Reservoir Formation Damage Episode 1 Part 9 docx

These models are applicable for special casesinvolving single-phase fluid systems in laboratory core tests.Porous media is considered in two parts: 1 the flowing phase, denoted by the su

Single-Phase Formation Damage

by Fines Migration and Clay Swelling*

Summary

A review of the primary considerations and formulations of the varioussingle-phase models for formation damage by fines migration and clayswelling effects is presented The applicability and parameters of thesemodels are discussed

Introduction

The majority of the formation damage models were developed forsingle phase fluid systems This assumption is valid only for very specificcases such as the production of particles with oil flow and for specialcore tests Nevertheless, it is instructive to understand these models beforelooking into the multi-phase effects Therefore, the various processesinvolving single-phase formation damage are discussed and the selectedmodels available are presented along with some modifications and criticalevaluation as to their practical applicability and limitations The method-ology for determination of the model parameters are presented Theparameters that can be measured directly are identified The rest ofthe parameters are determined by means of a history matching technique.The applications of the models and the parameter estimation method aredemonstrated using several examples

Parts reprinted by permission of the Society of Petroleum Engineers from Civan, ©1992SPE, SPE 23787 paper, and by permission of the U.S Department of Energy fromCivan, 1994

183

An evaluation and comparison of six selected models bearing directrelevance to formation damage prediction for petroleum reservoirs arecarried out The modeling approaches and assumptions are identified,interpreted, and compared These models are applicable for special casesinvolving single-phase fluid systems in laboratory core tests.

Porous media is considered in two parts: (1) the flowing phase, denoted

by the subscript /, consists of a suspension of fine particles flowing

through and (2) the stationary phase, denoted by the subscript s, consists

of the porous matrix and the particles retrained

The Thin Slice Algebraic Model Model Formulation

Wojtanowicz et al (1987, 1988) considered a thin slice of a porousmaterial and analyzed the various formation damage mechanisms assum-ing one distinct mechanism dominates at a certain condition Porous

medium is visualized as having tortuous pathways represented by N h tubes

of the same mean hydraulic equivalent diameter, Dh, located between the

inlet and outlet ports of the core as depicted in Figure 10-1 The sectional area of the core is A and the length is L The tortuosity factorfor the tubes is defined as the ratio of the actual tube length to the length

number of nonplugged tubes at any given time is denoted by N np and the

plugged tubes by N p , then the total number of tubes is given by:

Figure 10-1 Hydraulic tubes realization of flow paths in a core (after Civan, 1994; reprinted by permission of the U.S.

Department of Energy; after Civan 1992 SPE, reprinted by permission of the Society of Petroleum Engineers)

O

to d.

n

3 00

00

The Darcy and Hagen-Poiseuille equations given respectively by

(10-5)and

(10-6)are considered as two alternative forms of the porous media momentum

equations, q is the flowrate of the flowing phase and Ap is the pressure

differential across the thin core slice Thus, equating Eqs 10-5 and10-6 and using Eqs 10-1 and 10-2 the relationship between permeability,

K, and open flow area, A is obtained as:

Gradual pore reduction is assumed to occur by deposition of particlessmaller than pore throats on the pore surface to reduce the cross-sectionalarea, A, of the flow tubes gradually as depicted in Figure 10-2 Thus,

the number of tubes open for flow, Nnp, at any time remains the same as the total number of tubes, Nh, available Hence,

Then, using Eq 10-9 and eliminating A between Eqs 10-4 and 10-7leads to the following equation for the permeability to open flow arearelationship during the surface deposition of particles:

Surface deposits

Figure 10-2 Pore surface deposition in a core (after Civan, 1994; reprinted

by permission of the U.S Department of Energy; after Civan 1992 SPE;reprinted by permission of the Society of Petroleum Engineers)

somewhere along the tube, by a single particle to stop the flow throughthat particular tube Therefore, the cross-sectional areas of the individual

tubes, A h , do not change But, the number of tubes, N np , open for the flow

is reduced as depicted in Figure 10-3 The area of the tubes eliminatedfrom service is given by:

A p =N p A h (10-12)The number of tubes plugged is estimated by the ratio of the total volume

of pore throat blocking particles to the volume of a single particle of thecritical size

flowing suspension of particles Because A h is a constant, Eq 10-7 leads

to the following permeability to open flow area relationship:

Plugged tube

Nonplugged tube

^ ^ S~A

Figure 10-3 Pore throat plugging in a core (after Civan, 1994; reprinted by

permission of the U.S Department of Energy; after Civan 1992 SPE; reprinted

by permission of the Society of Petroleum Engineers)

of high concentration of particles in sizes larger than the size of the porethroats is injected into the core as depicted in Figure 10-4 The per-

meability, K c , of the particle invaded region decreases by accumulation

of particles But, in the uninvaded core region near the outlet, the

permeability of the matrix, K m , remains unchanged The harmonic mean permeability, K, of a core section (neglecting the cake at the inlet face) can be expressed in terms of the permeability, K c , of the L c long pore

filling region and the permeability, K m , of the L m long uninvaded region as

(10-16)

(10-17)which can be written as:

K(t) = L/[L c R c (t)

Figure 10-4 Pore filling and internal filter cake formation in a core (after Civan,

1994; reprinted by permission of the U.S Department of Energy; after Civan 1992SPE; reprinted by permission of the Society of Petroleum Engineers)

R c (t) and R m are the resistances of the pore filling and uninvaded regionsdefined by

The rate of increase of the filtration resistance of the pore filling particles

is assumed proportional to the particle mass flux of the flowing phaseaccording to:

dR c /dt = (k c /L c )(q/A)p pf

subject to the initial condition

Then, solving Eqs 10-20 and 21 yields:

The instantaneous porosity of a given cross-sectional area is given by:

ty = ty0-£p (10-23)

(|)0 and (() denote the initial and instantaneous porosity values, e is thefractional bulk volume of porous media occupied by the depositedparticles, given by

ep=mp/pp (10-24)

m p is the mass of particles retained per unit volume of porous media and

pp is the particle grain density For convenience, these quantities can be expressed in terms of initial and instantaneous open flow areas, Afo and Ay> and the area covered by the particle deposits, A , as

and (Ppf) are the particle mass concentrations in the flowing phase

at themlet and outlet of the core Wojtanowicz et al (1987, 1988) omittedthe accumulation of particles in the thin core slice and simplified Eq 10-30

to express the concentration of particles leaving a thin section by:

The rate of particle retention on the pore surface is assumed proportional

to the particle mass concentrations in the flowing phase according to:

r r =(dm p /dt) r =k r p pf (10-33)The rate of entrainment of the surface deposited particles by the flowingphase is assumed proportional to the mass of particles available on thepore surface according to:

Diagnostic Equations for Typical Cases

Wojtanowicz et al (1987, 1988) have analyzed and developed diagnosticequations for two special cases of practical importance:

1 Deposition of the externally introduced particles during the injection

of a suspension of particles

2 Mobilization and subsequent deposition of the indigeneous particles

of porous medium during the injection of a particle free solution

Deposition of Externally Introduced Particles

Three distinct permeability damage mechanisms are analyzed for agiven injection fluid rate and particle concentration As depicted inFigure 10-4, particles are retained mainly in the thin core section nearthe inlet face In this region the concentration of the flowing phase isassumed the same as the injected fluid (i.e., pp/ - ( P p f ) ).

Gradual pore reduction by surface deposition occurs when the particles

of the injected suspension are smaller than the pore constrictions Assumethat the surface deposition is the dominant mechanism compared to the

entrainment, that is, kr»ke.

Then, the solution of Eqs 10-35 and 36 yields:

Single pore blocking occurs when the size of the particles in the

injected fluid are comparable or bigger than the size of the pore strictions A substitution of Eqs 10-12, 13, and 14 into Eq 10-28 yieldsthe following diagnostic equation:

con-K/K0=l-C6t

in which the empirical constant is given by:

(10-41)

(10-42)

Cake formation near the inlet face of the porous media occurs when

the particles in the injected solutions are large relative to the pore sizeand at high a concentration Combining Eqs 10-22 and 17 yields thefollowing diagnostic equation:

(10-43)

in which

(10-44)

Mobilization and Subsequent Deposition of Indigeneous Particles

This case deals with the injection of a clear (particle free) solution into

a porous media A core is visualized as having two sections designated

as the inlet and outlet sides The particles of the porous media entrained

by the flowing phase in the inlet part are recaptured and deposited at theoutlet side of the core

Near the inlet port, the mobilization and entrainment of particles bythe flowing phase is assumed to be the dominant mechanism compared

to the particles retention (i.e., ke » kr} Thus, dropping the particle

reten-tion term, Eqs 10-35 and 36 yield the following solureten-tion for the mass

of particles remaining on the pore surface

Substituting Eq 10-45 and (pp/) = 0 , Eq 10-32 yields the followingexpression for the particle concentration of the flowing phase passingfrom the inlet to the outlet side of the core as

Depending on the particle concentration and size of the flowing phaseentering the core, the outlet side diagnostic equations for three permeabilitydamage mechanisms mentioned previously are derived next

Gradual Pore Reduction by Surface Deposition and Sweeping

Assume that the mass of the indigeneous or previously depositedparticles on the pore surface is m* Then, the area occupied by theseparticles is given by Eq 10-29 as

(10-47)and the area open for flow is given by Eq 10-28 as

(10-48)

Afg denotes the open flow area when all the deposits are removed.

If simultaneous, gradual pore surface deposition and sweeping areoccurring near the outlet region, then both the entrainment and retentionterms are considered equally important Thus, substituting Eq 10-46,

Eq 10-35 yields the following ordinary differential equation:

dm p /dt + k e m p = [k r k e ALm pg /q]exp(-k e t) (10-49)

The solution of Eq 10-49, subject to the initial condition mp=m*p

(previously deposited particles), is obtained by the integration factormethod as:

Eliminating Afo between Eqs 10-48 and 53, substituting Eqs 10-47 and

52 for A* and Ap, and then applying Eq 10-10 for Af and A*f yields

the following diagnostic equation:

Normally, mpg =m*p Wojtanowicz et al (1987) simplified Eq 10-54 by

substituting C7 = 0 when the mass of the particles initially available onthe pore surface is small compared to the mass of particles deposited later(i.e., m J s O )

If only pore sweeping occurs, then k r « k e Thus, substitute k r = 0

in Eq 10-56 to obtain C8 = 0 and Eq 10-54 becomes:

(K/K*f2 = 1 + C7[l - exp(-^r)] (10-57)

If only gradual surface deposition is taking place in the outlet region,

then k r » k e Therefore, dropping the particle retention term and stituting Eq 10-46, Eqs 10-35 and 36 for m pg = 0 are solved to obtain

sub-the amount of particles retained as:

Single Pore Blocking

If the permeability damage is solely due to single pore blocking, thensubstituting Eqs 10-46, 12, 13 and 14 into Eq 10-28 yields the followingdiagnostic equation:

A list of the diagnostic equations derived in this section are summarized

in Table 10-1 for convenience

Table 10-1 Diagnostic Equations for Typical Permeability Damage Mechanisms*

blocking KIK0=1-L1

\l-e~ket) Pore K /K -I C

filling 01 10

(l-e et)

Straight LinePlotting Scheme Eq #

vs. t ( t/ , \ ~\ 1 ) T\ 1 ln\\ + \\K K o \ — ll/C7> 11— /

The Compartments-in-Series Ordinary

Differential Model

Khilar and Fogler (1987) divided a core into n-compartments asdepicted in Figure 10-5 The contents of these compartments are assumedwell-mixed Therefore, the composition of the flow stream leaving thecompartments should be the same as the contents of the compartments.However, because particles having sizes comparable or larger than thepore throats are trapped within the porous media, the particle con-centration of the stream leaving a compartment will be a fraction, y, ofthe concentration of the fluid in the compartment, y is called the particletransport efficiency factor

Pore surfaces are considered as the source of in-situ mobilized particlesand the pore throats are assumed the locations of particle capture Aparticle mass balance over a thin slice yields

Figure 10-5 Continuously stirred compartments in series realization in a core

(after Civan, 1994; reprinted by permission of the U.S Department of Energy;after Civan 1992 SPE; reprinted by permission of the Society of PetroleumEngineers)

Substituting Eqs 10-66 and 67, and rearranging Eq 10-65 becomes:

m p = 0, t = 0

subject to

(10-72)The mass balance of indigeneous particles remaining on pore surfaces is

dm*p/dt = -re (10-73)

subject to

where mpo is the mass of particles initially available on pore surface.

The rate of particle entrainment by the flowing phase is assumed bothcolloidally and hydrodynamically induced

r = o c me c ' l p+oc,T h |T — T V c ) s Ifl (10-75)

T is shear stress a s is the specific pore surface area <*£, is the colloidallyinduced release coefficient given by (Khilar and Fogler, 1983):

(10-76)

P, is the capture coefficient.

Let ppfc be the critical particle concentration above which bridging at

pore throats occur and particles cannot travel between pore bodies If the

particle concentration is below p pfc , then no trapping at pore throats takesplace Therefore,

7 = 1, P ^ O f o r p ^ p ^ (10-81)

7 = 0, Pf* O f o rP / 7 /> pp / c (10-82)

The correlation between entrapment and permeability reduction is based

on the Hagen-Poiseuille flow assumption of flow through the pore throat

K/K0=[l-Bmp/m*po]2 (10-83) where B is a characteristic constant and K0 is the initial permeability.

Simplified Partial Differential Model

Cernansky and Siroky (1985) considered injection of a low particleconcentration suspension at a constant rate into porous media made of abed of filaments Neglecting the diffusion of particles and the contribution

of the small amount of particles in the flowing suspension, they expressedthe total mass balance of particles similar to Gruesbeck and Collins'(1982) simplified mass balance equation Thus, for incompressible liquid