Reservoir Formation Damage Episode 2 Part 9 pot

The permeabilities of the plugging and nonplugging pathways are givenby the following empirical relationships Civan, 1994: K = exp[-a,o - exp-aej 14-84 and 14-85Then, the average permeab

The rate of deposition in the nonplugging tubes can be expressed by(Civan, 1994, 1995, 1996):

subject to the initial condition

implemented this possibility in his calculational steps k d and k e are the

surface deposition and mobilization rate constants r\ e is the fraction ofthe uncovered deposits estimated by:

(14-80)

icr is the minimum shear stress necessary to mobilize the surface deposits

i w is the wall shear-stress given by the Rabinowitsch-Mooney equation(Metzner and Reed, 1995):

The permeabilities of the plugging and nonplugging pathways are given

by the following empirical relationships (Civan, 1994):

K = exp[-a(<|>,o - exp(-aej) (14-84)

and

(14-85)Then, the average permeability of the porous medium is given by:

Considering the simultaneous deposition of paraffins and asphaltenes, ep

and e np in Eqs 14-62 through 69 denote the sum of the paraffins andasphaltenes, that is,

= F° ~ + F ^

np, par ' ^np, asp

(14-90)(14-91)

Description of Fluid and Species Transport

The preceding treatment of the porous media impairment phenomenaimplies that the suspended particle and dissolved species concentrationsmay be different in the plugging and nonplugging pathways Then,separate sets of balance equations are required for the plugging and

nonplugging pathways Consequently, the numerical solution wouldrequire highly intensive computational effort However, this problem can

be conveniently circumvented by assuming that there is hydraulic action between these pathways (i.e., they are not isolated from each other)

inter-1 The mass balances are considered for the following four components

pseudo-a Gas

b Oil

c Suspended paraffins and asphaltenes

d Dissolved paraffins and asphaltenes

2 Total thermal equilibrium energy balance is considered

3 Non-Newtonian fluid description using the Rabinowitsch-Mooneyequation is resorted

4 The Forhheimer equation for the non-Darcy flow description is used

5 The average flow is defined as a volume fraction weighted linearsum of the flow through the plugging and nonplugging paths accord-ing to Gruesbeck and Collins (1982)

6 The average permeability is defined as a volume weighted linearsum of the permeabilities of the plugging and nonplugging pathsaccording to Gruesbeck and Collins (1982)

a In the plugging paths, a snowball deposition effect is represented

by an exponential decay function

b In the nonplugging paths, a gradual pore size reduction, sented by the power law function, is considered

repre-7 The precipitation of the asphaltene and paraffin is predicted, ing Chung's (1992) thermodynamic model for non-ideal solutions

apply-to determine the cloud point and the quantity of the precipitates apply-to

— (14-93)

for which Ring et al (1994) assumed w oL = 1.0 Considering that organic

precipitates only exist in the liquid phase, because it is the wetting phasefor these particles, the suspended paraffin and asphaltene particle massbalances are expressed by:

Note that both Ring et al (1994) and Civan (1996) neglected the term

on the right, representing the dispersion of particles

The mass balances of the paraffin and asphaltene dissolved in oil isgiven by:

d/dx(p L u L x iL )

= 9/9jt[(J)SL£>L3/3jt(pLJtlL)] : i = asphaltene or paraffin (14-97)

S is the saturation, p is the density, t is the time, x is distance, u is the volume flux, G p L is the volume fraction of the organic precipitates in

the liquid phase, w p<L denotes the mass fraction, x iL is the mole fraction

of organic dissolved in the oil, M i is the molecular weight and D iL isthe dispersion coefficient 3e,/3f represents the volume rate of retention

of organic deposits in porous media determined according to Eqs 14-68,

69, and 76

Assuming that the various phases are at thermal equilibrium at a

temperature of T v = T L = T s = T , the total porous media energy balance

+ B par k par + £ asp k asp

where U and H are the internal energy and enthalpy, respectively, q is the energy loss, p is pressure, k denotes the thermal conductivity, and T

con-The deposition of organic precipitates in porous media reduces the flowpassages causing the fluids to accelerate Therefore, Darcy's law ismodified as following, considering the inertial effects, according to theForchheimer equation (Civan, 1996):

'3jc : J = VorL (14-100)

where K is the permeability, PJ is the fluid pressure, and the non-Darcy

number is given by:

in which the porous media Reynolds number is given by:

where P is the inertial flow coefficient, and py and |iiy denote the density

and viscosity of a fluid phase J.

Note that the formulations presented here are applicable for

jnulti-dimensional cases encountered in the field if 3/3x is replaced by V • and

a vector-tensor notation is applied

Phase Transition

The source terms appearing on the right of Eqs 14-92 through 99 areconsidered a sum of the internal (rock-fluid and fluid-fluid interactions)and external (wells) sources When the oil is supersaturated, the internalcontribution to the source terms in Eq 14-94 is determined as the excessquantity of organic content of oil above the organic solubility at saturationconditions determined by Chung's (1992) thermodynamic model:

of the predicted and measured permeability impairments by paraffindeposition for below and above bubble point pressure cases

Note that, above the bubble point pressure, only the liquid phase existsand there is more severe formation damage Whereas, below the bubblepoint pressure, both the liquid and vapor phases exist and there is lesssevere formation damage

2 4

Pore Volume Throughput, PV

> Below bubble

point-Suttonand Robertsdata

i Above bubblepoint-Suttonand Robertsdata

— Below bubblepoint-

Simulation

- -Above bubblepoint-

Simulation

Figure 14-51 Comparison of the Sutton and Roberts (1974) experimental

data and simulation results for permeability reduction by organic depositionbelow and above bubble point pressure

Single-Porosity and Two-Phase Model for Organic Deposition

Ring et al (1994) developed a two-phase model considering only theparaffin precipitation They assumed that (1) oil is always saturated withthe paraffin, (2) the solution is ideal, (3) paraffin deposition obeys a firstorder kinetics, (4) pores undergo an irreversible continuous plugging, and(5) permeability reduction obeys a power law:

(14-105)

Ring et al (1994) determined that m = 8 for paraffin deposition.Wang et al (1999) developed an improved model considering thesimultaneous deposition of asphaltenes and paraffins Wang et al (1999)

model incorporates the features of Civan's (1995) dual-porosity model for

a single-porosity treatment The formulation of the Wang et al (1999)model and its experimental verification are described in the following

Formulation*

Wang et al (1999) used the ideal-solution theory to predict the solubilityand precipitation of paraffin and asphaltene in crude oil; an improved one-dimensional, three-phase, and four-pseudo-component model to representthe transport of paraffin and asphaltene precipitates; and a depositionmodel including the static and dynamic pore surface depositions and porethroat plugging to describe the deposition of the paraffin and asphaltene.The model was developed for analysis of the laboratory core flow tests.The gravity effects and the capillary pressure effects between vapor andliquid phases have been neglected The oil, gas, and solid phases wereassumed at thermal equilibrium

The oil, gas, paraffin, and asphaltene pseduo-components are denoted

by O, G, P, and A, respectively The vapor and the liquid phases are denoted by V and L, respectively Considering both the free and dissolved

gases, the gas component mass balance equation is given by:

and S L,pL,uL are the saturation, density and flux of the liquid phase,

respectively W GL represents the mass fraction of the dissolved gas in theliquid phase

Considering that the oil component exists only in the liquid phase, itsmass balance is given by:

(14-108)

where S P is the saturation of the suspended paraffin in the oil phase and

pp is the density of the paraffin W PL represents the mass fraction of

the dissolved paraffin in the liquid phase W SPL is the mass ratio ofthe paraffin precipitates suspended in the liquid phase to the liquid

phase £ p is the volume fraction of the deposited paraffin in the bulkporous media

The asphaltene mass balance equation is written similarly as:

+ PLULWAL) =

-PA

(14-109)

dt

in which S A is the saturation of the suspended asphaltene and p A is the

density of asphaltene W AL represents the mass fraction of the dissolved

asphaltene in the liquid phase W SAL is the mass ratio of the asphaltene

precipitates suspended in the liquid phase to the liquid phase £ A is thevolume fraction of the deposited asphaltene in the bulk porous media.The vapor and liquid phase volumetric fluxes are given by Darcy'sequation, respectively, as:

Kk RV

where K is the absolute permeability of the porous media, and k RV and

|LLV are the relative permeability and viscosity of the vapor phase,

respec-tively k RL and (J,L are the relative permeability and viscosity of the liquidphase, respectively Neglecting the capillary pressure between the vapor

and liquid phases, p represents the pressure of the pore fluids.

The total thermal equilibrium energy balance equation is expressed as:

(pvuvHv + PLULHL) =

(14-112)

where H v,HL,Hp,HAand HF are the enthalpies of the vapor, liquid,

paraffin, asphaltene, and porous media, respectively K V ,K L ,K P ,K A and K F are the thermal conductivities of the vapor, liquid, paraffin,

asphaltene, and porous media, respectively T is the equilibrium

tempera-ture of the system

The saturations of the vapor and liquid phases, and the paraffin andasphaltene precipitates suspended in the liquid phase, add up to 1:

The ideal-solution theory (Weingarten and Euchner, 1988; Chung,1992) is applied for the paraffin and asphaltene solubility predictions,respectively, as:

X AL = exP A / M l 1

where X PL and X AL indicate the mole fractions of the paraffin and

asphaltene dissolved in the oil, respectively, and X ps and X As are themole fractions of the paraffin and asphaltene at saturation A//P and A//Aare the latent heats of fusion of the paraffin and asphaltene, respectively

R is the universal gas constant T pm and T Am are the melting pointtemperatures of the paraffin and asphaltene, respectively However,improved models are available by Lira-Galeana and Firoozabadi (1996),Yarranton and Masliyah (1996), and Zhou et al (1996).

Applying Civan's model (1996), the paraffin and asphaltene depositionrates are given, respectively, by:

The static surface deposition is assumed to occur irrespective of thefluid flow However, the dynamic surface deposition is dominant dur-ing flow The plugging deposition is considered based on the follow-ing criteria:

(14-118)

j jf is the initial value of the plugging deposition rate constant ocy

represents empirically determined constants D p is the mean particles

diameter D pcr is the critical, mean pore throats diameter below whichpore throat plugging occurs, determined by Civan (1996):

(14-120)

where D pt is the mean pore throat diameter and A and B are empirical

constants C p is the suspended particle mass concentration in the oil.The instantaneous porosity is given by:

(14-121)

and the instantaneous permeability is estimated by (Civan et al., 1989):

where ((), and k t are the initial porosity and permeability of the porousmedia, respectively

Model Assisted Analysis of Laboratory Data

Wang et al (1999) solved the model equations using an implicit finitedifference method They determined the best estimates of the parameters

by history matching The data used in six test cases and the best estimates

of the model parameters are presented in Tables 14-7 and 14-8 by Wang

et al (1999)

Case 1

Sutton and Roberts (1974) first heated a Berea sandstone core saturatedwith a Shannon Sand crude oil to 54.4°C and then cooled the outlet ofthe core to 21.1°C for 2 hours without any flow The cloud point of theoil used in their experiment was 37.8°C The paraffin and asphaltenecontents of the crude oil were 4.1 and 0.7 weight percents, respectively.Then, they conducted a flow experiment by injecting the Shannon Sandcrude oil The temperature of the outlet of the core was kept at 21.1°C.Figure 14-52 shows that the simulated results are satisfactory They firstsimulated the static surface deposition during 2 hours of cooling withoutfluid flow Then, they simulated the damage during flow The pore throatplugging did not take place as indicated by the estimated values of rateconstants given in Table 14-7

Table 14-7 Parameters for Cases 1 and 2*

Temperature

rh,°cr-.°c

54.421.1

54.421.1Gas pseudo-component

M^g/gmolePiuc,g/cm3

16.00.00083

16.00.00083Oil pseudo-component

M0, g/gmole Pasc , g/cm3

104.110.72

122.510.75Paraffin pseudo-component

M r , g/gmol

**(%)PAc'S^

r °c'flu'A//p , cal/g-mole

522.44.10.8375.726,000

478.76.10.9871.723,600Asphaltene pseudo-component

M A , g/gmol

^(%)p^g/cm3r^°cA//,, , cal/g-mole

5,000.00.71.1178.744,400

5,000.00.11.1178.744,400Core Properties

30.52.50.250.314Data for Simulation

Ax, cm

At, secNumber of blocks

Ap, atm

2.5440.0120.68

2.5440.0120.68Deposition Parameters©

Ap = AA , I/sec

ap = o^,l/cm rPi=Y Ai ' l/cm af=aA

AB

0.00010.02450.00.00.00.0

0.0001230.01850.00.00.00.0

® Obtained by history matching: After Wang et al., ©1999 SPE; reprinted by permission

of the Society of Petroleum Engineers

CASE 3 4 5 6Oil Properties

T, °C(i (at 20 °C),cpAPI

^(%)

MA , g/gmol

5013295.36,000

5013295.36,000

5.2729.2953®

-6,000

5.2729.2953®

-6,000Core Properties

5.082.30.01220.243

26.52.560.0110.35

26.52.560.0110.35Data for Simulation

Ax, cm

At, secNumber of blocksFlow rate, ml/min

0.42100.012-

0.427200.012-

2.21500.0120.5

2.21500.0123.0Deposition Parameters®

0.00.00.00.00.00.00080.00.00.00.0

0.00.00.00.00.00.00390.00050.0180100

0.00.00.00.00.00.750.00.00.00.0

0.00.00.00.00.00.70.9160020095

CD The asphaltene content of the injected oil obtained bymixing an oil containing 3% (weight) asphaltene withasphaltene at the 60 to 40'volume ratio

© Obtained by history matching

* After Wang et al, ©1999 SPE; reprinted by sion of the Society of Petroleum Engineers

permis-Case 3

Wang et al (1997) simulated Minssieux's (1997) RUN GF 1 experimentswhere the Weyburn oil (Canadian oil) was injected into the Fontainebleausandstone at the reservoir temperature The asphaltene and resin contentsare 5.3% and 8.5% by weight Only the dynamic surface depositionoccurred as indicated by the rate constants given in Table 14-8 Figure14-54 shows satisfactory results

Injection Volume{pv) Figure 14-52 Simulation of the Sutton and Roberts (1974) experimental data

for Case 1 (after Wang et al., ©1999 SPE; reprinted by permission of theSociety of Petroleum Engineers)

ExperimentalData

Injection Volume(pv) Figure 14-53 Simulation of the Sutton and Roberts (1974) experimental data

for Case 2 (after Wang et al., ©1999 SPE; reprinted by permission of theSociety of Petroleum Engineers)