Reservoir Formation Damage Episode 1 Part 10 potx

The volume balance of particles indigeneous and/or external typesof the flowing suspension deposited in porous media is given as thedifference of the deposition by the pore surface and p

results in the pressure equation

k is a particle exchange rate coefficient A solution of Eqs 10-129

through 132 along with the particle deposition rate equations, Eqs

10-101 and 105, yields the particle volume fractions in the plugging andnonplugging flow paths

Model Considering the Clayey Formation Swelling and Indigeneous and External Particles

Civan et al (1989) and Ohen and Civan (1991, 1993) considered theformation damage by clayey formation swelling and migration of externallyinjected and indigeneous particles They assumed constant physicalproperties of the particles and the carrier fluid in the suspension Theyalso considered the effect of fluid acceleration during the narrowing ofthe flow passages by formation damage Ohen and Civan (1993) classifiedthe indigeneous particles that are exposed to solution in the pore space

in two groups: lump of total expansive (swelling, i.e total authigenic claythat is smectitic) and lump of total nonexpansive (nonswelling) particles,because of the difference of their rates of mobilization and sweepage fromthe pore surface They considered that the particles in the flowingsuspension are made of a combination of the indigeneous particles ofporous media entrained by the flowing suspension and the externalparticles introduced to the porous media via the injection of externalfluids They considered that the particles of the flowing suspension can

be redeposited and reentrained during their migration through porousmedia and the rates of mobilization of the redeposited particles shouldobey a different order of magnitude than the indigeneous particles of theporous media Further, they assumed that the deposition of the suspendedparticles over the indigeneous particles of the porous media blocks theindigeneous particles and limits their contact and interaction with theflowing suspension in the pore space They considered that the swellingclays of the porous media can absorb water and swell to reduce theporosity until they are mobilized by the flowing suspension They assumedthat permeability reduction is a result of the porosity reduction by netparticle deposition and formation swelling and by formation plugging bysize exclusion The Ohen and Civan (1993) formulation is applicable fordilute and concentrated suspensions, whereas, Gruesbeck and Collins'(1982) model applies to dilute suspensions

The mass balance equations for the total water (flowing plus absorbed)

in porous media and the total particles (suspended plus deposited) inporous media are given, respectively, by:

a/a1 [(4KJW + e w )Pvv ] + a/a* (owupw) = 0 (10-133)

(10-134)Thus, adding Eqs 10-133 and 134 yields the total mass balance equationfor the water and particles in porous media as:

In Eqs 10-133- through 135, <|) is the instantaneous porosity, p w and

pp are the densities of water and particles, u is the volumetric flux ofthe flowing suspension of particles, ew, ep, and e* represent the volumefraction of porous media containing the absorbed water, particles depositedfrom the flowing suspension, and the indigeneous particles in the porespace, respectively, and ow and Gp denote the volume fractions of thewater and particles, respectively, in the flowing suspension Thus,

(10-136)

According to Eq 10-135 the density of the flowing suspension is given

as a volumetric weighted sum of the densities of the water and ticles by:

par-(10-137)

For simplification purposes, assume constant densities for the water andparticles However, note that the density of suspension is not a constant,because it is variable by the particle and water volume fractions based

on Eq 10-137 Therefore, Eqs 10-134 and 135 can be expressed,respectively, as:

(10-139)

Considering the rapid flow of suspension as the flow passages narrowduring the formation damage, the Forchheimer equation is used as themomentum balance equation:

(10-140)where \j/ is the flow potential defined as:

where \k w is the viscosity of water.

Substituting Eq 10-140 into Eq 10-139 yields the following equationfor the flow potential:

(10-145)

The particle volume fraction and the flow potential can be calculated

by solving Eqs 10-138 and 145 simultaneously, using an appropriatenumerical method such as the finite difference method used by Ohen andCivan (1993), subject to the initial and boundary conditions given by:

t>0

(10-146)(10-147)(10-148)The volumetric rate of water absorption is estimated by (Civan etal., 1989):

The volume balance of particles (indigeneous and/or external types)

of the flowing suspension deposited in porous media is given as thedifference of the deposition by the pore surface and pore throat depositionprocesses and the re-entrainment rates by the colloidal and hydrodynamicprocesses as (Civan, 1996, 1996):

(10-155)

Let single and double primes denote the nonswelling and swellingclays The volume balances of the nonmobilized indigeneous nonswellingand swelling clays remaining in porous media is given in terms of thecolloidal and hydrodynamic mobilization rates, respectively, by:

= -*r'e'p T\' e (c' cr -c)-k (10-156)

(10-164)where <j)0 is the initial porosity.

The instantaneous permeability is estimated by means of the modifiedKozeny-Carman equation (see Chapter 5 for derivation)

where K 0 and § 0 are the initial permeability and porosity and 7 is theflow efficiency factor, which is a measure of the fraction of the porethroats remaining open (see Figure 10-3) Thus, when all the pore throuts

are closed, then y = 0 and K = Q, even if 0 ^ 0

The cumulative volume of fluid injected at x = 0, expressed in terms

of the initial undamaged pore volume, is given by

(10-166)

u 0 is the injection volumetric flux

The cumulative fines production at x =• L, in the effluent is

QpL

t

UL and G pL are the effluent volumetric flux and particle fraction, respectively

The harmonic mean average permeability of the core of length L is

x = ln(r/rw) (10-169)

r and rw denote the radial distance and the well bore radius, respectively

Model Assisted Analysis of Experimental Data

Without the theoretical analysis and understanding, laboratory work can

be premature, because the analyst may not exactly know what to lookfor and what to measure The theoretical analysis of various processesinvolved in formation damage provide a scientific guidance in designingthe experimental tests and helps in selecting a proper, meaningful set ofvariables that should be measured Having studied the various issuesinvolving formation damage by fines migration, we are prepared to

conduct laboratory experiments in a manner to extract useful information.Here, the analysis of experimental data by means of the mathematicalmodels developed in this chapter is illustrated by several examples takenfrom the literature.

Applications of the Wojtanowicz et al Model

In general, formation damage may be a result of a number of mechanismsacting together with different relative contributions But the Wojtanowicz

et al (1987, 1988) analysis of experimental data is based on the tion that one of the potential formation damage mechanisms is dominantunder certain conditions Therefore, by testing the various diagnosticequations given in Table 10-1 derived by Wojtanowicz et al for possiblemechanisms involving the laboratory core damage, the particular govern-ing damage mechanism can be identified They have demonstrated thatportions of typical laboratory data can be represented by different equa-tions, indicating that different mechanisms are responsible for damage.For example, as indicated by Figures 10-7 and 10-8, the initial and later

assump O T S S * 4 0 5 ma//

(SLOPE =-4.38x10

TSS=976 mq/4

(SLOPE'-5.69xlO~ 4 "min INTERCEPT = 9.83) TSS*l990mg/£

(SLOPE 30.5x10 INTERCEPT =1.01)

50 TIME ( Min) Figure 10-7 Diagnostic chart for gradual pore blockage by external particles

invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission ofthe Society of Petroleum Engineers and after Wojtanowicz et al., ©1988,reprinted by permission of the ASME)

Figure 10-8 Diagnostic chart for gradual pore blockage by external particles

invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission ofthe Society of Petroleum Engineers and after Wojtanowicz et al., ©1988,reprinted by permission of the ASME)

portions of the experimental data for damage by foreign particles invasionwith low particle concentration drilling muds (0.2%, 0.5%, and 1.0% byweight) can be represented by Eqs Tl-1 and 2, successfully, revealingthat the pore surface deposition and pore throat plugging mechanisms aredominant during the early and late times, respectively Figure 10-9 showsthat Eq Tl-3 provides an accurate straight-line representation of the coredamage with injection of suspensions of high concentration drilling muds(2% and 3% by weight) of foreign particles, revealing that the dominantformation damage mechanism should be the pore filling and internal cakeformation The data plotted in Figure 10-10 shows that the sizes andconcentrations of the particles of the injected suspension significantlyaffect the durations and extent of the initial formation damage by poresurface deposition (Eq Tl-1) and later formation damage by pore throatplugging (Eq Tl-2) mechanisms

Figure 10-11 shows that the damage of the core by a particle-freecalcium chloride-based completion fluid is due to the plugging of porethroats by particles mobilized by brine incompatibility, because Eq Tl-7can represent the data satisfactorily by a straight-line Figures 10-12 and

(text continued on page 218)

35 30

20

10

A TSS=3843mg/.«

(SLOPE= 0.389 '/min INTERCEPT = -5.36)

a TSS - 5790 mg/0

(SLOPE = 0.1396'/min INTERCEPT*-3.09)

50 TIME ( M i n )

100

Figure 10-9 Diagnostic chart for cake forming by external particles invasion

(after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society

of Petroleum Engineers and after Wojtanowicz et al., ©1988, reprinted bypermission of the ASME)

Figure 10-10 Diagnostic chart for transition from gradual pore blockage to

single pore blockage during external particles invasion (after Wojtanowicz et al.,

©1987 SPE; reprinted by permission of the Society of Petroleum Engineers andafter Wojtanowicz et al., ©1988, reprinted by permission of the ASME)

10 15 20 TIME (Min)

25 30

Figure 10-11 Diagnostic chart for cake forming by calcium chloride-based

completion fluid invasion (after Wojtanowicz et al., ©1987 SPE; reprinted bypermission of the Society of Petroleum Engineers and after Wojtanowicz etal., ©1988, reprinted by permission of the ASME)

Figure 10-12 Diagnostic chart for gradual pore blocking by ammonium

nitrate/alcohol-based completion fluid invasion (after Wojtanowicz et al.,

©1987 SPE; reprinted by permission of the Society of Petroleum Engineersand after Wojtanowicz et al., ©1988, reprinted by permission of the ASME)

(text continued from page 215)

10-13 show that the damage of the core by particle-free ammoniumnitrate/alcohol-based completion fluids are due to pore surface depositionand pore surface sweeping, because the data can be satisfactorily fitted

by straight-lines according to Eqs Tl-6 and 5, respectively Figure 10-14showing the plot of data for the combined effects of pore surfacedeposition and sweeping according to Eq Tl-4, indicates the effect ofthe flow rate on damage As can be seen, the rate of formation damageincreases by the flow rate Wojtanowicz et al (1987) explain this increasedue to about a fivefold increase in the value of the release coefficient,

ke, as a result of about a threefold increase in the flow rate from 3 to

10 ml/min

The best estimates of the intercept and slope values obtained by theleast-squares regression analysis for the cases analyzed are presented byWojtanowicz et al (1987) in Figures 10-7 through 10-14 Using theseparameters in the relevant equations representing these cases, the relative

retained permeability curves vs t are plotted in Figure 10-15 for

com-parison As can be seen, pore filling by cake formation causes the mostsevere damage

Figure 10-13 Diagnostic chart for gradual pore sweeping by ammonium

nitrate/alcohol-based completion fluid invasion (after Wojtanowicz et al.,

©1987 SPE; reprinted by permission of the Society of Petroleum Engineersand after Wojtanowicz et al., ©1988, reprinted by permission of the ASME)

Figure 10-14 Diagnostic chart for combined effects of gradual pore blocking

and sweeping (after Wojtanowicz et al., ©1987 SPE; reprinted by permission

of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988,reprinted by permission of the ASME)

Ki = 670 rodRV.= I8.3cc

q - 10 cc/min

GRADUAL BLOCKING

SCREENING

0 5 10 15 20 25 3O 35 40 45 50 CUMULATIVE INJECTION (Pore Volume P.V.)

Figure 10-15 Comparison of the effects of gradual blocking, screening, and

straining on permeability reduction (after Wojtanowicz et al., ©1987 SPE;reprinted by permission of the Society of Petroleum Engineers and afterWojtanowicz et al., ©1988, reprinted by permission of the ASME)

Applications of the Ceriiansky and Siroky Model

The objectives of the experimental studies were threefold:

1 Determine an empirical relationship between permeability andporosity in the form of Eq 10-92

2 Determine the values of the deposition and entrainment rate

con-stants, k p and k' e

3 Study the effects of the length of porous media and the rate and centration of the particle suspension injected into the porous media.The pressure difference across the porous media and the particleconcentration of the effluent were measured as functions of time duringthe injection of a suspension of finely ground limestone particles at agiven concentration and rate

con-The porous material was prepared by using nonwoven felt of filaments

of polypropylene The porous material samples of 4 cm diameter and 0.5,1.0, 1.5, and 2.0 cm lengths were used The particle suspension wasprepared using finely ground limestone of 2,825 kg/m3 density in water.The porosity was determined by the weighting method The discretetimes at which measurements are taken are denoted by the subscript

i = 2,3, ,N and the initial time is denoted by / = 1 The permeability

was determined by Darcy's equation by neglecting the effect of gravityfor short samples:

The E and G parameter values were determined as a function of the

length of porous media by nonlinear regression of Eq 10-92 to the data

given in Figure 10-16 Plot of E and G vs the length are given in Figure 10-17 Exponential regressions of these data indicate that E = 0.14 and

G = 5.3 as the length approaches zero, although the data are of low

Figure 10-16 Correlation of the deposit fraction and permeability for material

POP 1, using a 5 mm thick porous material and c 0 = 0.1 kg/m3 (Cerhansky, A.,

& Siroky, R., 1985; reprinted by permission of the AlChE, ©1985 AlChE, all rightsreserved, and after Cerhansky and Siroky, 1982, reprinted by permission)

BEavg G/Gavg Linear (E/Eavg) Linear (G/Gavg)

0 5 10 15 20 25

Length, L (mm)

Figure 10-17 Correlation of the Cernansky and Siroky (1985) data for

variation of the E and G parameters by the length of porous media formaterial POP 1 = 0.179andGavg = 5.66

quality, as indicated by the coefficients of regressions R 2 = 0.78 and

R 2 = 0.18, respectively.

The mean diameter of particles is 20.2jam and the estimated dimension

of pores 50jom The porosity is (j>0=0.31 The concentration of the

injected suspension is c in =0.lkg/m 3 or <5 in = (0.1&g/ra3)/(2,825£g/m3)

= 3.54xlO-5m3/m

To determine the deposition and entrainment rate constants, in Eq 10-91

i' cr - 0 was substituted and the derivative with respect to time was

evaluated numerically using the central and backward finite differenceapproximations given below, respectively, for the interior and the finalpoints (see Chapter 16 for derivation):

(10-175)