calculate gasand condensate relative permeability as a function of condensate saturation.


(l) Continue the condensation process until the gas flow is interrupted, that is, thegas relative permeability becomes zero.

(m) Repeat the simulation, Steps (a)–(l), for new samplings of the pore size distri-bution. Report results with a confidence interval of 95%.

6. Simulations

The mechanistic model of condensation in pore networks is used here to find thepore-level distribution of gas and condensate, their corresponding relative per-meabilities, and their sensitivities to network size, network dimensionality and poresize distribution (width, variance and skewness). The results are based on conden-sate density of 800 kg/m3, gas–condensate interfacial tension of 0.001 dynes/cm,receding contact angle of 10◦ and advancing contact angle of 30◦. Pore segmentsare rectilinear with square cross-sections circumscribing circles of given radii. Poresegment radius is randomly assigned according to the uniform, truncated uniformand truncated log-normal distributions shown in Table I and Figure 10. Pore lengthis 300 µm. Condensate is added in increments of 2000 µm3.

In companion papers we use the mechanistic model of pore-level condensa-tion in pore networks presented here to determine gas and condensate relativepermeabilities as a function of condensate saturation and to explore their sensi-tivities to pore parameters, such as shape; pore-fluid parameters, such as surfacewettability; and gas–condensate parameters, such as interfacial tension.

A combination of cycling mechanisms operates as the gas relative permeabilitydecreases. Above we mentioned the condensate filling–emptying cycles in verticalpore segments. We also indicated the reconnection of gas-saturated single pores

Table I. Pore size distributions and parameters. Figure 10 shows graphical representations. B andD are adjustable parametersa

Distribution function Parameters Graphical

Uniform rt,min = 10 µm rt,max = 150 µm Figure 10(a)1

rt,max−rt,minrt,min = 60 µm rt,max = 100 µm Figure 10(b)

Normal µ = 80 µm σ = 30 µm Figure 10(c)

1σ√

2πexp

[− 1

2

(rt−µσ

)2]

σ = 15 µm Figure 10(d)

σ = 5 µm Figure 10(e)

Log-normal µ = 55 µm σ = 50 µm Figure 10(f)

1Brt

√2π

exp

[− 1

2

(ln(rt)−D

B

)2]

σ = 30 µm Figure 10(g)

µ = exp[D + B2

2

]; σ = 15 µm Figure 10(h)

σ 2 = exp(2D + B2)(exp(B2) − 1) σ = 8 µm Figure 10(i)

a rt,min = minimum pore radius; rt,max = maximum pore radius; µ= mean pore radius;σ = standard deviation.


Figure 10. Truncated pore size distributions. Parameters are listed in Table I.

or clusters of pores. However, simultaneously, in various other places of the net-work, isolation of gas-saturated single pores and/or clusters of pores occur. Thegas reconnection–isolation balance determines a smooth and monotonic decreaseof the gas relative permeability as the condensate saturation increases, as we showlater.

Figure 11 shows a Monte Carlo generated sequence of gas–condensate distribu-tion as condensate inventory increases at a fixed capillary pressure. The network istwo-dimensional and 10 × 10 nodes per side. Pore size distribution corresponds tothe truncated log-normal distribution in Figure 10(g). Figure 11 displays conden-


Figure 11. Monte Carlo generated sequence of gas (white) and condensate (black) distributionin a two-dimensional network of 10 × 10 nodes per side. Condensate filling–emptying cyclesin vertical pores 1 and 3. Condensate filling process in horizontal pore 2.

sate filling–emptying sequences in a vertical pore segment circled and numbered1. For this sequence, Frame 1 shows the appearance of a small condensate bridge,enough to prevent the flow of gas through the pore. Condensate accumulation


continues and the bridge in Frame 1 reaches a critical height or weight and be-comes unstable. The condensate column flows to the lower end of the vertical poreas Frame 2 shows. The condensate column becomes pinned at the lower end ofpore 1 allowing for additional condensate accumulation as Frames 3–5 show. Thecondensate column reaches its critical height in Frame 5, and condensate drippingbegins. Pore 1 becomes empty in Frame 6 and the gas phase becomes reconnectedthrough it again. Frame 7 shows the appearance of a new condensate bridge. Theprocess continues in Frames 8–12. The condensate filling–emptying sequence ispresented in Figure 11 for a different vertical pore, circled and numbered 3. It isimportant to notice that in this case pore emptying occurs in Frame 11, later in thesequence if compared with pore 1. This only reveals that pore segment 3 is largerthan pore segment 1. Figure 15 also shows the filling sequence of a horizontalpore segment circled and numbered 2. Accumulation of condensate begins with theformation of a bridge and continues until the pore becomes completely saturated,this occurs in Frame 10. Condensate generation continues in pore 2 overflowing toadjoining pores. A feature likely confusing in our simulations is that condensatesaturation may decrease temporarily although condensate increments continue asTable II shows. This can be explained on the bases of capillary equilibrium. At aset capillary pressure Pc, the population of pore segments with Ps �Pc experiencesthe condensate filling–emptying cycle continuously. The unstable condensate dripstowards adjoining pores that may retain it only if they do so at the set capillary

Table II. Gas and condensate distributions in a two-dimensional network of 10 × 10 nodes as con-densate increases at a constant capillary pressure of 0.382622 Pa. Condensate saturation, Sc, gasrelative permeability, krg, condensate relative permeability, krc, number of condensate increments oriterations, and frames identification

Sc krg krc Iteration Frames in

numbera Figure 11

0.15645 0.9581146 0.0002711 1 1

0.15806 0.9581262 0.0002729 31 2

0.15797 0.9584726 0.0002716 38 3

0.15902 0.9581239 0.0002734 46 4

0.15994 0.9581199 0.0002744 61 5

0.15932 0.9581637 0.0002721 64 6

0.15952 0.9581189 0.0002724 65 7

0.15901 0.9581122 0.0002708 76 8

0.16124 0.9581262 0.0002737 106 9

0.16204 0.9581217 0.0002747 121 10

0.16061 0.9581405 0.0002709 129 11

0.16147 0.9581193 0.0002725 136 12

a Condensate increments at the set capillary pressure.


pressure, that is limited to pore segments with Ps �Pc and in the filling leg ofa filling–emptying cycle. In cases, the amount of condensate to be redistributedexceeds the amount that can be retained by the pores then the excess hyperdispersesout of the network and the condensate saturation decreases temporarily. Variousexamples are shown in Table II. The excess of condensate will not go to wedgesbecause these are also in capillary equilibrium.

Figure 12 displays a Monte Carlo generated three-step sequence of a gasisolation–reconnection–isolation cycle as condensation occurs at a set capillarypressure. Pore network is two-dimensional, size is 20 × 20 nodes per side, andpore size distribution is truncated log-normal (see Figure 10(g)). For the set capil-

Figure 12. Monte Carlo generated sequence of gas (grey and black) and condensate(white) distribution in a two-dimensional network of 20 × 20 nodes per side. Gasisolation–reconnection–isolation cycle.


Table III. Gas and condensate distributions in a two-dimensional network of 20 × 20 nodes perside as condensation progresses at a constant capillary pressure of 0.111118 Pa. Condensate satura-tion, Sc, gas relative permeability, krg, condensate relative permeability, krc, number of condensateincrements or iterations, and frames identification

Sc krg krc Iteration Frames in

numbera Figure 12

0.18331 0.181458 0.017102 172 1

0.18322 0.293954 0.017096 173 2

0.18323 0.181085 0.017099 174 3

a Condensate increments at the set capillary pressure.

lary pressure, Table III summarizes the corresponding condensate saturation andrelative permeabilities of gas and condensate. Left, middle and right columns inFigure 12 display the gas–condensate distributions attained, respectively, after 172,173 and 174 condensate increments at the set capillary pressure. Figure 12, top row,shows the gas cluster; Figure 12, middle row, its sample spanning portion, withthe gas islands removed; and Figure 12, bottom row, its backbone, with danglingends and loops, not contributing to the gas permeability, removed. Figure 12, leftcolumn, signals a gas-free pore, circled, preventing the connection of two gasfiords. Figure 12, central column, displays the same pore now gas-occupied bythe process described in Figure 11, and thus allowing the gas fiords connection.Figure 12, right column, signals the same pore segment again free of gas, breakingthe connection of the gas fiords. As we mentioned, the gas connection–isolationbalance determines the shape of the gas relative permeability as the condensatesaturation increases.

Finally, Figure 13 shows a Monte Carlo generated sequence of gas–condensatedistributions as condensation progresses in a large two-dimensional network of60 × 60 nodes per side. Again, we use the pore size distribution shown inFigure 10(g). For each frame in Figure 13, Table IV summarizes the correspondingdata on capillary pressure, condensate saturation, and relative permeabilities ofgas and condensate. This table indicates that as gas pressure decreases, capillarypressure decreases, condensate saturation increases, gas relative permeability de-creases and condensate relative permeability increases. Frame 1 shows the gasdistribution at the highest pressure in Table IV, condensate is confined to porecorners and gas can flow freely through the network. As pressure decreases moreliquid drops out, condensate saturation increases, and the gas relative permeabil-ity decreases. Frames 2 through 5 show an increasing number of pore segmentsoccupied by condensate, gas flow paths are less and increasingly tortuous. Frame6 shows the gas distribution very close to the gas percolation threshold; gas isweakly connected. Liquid dropout continues as pressure decreases. Frames 7 and 8


Figure 13. Monte Carlo generated sequence of gas (white) and condensate (black) distributionin a large two-dimensional network of 60 × 60 nodes per side.

show disconnected gas clusters only, the relative permeability of gas is zero. Whenconditions of Frame 6 are attained in practice, then gas production may not bepossible anymore.


Table IV. Gas and condensate distributions in a two-dimensional network of 60 × 60 nodes per sideas the capillary pressure decreases. Capillary pressure, Pc, condensate saturation, Sc, gas relativepermeability, krg, condensate relative permeability, krc, and frames identification

Pc(Pa) Sc krg krc Frames in

Figure 13

0.633718 0.015495 0.97582 0.000171 1

0.358920 0.048922 0.92297 0.001658 2

0.261664 0.097404 0.82085 0.005594 3

0.221628 0.143360 0.65692 0.010086 4

0.192218 0.199284 0.40845 0.015944 5

0.169699 0.260464 0.06017 0.022630 6

0.155992 0.308699 0.00000 0.028117 7

0.144334 0.355840 0.00000 0.033483 8

6.1. NETWORK SIZE

To explore the effect of network size on the relative permeabilities of gas andcondensate we use square networks of dimension L2 and simple cubic networksof dimension L3 as underlying lattices, with the log-normal distribution shown inFigure 10(g). In two dimensions we use L values up to 128 nodes and in three di-mensions values up to 32 nodes. As expected the variance of both the gas and con-densate relative permeability, calculated on the bases of six realizations, decreasesas the size of the network increases. In two dimensions the variance vanished forL = 64 nodes per side of the network and in three dimensions it vanished forL = 20 nodes per side of the network. In what follows we use networks of 64 × 64nodes, with 8192 pore segments, in two dimensions, and 20 × 20 × 20 nodes, with24,000 pore segments, in three dimensions.

6.2. NETWORK DIMENSION

The relative permeabilities of gas and condensate as a function of condensatesaturation in two- and three-dimensional optimum-sized pore networks with thelog-normal pore size distribution of Figure 10(g) are shown in Figure 14. As ex-pected, the gas relative permeability is always higher in networks embedded inthree dimensions than in two dimensions. In both cases, the relative permeabil-ity of the gas phase approaches a typical inverted s-shaped form and shows twodistinct regimes. In the first regime, for low condensate saturations, the gas rela-tive permeability decreases slowly and linearly with saturation; condensate existsmainly as liquid wedges in pore corners. These liquid wedges decrease the gasconductance but do not prevent the gas from flowing. The linear regime ends whencondensate wedges turn to condensate bridges that prevent the flow of gas. In thesecond regime, for condensate saturations medium and high, as pore segments


Figure 14. Relative permeabilities of gas and condensate in optimum-sized two- andthree-dimensional networks.

become saturated with condensate, the gas relative permeability decreases faster,abruptly in the two-dimensional case. Near the percolation threshold of sample-spanning paths of gas-filled pore-segments, the gas relative permeability recoversits low decreasing pace, until its flow is stopped and its effective permeabilitybecomes zero. The gas relative permeability near the percolation threshold for thegas phase is knee-shaped, in agreement with experimental data discussed in theliterature (Toledo and Firoozabadi, 1998; Li and Firoozabadi, 2000). Regardingthe condensate relative permeability, Figure 14 shows that in the first regime itis very low and barely increases as condensate wedges become thicker. In thesecond regime the relative permeability of the condensate phase is higher thanin the first regime. Figure 14 also shows that the condensate relative permeabil-ity is not affected significantly by the dimensionality of the pore network. Thisis also expected as condensate wedges in pore corners and small vertical porescontrol the permeability of the condensate at any condensate saturation, a con-dition in our model. It should be noticed in Figure 14 that after the flow of gasis stopped condensate dropout continues. In our simulations condensate dropoutcontinues until every horizontal pore segment becomes saturated with condensate.In three-dimensional networks, at any given saturation, the gas phase has morepaths available for flow. Therefore the gas flow is stopped, and thus the gas effective


permeability decreases to zero, at much higher condensate saturation in three-dimensional networks. In the literature, thus far, most of the pore-level modelingof gas–condensate relative permeabilities is based on two-dimensional networks.Our results show that modeling the behavior of three-dimensional networks ismandatory in attempting a comparison with experimental data.

6.3. PORE SIZE DISTRIBUTION

Figures 15–17 show calculated gas and condensate relative permeabilities for 20×20×20 pore networks with pore sizes drawn, respectively, from uniform, truncatednormal and truncated log-normal distributions (see Figure 10). Results correspondto 95% confidence intervals around the mean of six realizations of each pore sizedistribution. The wider and/or skewer the pore-size distribution the narrower thecondensate saturation range where the linear regime exists. How abruptly the gasrelative permeability decreases during the second regime clearly depends on poresize distribution for all other parameters fixed as Figures 15–17 show. The widerand/or skewer the pore-size distribution is the more abrupt the gas permeabilitydecreases. In all cases of pore size distribution, the smaller pore segments controlthe conductance of the condensate network. Large pore segments saturated with

Figure 15. Relative permeabilities for networks with uniform distribution of pore size. Effectof pore size width. Pore segments with square cross-section.


Figure 16. Relative permeabilities for networks with truncated normal distributions of poresize. Effect of pore size variance. Pore segments with square cross-section.

condensate, if exist, act merely as connectors of condensate but do not control theoverall condensate conductance. After the flow of gas is stopped condensate drop-out continues. In this case, condensate relative permeability increases and does itfaster as the width and skewness of the pore-size distribution increase. This featureis clearer in Figures 15 and 17 for uniform and truncated log-normal distributionsof pore sizes.

6.4. PORE SIZE WIDTH

Figure 15 shows the effect of the pore size width of uniform distributions (Fig-ure 10(a) and (b)) on the relative permeability of gas and condensate. Mean poresize is kept fixed at 80 µm. Figure 15 shows that during the first regime, the gasrelative permeability displays no significant sensitivity to the width in pore sizes,except that the higher the width the narrower the condensate saturation range forthe first regime. However, the same figure shows that the width of pore sizes has aprofound impact in the second regime. At any given saturation during the secondregime, the lower the width in pore sizes the higher the relative permeability ofthe gas phase. Pore networks decorated with pore size distributions exhibitinghigh pore size width attain condensate saturations with a higher fractions of small


Figure 17. Relative permeabilities for networks with truncated log-normal distributions ofpore size. Effect of pore size skewness. Pore segments with square cross-section.

condensate-saturated pore segments as compared to distributions with low poresize width. At the end of the second regime, the flow of gas is stopped and its rela-tive permeability becomes zero. Figure 15 also shows that the relative permeabilityof the condensate phase in the first and second regime exhibits no sensitivity topore size width. This is easily explained in the light of percolation theory. In allcases of size width, the smaller pore segments, available in large numbers, controlthe conductance of the condensate network.

6.5. PORE SIZE VARIANCE

Figure 16 shows the effect of pore size variance of truncated normal distributions(Figure 10(c)–(e)) on the relative permeability of gas and condensate. At low sat-urations of the condensate phase, pore segments holding liquid wedges in corners,though remaining open to the flow of gas, control the relative permeability of thegas phase. Figure 16 shows that during the first regime, the gas relative permeabilitydisplays no significant sensitivity to the variance in pore sizes. However, the samefigure shows that the impact of the pore size variance is significant during thesecond regime. At any given saturation during the second regime, the lower thevariance in pore sizes the higher the relative permeability of the gas phase. Pore


networks decorated with pore size distributions exhibiting high pore size varianceattain condensate saturations with higher fractions of small condensate-saturatedpore segments as compared to distributions with low pore size variance. Thus, fora given condensate saturation the gas relative permeability is lower for the networkwith higher variance in pore sizes. For example, Figure 16 shows at a condensatesaturation of 40% that the gas relative permeability is 45% for a standard deviationof 5 µm, 40% for a standard deviation of 15 µm, and 31% for a standard deviationof 30 µm. At the end of the second regime the relative permeability becomes zero.Another way to analyze the results of Figure 16 for the second regime is to seta gas relative permeability and look for the condensate saturation correspondingto various pore size variances. For a given gas relative permeability, Figure 16shows that pore networks with high pore size variance have a lower saturation ofcondensate than pore networks with low pore size variance. The lower saturationof condensate is explained by the larger fraction of small pore segments availablefor condensation in networks with higher pore size variance. Figure 16 also revealsthat the relative permeability of the condensate phase in the first and second re-gime shows no sensitivity to pore size variance. In all cases of size variance, thesmaller pore segments, available in large numbers, control the conductance of thecondensate network. Large pore segments saturated with condensate act simply asconnectors of condensate but do not control the overall condensate conductance.Figure 16 also shows that for a given pore size distribution and after the flow of gasis stopped, condensate dropout continues and does it, according to our model, untilevery horizontal pore segment becomes saturated.

6.6. PORE SIZE SKEWNESS

Figure 17 shows the effect of the pore size skewness towards the smaller-sizedpores of truncated log-normal distributions on the relative permeability of gas andcondensate. A similar analysis as for Figure 16 applies with few additions. Thefirst regime, where wedges dominate the condensate saturation and the relativepermeability of gas and condensate, comprises a narrower range of condensatesaturation. The second regime is characterized by a more pronounced spread ofthe gas relative permeability as the pore size skewness increases. At any givensaturation during the second regime, the lower the skewness in pore sizes the higherthe relative permeability of the gas phase. Pore networks decorated with pore sizedistributions exhibiting high pore size skewness attain condensate saturations withhigher fractions of small condensate-saturated pore segments as compared withdistributions exhibiting low pore size skewness. The effect of the skewness iseven more pronounced than the effect of the variance of the pore size distribution.Thus, for a given condensate saturation the gas relative permeability is lower forthe network with higher skewness in pore sizes. For example, Figure 17 showsat a condensate saturation of 35% that the gas relative permeability is 52% forthe lowest pore size skewness and 6% for the highest pore size skewness. Again,


at the end of the second regime, the flow of gas is stopped and its relative per-meability becomes zero. Figure 17 also shows that at high condensate saturationsthe relative permeability of the condensate phase is higher for networks decoratedwith pore size distributions exhibiting higher skewness. Results here show thatabrupt drops in gas relative permeability, in the absence of vaporization, belongto porous media with high pore size width, variance and skewness. Experimentaldata in the literature for various sedimentary rocks support our findings. Gravieret al. (1986) and Chen et al. (1995) observed an abrupt decrease near critical gassaturation, the point where gas ceases to flow. On the contrary, Mukenrud (1989)and Haniff and Ali (1990) did not observe such an abrupt decrease but instead asmooth, monotonic decrease. Our results here suggest explanations for these ap-parently contradictory experimental observations. We hope our model and resultshelp guiding interpretation of new experimental data.

7. Conclusions

A mechanistic model of the retrograde condensation process in two- and three-dimensional pore networks under gravitational forces is used to determine thepore-level distribution of gas and condensate, their corresponding relative per-meability and their sensitivities to network size and dimensionality. The three-dimensional model is used to determine the relative permeabilities of gas andcondensate phases and their sensitivities to pore size distribution, that is, poremean, width and skewness.

Condensate connectivity is always maintained although in cases through con-densate wedges in pore corners only. Condensate wedges contribute to both con-densate saturation and overall conductance. Complex condensate filling–emptyingcycles in pore segments and gas connection–isolation cycles have been includedin our simulations of condensation. These mechanisms determine the shape of theresulting relative permeabilities. Using Monte Carlo simulation of the condensationprocess we find the optimum-sized network for studying the condensation processand minimizing permeability variances to pore size distribution samplings.

The relative permeability of the gas phase approaches a typical inverteds-shaped form and shows two distinct regimes. In the first regime, the gas relativepermeability decreases slowly at low saturations of condensate, while condensateexists mainly as liquid wedges in pore corners. In the second regime, as poresegments become bridged by condensate, the gas relative permeability decreasesfaster, abruptly in cases, reaching a very low value. Near the percolation thresholdof sample-spanning paths of gas-filled pore-segments, the gas relative permeabi-lity recovers its low decreasing pace, until its flow is stopped and its effectivepermeability decreases to zero.

According to the results, the way the gas relative permeability decreases asfunction of the condensate saturation clearly depends on pore shape and poresize distribution, width, variance and skewness, for all other parameters fixed. Our


results show that the higher the skewness of the pore size distribution, towards thesmaller pores, the more pronounce is the drop of the gas relative permeability.

We also show that the relative permeability of the condensate phase in the firstand second regime exhibits little sensitivity to pore size width, variance and neg-ative skewness. We hope our model and results stimulate new experimental workand help guiding interpretation of the data.

Acknowledgements

We thank CONICYT of Chile for financial support through projects FONDECYTNo. 2990056 and 1980462 and University of Concepción through Project DIUCNo. 97.096.049-1.0. PGT became interested in the subject while visiting theRecovery Engineering Research Institute at Palo Alto, California.

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