Reservoir Formation Damage Episode 1 Part 5 potx

The Flow Efficiency Concept Rajani 1988 concluded that permeability function can be separatedinto and expressed as a product of a function incorporating the poregeometry and a function o

n (5-8)

(5-9)

A" is the intrinsic permeability of porous media The cross-sectional area

of porous media open for flow can be expressed by:

Permeability Relationships 83

Bourbie et al (1986) determined that n = 1 for ()><0.05 and n = 3 for 0.08

<(j)<0.25 In view of this evidence and Eq 5-14, the Carman-Kozenyequation appears to be valid for the 0.08 < <)) < 0.25 fractional porosityrange Reis and Acock (1994) warn that these exponents may be low

"because the permeabilities were not corrected for the Klinkenberg effect."

The Modified Carman-Kozeny Equation

Incorporating the Flow Units Concept

The derivation of the Carman-Kozeny equation presented in the ceding section inherently assumed uniform diameter cyclindrical flowtubes analogy Therefore, for applications to nonuniform diameter flowtubes, the Carman-Kozeny equation has been modified by inserting a

pre-geometric shape factor, F s (Amaefule et al., 1993), as:

(5-16)

Hearn et al (1984, 1986) introduced the "flow units" concept and Amaefule

et al (1993) defined a lumped parameter as following, called the "flow

zone indicator" to combine the three unknown parameters, F s , i and Zg,into one unknown parameter:

(5-17)

Therefore, a plot of experimental data based on the logarithmic form of

Eq 5-16 (Amaefule et al., 1993)

should yield a straightline with a slope of two Hence, the FZI2 value

can be obtained as the value of K/§ at the (|) = 0.5 value.

Implicit in Eq 5-18 is the assumption that formations with similar flowcharacteristics can be represented by the same characteristic flow zoneindicator parameter values Consequently, formations having distinct flowzone parameters can be identified as different flow units

The Modified Carman-Kozeny Equation

for Porous Media Altered by Deposition

Based on the Carman-Kozeny model, Eq 5-14, Adin's (1978) relation of experimental data leads to a permeability-porosity model as:

where oc and n are some empirical parameters Arshad's equation accounts

for the formation of the dead-end pores during deposition, which do notconduct fluids

The Flow Efficiency Concept

Rajani (1988) concluded that permeability function can be separatedinto and expressed as a product of a function incorporating the poregeometry and a function of porosity as:

(5-23)

sig-This problem can be alleviated by introducing a flow efficiency factor,

y, in view of Eq 5-19 (Ohen and Civan, 1993; Chang and Civan, 1991,

1992, 1997) Hence, the permeability variation can be expressed by(Chang and Civan, 1997):

K

(5-24)

where a, b, and c are some empirically determined parameters and K 0 and § 0 denote the permeability and porosity at some initial or referencestate The flow efficiency factor, y, can be interpreted as a measure ofthe fraction of the open pore throats allowing fluid flow Thus, when the

pore throats are plugged, then y = 0, and therefore K = 0, even if <|) * 0.

This phenomenon is referred to as the "gate or valve effect" of the porethroats (Chang and Civan, 1997; Ochi and Vernoux, 1998)

In order to estimate the flow efficiency factor, Ohen and Civan (1993)assumed that, although the pore throat sizes vary with time, they alwaysremain log-normally distributed:

in the range of dl <y<d h , where s d is the standard deviation and dt is

the mean pore throat diameter

Then, assuming that the pore throats smaller than the size, dp , of the

suspended particles will be plugged, the flow efficiency factor is estimated

by the fraction of pores remaining open at a given time:

"I l d \

where Ep is the plugging efficiency factor Particles that are sticky and

deformable can mold into the shape of pore throats and seal them Then,

the plugging is highly efficient and E p is close to unity Particles that arerigid and nonsticky cannot seal the pore throats effectively and still allow

for some fluid flow Thus, E < 1 for such plugs.

The lower and upper bounds of the pore throat size range are estimated

by a simultaneous solution of the non-linear integral equations given by:

where k6 is a rate constant and ep is the volume of deposition per unit bulk

volume, subject to the initial mean pore throat diameter, either determinedfrom the initial pore throat size distribution using Eq 5-28, or estimated as

a fraction of the mean pore diameter using:

(5-30)

Note that r\ is not a fraction because it is a lumped coefficient including

the mentioned fraction, some unit conversion factors, and the shape factor.Chang and Civan (1991, 1992, 1997) considered that the pore throatand particle diameters can be better represented by bimodal distributionfunctions over finite diameter ranges, given by Popplewell et al (1989) as:

(5-31)

where w is an adjustable weighting factor in the range of 0 < w < 1, and./i(y) and/2(j) denote the distribution functions for the fine and coarsefractions, each of which are described by:

(5-32)

Permeability Relationships 87

with different values of the parameters a, ra, dt , and d h Chang and Civan

(1991, 1992, 1997) used the critical particle diameter, \dp ] , necessary

for pore throat jamming, determined according to the criteria described

in Chapter 8

For applications with multiphase flow systems, Liu and Civan (1993,

1994, 1995, 1996) used a simplified empirical equation for permeabilityreduction in porous media as:

(5-33)

where K 0 and <|>0 are the reference permeability and porosity, K f, is theresidual permeability of plugged formation, and / is a flow efficiencyfactor given by:

(5-34)

where i and / denote the species and phases, kf are some rate constants

and eu are the quantity of the pore throat deposits The instantaneousporosity is given by:

(5-35)

where (e, / ) is amount of surface deposits

The Plugging-Nonplugging Parallel Pathways Model

The porous media realization is based on the plugging and nonpluggingpathways concept according to Gruesbeck and Collins (1982) Relativelysmooth and large diameter flowpaths mainly involve surface depositionand are considered nonplugging Flowpaths that are highly tortuous andhaving significant variations in diameter are considered plugging In theplugging pathways, retainment of deposits is assumed to occur by jam-ming and blocking of pore throats when several particles approach narrowflow constrictions Deposits that are sticky and deformable usually sealthe flow constrictions (Civan, 1990, 1994, 1996) Therefore, conductivity

of a flow path may diminish without filling the pore space completely.Fluid seeks alternative flow paths until all the flow paths are eliminated

Then the permeability diminishes even though the porosity may benonzero Another important issue is the criteria for jamming of porethroats As demonstrated by Gruesbeck and Collins (1982) experimentallyfor perforations, the probability of jamming of flow constrictions dependsstrongly on the particle concentration of the flowing suspension and theflow constriction-to-particle diameter ratio.

The pore plugging mechanisms are analyzed considering an tesimally small width slice of the porous core The total cross-sectional

infini-area, A, of the porous slide can be separated into two parts: (1) the area

A p , containing pluggable paths in which plug-type deposition and pore

filling occurs, and (2) the area, Anp , containing nonplugging paths in

which nonplugging surface deposition occurs Thus, the total area ofporous media facing the flow is given by:

As explained in Chapter 8, the pore size distribution of the porousmedium and the size distribution of the particles determine its value.However, its value varies because the nonplugging pathways undergo atransition to become plugging during formation damage

The volumetric flow rate, q, can also be expressed as a sum of the flow rates, qp and qnp , through the pluggable and nonpluggable paths as:

Thus, by means of Eqs 5-36 through 43 the total superficial flow isexpressed as (Gruesbeck and Collins, 1982):

The total deposit volume fraction and the instantaneous available porosityare given by:

The permeabilities of the plugging and nonplugging pathways are given

by the following empirical relationships by Civan (1994) by generalizingthe expressions given by Gruesbeck and Collins (1982):

and

(5-59)

Permeability Relationships 91

where n{ and n2 are the permeability reduction indices, a is a coefficient and K and Knpg are the permeabilities at the reference porosities §PO and §np of the plugging and nonplugging pathways, respectively Eq.

5-58 represents the snow-ball effect of plugging on permeability, while

Eq 5-59 expresses the power-law effect of surface deposition on ability Eqs 5-58 and 5-59 have been also verified by Gdanski andShuchart (1998) and Bhat and Kovscek (1999), respectively, using experi-mental data Bhat and Kovscek (1999) have shown that the power-lawexponent in Eq 5-59 can be correlated as a function of the coordinationnumber and the pore body to throat aspect ratio, applying the statisticalnetwork theory for silica deposition in silicaous diatomite formation Note

perme-that, for n2 < 0 and £np /$ npg «1, Eq 5-59 simplifies to the expression

given by Gruesbeck and Collins (1982):

(5-64)expG -1

In view of the Gruesbeck and Collins (1982) plugging and nonplugging

pathways approach, Civan (2000) concluded that the E coefficient can

be analogous to the fraction of the nonplugging pathways and Eqs 5-63and 64 can be attributed to the nonplugging and plugging pathways inporous media, respectively

Multi-Parameter Regression Models

Efforts for development of empirical correlations and theoreticalmodels for prediction of the permeability of porous media are beingpursued by many investigators because the applications of the theoreticalmodels in the formation damage prediction have had limited success.Because of their inherent simplifications, these models are not able torepresent the complicated nature of the relationship of the permeability

to the petrographical, petrophysical, and mineralogical parameters ofgeological porous materials Empirical models, such as by Nolen et al.(1992) have been shown to incorporate such parameters to accuratelypredict permeability However, the mathematical form of such modelsvaried in the literature Extending the Nolen et al approach, Civan (1996)proposed two general empirical correlations:

(5-65)

(5-66)

in which xi :i = l,2, ,m represent the various petrographical,

petro-physical, and mineralogical variables, and b and a,-:/ = l,2, ,m are

empirically determined parameters

Network Models

Network models facilitate representations of porous media by scribed networks of nodes (pore bodies) connected with bonds (porethroats) Network models have been used by many researchers, including

pre-Permeability Relationships 93

Sharma and Yortsos (1987), Rege and Fogler (1987, 1988), and Bhat and

Kovscek (1999) Although, network models may serve as useful research

tools, their implementation in routine simulations of formation damage

problems may be cumbersome and computationally demanding Therefore,

they are not included in this chapter

Modified Fair-Hatch Equation

Liu et al (1997) formulated the texture, porosity, and permeability

relationship for scale formation Here, their approach is presented in a

manner consistent with the formulation given in this section By definition

of fractional volumes <|), <|>s, and § r occupied in the bulk volume,

respectively, by the pore space, deposited scales, and the non-reacting

rock grains, we can write

(5-67)

If the mineral grains forming the scales and the rock are assumed of

spherical shapes, the /t h grain volume can be approximated by:

Consider that there are a total of n, of the zt h grains and the number of

different mineral grains is N m Therefore, Liu et al (1997) express Eq

5-67 as:

(5-69)

and use a modified form of the Fair-Hatch equation (Bear, 1972, p 134)

to relate the texture, porosity, and permeability as:

Power-Law Flow-Units Equation

Civan (1996, 2000) expressed the mean-pore diameter as a parameter power-law function of the pore volume to solid volume ratio:

in which a, P and y are empirical parameters, and usually a = 1 P and

y depend on the pore connectivity and can be correlated as a function ofthe coordination number, Z, respectively, by:

p-1/p-1.0=l-exp(-CZ + D)

= i - exp(-AZ + fl)

(5-72)(5-73)The interconnectivity parameter can also be approximated by a power lawfunction of porosity as (Civan, 1996):

y = ctyn (5-74)

in which c and n are empirical parameters, y is zero when the pores are

blocked by deposition Civan (2000) verified the validity of Eqs 5-71through 74 using the data by Rajani (1988), Verlaan et al (1999), andBhat and Kovscek (1999)

Effect of Dissolution/Precipitation on

Porosity and Permeability

Civan (2000) expressed the precipitation/dissolution rate by:

where, t is time, k } is a rate constant, (|)0 is the initial porosity, e is

the volume fraction of deposits in porous media, F is the solution

Permeability Relationships 95

saturation ratio, and n is a process rate exponent The porosity, (|), andthe volume fraction of the deposits in the pore volume, a, are related,respectively, by:

§o =<)> + £ (5-77)

Once the porosity is calculated, the permeability can be determined by

Eq 5-71 Civan (2000) verified this approach using the Koh et al (1996)data for permeability impairment by silica deposition

Effect of Deposition/Dissolution and

Stress on Porosity and Permeability

Civan (2000) modified the equations of Adin (1978), Arshad (1991),Tien et al (1997), and Civan (1998) as:

(5-79)

in which, the subscripts o and °° indicate the initial and terminal porosity

and permeability values, oc(:« = 1,2, ,8 are empirical parameters, and

p eff denotes the effective overburden stress given according to Nieto et

al (1994) and Bustin (1997) as:

in which a is Biot's constant, and p ob and p pf are the overburden stressand pore fluid pressure, respectively Civan (2000) verified these equationsusing the data by Nieto et al (1994) and Bustin (1997)

Effect of Temperature on Porosity and Permeability

Gupta and Civan (1994) and then Civan (2000) formulated the tion of porosity and permeability by temperature Representing thevolumetric thermal expansion coefficient of the porous media grains bythe linear equation

where K is the permeability, p is the fluid pressure, and the

non-Darcy number is given by:

empir-Permeability Relationships 97

8.91xl08T

(5-88)

v is the kinematic viscosity of the fluid Determine the expression

of the average permeability to be used instead of Eq 5-49, which

considers Nnd = 1 for Darcy flow.

Gdanski and Shuchart (1998) have correlated their permeability vs.porosity measurements obtained during sandstone-acidizing by:

By comparison of Eqs 5-89 and 5-90, show that the parameter

values are Kpg = 0.01 md, (^ = 0, a = 15 and n { = 0.29 (Civan, 2000).

3 Show that Eq 5-60 can be derived by a truncated series

approxi-mation of Eq 5-59 for n2 < 0 and £np /$ npo «l (Civan, 1994).

References

Adin, A., "Prediction of Granular Water Filter Performance for Optimum

Design," Filtration and Separation, Vol 15, No 1, 1978, p 55-60.

Adler, P M., Jacquin, C G., & Quiblier, J A., "Flow in Simulated Porous

Media," Int J Multiphase Flow, Vol 16, No 4, 1990, pp 691-712.

Amaefule, J O., Altunbay, M., Tiab, D., Kersey, D G., & Keelan, D.K., "Enhanced Reservoir Description: Using Core and Log Data toIdentify Hydraulic (Flow) Units and Predict Permeability in UncoredIntervals/Wells," SPE 26436, Proceedings of the 68th Annual TechnicalConference and Exhibition of the SPE held in Houston, TX, October3-6, 1993, pp 205-220

Arshad, S A., "A Study of Surfactant Precipitation in Porous Media withApplications in Surfactant-Assisted Enhanced Oil Recovery Processes,"Ph.D Dissertation, University of Oklahoma, 1991, 285 p

Bear, J., Dynamics of Fluids in Porous Media, American Elsevier Publ.

Co., Inc., New York, New York, 1972, 764 p

Bhat, S K., & Kovscek, A R., "Statistical Network Theory of Silica

Deposition and Dissolution in Diatomite," In-Situ, Vol 23, No 1, 1999,

pp 21-53