SPE-97739-MS

CANADIAN HEAVY OIL ASSOCIATION

SPE/PS-CIM/CHOA 97739 PS2005-329

Numerical Studies of Gas Exsolution in a Live Heavy-Oil ReservoirM. Uddin, Alberta Research Council Inc.

Copyright 2005, SPE/PS-CIM/CHOA International Thermal Operations and Heavy Oil Symposium This paper was prepared for presentation at the 2005 SPE International Thermal Operations and Heavy Oil Symposium held in Calgary, Alberta, Canada, 1–3 November 2005. This paper was selected for presentation by an SPE/PS-CIM/CHOA Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers, Petroleum Society–Canadian Institute of Mining, Metallurgy & Petroleum, or the Canadian Heavy Oil Association and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the SPE/PS-CIM/CHOA, its officers, or members. Papers presented at SPE and PS-CIM/CHOA meetings are subject to publication review by Editorial Committees of the SPE and PS-CIM/CHOA. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the SPE or PS-CIM/CHOA is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435. Abstract Gas phase formation in supersaturated live heavy oil occurs by bubble nucleation and growth. Modeling the dynamics of these processes in cold production is essential, since they are key mechanisms in determining oil recovery. The greatest challenge in field scale simulations of cold production is to quantify the spatial and temporal changes of the gas exsolution and transport processes.

This paper describes a new kinetic model which, when coupled with a thermal reservoir simulator, can simulate the dynamics of gas exsolution and transport processes in a heavy oil reservoir. In this model, two relatively simple types of mass transfer equations predict bubble nucleation and growth in a live heavy reservoir. The model structure and parameters were investigated in comparison with a previously published model.

The capability of the proposed kinetic model to handle the dynamics of gas phase formation in a heavy oil reservoir was explored in two sets of laboratory experimental data. In set 1, numerical history matches of pressure data were performed for eight constant withdrawal rate experiments. In set 2, numerical history matches of oil and gas production data were performed for four pressure depletion experiments. A close agreement was achieved between numerical simulation and experimental results. The model can be applied in the field scale simulations of cold production to predict gas exsolution and gas builds up in an oil reservoir. Introduction Gas formation in a live heavy oil reservoir occurs by gas bubble nucleation and growth. Bubble nucleation can be

initiated in a flow system when the reservoir pressure is reduced below the saturation pressure (i.e., supersaturation conditions). Initially, gas bubbles remain dispersed in the oil phase. Gradually, these bubbles grow and become free gas.

Published laboratory CT experiments data have shown a very wide distribution of gas bubbles and bubble clusters in a live heavy oil flow system (Treinen et al.1; Sahni2). It is well known that, at an early stage of primary cold production, the flow system contains a large number of fairly uniform dispersed small gas bubbles (bubbles with diameters less than the pore throat size) and, at a later stage, the flow system appears as a highly heterogeneous distribution of small bubbles, large bubbles and gas bubble clusters. The reservoir engineer needs the capability to quantify the spatial and temporal dynamics of gas bubble accumulation in field scale cold production simulation.

Previously, several papers discussed the kinetics of bubble nucleation and bubble growth mechanisms in a live heavy oil porous media (for example, Moulu3; Yortsos and Parlar4; Kashichiev and Firoozabadi5). The field scale application of the kinetics approach with an existing numerical simulator has remained a big challenge with its need to deal with large number of process mechanisms (for example, geological parameters, well operating constrains) using larger size of grid blocks.

In this paper, alternative formulations for bubble nucleation and growth have been presented. The formulations can easily couple with the numerical simulator via a set of pseudo kinetic reactions. The numerical simulator can then describe the dynamics of bubble nucleation and growth processes in a heavy oil reservoir. The numerical predictive capability of the proposed nucleation and growth model was evaluated against laboratory gas exsolution data. Background Bubble Nucleation. Gas bubble nucleation occurs spontaneously in supersaturated oil when a thermodynamic fluctuation of sufficient magnitude occurs to form a cluster of molecules bigger than a certain critical size (Moulu3; Wall and Khurana6; El-Yousfi et al.7). Several parameters such as supersaturation, oil viscosity, gas-oil interfacial tension, gas solubility and some other petrophysical parameters play an important role in the overall nucleation in an oil reservoir.

2 SPE/PS-CIM/CHOA 97739

Nucleation in an oil reservoir takes place on foreign matter, such as the surfaces of the particles in porous media, poorly wetted cavities, and existing bubbles trapped in cavities (El-Yousfi et al.7; Bernath8; Danesh et al.9; Li and Yortsos10; Cole11; Kamath and Boyer12; Fisch et al.13). Claridge and Prats14 suggested that the adsorption of asphaltenes on bubble surfaces could stabilize the bubbles at a very small size. This view was later repudiated by Maini15.

In viscous foamy oil flow, pore geometry and low diffusivity restrict bubble growth. Hence, supersaturation is not easily relieved. Higher supersaturation levels would cause more sites to be nucleated. Sites available for nucleation could exist in foamy oil as suspended solids Maini et al.16. Wieland and Kennedy17 noted that the rate of diffusion would affect the number of bubbles formed. If the diffusion coefficient is relatively high, the dissolved gas in the oil surrounding a single bubble will tend to diffuse into the bubble rather then forming other bubbles. On the other hand, the formation of a larger number of bubbles will tend to occur if the diffusion coefficient is very low.

Danesh et al.9 suggested that the presence of connate water might delay bubble nucleation, because diffusion of the light components of the oil into the water phase reduces the population of low-density clusters in the oil and prevents the formation of bubbles. In contrast, Kamath and Boyer12 suggested, as oil has a lower surface energy compared to water, it is possible that there are more hydrophobic sites than oleophobic sites. Hence, nucleation may be easier when cores contain connate water saturation. Bubble Growth. Gas bubble growth in a flow system is generally controlled by mass, momentum and/or heat transfer across the bubble-liquid interface (Theofanous et al.18; Szekely and Martins19; Szekely and Fang20; Kashchiev and Firoozabadi5). Mass transfer occurs primarily by diffusion, coalescence, and evaporation/condensation at the gas-oil interface. Momentum transfer is governed by hydrodynamic forces in connection with the capillary pressure, liquid inertia and viscosity. Heat transfer takes place by the flow of heat between the liquid and the bubble.

Lillico et al.21 described the growth of bubbles due to coalescence as a result of collisions caused by Brownian motion and shearing flow. This process eventually leads to the formation of a wide distribution of bubble sizes within the porous medium. At the beginning of the process, coalescence is less likely to occur, since the bubbles are relatively small and sufficiently far from each other. The study also found the effect of oil viscosity on bubble distributions was significant with the heavier oil having a tendency to allow more small bubbles to survive at high gas saturations. In heavy oil reservoirs, the effect of viscous forces is expected to be significant in controlling bubble growth. The expansion of the gas phase by diffusion must overcome the resistance caused by the surrounding viscous oil. The growth rate is likely to decrease with increasing oil viscosity.

The non-equilibrium effects such as liquid inertia, surface tension and viscous forces play an important role in controlling the gas bubble growth in a supersaturated liquid where the bubble growth is not solely determined by heat or mass transfer processes (Szekely and Martins19). The gas bubbles within a porous medium do not grow spherically because of viscous resistance and pore geometry constraints. The competition between growing clusters in a porous medium is different from that in the bulk. In the bulk, the competition is based on the dependence of solubility on bubble curvature, and the larger bubbles grow at the expense of smaller ones. In porous media, cluster growth is typically controlled by the porous medium capillary characteristics. The dependence of solubility on growth radius is an insignificant factor (Li and Yortsos10). It is generally accepted that bubble growth by diffusion follows a percolation pattern when the pressure decline rate is low. However, when viscous forces become dominant, the bubble growth pattern will deviate from percolation (Li22). Dominguez et al.23, based on Hele-Shaw cell and micro-model studies, showed that the gas cluster growth became elongated in the direction of gravity when the Bond number (relative importance of gravity and capillary forces) increases. Model Development A new gas exsolution kinetic model using a non-equilibrium mass transfer approach is presented in this study. In this kinetic model, a system of first order rate equations is formulated in order to capture dynamic of gas bubble nucleation and growth in a heavy oil reservoir. The rate equations can then be coupled as a source term in the governing flow equations of the numerical simulator (for example, CMG STARS24). An earlier development of the present gas exsolution model was presented in AERI/ARC Core Industry Research Program reports (Uddin et al.25, 26). Governing Equation. The assumptions imposed when developing the flow equations are: (i) local thermodynamic equilibrium, (ii) immobile solid phases, (iii) slightly compressible rock and fluids, (iv) Fickian dispersion, and (v) Darcian flow. The governing mass conservation equation of any flowing component, i, under these assumptions in a porous media grid block can be defined as:

( )∑∑

∑∑∑

==

= ==

−++

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛∆+∆Φ=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

∂∂

r

F

n

kkkiki

jijkj

n

nijjiv

jjijjiv

jijjfl

rssVjq

jDjTAdjSt

V

1

'3

1

1

3

1

3

1

)(

ρ

ρφρφρφ

(1)

where the left hand side of the equation is the rate of change of accumulation, the first right hand term is the net rate of inflow, the second right hand term is the external source/sink (for example, well injection or production) and the third right hand term is the internal source/sink (for example, chemical reaction). An analogous equation holds for heat flow, with the thermal conductivity term replacing the diffusive flux term.

SPE/PS-CIM/CHOA 97739 3

The variables are listed in the nomenclature.

An important aspect of the current gas exsolution kinetic model is the treatment of the internal source/sink term. In this case, reaction rate, rk, in the source/sink term represents the bubble nucleation and growth rates. Kinetic Model. A five component kinetic model was developed to simulate gas bubble nucleation and growth in a live heavy oil reservoir. The components are water (H2O) in aqueous phase, heavy oil (HC), dissolved gas (gd) and dispersed gas bubbles (bd) in oleic phase and connected gas bubbles (bc) in gaseous phase. The present gas exsolution model can be organized into a component-phase chart as in Table 1.

Table 1: Components and phases – Gas exsolution model

Phase Component Aqueous Oleic Gaseous Water (H2O) Heavy oil (HC) Dissolved gas (gd) Dispersed bubble (bd) Connected bubble (bc)

X

X X X

X

It should be noted that a phase is a physical manifestation

of (and is composed of) one or more components. For example, the component water may be found in liquid, gaseous and solid phases. All physical properties are assigned to a model component in terms of the phases in which that component may be found. The necessary component properties can be obtained from the literature. In this kinetic model, all components except dispersed bubble (bd) retain their usual definitions. Here, we assume that the dispersed bubble is a pseudo component in the oleic phase when gas bubble size is smaller than the pore throat size in a porous medium. As bubble growth continues and their size increases to the pore throat size, the dispersed bubble then becomes a connected bubble (bc) in the gaseous phase. The physical characterization and quantification of the basic properties of the pseudo dispersed bubble component are described in the following section.

Dispersed Bubble. Gas bubbles ranged from critical size to pore throat size are assumed to be dispersed bubbles. The critical size is a meta-stable equilibrium such that only a bubble greater than the critical radius continues to grow to a large bubble. According to the nucleation theory, the critical bubble radius, rcr, and the nucleation energy, ∆E, can be expressed as, rcr = 2σ/(f∆P), and ∆E = 16πσ3/3(f∆P)2. Here, σ is the gas-oil interfacial tension and f∆P is the fraction of the excess energy (see Appendix A). This illustrates that the distribution of the critical bubbles in an oil reservoir will be dictated by the oil properties, viscous force and characteristics of the porous media.

The micro-model and CT experiment data in Figure 1 observed a large number of persistent small bubbles – most of them of 3 to 5 µm size. The peak visible bubble counts in micro-model oil of 12,780 mPa.s were 48 and 34 per pore body at 0.03 m/day and 0.001 m/day, respectively. Bubble moving through pore throat

Sand grainOilBubble withinpore body

Stage I Stage II

Bubble moving

Stage III Stage IV

Bubbles coalescence Bubbles coalescence

Bubble moving through pore throat

Sand grainOilBubble withinpore body

Stage I Stage II

Bubble movingBubble moving through pore throat

Sand grainOilBubble withinpore bodyBubble withinpore body

Stage I Stage II

Bubble moving

Stage III Stage IV


Stage III Stage IV


Figure 1: Gas bubble visualization within heavy oil in a dual depth micro-model.

The above studies showed an existence of many possible physical scenarios of dispersed bubbles in an oil reservoir ranged from critical bubble size to pore throat size. However, as the porous media porosity is statistically a normally distributed parameter, one can assign some statistical properties for the distribution of dispersed bubbles in a given flow system.

In this study, a simple approach was used to quantify the dispersed bubbles in a given flow system. Let us assume a single representative size (for example, an average value of the dispersed bubbles size) of the dispersed bubbles in a grid block which is greater than the critical bubble size and less than the mean pore throat size. Further, the compressibility of this dispersed component is assumed to be gas like compressibility. Using the ideal gas law, the volumetric mole density can then be estimated to approximate flow system expansion resulting from the appearance of the dispersed bubbles. All other necessary basic thermodynamic properties can be assigned to the properties of the dissolved gas component.

Bubble Nucleation. The gas bubble nucleation equation, dN/dt ∝ exp(-∆E/kBT), has been widely discussed in literature. This is similar to the Maxwell statistical approach to find a critical gas bubble of energy, ∆E = 16πσ3/3(f∆P)2, in an excess energy field. The parameters kB and T are the Boltzmann constant and energy field temperature, respectively.


In the present gas exsolution model, two forward pseudo kinetic reactions are used to describe an alternate form of bubble nucleation equation, Equation A-9. Reaction 1 [ ] [ ]dd bg →

Reaction 2 [ ] [ ] [ ]ddd bbg 2→+

Here, [gd] and [bd] are the dissolved gas and dispersed bubble components in the oleic phase, respectively. The mass transfer rates for the above kinetic reactions can be defined as:

( gdgdd ccNr ′−= 1)1( )

bdgdgdd cccNr 22

)2( ′−=

)

(2)

( ) ( ) (3) Where, rd

(1) and rd(2) are the mass transfer rates per unit

volume per unit time of dissolved gas into dispersed bubbles, N1 and N2 are the transfer coefficients or nucleation parameters, cgd is the actual dissolved gas concentration estimated as cgd =(φSoρo)(xgd), where, S0 is the oil saturation, ρo is the oil phase mole density and φ is the porosity, xgd is the dissolved gas mole fraction, c′gd is the equilibrium dissolved gas concentration defined by c′gd = (φSoρo)(x′gd), where, x′gd is the equilibrium dissolved gas mole fraction estimated as x′gd = y′/K, where K is the phase equilibrium constant and y′ is the gas mole fraction.

Bubble Growth. When a gas bubble has surpassed the critical radius (i.e., bubble radius, r ≥ rcr), it can grow irreversibly with a certain growth rate. Here, several mechanisms play a critical role in controlling bubble growth such as evaporation and condensation at the gas-oil interface, diffusion, hydrodynamic forces (capillary pressure, liquid inertia and viscosity) and heat transfer. Several studies proposed a simple power-type growth equation.

In the present model, two forward pseudo kinetic reactions are used to describe an alternate form of bubble growth equation. Reaction 3 [ ] [ ]cd bb →

Reaction 4 [ ] [ ] [ ] [ ]cddd bgbg +→+

Here, [gd] and [bd] are the dissolved gas and dispersed bubble components in the oleic phase, respectively, and [bc] is the connected bubble component in the gaseous phase. The component [gd] in reaction 4 is used to define supersaturation and will not participate in reaction. Now, using a simple probabilistic approach, the mass transfer rates for the above kinetic reactions can be formulated as:

( bdc cGr 1)1( = (4)

( ) ( )bdgdgdc cccGr 22

)2( ′−= (5) Where, rc

(1) and rc(2) are the transfer rates of dispersed bubbles

into connected bubbles, G1 and G2 are the transfer coefficients or growth parameters. All other parameter definitions have

been given previously. It should be noted that the present form of the growth kinetics agrees with the power-form of bubble growth rate as mentioned in the literature. Model Application The gas exsolution model coupled with the numerical simulator-CMG STARS was applied in history matches of several laboratory rate and pressure depletion test data. The best fitted nucleation (N1 and N2) and growth parameters (G1 and G2) were obtained. The sensitivity of these kinetic parameters under different flow regimes was studied. Experimental Data. During this study, a large number of published laboratory gas exsolution data including CT experiment were compiled (for example, Sahni2; Tang and Firoozabadi27; Lillico et al.21; Kumar et al.28; Pooladi-Darvish and Firoozabadi29). This data showed some common trends such as gas saturation and oil recovery increase with pressure depletion rate and a wide distribution of gas bubbles and bubble clusters in a flow field.

Several studies have also shown relatively higher critical gas saturation for faster depletion experiments. Previous studies claimed that the higher critical gas saturation obtained in faster depletion experiments occured due to a greater number of gas bubbles nucleation in the flow system. Some studies in an external energy drive (i.e., non-exsolution process) flow showed that the residual gas saturation or the critical gas saturation decreases with increased viscous force or pressure depletion rate (for example, Delshad30).

The previous studies emphasized that the relative permeability and the PVT data will play a critical role in numerical history matches of gas exsolution data. Hence, the laboratory gas exsolution experimental data were carefully selected depending on the availability and reliability of the PVT data and the relative permeability data.

Rate Experiments. The laboratory gas exsolution tests in the live heavy oil sand pack were conducted at Alberta Research Council. Alberta heavy oils produced from Lloydminster and Cold Lake areas with dead oil viscosity of 1,367 mPa.s, 32,380 mPa.s, and 51,615 mPa.s at 20 °C were used. The oil was dewatered, filtered to remove fines, and saturated with methane, achieving a GOR (gas to oil ratio) of 8.0 to 8.5 std. m3/m3.

The gas exsolution tests were conducted in a vertical sand pack which was 27.50 cm long and 7.74 cm in diameter. A schematic diagram of the laboratory experimental setup is shown in Figure 2 (I). Further details of the experimental setup and procedure can be found in the literature (Lillico et al.21).


Figure 2. Schematic diagrams of the experimental setups – I. Rate experiment and II. Pressure depletion experiment.

The experimental tests were conducted in +150 –200 mesh Ottawa sand, packed to a porosity of 35.4% to 37.5% and a permeability of 2.0 to 4.0 Darcy. The sand pack volumes varied from 1,014 cm3 to 1,053 cm3. The sand pack was saturated first with water, then with live heavy oil, leaving water saturations of 2.5% to 10.0%. Mercury was then injected into the bottom of the pack and the live oil was displaced out through the top of the pack. The injection continued until mercury was produced at the top end. The oil and mercury saturations were slightly different from test to test. The key parameters for eight mercury withdrawal tests are summarized in Tables 2 and 3.

Table 2: Test parameters - Mercury withdrawal experiments

Test # Initial conditions R1 R2 R3 R4

Main Visc. (mPa.s)(1)

Temperature (°C) W. rate (cm3/min) Others Porosity Oil saturation Mercury saturation Water saturation GOR (std m3/m3) Pressure (kPa ga)

32,380

20 0.0019

0.356 0.472 0.428 0.100 7.970 3980

32,380

20 0.0077

0.356 0.590 0.353 0.057 8.210 4020

32,380

20 0.0303

0.355 0.639 0.321 0.040 8.200 3975

32,380

20 0.1200

0.355 0.565 0.372 0.063 8.070 3950

Table 3: Test parameters - Mercury withdrawal experiments

II. Pressure depletion experiment

Sand pack(Length - 300 cmB. diameter - 1 cmT. diameter - 9.1 cm)

P &

T re

cord

ing

Outlet pressuresystem

Top end pressuresystem

Isco pump

Oil

WaterMercury

Transparenttubing

BPR, Accumulator

Isco pump

Sand pack(Length – 27.50 cmDiameter – 7.74 cm)

I. Rate experiment

CH4 gas abovesaturation pressure

II. Pressure depletion experiment

Sand pack(Length - 300 cmB. diameter - 1 cmT. diameter - 9.1 cm)

P &

T re

cord

ing

Outlet pressuresystem

Top end pressuresystem

Isco pump

Oil

WaterMercury

Transparenttubing

BPR, Accumulator

Isco pump

Sand pack(Length – 27.50 cmDiameter – 7.74 cm)

I. Rate experiment

CH4 gas abovesaturation pressure

Test # Initial conditions

R5 R6 R7 R8 Main Visc. (mPa.s) Temperature (°C) W. rate (cm3/min) Others Porosity Oil saturation Mercury saturation Water saturation GOR (std m3/m3) Pressure (kPa ga)

32,380(1)

20 0.4974

0.354 0.512 0.388 0.100 8.170 3,980

32,380

50 0.0303

0.369 0.692 0.283 0.025 7.800 6,448

1,367

20 0.0303

0.375 0.583 0.352 0.066 7.800 3,892

51,615

20 0.0303

0.374 0.580 0.378 0.042 8.030 5,903

(1) Oil viscosities 2,735 mPa.s at 50 oC, 349 mPa.s at 80 oC,

and oil density 0.991 gm/cm3

During the test, the top of the pack was shut-in and the

mercury was withdrawn at the bottom of the pack. The mercury withdrawal continued at a constant rate until the mercury phase became discontinuous and substantial volumes of gas were produced. During this withdrawal period, the top and bottomhole pressures were continuously monitored and recorded. All of the tests were conducted under isothermal conditions at 20 °C except one (Test R6), which was conducted at 50 °C.

The concept behind the mercury withdrawal experiments was to quantify the exsolution of dissolved gas from live heavy oil within a porous medium. The mercury withdrawal increased the pore volume available to the live oil, allowing it to expand entirely within the porous media.

Pressure Depletion Experiments. The pressure depletion tests were conducted in a large scale laboratory model at Alberta Research Council. Alberta heavy oil produced from Lloydminter area with dead oil viscosity of 39,320 mPa.s at 20 °C was used.

The laboratory physical model was a vertical 300 cm long stepped cone which was 1 cm in diameter at the bottom production end and 9.1 cm at the top. The purpose of this large scale irregular physical model was to evaluate solvent based recovery processes. This topic is outside the scope of this paper and will not be discussed here. A schematic diagram of the laboratory experimental setup is shown in Figure 2 (II).

The overall material and method (for example, packing and recording systems) were similar to the rate experiments as discussed in the previous section. The stepped cone was packed with clean reservoir sand of absolute permeability of 3.5 to 4.5 Darcy. During sand packing, the physical model was oriented vertically with the narrow end at the top in order to get a more densely packed sand pack. During flooding with water and oil, the physical model was inverted so that the narrow end was at the bottom. The key parameters for the four selected tests are summarized in Table 4.


Table 4: Test parameters - Pressure depletion experiments

(1) Oil viscosities 343 mPa.s at 75 oC, 48.4 mPa.s at 120 oC,

and oil density 0.991 gm/cm3

The oil and gas production at the bottom end of the sand pack were continuously monitored and recorded. The temperature and pressure were measured at several locations along the sand pack. Numerical Data. The chemical reaction term available with the numerical simulator-CMG STARS (i.e., source/sink term in the governing flow equation, Equation 1) was utilized to represent the formulated gas bubble nucleation and growth processes in the sand pack.

In the present five-component kinetic model, the components are: one water component, three components in the oleic phase (heavy oil, dissolved CH4, and dispersed bubble), and one component in the gaseous phase (connected bubble). The dispersed bubbles in the flow system can be treated as small bubbles (diameters less than pore throat), which are carried with the continuous oil phase. Usually, this component will have the same chemical properties as the dissolved CH4, but will contribute to high oil phase compressibility. In the simulator, the gas like compressibility with an appropriate partial molar density can be used.

Component Properties. The necessary basic properties of the model components (water, oil, dissolved CH4 and free CH4) such as molecular mass Mm, molar density ρ, compressibility cp, critical pressure Pcr, and critical temperature Tcr, were obtained from the literature. Values of the component properties are summarized in Table 5.

Table 5: Component basic properties – Gas exsolution experiment

Component Mm

(kg/gmol) ρ

(g mol/cm3) cp

(kPa-1) Pcr

(kPa) Tcr

(oC)

Water Oil Diss. CH4Disp. Bub. Free CH4

0.201 0.511 0.016 0.016 0.016

6.76x10-2

1.92x10-3

18.72x10-3

4.50x10-4

-

4.0x10-8

1.0x10-6

7.5x10-6

7.5x10-6

gas

22,107 1,158 4,600 4,600 4,600

374

913.46 -82.5 -82.5 -82.5

With the exceptions that are discussed, the properties of the dispersed bubble (pseudo component) were assumed to be the same as dissolved CH4. The molar density of the dispersed bubbles was assigned to be 4.5x10-4 gmole/cm3, while the dissolved CH4 molar density was assigned to be that of liquid CH4 of 18.723x10-3 gmole/cm3 at a temperature of 20 °C and pressure of 101 kPa. A slightly different value of molar density can be used as well. In that case, the numerical history match parameters (i.e., nucleation and growth rates parameters) will be adjusted accordingly. The dissolved CH4 compressibility was obtained by numerically history matching the initial pressure decline of the mercury withdrawal experiments. The best fitted compressibility for dissolved CH4 was found to be 7.50 x 10-6 kPa-1. The other necessary thermal properties such as heat capacity and heat conductivity were obtained from the CMG STARS manual.

Test # Initial conditions P1 P2 P3 P4 Main Oil visc. at 20 oC (mPa.s)(1)

Temperature (°C) Others Porosity Oil saturation Water saturation GOR (std m3/m3) Pressure (kPa ga)

39320 22.9

0.37 0.88 0.12 7.39 3260

39320 19.9

0.36 0.88 0.12 8.30 3425

39320 20.7

0.39 0.88 0.12 8.30 3358

39320 20.7

0.38 0.88 0.12 8.02 3500

Phase Viscosities. The oil phase viscosity was obtained

by a logarithmic mixing, ln(µo) = ∑xi⋅ln(µoi), where, µoi is the measured or estimated viscosity of the liquid phase components (dead oil, water, dissolved gas and dispersed bubble), and xi is the components mole fraction. The gas phase (i.e., connected bubble) viscosity was defined by a correlation, µg = 0.0136+3.8x10-5T, where, T is the temperature in °C. It should be noted that gas phase viscosities usually are much smaller than liquid phase values, and hence will tend to dominate the flow when gas bubble is mobile.

K-Value. The gas/oil equilibrium K-value of dissolved CH4 is the main driving parameter in the present gas exsolution model. At a given temperature and pressure, this parameter can be defined by K = y′/x′, where, x′ and y′ are the dissolved CH4 equilibrium mole fraction in the oil and gas phases, respectively.

The equilibrium K-values were calculated from the laboratory PVT data. The computed K-values as a function of temperature and pressure are given in Figures 3 and 4.

0

100

200

300

0 1000 2000 3000 4000

Pressure (kPa)

K

Lloydminster & Cold Lake Heavy Oil @ 20 0C

(1) - 1,367 mPa.s(2) - 5,274 mPa.s(3) - 32,380 mPa.s(4) - 51,615 mPa.s

(1)

(2)

(3)

(4)

0

100

200

300

0 1000 2000 3000 4000

Pressure (kPa)

K

Lloydminster & Cold Lake Heavy Oil @ 20 0C

(1) - 1,367 mPa.s(2) - 5,274 mPa.s(3) - 32,380 mPa.s(4) - 51,615 mPa.s

(1)

(2)

(3)

(4)

Figure 3. CH4 gas/oil K-value obtained from PVT data: Lloydminster and Cold Lake heavy oils at a temperature of 20 0C.


00 1000 2000 3000

Pressure (kPa)

K

Lloydminster oil (32,380 mPa.s at 200C)

20 0C

50 0C

80 0C

Figure 4. CH4 gas/oil K-value obtained from PVT data: Lloydminster heavy oil at temperatures of 20, 50 and 80 0C. Results and Discussions Rate Experiments. The gas exsolution model was applied to history matches of eight mercury withdrawal experiments, five at different withdrawal rates, and three with different oil viscosities and temperatures. Depending on the withdrawal rates, the sand pack flow system can be categorized into relatively slower and faster flow regimes.

Figures 5 to 8 show the comparisons between the experimental and numerical pressure responses at the bottom of the sand pack. The numerical history matches were organized into slower rate tests in Figures 5, 7 and 8 and faster rate tests in Figure 6. The numerical predictions showed a good agreement with the experimental results.

The general shape of the pressure responses can be characterized as a steep initial draw down, followed by a rebound in pressure, and then by a steady decrease in pressure that slowly approaches the equilibrium behavior. This process can be explained by the volume expansion due to increased bubble nucleation and growth. The steep initial draw down period shows only miniscule amounts of gas being generated by the slow nucleation of tiny bubbles. This behavior also illustrates the dependency of the degree of non-equilibrium on the mercury withdrawal rate. Higher gas suppression can be noticed at higher rates of mercury withdrawal. Due to greater supersaturation at higher withdrawal rates, the volume expansion within the sand pack is much greater than the amount of gas bubble nucleation, thus it takes longer for the pressure to approach equilibrium.

In numerical methodology, first a detailed numerical history match was performed for Test R3 (relatively moderate rate test, average flow velocity of 0.0093 m/day) by tuning the nucleation (N1 and N2) and growth (G1 and G2) parameters. The best-fitted nucleation parameters N1 and N2 were 2.0x10-5 min-1 and 2.0x108 (gmole/cm3)-2min-1, respectively, and the growth parameters G1 and G2 were 0.0 and 1.6x108 (gmole/cm3)-2min-1, respectively. Numerical history matches were then performed for all other tests with minor adjustment of the above nucleation and growth parameters.

Figure 5. Experimental (line) and numerical (marker) pressure responses for three relatively slower rate mercury withdrawal tests Figure 6. Experimental (line) and numerical (marker) pressure responses for two relatively faster rate mercury withdrawal tests.

The numerical history matches in Figures 5, 7 and 8 illustrate that the gas exsolution model calibrated with Test R3 showed good matches for all the relatively slower to moderate withdrawal tests for wide ranges of temperature (20 - 50 0C) and viscosity (1,367 - 51,615 mPa.s) flow regimes. The history match also showed that the growth parameter G1 is not a sensitive parameter for relatively slow to moderate rate experiments. Here, the parameter G1 was assumed to be negligible (i.e., Reaction 3 was not important)

Figure 6 shows the nucleation parameter N1 and the growth parameter G1 increased significantly for relatively faster rate experiments (Tests R4 and R5). The parameter G1 for Tests R4 and R5 were found to be 1.0x10-3 min-1 and 3.0x10-2 min-1, respectively. The role of the growth parameter G1 can be explained by bubble growth due to coalescence from shear or Brownian motion. The value of the parameter G1 should be increased to match the higher withdrawal rate experiments because of higher shear flow. The laboratory micro-model data in Figure 1 demonstrated that a significant number of dispersed bubbles get connected by overcoming the Capillary force when the flow system is viscous dominated.

100

200

300

00 1000 2000 3000

Pressure (kPa)

K

Lloydminster oil (32,380 mPa.s at 200C)

20 0C

50 0C

80 0C

100

200

300

0

1000

2000

3000

4000

5000

0 1,000 2,000 3,000 4,000 5,000

Time (min)

Pre

ssur

e (k

Pa)

N1=2.0x10-5 min-1, N2=2.0x108 (gmole/cm3)-2 min-1

G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

0.0303 cm3/min

0.0077 cm3/min

0.0019 cm3/min

T = 20 0C, µ = 32,380 mPa.s at 20 0C0

1000

2000

3000

4000

5000

0 1,000 2,000 3,000 4,000 5,000

Time (min)

Pre

ssur

e (k

Pa)


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

0.0303 cm3/min

0.0077 cm3/min

0.0019 cm3/min

T = 20 0C, µ = 32,380 mPa.s at 20 0C

0

1000

2000

3000

4000

5000

0 50 100 150 200 250

Time (min)

Pre

ssur

e (k

Pa)

0.4974 cm3/min

0.1200 cm3/min

T = 20 0C, µ= 32,380 mPa.s at 20 0C

N1=1.0x10-3

N2=2.0x108

G1= 3.0x10-2

G2=1.6x108

N1=3.5x10-5

N2=2.0x108

G1= 1.0x10-3

G2=1.6x108

0

1000

2000

3000

4000

5000

0 50 100 150 200 250

Time (min)

Pre

ssur

e (k

Pa)

0.4974 cm3/min

0.1200 cm3/min

T = 20 0C, µ= 32,380 mPa.s at 20 0C

N1=1.0x10-3

N2=2.0x108

G1= 3.0x10-2

G2=1.6x108

N1=3.5x10-5

N2=2.0x108

G1= 1.0x10-3

G2=1.6x108


Figure 7 shows a very similar trend of the pressure curves for the tests at temperatures 20 °C and 50 °C, except that the pressure curve at 50 °C is shifted upward. For the limited temperature measurements studied the kinetic parameters appear approximately constant. Numerical simulation illustrated that temperature had little effect on the parameters, N1, N2 and G2. However, G1 was negligible at lower temperature and increased at higher temperature. Figure 7. Experimental (line) and numerical (marker) pressure responses for experiments conducted at two different temperatures.

Figure 8 shows a good numerical history matches for two different types of heavy oils, with viscosities of 1,367 mPa.s and 51,615 mPa.s at 20 °C. An important trend can be observed that the non-equlibrium pressure rebounds in the pressure curves are closed to each other. It should be noted that no noticeable changes in the overall composition of the test oils were observed. At pressure > 1,000 kPa, the gas/oil equilibrium K-values in Figure 3 are not significantly different. Figure 8. Experimental (line) and numerical (marker) pressure responses for two different oil viscosity tests.

A detailed sensitivity analysis was conducted to demonstrate the role of the gas exsolution parameters on a heavy oil flow system. It was observed that the nucleation

parameter N1 mainly controls the early part of the pressure curves, whereas the effect of parameter N2 can be seen over the entire run. The growth parameters G1 and G2 control the free gas production. In both cases, the system pressure gradually increases with an increase in values of the parameters. The sensitivity results depend upon the input parameters such as compressibility and molar density. Pressure Depletion Experiments. Figures 9 to 12 show numerical predictions of oil and gas production in comparison with the tests results. In the pressure depletion tests, the average pressure drawdown conditions were 33 days at 0.058 kPa.min-1 for Test P1, 25 days at 0.085 kPa.min-1 for Test P2, 33 days at 0.087 kPa.min-1 for Test P3 and 18 days at 0.116 kPa.min-1 for Test P4. The initial pressures ranged from 3,260 kPa in Test P1 and 3,500 kPa in Test P4. In comparison with the rate control experiments in the previous section, all of these four pressure depletion tests can be categorized into relatively slower rates experiments. Here, the gas ex-solution parameters for the slower rate mercury withdrawal tests were chosen. 0

The results showed a good agreement between the numerical prediction and experimental data for all four tests. At the later stages, some deviation can be observed in the numerical history matches, particularly, oil prediction in Figure 9 and gas prediction in Figure 10. This deviation could be related to the deviation of relative permeability end points, or some other flow resistance factors. Figure 9. Cumulative gas and oil production results for Test P1 - Experimental (marker) and numerical (line). Figure 10. Cumulative gas and oil production results for Test P2 - Experimental (marker) and numerical (line).

0

4000

8000

12000

0 10000 20000 30000 40000 50000

Time (min)

Cum

gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300

Cum

oil

(cm

3 )


G1 = 0.0 , G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

0

4000

8000

00

0 10000 20000 30000 40000 50000

Time (min)

Cum

gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300

Cum

oil

(cm

3 )


G1 = 0.0 , G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

120

1000

00

00

00

00

00

00

0 200 400 600 800 1,000

Time (min)

Pre

ssur

e (k

Pa)


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

20 0C

50 0C

q = 0.0303 cm3/min, µ = 32,380 mPa.s at 20 0C

20

30

40

50

60

70

0

00

00

00

00

00

00

00

0 200 400 600 800 1,000

Time (min)

Pre

ssur

e (k

Pa)


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

20 0C

50 0C

q = 0.0303 cm3/min, µ = 32,380 mPa.s at 20 0C10

20

30

40

50

60

70

0

0

0

0

0

0

0

0

0 1,000 2,000 3,000

Time (min)

Pre

ssur

e (k

Pa)


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

1367 mPa.s at 20 0C

51615 mPa.s at 20 0CT = 20 0C, q = 0.0303 cm3/min

0

4000

8000

12000

0 10000 20000 30000 40000 50000

Time (min)

Cum

. gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300

Cum

oil

(cm

3 )


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

0

4000

8000

12000

0 10000 20000 30000 40000 50000

Time (min)

Cum

. gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300C

um o

il (c

m3 )


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

100

200

300

400

500

600

700

0

0

0

0

0

0

0

0

0 1,000 2,000 3,000

Time (min)

Pre

ssur

e (k

Pa)


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

1367 mPa.s at 20 0C

51615 mPa.s at 20 0CT = 20 0C, q = 0.0303 cm3/min

100

200

300

400

500

600

700


Figure 11. Cumulative gas and oil production results for Test P3 - Experimental (marker) and numerical (line). Figure 12. Cumulative gas and oil production results for Test P4 - Experimental (marker) and numerical (line). Conclusions A five component kinetic model has been developed for simulating gas exsolution in a live heavy oil reservoir. The components are water (H2O) in aqueous phase, heavy oil (HC), dissolved gas (gd) and dispersed gas bubbles (bd) in oleic phase and connected gas bubbles (bc) in gaseous phase. The kinetic model contains two types of mass transfer equations: one nucleation equation transfers dissolved gas into dispersed bubbles and one growth equation transfers dispersed bubbles into connected bubbles.

The gas exsolution model was applied to history matches of eight rate experiments and four pressure depletion experiments. By tuning the nucleation and growth parameters, good agreement was obtained between the numerical simulations and experimental results. The numerical history matches for all slower rate experiments, flow rate ≤ 0.0093 m/day, showed no noticeable changes in the model parameters. The gas exsolution model coupled with a

compositional thermal simulator can be applied in field cold production simulation.

0

4000

8000

12000

0 10000 20000 30000 40000 50000

Time (min)

Cum

gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300

Cum

oil

(cm

3 )


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

0

4000

8000

12000

0 10000 20000 30000 40000 50000

Time (min)

Cum

gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300

Cum

oil

(cm

3 )


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

Acknowledgements The author would like to thank Dr. J. Ivory and Dr. D. Coombe for their helpful technical discussions and Dr. D. Lillico, R. Coates and E. Jossy for their useful discussions about the experimental methodology. The financial support provided by the AERI/ARC Core Industry Research Program and the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. Nomenclature Adi = adsorbed component per unit mass of rock ci = component actual concentration in oil c′i = component equilibrium concentration in oil cp = compressibility Dij = total dispersion coefficient Ea = gas bubble activation energy

0

G1 & G2 = gas bubble growth rate parameters ji = phase mole fraction k = absolute permeability kr = relative permeability kB = Boltzmann constant Mm = molecular mass nF = number of neighbouring grid block faces N1 & N2 = gas bubble nucleation rate parameters P = actual pressure Pe = equilibrium pressure qjk = volumetric well phase rate r = gas bubble radius rk = volumetric reaction rate ski = reactant stoichiometric coefficient s′ki = product stoichiometric coefficient Sj = saturation t = time T = temperature Tj = transmissibility between two regions V = total phase volume xi = component actual mole fraction in oil phase x′i = component equilibrium mole fraction in oil phase yi = component actual mole fraction in gas phase y′i = component equilibrium mole fraction in gas phase ∆E = gas bubble nucleation energy ∆Ji = mole fraction difference between nodes ∆Φj = potential difference between adjacent regions ρj = phase mole density σ = gas-oil interfacial tension φv = void porosity φfl = fluid porosity µj = phase viscosity Subscripts: cr = critical F = face i = component

2000

4000

6000

0 10000 20000 30000 40000 50000

Time (min)

Cum

gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300C

um o

il (c

m3 )


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas

00 10000 20000 30000 40000 50000

Time (min)

Cum

gas

(cm

3 ) & P

bh (k

Pa)

0

50

100

150

200

250

300C

um o

il (c

m3 )


G1= 0.0, G2=1.6x108 (gmole/cm3)-2 min-1

Pbh

Oil

Gas2000

4000

6000


j = phase (typically, water, oil and gas) k = reaction References 1. Treinen, RJ, et al.: “Hamaca: solution gas drive recovery in a

heavy oil reservoir, experimental results,” Paper SPE 39031 presented at the 1997 Latin American and Caribbean Petroleum Engineering Conference and Exhibition, Rio de Janeiro, Brazil, Aug. 30 – Sept. 3.

2. Sahni, A. at el.: “Experiments and analysis of heavy oil solution

gas drive,” SPE 71498, presented at the 2001 Annual Technical Conference and Exhibition, New Orleans, Louisiana, Sept. 30 – Oct. 3.

3. Moulu, J.C.: “Solution gas drive: experiments and simulation,”

J. Pet. Sci. Eng. (Oct. 1989) 2, 379. 4. Yortsos, Y.C. and Parlar, M.: “Phase change in binary systems

in porous media: application to solution-gas drive,” Paper SPE 19697 presented at the 1989 Annual Technical Conference and Exhibition of SPE of AIME, San Antonio, TX, Oct. 8-11.

5. Kashchiev, D. and Firoozabadi, A.: “Kinetics of the initial stage

of isothermal gas phase formation,” J. Chem. Phys. (Mar. 1993) 98, 4690.

6. Wall, C.G. and Khurana, A.K.: “The effect of rate pressure

decline and liquid viscosity on low-pressure gas saturations in porous media,” J. Inst. Pet. (Nov. 1972) 58, 335.

7. El-Yousfi, A. at el.: “Mechanisms of solution gas drive

liberation during pressure depletion in porous media,” C.R. Acad. Sci. Paris (1991) 313, Serie II, 1093.

8. Bernath, L.: “Theory of bubble formation in liquids,” Ind. Eng.

Chem. (Jun. 1952) 1310. 9. Danesh, A. at el.: “Pore level visual investigation of oil recovery

by solution gas drive and gas injection,” paper SPE 16956 presented at the 1987 SPE Annual Technical Conference and Exhibition, Dallas, TX, Sept. 27-30.

10. Li, X. and Yortsos, Y.C.: “Visualization and numerical studies

of bubble growth during pressure depletion,” Paper SPE 22589 presented at the 1991 SPE Annual Technical Conference and Exhibition, Dallas, TX, Oct. 6-9.

11. Cole, R.: “Boiling Nucleation,” Adv. Heat Trans. (1974) 10, 85. 12. Kamath, J. and Boyer, R.E.: “Critical gas saturation and

supersaturation in low-permeability rocks,” Paper SPE 26663 presented at the 1993 Annual Technical Conference and Exhibition, Houston, TX, Oct. 3-6

13. Fisch, J.C. at el.: “Nucleation,” J. Appl. Phys. (1948) 19, 775. 14. Claridge, E.L. and Prats, M.: “A proposed model and

mechanism for anomalous foamy heavy oil behavior,” Paper SPE 29243 presented at the 1995 International Heavy Oil Symposium, Calgary, AB, Jun. 19-21.

15. Maini, B.B.: “Foamy-oil Flow,” Paper SPE 68885, J. Cdn. Pet.

Tech. (Oct. 2001) 54

16. Maini, B.B. at el.:, “Significance of foamy-oil behaviour in primary production of heavy oils,” J. Cdn. Pet. Tech. (Nov. 1993) 32, 50.

17. Wieland, D.R. and Kennedy, H.T.: “Measurement of bubble

frequency in cores,” Trans. AIME (1957) 210, 122. 18. Theofanous, T. at el.: “Theoretical study on bubble growth in

constant and time-dependent pressure fields,” Chem. Eng. Sci. (1969) 24, 885.

19. Szekely, J. and Martins, G.P.: “Non-equilibrium effects in the

growth of spherical gas bubbles due to solute diffusion,” Chem. Eng. Sci. (1971) 26, 147.

20. Szekely, J. and Fang, S.D.: “Non-equilibrium effects in the

growth of spherical gas bubbles due to solute diffusion –II: The combined effects of viscosity, liquid inertia, surface tension, and surface kinetics,” Chem. Eng. Sci. (1973) 28, 2127.

21. Lillico, D.A. at el.: “Gas bubble nucleation kinetics in a live

heavy oil,” Colloids and Surfaces, A: Physicochemical and Engineering Aspects (2001) 192, 25.

22. Li, X.: Bubble growth during pressure depletion in porous

media, Ph.D Dissertation, University of Southern California, Los Angeles, CA (May 1993).

23. Dominguez, A. at el.: “Gas cluster growth by solute diffusion in

porous media. Experiments and automaton simulation on pore network,” International Journal of Multiphase Flow (Jan. 2000) 26, 1951.

24. CMG STARS: Advanced Process and Thermal Reservoir

Simulator, Computer Modelling Group Ltd., Office #200, 3512-33 Street N.W. Calgary, Alberta, Canada T2L 2A6, (1999).

25. Uddin, M. et al.: “Modelling of gas exsolution in a live heavy

oil,” AERI/ARC Core Industry Research Program, Report No. 0203-2, Alberta Research Council, Alberta, (Oct. 2002).

26. Uddin, M. et al.: “Parameter estimation of the foamy oil model

for different flow regimes,” AERI/ARC Core Industry Research Program, Report No. 0304-3, Alberta Research Council, Alberta, (Jul. 2003).

27. Tang, G.Q. and Firoozabadi, A.: “Gas and liquid-phase relative

permeabilities for cold production from heavy oil,” SPE 56540, presented at the 1999 SPE Annual Technical Conference and Exhibition, Houston, Oct. 3-6.

28. Kumar, R. at el.: “An Investigation into enhanced recovery

under solution gas drive in heavy oil reservoirs,” SPE 59336 presented at the 2000 SPE Improved Oil Recovery Symposium, Tulsa, OK, Apr. 2-5.

29. Pooladi-Darvish, M. and Firoozabadi, A.: “Solution gas drive in

heavy oil reservoirs,” J. Cdn. Pet. Tech., (Apr. 1999) 38, 54. 30. Delshad, M.: Trapping of micellar fluids in berea sandstone,

Ph.D. dissertation, University of Texas, Austin, (1990). Appendix A: Bubble Nucleation in Heavy Oil Non-equilibrium Pressure. Gas bubble nucleation in live heavy oil occurs when the flow system pressure is reduced


below the equilibrium pressure. The deviation of the system pressure from the equilibrium value can be considered as an excess energy density in the live heavy oil. This non-equilibrium pressure deviation was appeared in constant rate gas exsolution tests in live heavy oil sand pack. The pressure responses versus gas saturation for two constant rate experiments are shown in Figure A-1. Figure A-1. Non-equilibrium pressure responses in the live heavy oil sand pack. Nucleation Energy. Let us assume Pe is the equilibrium pressure and P is the actual pressure in an isothermal flow system at temperature T. The pressure difference, ∆P =Pe – P, can be considered as the excess energy density in the live heavy oil. Since the system is energy dissipative, only a fraction of excess energy, f∆P (f is the fraction coefficient), will be used for gas bubble nucleation and the remaining fraction (1-f)∆P will be consumed by viscous dissipation as the heavy oil expands (Lillico et al.21).

Consider a spherical gas bubble of radius r in an isothermal flow system of excess energy ∆P. In this flow field, the following energy conservation equation holds for a spherical gas bubble:

σππ 23 434 rPfrE +∆−=∆ (A-1)

where, ∆E is the free bubble energy and σ is the gas-oil interfacial tension. Under this condition, the critical bubble size can be determined by the size at which the bubble energy ∆E reaches a maximum. This condition can be satisfied as:

0)(=

∆dr

Ed (A-2)

By combining Equations A-1 and A-2, the critical bubble radius rcr can be obtained as:

Pfrcr ∆

= σ2 (A-3)

By substituting Equation A-3 into Equation A-1, the maximum bubble energy or nucleation energy can be obtained as:

σσπσπ23

24234

⎟⎟⎠

⎞⎜⎜⎝

⎛∆

+∆⎟⎟⎠

⎞⎜⎜⎝

⎛∆

−=∆Pf

PfPf

E (A-4)

2

3

)(316

PfE

∆=∆ πσ (A-5)

1000

1500

2000

2500

3000

0 0.05 0.1 0.15 0.2

Gas Saturation

Pre

ssur

e (k

Pa)

`

Equilibrium

0.0077 cm3/min0.0303 cm3/min

Bubble point

∆P

(P, T)

(Pe, T)

1000

1500

2000

2500

3000

0 0.05 0.1 0.15 0.2

Gas Saturation

Pre

ssur

e (k

Pa)

`

Equilibrium

0.0077 cm3/min0.0303 cm3/min

Bubble point

∆P

(P, T)

(Pe, T)

Nucleation Rate. The nucleation in porous media depends upon several key factors such as, the available excess energy, fluid densities, gas-oil interfacial tension, and gas solubility. Using the well known Maxwell statistical distribution, gas bubble population or bubble nucleation in a live heavy oil flow system can be defined as:

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−∝∂∂

TkE

tN

Bexp (A-6)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆−=

∂∂

2)(exp

P

BAtN (A-7)

Where, A (cm-3min-1) and B (kPa2) are the nucleation parameters, kB is the Boltzmann constant and T is the temperature.

If suppose the dissolved gas equilibrium concentration c′gd is known from PVT data, and then a given pressure supersaturation ∆P can be translated into an equivalent concentration supersaturation, ∆C = (cgd-c′gd).

Now, consider a grid block containing dispersed gas bubbles with concentration cbd at time t. Further, assume concentration supersaturation ∆C is an alternate gas exsolution driving force in the flow system. Then using a simple statistical approach, the dispersed bubbles concentration rate can be defined as:

( ) ( ) bdB

a

B

abd cCTk

EnC

TkE

nt

c 221 )exp()exp( ∆−+∆−=

∂∂

(A-8)

Where, n1 and n2 are the mass transfer coefficients, and Ea is the activation energy. After simplifying Equation A-8, the overall nucleation rate can be written as:

( ) ( ) bdcCNCNtN 2

21 ∆+∆=∂∂ (A-9)

Where, N1 (min-1) and N2 ((gmole/cm3)-2min-1) are the nucleation parameters defined as N1, N2= (n1, n2) exp(-Ea/kBT).

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SPE-97739-MS