numerical analysis of multi-mechanistic flow effects in naturally

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Numerical analysis of multi-mechanistic flow effects in naturally

fractured gas-condensate systems

Luis F. Ayala H. ⁎, Turgay Ertekin, Michael Adewumi

Petroleum and Natural Gas Engineering Program, The Pennsylvania State University 122 Hosler Building,

University Park, Pa 16802-5001, USA

Received 9 September 2006; received in revised form 13 November 2006; accepted 14 November 2006

Abstract

The study of depletion performance of naturally fractured reservoirs has gained wide interest in the petroleum industry during the

last few decades and poses a challenge for the reservoir modeler. The presence of a retrograde gas-condensate fluid incorporates an

additional layer of complexity to the performance of this class of reservoirs. Upon depletion, reservoir pressure may fall below the

dew-point of the hydrocarbon mixture which results in liquid condensation at reservoir conditions. In the case of fractured gas-

condensate reservoirs, condensate will first appear in the high-conductivity channels supplied by the fracture network and around the

external edges of the matrix blocks which are the zones prone to faster depletion. Since the bulk of hydrocarbon storage resides inside

the matrix, it is critical to answer the question whether thecondensate formed in the matrix edges would irreversibly trap the gas found

in the inner-most portions of the matrix. It is believed that the interplay of Darcian-type flow and Fickian-type flow (multi-mechanistic

flow) is the key to answering the questions about depletion performance and ultimate recovery in these reservoirs. This studyinvestigates the recovery mechanisms from a single matrix block surrounded by an orthogonal matrix network, as the fundamental

building block for the full-scale system. In this work, we show the dominant flow processes and recovery mechanisms taking place in

naturally fractured gas-condensate reservoirs and describe the depletion performance of these systems, which provides guidance for

the development and analysis of this class of reservoirs.

© 2006 Elsevier B.V. All rights reserved.

Keywords: Gas-Condensate Reservoirs; Retrograde Condensation; Diffusion; Darcy's Law

1. Introduction

Gas-condensate reservoirs have been the subject of

intensive research for many years as they represent an

important class of the world's hydrocarbon reserves. In a

gas-condensate reservoir, initial reservoir conditions are

located between the critical point and cricondenthem of

the reservoir fluid, as shown in Fig. 1. In general, hydro-

carbons in a gas-condensate reservoir are either wholly

or predominantly in the vapor phase at the time of

discovery. Upon isothermal depletion, once the reservoir pressure falls below the dew-point of the hydrocarbon

mixture, a liquid hydrocarbon phase is developed. Ap-

pearance of a liquid phase upon vapor expansion (under

isothermal conditions) is not possible for pure substan-

ces; thus, this behavior is categorized as “retrograde” for

this particular type of mixtures. The retrograde liquid

may revaporize if depletion continues. The major con-

cern while producing a gas condensate reservoir has to

do with the loss of this valuable liquid to the reservoir

and the associated impairment in gas productivity. The

Journal of Petroleum Science and Engineering 58 (2007) 13–29

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⁎ Corresponding author. Tel.: +1 814 8654053; fax: +1 814 8653248.

E-mail address: [email protected] (L.F. Ayala H.).

0920-4105/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2006.11.005

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http://dx.doi.org/10.1016/j.petrol.2006.11.005

http://dx.doi.org/10.1016/j.petrol.2006.11.005

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desirable outcome of a simulation study for gas con-

densate reservoirs is the development of the best opera-tional production scheme that maximizes recovery with

the minimum loss of condensate at reservoir conditions.

Naturally fractured reservoirs contribute in a large

extent to the worldwide production of oil and gas.

According to Pápay (2003), more than 50% of the world

petroleum production comes from fractured reservoirs.

Aguilera (1995) defined a naturally fractured reservoir

as a reservoir that contains fractures created by nature

that have either a positive or a negative effect on fluid

flow. The presence of fractures separates the rock in

several matrix blocks and provides high-permeability

channels for fluid mobilization. The numerical simula-tion of fractured reservoirs is a challenge for reservoir

engineers because they present an extreme property

contrast between the two domains it comprises: matrix

and fractures.

The study of naturally fractured gas-condensate re-

servoirs inherently brings together one of the more

complex thermodynamic and hydrodynamic phenome-

na within the solution domain. This study is based upon

the hypothesis that multi-mechanistic flow may be a

decisive recovery mechanism in naturally fractured gas-

condensate reservoirs. When matrix permeability is verysmall (very tight matrices), diffusion may be responsible

for an important fraction of fluid flow, rather than bulk

flow driven by pressure gradients. If this is the case,

fluid flow results from the combined action of transport

driven by both concentration and pressure fields. This is

basis of the multi-mechanistic flow concept introduced

by Ertekin et al. (1986). In the case of naturally fractured

gas-condensate reservoirs, the condensate dropout may

create a liquid barrier for gas flow that might enhance

diffusion as the crucial mechanism for fluid transport.

The objective of this study is the development of a

numerical model that can capture the dominant flow

processes and recovery mechanisms of naturally frac-

tured gas-condensate reservoirs. To date, the work of

Castelijns and Hagoort (1984) is the one available study

dedicated to the analysis of retrograde condensation in

naturally fractured gas-condensate reservoirs. Their ana-

lysis was applied to assess the potential of condensaterecovery in the Waterton reservoir (Alberta, Canada)

and it was restricted to the properties of the reservoir

under consideration. In their study, analytical flow

models were presented to calculate the possibility of

recovering part of the condensate by gravity drainage. In

the present study, we have concentrated our efforts in the

preliminary understanding of condensate build-up with-

in the matrix at saturations below critical, with special

emphasis in the limiting case where matrix permeabil-

ities are extremely low and the cases where the effects of

multi-mechanistic flow cannot be ignored.

1.1. Depletion performance of naturally fractured gas-

condensate reservoirs

Fig. 2 depicts a typical naturally fractured reservoir.

Naturally fractured reservoirs are ideally depicted as a

large number of matrix blocks connected by a fracture

network. This model is usually referred to as the “sugar-

cube” model and it was first proposed by Warren and

Root (1963). Fig. 3 illustrates the corresponding repre-

sentation commonly used for numerical modeling pur-

poses. For this idealization to be useful, it is claimed that the stack of rectangular blocks in Fig. 3 stores most of

the hydrocarbons found in the actual configuration shown

in Fig. 2. In addition, the horizontal and vertical inter-

secting fractures supply a network with an equivalent

fluid conductivity as that provided by the original frac-

tures. Barrenblatt et al. (1960) were the first to formulate

the equations for single phase flow in double porosity

reservoirs. The actual composite medium of blocks and

fractures is replaced by two overlapping continua: the

continuum of the matrix blocks and the continuum of the

Fig. 1. Phase envelope for a typical gas-condensate fluid.

Fig. 2. A naturally fractured reservoir.

14 L.F. Ayala H. et al. / Journal of Petroleum Science and Engineering 58 (2007) 13 – 29



law. Permeability of the system controls the prevailingmechanism, as flow through an extremely tight matrix

expected to occur due to Fickian flow alone. The present

work attempts to assess the impact of multi-mechanistic

flow on the isothermal depletion of tight, naturally

fractured retrograde gas reservoirs. The main objective

of this study is the development of an isothermal

compositional simulator for the modeling of depletion

performance of naturally fractured gas-condensate

reservoirs in the multi-mechanistic flow domain.

2. The numerical model

Simulation studies are conducted using a fully impli-

cit, in-house compositional simulator (Ayala H. et al.,

2006). The multi-mechanistic compositional material

balance that the numerical simulator solves for each

component at each discrete gridblock of the system is

written as follows:

jd xm

kk ro

lo

Pqo

ðj po−gojG Þ þ ym

kk rg

lg

Pqg

!"

ðj pg−ggjG Þ þ ym/S g Deff jPqg

þ M ⁎m ¼ A

At ½/ð xm

PqoS o þ ymPqgS gÞ m ¼ 1; 2; N nc

ð1Þ

Eq. (1) assumes that all hydrocarbon components are

found both in the liquid and gas phases. In addition, the

following standard assumptions are taken: hydrocarbons

reach instantaneous equilibrium at any spatial point and

at any time in the reservoir (i.e., thermodynamic equili-

brium is reached faster compared to the velocity of fluid

flow), reservoir is isothermal and hence geothermal

gradients are negligible, the multi-mechanistic phenom-

enon only takes place in the gas phase, while the flow of

the liquid phases is only due to flow potential gradients.

In Eq. (1), the inclusion of diffusion using a Fick's law

with a porosity-saturation multiplier and advection

calculated by Darcy's law has been called the advec-tive-diffusive model (ADM). Other authors, such as

Webb and Pruess (2003), have also presented the dusty-

gas model (DGM) approach, where the coupling effects

between ordinary diffusion, Knudsen diffusion, and

advection are studied. In this work, the multimechanic

approach (e.g., ADM) is examined as it retains the

convenient form of the classical diffusivity equation for

porous media for added ease in implementation. This

model could be easily modified to introduce component-

specific diffusion coefficients rather than an effective

coefficient for the gas phase alone. With such approach,the effect of concentration on diffusion coefficients and

molecular stripping within the gas phase could be

examined in a future work. Thermal and pressure-driven

diffusion could be also incorporated.

In the system under consideration, water exits at its

irreducible saturation and thus a maximum of two mobile

phases exist in the reservoir at a given time (gas and

condensate). In addition, water does not have hydro-

carbons in solution and does not affect hydrocarbon

equilibrium; thus, water does not appear either in the gas

phase or in the condensate phase. The Peng and Robin-

son (1976) equation of state is utilized to model the PVT behavior and fluid equilibria. This work uses a 6-com-

ponent lumped version of the retrograde gas presented

by Kenyon and Behie (1987) in their SPE Third Compa-

rative Solution Project paper. Their relative permeability

and capillary pressure values are utilized. The phase

envelope of this fluid is presented in Fig. 8. A validation

of the proposed tool against other commercial simulators

Fig. 7. Condensate appearance in naturally fractured gas-condensate

reservoirs.

Fig. 8. Phase envelope of the gas-condensate reservoir fluid.




and Kenyon and Behie's published data has been inclu-

ded in the work of Ayala H. et al. (2006).

2.1. Single-block matrix studies for maximum scrutiny

Naturally fractured reservoirs do occur in diverse andcomplex reservoir structures whose details may remain

unknown. Nonetheless, the discrete nature of reservoir

simulation requires the use of theoretical models that are

able to reproduce its overall (bulk) performance. Matrix

blocks may contain over 90% of the total hydrocarbon

in place; therefore, the main concern is the extraction of

hydrocarbon from the matrix and not from the fracture.

This suggests that some effort must be spent in under-

standing the mechanisms which take place within the

rock matrix blocks.

In a naturally fractured reservoir, the fractured system provides the preferential path for fluid flow throughout

the reservoir. The fracture network is bounded by large

surface areas of matrix rock through which fracture

pressure changes are communicated to the fluids inside

the matrix. Under these conditions, the fractures define

the boundary conditions for the matrix rock they

surround. The type of environment around the matrix-

rock block drives performance and hydrocarbon recov-

ery from the particular matrix block. In addition, as the

entire fractured reservoir is depicted as a stack of several

single-blocks of matrix rock, ultimate recovery from the

entire reservoir is controlled by the behavior of each of the single-matrix block in the stack. Therefore, as

depicted in Fig. 9, understanding the phenomena taking

place in a single matrix block can provide some key

information to our understanding of the overall behavior

of naturally fractured reservoirs. In single-block matrix

studies, the flow in the fracture network is of secondary

importance. The fracture is “felt ” by the matrix rock by

imposing an appropriate boundary condition to the flow

equations of the matrix. This analysis is usually

performed for different conditions (i.e., environments

or boundary conditions) in the fracture. Over the years,

single-matrix block studies have been considered

essential in the literature for the understanding of natu-

rally fractured reservoir dynamics (Yamamoto et al.,

1971; Peaceman, 1976; Van-Golf-Racht, 1982; Saidi,

1987; da Silva and Belery, 1989).

In the single-block matrix studies carried out in thiswork, all six block faces are surrounded by a highly

conductive fracture network that maintains the same

environment (in terms of fracture pressure) in matrix

block faces that are parallel to each other. Because flow is

symmetrical with respect to the x- and y-axes, only a

quarter of the matrix block is sufficient for simulation.

Symmetry planes are represented by no-flow boundaries.

Presence of gravitational forces does not allow symmetry

with respect to the z -axis. In this work, both matrix and

surrounding fracture are discretized. Each gridblock of

matrix block that is found at the edges of the matrix hasone neighboring fracture gridblock cell whose pressure is

known—therefore, fracture gridblocks do not take part in

the Newton–Raphson updating. Conditions at the fracture

are defined at the upper-most layer. Gas gravity gradients

define the pressure conditions for all other fracture layers.

In addition, overall composition of the fracture fluid is

assumed to be equal to the composition of the gas that is

present in the neighboring gridblock of matrix rock.

2.2. Parametric study of multi-mechanistic flow

Parametric studies were conducted in order to illu-strate the influence that several variables (such as block

size or fracture spacing, matrix permeability, diffusion

strength, different environment conditions or fracture

depletion schemes) have on the depletion behavior of a

matrix block of the reservoir rock under multi-mecha-

nistic flow conditions. Table 1 lists variables which were

not part of the parametric studies conducted in this

work. It is important to indicate that these values pro-

vided a frame of reference for the parametric studies;

yet, the developed model is general and capable of

utilizing different sets of input data. Therefore, irreduc-ible saturation values, fracture permeability and aperture,

compressibilities, relative permeabilities, etc. presented

in this table can be modified accordingly. Qualitative

trends presented and discussed in paper are expected to

remain valid for other input data combinations.

Table 2 presents the different ranges of matrix

permeability (k m), effective diffusion coefficient ( Deff ),

matrix block size (h = d = w), and fracture depletion rate

employed in this work. The selection of matrix perme-

abilities shown in Table 2 is based on the classification

for matrix permeability of tight formations reported in

Table 3. Matrix block dimensions (i.e., fracture spacing)Fig. 9. A single-block of matrix rock of a naturally fractured reservoir.




are within the order of magnitude of the ones used in

previously published single-block studies (Yamamoto

et al., 1971; Van-Golf-Racht, 1982), even though some

authors consider that fracture spacing of about 10 m

(33 ft) can be actually considered as large (Carlson, 2003).

This study employs effective diffusion coefficients ran-

ging from 0 to 20 ft 2/day. Cussler (2001) explains that the

typical values of gas diffusion coefficients range between

0.1 (9.3 ft 2/day) and 1 cm2/s (93 ft 2/day) at standard

conditions. Influence of pressure and temperature dra-matically changes this range. Reported values of diffusion

coefficient for gas reservoir fluids (Katz et al., 1959;

Sigmund, 1976; Ertekin et al., 1986; da Silva and Belery,

1989) suggest that typical diffusion coefficient values can

be found between 0 to 20 ft 2/day—where the latter re-

presents a rather large value of gas diffusion coefficient

for reservoir fluids.

2.3. Behavior of extremely tight systems when diffusion

is neglected

The present section discusses the behavior of natu-

rally fractured gas-condensate reservoirs with extremely

tight matrices when diffusion either does not take place

in the system or has been neglected—i.e., all fluid flow

strictly follows Darcy's law. Table 4 presents the sce-

nario considered in this section and Fig. 10 reveals the

corresponding predictions for gas recovery, total

hydrocarbon HC recovery (or molar recovery), and

condensate recovery expected for such a system after a

period of 10 years of production. Recovery trends exhi-

bited in Fig. 10 are typical of gas-condensate systems, in

terms of the relative value of gas recoveries to conden-

sate recoveries. Fig. 11 presents the prediction of con-

densate appearance at reservoir conditions for theextremely tight system under study and Fig. 12 illus-

trates fracture depletion condition with time. After about

840 days of production, fracture depletion levels

(Fig. 12) have induced fluid dew point pressure con-

ditions (see Fig. 8) around the matrix block and thus

condensate starts showing up at the edges of the matrix.

In Fig. 10, gas and condensate recoveries (and thus

molar hydrocarbon recovery) are the same as long as the

hydrocarbon fluid maintains its integrity as a single-phase

gas. Fig. 10 shows that surface gas, molar hydrocarbon,

and condensate recovery start departing significantly fromeach other soon after condensate appears at reservoir

conditions. Fig. 10 demonstrates that most hydrocarbons

produced from this reservoir show at the surface as a gas.

After 10 years of production, when production operations

are halted, the operator of this hypothetical field has been

Table 3

Permeability ranges for systems under consideration

Extremely tight systems k mb0.01 md

Tight systems 0.01≤k mb0.1 md

Moderately permeable systems 0.1≤k mb1 md

Highly permeable systems k m≥1 md

Table 1

Variables maintained constant during parametric studies

Matrix Matrix porosity (ϕf ) 0.13

Initial pressure ( pi) (at center block, gravitationally equilibrated) 4000 psia

Temperature (T ) 200 F

Rock compressibility (cϕ) 1 × 10−6 cp−1

Fracture Fracture width (aperture) 0.01 ft

Fracture permeability (k f ) 2000 md

Initial fracture pressure (at upper-most layer) 4000 psia

Minimum fracture pressure ( pf,min) 600 psia

Temperature (T ) 200 F

HC fluid Modified 6-component SPE fluid

Water Water saturation (S wirr ), immobile 0.16

Water compressibility (cw) 1 × 10−6 psia−1

Relative permeabilities SPE Third Comparative Solution Project (Kenyon and Behie, 1987)

Capillary pressures SPE Third Comparative Solution Project (Kenyon and Behie, 1987)

Separation train conditions Primary separator 315 psia, 60 F

Second separator 65 psia, 60 F

Stock tank 14.7 psia, 60 F

Simulation time 10 years

Table 2

Variable values used for parametric studies

Matrix permeability

(isotropic and homogeneous) (k m)

1×10−3, 1×10−2, 1×10−1

and 1 md

Matrix block dimensions (d , w, h) 500, 200, and 40 ft

Effective diffusion coefficient ( Deff ) 0, 5, and 20 ft 2/day

Fracture depletion rate 1, 2, 5, 20 psi/day




able to recover 54% of the original surface gas in place but

only 24% of the original condensate in place, which

makes up the recovery of about 50% of the original

hydrocarbon in place. Condensation at reservoir condi-

tions, illustrated in Fig. 11 for the present scenario, is

undesirable because the heavy components that make up

the valuable surface condensate are lost to the formation.

Very low condensate recoveries, compared to gas reco-

veries, are typical of gas-condensate systems when

hydrocarbon condensation takes place at the reservoir rather than the surface. If no condensation takes place in

the formation,gas and condensate recoveries would be the

same—which is the case during the first 840 days of

production. Nevertheless, gas recoveries are still low

(roughly about 50%) when it is considered that the field

has been producing for 10 years.

Fig. 13 reveals condensate saturation (S o) cross-

sectional snapshots taken vertically at center of the matrix

block. The results obtained for the quarter-block simula-

tion are used to reproduce the behavior of the entire block

based on the existing symmetry with respect to the x-axis.

The first condensation snapshot in Fig. 13 is shown for t =900 days, close after condensate has appeared at

reservoir conditions (see overall reservoir condensation

profile in Fig. 11). In this snapshot, condensate first

appears at the top corners of the matrix block, where

pressure is lowest due to gravity gradients (simulations

assumed gravitational equilibrium at t =0). Condensate

builds up around the edges of the block as time progresses,

thus creating a coating of immobile liquid that traps the

gas inside the block (represented as blue areas with zerocondensate, S o= 0). Maximum condensate saturations

are about 20%, while the critical condensate saturation

for mobilization is 24%. Therefore, all condensate is

virtually immobile. The thickness of this barrier of im-

mobile condensate increases with time; but it is not until

after 8 years that inner-most gas is finally reached

(when pressure transients are able to reach the inner

portions of the matrix rock). It is important to note that

condensate profiles are virtually symmetrical with res-

pect to the z -axis, rendering negligible the effect of

gravity. The reason for this is two-fold: the matrix is

extremely tight and hence limits condensate movement inthe z -direction, but condensate never attains mobility

nonetheless (S obS oc). This suggests that simulations

could have been done using a 1/8 of a matrix block,

assuming flow symmetry with respect to the z -axis,

without any great loss of accuracy. Even a 1/16 of a

matrix block could be simulated using the existing sym-

metry with respect to the 45° vertical plane that cuts the

Fig. 11. Reservoir condensation in extremely tight matrices when no

diffusion takes place.

Fig. 12. Fracture pressure (top layer) for a depletion rate of 1 psi/day.

Table 4

Extremely tight matrix base case

Matrix block dimensions (d = w = h) 500 ft

Matrix permeability, k m (isotropic and homogeneous) 1× 10−3 md

Effective diffusion coefficient ( Deff ) 0 ft 2/day

Fracture depletion rate 1 psi/day

Fig. 10. Recoveries for extremely tight matrices without diffusion.




Fig. 13. Condensate saturation maps for extremely tight matrices with no diffusion.




x– y plane in half. However, this cannot be done using the

rectangular gridblocks employed in this work.

Re-vaporization of condensate is evident in the areas

subjected to the most extensive depletion in Fig. 13—

i.e., the corners of the matrix rock. It is observed that

condensate saturation at the corners (the first areas tohost condensate, as seen in the snapshot at t =1000 days)

reach maximum condensation, and then, upon further

depletion, begin re-vaporizing the condensate. Local re-

vaporization phenomena is seen, for instance, in the

snapshot of t = 3000 days, where condensation saturation

level at the corners (which are extensively depleted

areas) is about the same than condensate levels at the

inner portions of the block (a poorly depleted area). The

value of S o in those areas is about 13%, shown in yellow.

Global condensate re-vaporization is depicted in Fig. 11,

where the amount of reservoir condensate reaches themaximum of 18 reservoir barrels per thousand cubic feet

of total porous volume–after about 2850 days of pro-

duction–and then steadily declines.

2.4. Role of gas diffusion in extremely tight matrix

blocks

In the previous section, it was assumed that Darcy's

law controlled fluid flow in an extremely tight system

and no diffusion was taking place. However, the main

hypothesis driving this study is that the presence of

fractures around a matrix block of extremely low permeability would create molecular concentration

gradients in the gas phase large enough as to trigger a

significant amount of gas diffusion (i.e., multimecha-

nistic flow). This second recovery mechanism–or

Fickian flow–could considerably overcome the flow

impairment to gas flow that is posed by the eventual

appearance of a condensate barrier around the edges of

the block. As a result, the recovery of gas stored in the poorly depleted inner-matrix areas of the matrix block

may be largely driven by concentration gradients. In this

work, we examine the role of gas diffusion in the reco-

very of gas from the inner portions of extremely tight

matrix represented by values of 5 and 20 ft 2/day of

effective diffusion coefficient. As discussed previously,

effective diffusion coefficients in the gas phase may take

values between 0 and 20 ft 2/day. Figs. 14 and 15 display

the effect that multi-mechanistic flow has on the recovery

performance of this system. Both figures reproduce the

base case ( Deff = 0) for comparison purposes. These two

new scenarios are subjected to the same boundarycondition depicted in Fig. 12 and other conditions of

Table 4—with the exception of the values of Deff .

Figs. 14 and 15 effectively demonstrate that the

concentration gradients that are established within

the matrix block can be responsible for a large fraction

of fluid recovery. Fig. 14 shows that diffusion alone

Fig. 14. Effect of diffusion on gas and condensate recoveries for extremely tight matrices.

Fig. 15. Effect of diffusion on hydrocarbon molar recovery for

extremely tight matrices.

Fig. 16. Effect of diffusion on reservoir condensation for extremelytight matrices.




Fig. 17. Effect of diffusion on condensate saturation for Deff =5 ft 2/day.




can be responsible for up to 26% of gas and 12% of

condensate ultimate recoveries—an increase of about

25% in overall hydrocarbon recovery as shown in

Fig. 15. Multi-mechanistic flow certainly takes place

and Darcy's law alone cannot fully predict system

behavior due to the considerable concentration gra-

dients that develop during depletion. In extremely tight

systems, diffusion allows for a considerably larger

withdrawal of in-situ fluids than what is predicted by

Darcy's law. This larger fluid withdrawal translates

Fig. 18. Effect of diffusion on condensate saturation maps for Deff =20 ft 2/day.




into a larger system depletion that, for a gas-conden-

sate system, signifies that a larger condensate dropout would be obtained at reservoir conditions. This is

clearly depicted in Fig. 16. As diffusion coefficient

increases, more fluid is withdrawn out of the reservoir

and hence more condensation is to be expected at

reservoir condition.

Figs. 17 and 18 describe the effect of gas diffusion in

the development of condensate at reservoir conditions at

different depletion stages. These two figures should be

compared among themselves and against Fig. 13, which

presented the same prediction for the case where diffusion

was considered negligible. The effect of diffusion on the

formation of condensate is dramatic. Diffusion doesenable the “trapped” inner-most gas to flow out driven by

concentration gradients. Larger depletion at the inner-

most portion of the matrix blocks allows larger recoveries

and larger condensate dropouts. Condensate appearance

at the inner-most portion of the matrix occurs at much

shorter times than shown in Fig. 16. However, condensate

saturation never reaches the critical value needed for

mobilization (S oc=24%) and, as a result, condensate

remains immobile at reservoir conditions.

2.5. Effect of matrix permeability: the multi-mechan-

istic flow domain

Figs. 19, 20 and 21 present the prediction for hydro-

carbon recovery for the cases of tight matrices

(k m=1×10−2 md), moderately permeable matrices

(k m=1×10−1 md), and permeable matrices (k m=1 md).

It is evident that the contribution of diffusion to recoveries

and overall system behavior becomes negligible as the

permeability of the matrix (k m) increases, especially when

one compares these three figures to Figs. 14 and 15 of thecase of extremely tight systems.

Thegeneral observation is that, when the matrix rock is

permeable enough, recovery is controlled by fracture

depletion only. In other words, a permeable matrix is able

to deliver the amount of fluids as prescribed by the

fracture boundary condition without restriction. This is

what is observed in Figs. 20 and 21, which present

Fig. 19. Recoveries for tight matrices (k m= 1× 10−2 md).

Fig. 20. Recoveries for moderately permeable matrices (k m=0.1 md).

Fig. 21. Recoveries for permeable matrices (k m=1 md).

Fig. 22. Effect of permeability on reservoir condensation.




identical recovery values in time. In the case of tight

matrices (Fig. 19), Fickian flow contributes to hydrocar-

bon recovery, albeit in a limited extent. The effect of

matrix permeability on the extent of reservoir condensate

appearance is described in Fig. 22. Predictions for the

more permeable systems (k m= 0.1 and 1 md) lie on top of

each other in this figure. In general, larger matrix

permeabilities foster easier and more evenly spread

reservoir condensation throughout the matrix rock.Fig. 22 shows that the more permeable the system, the

sooner the condensate spreads and appears in the matrix

block once dew point conditions are reached. For the case

of the more permeable systems, condensation reaches a

constant value once fracture depletion stops (minimum

fracture pressure is reached) and no more pressure

changes are imposed by the fracture (see Fig. 12).

In summary, in naturally fractured gas-condensate re-

servoirs, Fickian flow can be responsible for a con-

siderable amount of fluid recovery for extremely low

matrix permeabilities—with values in the order of

1×10−3 md or less. Fluid flow in more permeable

matrices— permeabilities of 0.1 md or more-obeys

Darcy's law and thus it is said that the Darcian com-

ponent of the dual-mechanistic flow prevails, while the

contribution of Fickian flow is negligible. In other words,

flow driven by gas concentration gradients prevails at

very low values of matrix permeability, while flow driven

by pressure gradients dominates at large values of matrix permeability. The multi-mechanistic flow mapping pre-

sented by Ertekin et al. (1986) has been corroborated for

the case of naturally fractured gas-condensate reservoirs

for the system under study. This mapping is shown

in Fig. 23. To reinforce the multi-mechanistic flow

spectrum shown in Figs. 23 and 24 demonstrates that for

extremely tight systems with k mb1×10−3 (the example

Fig. 23. The multi-mechanistic flow spectrum.

Fig. 25. Top fracture pressure evolution in time as a function of depletion rate.

Fig. 24. Role of diffusion in extremely tight matrices (k m=1×10−4 md).

Fig. 26. Effect of depletion rate on recovery for extremely tight

systems.

Fig. 27. Effect of depletion rate on recovery for tight systems.




uses k mb1×10−4 md) Fickian flow is–in fact –the

mechanism responsible for recovery. Ultimate gas

recoveries can be up to 4 times larger than the recovery

obtained by Darcian flow alone—i.e., Fickian flow being

responsible for the recovery of more than 2/3 of the gas

brought to the surface. It is important to highlight that

some potential shifts in the flow domain regions described

in Fig. 24 may occur as a result of the compositional

characteristics of the gas-condensate reservoir, which are

not studied in detail in this work. Different gas-condensate

fluid characterization should be examined in order to

confirm or refine such mapping.

2.6. Effect of fracture depletion rate

Fig. 25 presents the four different depletion rates

(boundary conditions) that were imposed to the matrix

block under study. Fig. 25 also presents the value of the

dew point of the gas-condensate system under consider-

ation. In general, the higher the depletion rate, the sooner

the system reaches dew point conditions ( psat = 3158 psia)

and thus the sooner the condensate is developed at reser-

voir conditions. In general, the effect of depletion rate on

recoveries is intuitive. The higher the depletion rate, the

sooner the hydrocarbons arebrought to the surface and the

higher the recoveries at a given point in time. Figs. 26, 27

and 28 display the effect of flow rate (depletion rate) on

recovery for extremely tight, tight, and permeablematrices. Molar recoveries are presented, although the

same behavior is seen for gas and condensate recoveries.

In these figures, an upper bound ( Deff =20 ft 2/day) and a

lower bound ( Deff =0 ft 2/day) are shown for different

depletion rates that delineate a region of influence for

Fickian flow. That is, the impact of Fickian flow on

recovery is represented by the distance between the lower

and upper bound for a given depletion rate. From these

figures, it is clearly seen that Fickian flow is favored by

higher depletion rates. The role of diffusion in the

recovery of hydrocarbon from the system becomes more pronounced as depletion rates are increased. The larger

the depletion rate, the earlier Fickian flow takes place in

the system. This is to be expected, because a larger

depletion rate creates larger pressure and concentration

gradients around the matrix block, triggering multi-

mechanistic flow at earlier times. It is also observed that

the mapping presented in Fig. 23 remains applicable. The

role of diffusion diminishes with increased matrix perme-

ability, but depletion rates show a more prominent role of

Fickian flow than lower depletion rates do—for a given

permeability. Fig. 28 demonstrates that the role of diffu-

sion in permeable systems is insignificant —regardless of flow rate. In summary, matrix permeability controls the

relative contribution of Fickian and Darcian flows at any

depletion rate level, while the significance of Fickian flow

is emphasized by larger flow rates.

2.7. Effect of fracture spacing and matrix block size

Matrix block size is considered one parameter that

can greatly influence behavior of a naturally fractured

reservoir. Matrix block size is determined by fracture

Fig. 28. Effect of depletion rate on recovery for permeable and

moderately permeable systems.

Fig. 29. Matrix block size and fracture spacing.




spacing; thus, these two terms can be used interchange-ably in the context of the “sugar-cube” model. Fig. 29

shows that as larger number of fractures are found in the

system, the spacing among them decreases and thus the

size of the matrix blocks. Changes in fracture spacing

are expected to have important effects on the behavior of

the system, as the naturally fractured reservoir becomes

more highly or poorly interconnected. In this work,

matrix block sizes of 500 ft, 200 ft, and 40 ft are

investigated. Fig. 30 presents the effect of matrix block

size (fracture spacing) on molar recovery for the case of

extremely tight systems and the depletion rates of 1 psi/

day and 20 psi/day. At any point in time, molar recoveryvalues are higher as matrix block sizes become smaller

for a given depletion rate. This is reasonable because,

when fracture spacing and matrix block size are redu-

ced, it is easier to extract the fluid out of the reservoir;

thus, molar recoveries would increase when a larger

number of fractures is found in the system (i.e., when

fracture spacing is reduced). In general, the innermost

matrix fluids in small but extremely tight matrix block sizes are more prone to faster depletion than the innermost

matrix fluids stored in large and extremely tight matrix

blocks. It is also clear that fracture depletion rate has an

important role on fluid recovery, with the combination of

the largest depletion rate (20 psi/ft) and smallest matrix

block size (40 ft) reaching maximum recovery in less than

a year of production.

The enhanced fluid recovery that takes place in

extremely tight but small matrices has an important

additional effect on the multi-mechanistic flow mapping

presented in Fig. 23. Figs. 31 32 and 33 reveal the effect

of block size on the role of diffusion for extremely tight systems (k m=1×10−3 md). These figures demonstrate

that diffusion effects (represented by the width of the

shadowed areas) become progressively less significant

as matrix block size decreases. As a result, diffusion

contributes very little to molar recovery–if at all–for

small matrix blocks ( L =40 ft), even though the perme-

ability of the matrix is very small (k m= 1× 10−3 md).

Fig. 31. Effect of block size ( L =500 ft) on the role of diffusion inextremely tight matrices.

Fig. 30. Effect of block size on molar recovery for extremely tight

matrixes ( Deff =0 ft 2/day).

Fig. 32. Effect of block size ( L =200 ft) on the role of diffusion in

extremely tight matrices.

Fig. 33. Effect of block size ( L =40 ft) on the role of diffusion inextremely tight matrices.




This immediately suggests that the mapping presented

in Fig. 23 is not valid for heavily fractured systems with

small fracture spacing and matrix block sizes. The

reduction of size of the matrix block has a similar effect

than the increase of matrix block permeability studied

before. To some extent, the ease for fluid withdrawal in anaturally fractured reservoir (or “overall permeability”

of the system as a whole) is both described by matrix

block permeability and how highly interconnected the

fracture network is.

3. Summary and conclusions

This study has verified that hydrocarbon recovery in a

naturally fractured gas–condensate reservoir can be

largely driven by concentration gradients–rather than

pressure gradients alone as predicted by Darcy's law.Reservoir condensate appearance in a naturally fractured

gas-condensate reservoir creates a “coat ” around the

matrix block that encloses the inner-most gas and the

extent of this gas flow barrieris dependent upon the type of

fluid, matrix permeability, and the rate of fracture pressure

depletion. The occurrence of Fickian flow as an additional

recovery mechanism can considerably overcome the flow

impairment to gas flow that is posed by the eventual

appearance of this condensate barrier. It has been shown

that the contribution of Fickian flow to ultimate recovery

may be responsible for up to 2/3 of the total hydrocarbon

recovered from systems composed of large and extremelytight matrix blocks such as the one presented in this study

(k mb1×10−3 md, L = 500 ft). For the systems under

study, the relative contribution of the Fickian and Darcian

components in multi-mechanistic flow in naturally

fractured gas-condensate reservoirs was largely controlled

by the permeability of the matrix and fracture spacing. For

large fracture spacing with large matrix block sizes (200 ft

and above, for the conditions depicted in this study), the

contribution of Fickian flow can largely surpass that of

Darcian flow when the matrix block is extremely tight

(k mb1×10−3

md). The contribution of Fickian flow isnegligible, for all scenarios investigated in this study,

when matrix blocks are permeable or moderately perme-

able (k mN0.1 md). Fickian flow contribution can be

ignored if the fracture spacing is small (40 ft or less, for the

fluid and system under study). While matrix permeability

and fracture spacing control the relative contribution of

Fickian and Darcian flow, the significance of Fickian flow

is emphasized at higher flow rates. In general, the impact

of Fickian flow on recovery was favored by larger fracture

depletion rates. Additional work is underway to study the

effect of capillary imbibition at the fracture/matrix

interface, which has been demonstrated can have

important effects on recovery under certain conditions

(Castelijns and Hagoort, 1984). Other areas open to further

examination in this topic include: effects of changes in

fracture permeability and aperture, multi-mechanistic flow

in the liquid phase, effects of molecular stripping within

the gas phase, thermal and pressure-driven diffusion,effect of concentration on diffusion coefficients, and effect

of changes in fluid composition, capillary and relative

permeability values.

Nomenclature

cw water compressibility (psi−1)

cϕ rock compressibility (psi−1)

d depth of a typical matrix block in a naturally

fractured reservoir (ft),

Deff effective gas diffusion constant (ft 2/day),

G downward depth (ft),h thickness of a typical matrix block in a naturally


k permeability (md),

k m matrix permeability (md),

k f fracture permeability (md),

k rg gas relative permeability (unitless),

k ro condensate relative permeability (unitless),

L size of a rectangular matrix block d = w = h,(ft),

M m⁎ production/injection source term for the m-th

component, ( lbmol/day/ft 3)

nc total number of hydrocarbons in the system,

pI initial pressure (psia), pf fracture pressure (psia),

pg gas phase pressure (psia),

po condensate phase pressure (psia),

t time (days),

T temperature (F),

S g gas saturation (fraction),

S o condensate saturation (fraction),

S wirr irreducible water saturation (fraction),

w width of a typical matrix block in a naturally


xm molar fraction of the m-th component in thecondensate phase (fraction),

ym molar fraction of the m-th component in the gas

phase (fraction),

x, y, z cartesian coordinates,

ϕ porosity (fraction),

ϕf matrix porosity (fraction),

γg gas specific gravity (unitless),

γo condensate specific gravity (unitless),

ρ ¯ g molar density of the gas phase (lbmol/ft 3),

ρ ¯ o molar density of the condensate phase (lbmol/ft 3),

μg

viscosity of the gas phase (cp),

μo viscosity of the condensate phase (cp).




Acknowledgements

The authors would like to express their sincere

appreciation to the Petroleum and Natural Gas Section at

The Pennsylvania State U. for providing the support and

computational facilities required for the completion of this work. An initial version of this paper was presented

as SPE 90010 at the 2004 SPE International Petroleum

Conference in Mexico and the feedback obtained from

colleagues in this forum is greatly appreciated.

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