Reservoir Formation Damage Episode 1 Part 7 potx

132 Reservoir Formation Damage7-21Several other relationships, which may be convenient to use in the for-mulation of the transport phenomena in porous media, are given inthe following: T

132 Reservoir Formation Damage

(7-21)Several other relationships, which may be convenient to use in the for-mulation of the transport phenomena in porous media, are given inthe following:

The volume flux , My, and the velocity , Vy, of a phase j are related by:

(7-22)

where e jr is the volume fraction of the irreducible phase j in porous media When an irreducible residual fluid saturation, S jr exists in porousmedia, Eq 7-22 should be substituted into Eq 7-15 for the flowing phasevolume flux as:

(7-23)

In deforming porous media, the volumetric flux of the solid phase can

be expressed in terms of the velocity according to the following equation:

«,=6,v, (7-24)where e5 and v s denote the solid phase volume fraction and velocity,respectively

Substituting Eq 7-14, Eq 7-24 becomes:

",=(l-4>)v, (7-25)

Accounting for the immobile fluid fraction, e jr , in deforming porous

media, the volumetric flux of the fluid relative to the deforming solidphase is given by Civan (1994, 1996):

(7-26)

The volume fraction of species i of phase j in the bulk system is

given by:

(7-27)

Multi-Phase and Multi-Species Transport in Porous Media 133

«</=<Vry (7-31)

where u r - is the volume flux of phase j.

The mass flux of species i in phase j is given by:

Multi-Species and Multi-Phase Macroscopic Transport Equations

The macroscopic description of transport in porous media is obtained

by elemental volume averaging (Slattery, 1972) The formulations of themacroscopic equations of conservations in porous media have been carriedout by many researchers A detailed review of these efforts is presented

by Whitaker (1999) The mass balances of various phases are given by(Civan, 1996, 1998):

, p,) + V - ( 9 j u j ) = V• (e, Dj • Vpy)+ (7-33)

where u rj is the fluid flux relative to the solid phase, t is the time and

V • is the divergence operator p; is the phase density, m • is the net mass

rate of the phase j added per unit volume of phase j Dj is the hydraulic

dispersion coefficient which has been omitted in the petroleum ing literature

engineer-134 Reservoir Formation Damage

The species i mass balance equations for the water, oil, gas and solid

phases are given by:

where D i} is the coefficient of dispersion of species i in the j th phase, k

is the Boltzmann constant, and T is temperature The first term represents

the ordinary dispersive transport by concentration gradient For particulatespecies of relatively large sizes the first term may be neglected Thesecond term represents the dispersion induced by the gradient of thepotential interaction energy, <E>(y When the particles are subjected touniform interaction potential field then the second term drops out Thethird term represents the induced dispersion of bacterial species bysubstrate or nutrient, 5, concentration gradient due to the chemotaxis

phenomena (Chang et al., 1992) D sj is the substrate dispersion coefficient.Incorporating Eq 7-33 into Eq 7-34 leads to the following alternativeform:

Multi-Phase and Multi-Species Transport in Porous Media 135

purposes The Forchheimer equation for dimensional and

multi-phase fluids flow can be written for the j th phase as (Civan, 1994; Tutu

et al., 1983; Schulenberg and Miiller, 1987):

fluid with the pore surface, g is the gravitational acceleration, g(z-z 0 )

is the potential of fluid due to gravity, z is the positive upward distance measured from a reference at z 0 , and Q is the overburden potential, which

is the work of a vertical displacement due to the addition of fluid intoporous media (Smiles and Kirby, 1993)

K and |3 denote the Darcy or laminar permeability and the non-Darcy

or inertial flow coefficient tensors, respectively K rj and pr; are therelative permeability and relative inertial flow coefficient, respectively

Eq 7-38 can be written as, for convenience

in which v is the kinematic viscosity (or momentum diffusivity) given by

and N nd is the non-Darcy number for anisotropic porous media given by,

neglecting the interfacial drag force Fji

where / denotes a unit tensor and Re^ is the tensor Reynolds number

for flow of phase j in an anisotropic porous media given by

(7-43)

136 Reservoir Formation Damage

The permeability and inertial flow coefficient for porous materials aredetermined by means of laboratory core flow data and thus correlatedempirically (Civan and Evans, 1998) Liu et al (1995) give:

8.91xl08T

(7-45)

where (3 is in ft \ k is in mD, and 0 is in fraction.

The energy balance equations for the water, oil, gas, and solid phasesare given by:

<x=l

(7-46)

q j and q ja denote the external and interface heat transfer to the phase

j per unit volume of phase y; &y is the thermal conductivity of phase j, Note that the enthalpy Hj and internal energy Uj per unit mass of phase

j are related according to:

When the system is at thermal equilibrium (i.e T w = T 0 = T g = T s = T}

then Eq 7-48 can be written for each phase and then added to obtainthe total energy balance equation as:

Multi-Phase and Multi-Species Transport in Porous Media 137

(7-50)

The equation of motion given by Chase and Willis (1992), for deformingporous matrix can be written as following:

(7-51)

where T5 is the shear stress tensor for the solid matrix

The jump mass balance equations, given by Slattery (1972) can besimplified to express the boundary conditions as:

(7-52)

(7-53)

(7-54)

The superscript a denotes a quantity associated with the dividing surface,

which is moving at a macroscopic velocity of w°, and n a is the unit

vector normal to the dividing surface r°, rf and r? are the rates of addition of mass of the porous matrix, the th phase, and the species i

138 Reservoir Formation Damage

in the j th phase, respectively [I I] denotes a jump in a quantity across

a dividing surface defined by:

where the signs + and - indicate the post and fore sides, respectively, ofthe dividing surface

Exercises

1 Show that the balance of species / in phase j can also be expressed

in the following forms:

Blick, E F., & Civan, F., "Porous Media Momentum Equation for Highly

Accelerated Flow," SPE Reservoir Engineering, Vol 3, No 3, 1988,

Multi-Phase and Multi-Species Transport in Porous Media 139

Civan, F., "Waterflooding of Naturally Fractured Reservoirs—An EfficientSimulation Approach," SPE Production Operations Sympsoium, March21-23, 1993, Oklahoma City, Oklahoma, pp 395-407

Civan, F Predictability of Formation Damage: An Assessment Study andGeneralized Models, Final Report, U.S DOE Contract No DE-AC22-90-BC14658, April 1994

Civan, F "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter CakeFormation Model," SPE Paper No 28709, Proceedings of the SPEInternational Petroleum Conference & Exhibition of Mexico, October10-13, 1994, Veracruz, Mexico, pp 399-412

Civan, F "A Multi-Purpose Formation Damage Model," SPE 31101 paper,Proceedings of the SPE Formation Damage Symposium, Lafayette, LA,February 14-15, 1996, pp 311-326

Civan, F., "Quadrature Solution for Waterflooding of Naturally Fractured

Reservoirs," SPE Reservoir Evaluation and Engineering, April 1998,

pp 141-147

Civan, F, & Evans, R D., "Determining the Parameters of the Forchheimer Equation from Pressure-Squared vs Pseudopressure Formulations," SPE Reservoir Evaluation and Engineering, February 1998, pp 43-46 Forchheimer, P., "Wasserbewegung durch Boden," Zeitz ver Deutsch Ing.

Vol 45, 1901, pp 1782-1788

Liu, X., Civan, F., & Evans, R D "Correlation of the Non-Darcy Flow

Coefficient, J of Canadian Petroleum Technology, Vol 34, No 10,

Schulenberg, T., & Miiller, U., "An Improved Model for Two-Phase Flow

Through Beds of Coarse Particles," Int J Multiphase Flow, Vol 13,

No 1, 1987, pp 87-97

Slattery, J C Momentum, Energy and Mass Transfer in Continua,

McGraw-Hill Book Co., New York, 1972, pp 191-197

Smiles, D E., & Kirby, J M., "Compressive Cake Filtration—A

Com-ment," Chem Eng ScL, Vol 48, No 19, 1993, pp 3431-3434.

Tutu, N K., Ginsberg, T., & Chen, J C., "Interfacial Drag for Two-Phase

Flow Through High Permeability Porous Beds," Interfacial Transport Phenomena, Chen, J C & Bankoff, S G., (eds.), ASME, New York,

pp 37-44

Whitaker, S., The Method of Volume Averaging, Kluwer Academic Publishers,

Boston, 1999, 219 p

2 Mobilization of in-situ formation particles due to the incompatibility

of the fluids injected into porous media and by various rock-fluidinteractions, and

3 Production of particulates by chemical reactions, and inorganic andorganic precipitation

Fluids injected into petroleum reservoirs usually contain iron colloidsproduced by oxidation and corrosion of surface equipment, pumps, steelcasing, and drill string (Wojtanowicz et al., 1987) Brine injected forwaterflooding may contain some fine sand and clay particles Mud finescan invade the formation during overbalanced drilling These are someexamples of the externally introduced particles

Petroleum bearing formation usually contains various types of clay andother mineral species attached to the pore surface These species can be

140

Paniculate Processes in Porous Media 141

released by colloidal forces or mobilized by hydrodynamic shear of thefluid flowing through porous media Fine particles can also be generated

by deformation of rock during compression and dilatation This is due

to variation of the net overburden stress and loss of the integrity of rockgrains Fine particles are unleashed and liberated because of the integrityloss of rock grains by chemical dissolution of the cementing materials

in porous rock, such as by acidizing or caustic flooding These are thetypical internal sources of indigenous fine particles

Paniculate matter can be produced by various chemical reactions such

as the salt formation reactions that occur when the seawater injected forwaterflooding mixes with the reservoir brine, and formation of elementalsulfur during corrosion Paniculate matter can also be produced byprecipitation due to the change of the thermodynamic conditions and ofthe composition of the fluids by dissolution or liberation of light gases(Amaefule et al., 1988) These are typical" mechanisms of particle pro-duction in porous media

Once entrained by the fluids flowing through porous media, the variousparticles migrate by four primary mechanisms (Wojtanowicz et al., 1987):

As the fine particles move along the tortuous flow pathways existing

in porous media, they are captured, retained, and deposited within theporous matrix Consequently, the texture of the matrix is adversely altered

to reduce its porosity and permeability Frequently, this phenomena isreferred to as formation damage measured as the permeability impairment

Particulate Processes

The various particulate processes, schematically depicted in Figure 8-1,can be classified in two groups as the internal and external processes

142 Reservoir Formation Damage

Hydrodynamlc mobilization

Colloidal expulsion

Liberation of particles by cement dissolution

Surface deposition

Pore throat plugging

Internal cake formation by small particles Internal and External cake formation by small particles External cake formation by large particles

Figure 8-1 Various particulate processes.

Paniculate Processes in Porous Media 143

The external processes occur over the formation face and are discussed

in Chapter 12 The internal processes occur in the porous media and can

be classified in three groups as (Civan, 1990, 1994, 1996):

1 Pore Surface Processes

a Deposition

b Removal

2 Pore Throat Processes

a Plugging (screening, bridging, sealing, Figure 8-2)

b Unplugging

3 Pore Volume Processes

a In-Situ Cake Formation

b In-Situ Cake Depletion

c Migration

d Generation and Consumption (chemical reactions, rock mation and crushing, liberation of fine particles by chemicaldissolution of cement, coagulation/disintegration)

defor-e Interphase Transport or Exchange

(c)

Figure 8-2 Mechanism of pore throat blocking: a) plugging and sealing,b) flow restriction, c) bridging (after Civan, 1994; reprinted by permission ofthe U.S Department of Energy)

144 Reservoir Formation Damage

The net amount of particles deposited in porous media is expressed by:Instantaneous amount of particles in porous matrix = initial amount ofparticles in porous matrix + net amount of particles deposited on poresurface + net amount of particles deposited behind pore throats

The various particulate processes are depicted in Figure 8-3

The fundamental particle generation mechanisms are:

1 Hydrodynamic mobilization

2 Colloidal expulsion

3 Liberation of particles due to the loss of integrity of rock grains

by chemical dissolution of cement or by rock compression, crushing,and deformation

4 Chemical and physico-chemical formation

Unswollen Particles

in the Porous Matrix

Swollen Particles Extending from Pore Surface

Deposited Particles

Particles

Suspension

T

DEPOSITION FLOWING PHASE

Plague ENTRAPMENT

&

Figure 8-3 Particulate processes in porous media (after Civan, 1994;

reprinted by permission of the U.S Department of Energy, modified after

Civan et al 1989, from Journal of Petroleum Science and Engineering, Vol.

3, "Alteration of Permeability by Fine Particle Processes," pp 65-79, ©1989,with permission from Elsevier Science)

Paniculate Processes in Porous Media 145

The fundamental particle retention mechanisms are:

1 Surface deposition (physico-chemical)

2 Pore throat blocking (physical jamming)

3 Pore filling and internal filter cake formation (physical)

4 Screening and external filter cake formation (physical)

Forces Acting Upon Particles

Ives (1985) classified the various forces acting on particles in a ing suspension in three categories as (a) forces related to the transportmechanisms, (b) forces related to the attachment mechanisms, and(c) forces related to the detachment mechanisms, and characterized them

flow-in terms of the relevant dimensionless groups

Forces Related to the Transport Mechanisms

The important relevant quantities governing the particle behavior in a

suspension can be summarized as following: d and D are particle and

porous media grain diameters, respectively; p5 is the density of particles;

p and \JL are the density and viscosity of the carrier liquid, respectively;

va is the convective velocity; g is the gravitational acceleration ficient; and T is the absolute temperature.

coef-Inertia Force The inertia of a particle forces it to maintain motion in

a straight line The inertia force can be expressed by the dimensionlessgroup as (Ives, 1985):

(8-1)

Gravity Force As a result of the density difference between the particle

and the carrier liquid, particles tend to move in the gravity directionaccording to Stokes' law The velocity of a spherical particle undergoing

a Stokes' motion is given by:

(8-2)

The gravity force acts upward when particles are lighter and, therefore,buoyant The gravity force acts downward when particles are heavierand, therefore, tend to settle The gravity force can be expressed by a