SPE 122241 PA (Formation Damage and Well Productivity Simulation)

September 2010 SPE Journal 751

Formation-Damage and Well-Productivity Simulation

Arild Lohne, SPE, and Liqun Han,* SPE, International Research Institute of Stavanger; Claas van der Zwaag, SPE, Statoil; Hans van Velzen, Nederlandse Aardolie Maatschappij (NAM);

Anne-Mette Mathisen, SPE, Statoil; Allan Twynam, SPE, BP; Wim Hendriks, SPE, Shell; Roman Bulgachev, SPE, BP; and Dimitrios G. Hatzignatiou, SPE, International Research Institute of Stavanger

*Formerly at the International Research Institute of Stavanger (IRIS), now with Halliburton.Copyright 2010 Society of Petroleum Engineers

This paper (SPE 122241) was accepted for presentation at the 8th European Formation Damage Conference, Scheveningen, The Netherlands, 2729 May 2009, and revised for publication. Original manuscript received for review 3 April 2009. Revised manuscript received for review 11 December 2009. Paper peer approved 23 December 2009.

SummaryIn this paper, we describe a simulation model for computing the damage imposed on the formation during overbalanced drilling. The main parts modeled are filter-cake buildup under both static and dynamic conditions; fluid loss to the formation; transport of solids and polymers inside the formation, including effects of pore-lining retention and pore-throat plugging; and salinity effects on fines stability and clay swelling. The developed model can handle multicomponent water-based-mud systems at both the core scale (linear model) and the field scale (2D radial model). Among the computed results are fluid loss vs. time, internal damage distribu-tion, and productivity calculations for both the entire well and individual sections.

The simulation model works, in part, independently of fluid-loss experiments (e.g., the model does not use fluid-leakoff coef-ficients but instead computes the filter-cake buildup and its flow resistance from properties ascribed to the individual components in the mud). Some of these properties can be measured directly, such as particle-size distribution of solids, effect of polymers on fluid viscosity, and formation permeability and porosity. Other properties, which must be determined by tuning the results of the numerical model against fluid-loss experiments, are still assumed to be rather case independent, and, once determined, they can be used in simulations at altered conditions as well as with different mud formulations. A detailed description of the filter-cake model is given in this paper.

We present simulations of several static and dynamic fluid-loss experiments. The particle-transport model is used to simulate a dilute particle-injection experiment taken from the literature. Finally, we demonstrate the models applicability at the field scale and present computational results from an actual well drilled in the North Sea. These results are analyzed, and it is concluded that the potential effects of the mechanistic modeling approach used are (a) increased understanding of damage mechanisms, (b) improved design of experiments used in the selection process, and (c) bet-ter predictions at the well scale. This allows for a more-efficient and more-realistic prescreening of drilling fluids than traditional core-plug testing.

IntroductionA simulation tool, referred to as Maximize, has been developed for the purpose of investigating fluid loss to the formation during overbalanced drilling and the impairment imposed on the forma-tion by the invading fluid. The objective of the program is to serve as a tool for supporting well planners decisions related to the choice of well fluids, and integrating, analyzing, and interpreting laboratory and field formation-damage data.

The filtration properties of the mudcake, forming at the wellbore surface, have been investigated by several authors over the years;

see, for example, Ferguson and Klotz (1954), Outmans (1963), Bezemer and Havenaar (1966), Arthur and Peden (1988), Fordham et al. (1991), and Dewan and Chenevert (2001). The common understanding is that, once the filter cake has been formed, it will control the filtration rate independently of the formation properties, except at very low permeability where the flow resistance offered by the formation is comparable to the filter-cake resistance.

The properties of the filter cake depend only on its composition, the pressure drop over the cake p, and the shear stress acting on the cake surface by the circulating mud. Under dynamic conditions, the filtration rate will approach asymptotically a limiting steady-state rate, which depends only on the shear stress at the cakes surface.

Semmelbeck et al. (1995) combined a filter-cake-building-and-filtration model with a fluid-flow simulator for computing the fluid-invasion profile along the well. Further improvement of the filter-cake model was presented by Dewan and Chenevert (2001), who demonstrated the derived models capability to repro-duce complex laboratory experiments with sequential changes in dynamic shear rate and overbalance pressure. Others also have investigated filtrate invasion by numerical simulations [e.g., Ding et al. (2002), Wu et al. (2004), and Suryanarayana et al. (2007)].

Knowledge of the filtrate profile along the wellbore is important for well log interpretations and for evaluating the damage imposed on the formation by the invading fluid. A number of potentially damaging mechanisms and modeling of these are described in detail by Civan (2000). For water-based muds, one of the main damage mechanisms is particulate plugging of the formation, either by externally introduced particles or by in-situ fines mobi-lized by the invading fluid. The mechanisms of particle transport in porous medium have been investigated by Gruesbeck and Collins (1982), Sharma and Yortsos (1986), Wennberg and Sharma (1997), and Al-Abduwani et al. (2005).

Traditionally, the computation of fluid loss has relied on using experimental filtration rates or, as in the more-sophisticated mod-els, experimental cake permeability and porosity obtained for a specific mud. In our approach, we bring this one step forward by computing the cake properties from its composition. The proper-ties from which the cakes porosity and permeability are computed are ascribed to the individual components. In Ding et al. (2002) and Suryanarayana et al. (2007), the damage in invaded zones is simulated on the basis of experimental return permeability and an assumed damage profile. We make no such assumptions but model the flow and retention of both particles and polymers inside the formation and compute the permeability reduction from the amount of trapped material.

Models that are used should be consistent with the physics involved at a macroscopic level. Input parameters should ideally be of three kinds: (a) properties of involved components (e.g., viscosity, density, size of particles), (b) properties of the formation (e.g., permeability, porosity), and (c) model parameters describing events that are independent of Parameters a and b. The benefit of this approach is that the amount of empirical input parameters is reduced and the ones used are more universal and show less vari-ance among different experiments. Finally, the proposed approach will increase the use of earlier experience because, ideally, all experiments should be matched by a single data set, provided adequate descriptions of the experiments are available.

752 September 2010 SPE Journal

Simulation ModelThis section describes the various modules that constitute the newly developed simulation model. The two main parts of the Maximize program are (1) the filter-cake model handling filter-cake buildup and controlling flow into the formation and (2) the reservoir-flow model handling flow inside the formation. The fluids introduced to the formation contain a number of dissolved and dispersed com-ponents, which, in turn, may change the original flow properties of the formation through various chemical and physical processes. The retention of solids and polymer is split into a pore-throat-trap-ping model and a pore-lining-adsorption model. Brine interaction with the rock surface and clays is described by a multicomponent cation-exchange model, which provides input to processes such as fines migration and clay swelling.

Flow Model. The ow model handles two-phase ow in 1D linear (typically used for cores), 2D rectangular, and 2D radial models (appropriate for the eld scale). We use two boundaries (wells) constrained by constant rate or constant pressure. The ow equation is solved using the implicit-pressure/explicit-saturation (IMPES) method (implicit in pressure and explicit in saturations with respect to time); see Aziz and Settari (1979) for further details on reservoir simulators.

The phase flow between grid cells is computed from the pressure solution. Knowing the flow, local concentrations of oil, water, and dissolved or suspended species are updated for the new timestep. Chemical and physical processes are decoupled from the flow equation. Components are first moved between cells, and then changes over the timestep caused by physical and chemical processes are computed locally in each grid cell.

Rock properties, such as absolute permeability k, porosity , relative permeabilities, and clay content are allowed to vary between layers (sections). Moderate compressibility of both the formation and the fluid components is handled by the model. Corey-type relative permeabilities are used.

k k S SS SS S

j o wrj rje jnEj jn j jrwr or

= =

=, , ,

1. . . . . . . . . . . . . . . . . (1)

As stated earlier, this is a multicomponent simulator. Properties of components, such as viscosity or component effect on viscosity, are described in the following sections.

In simulations with mud filtration, the rate will change very rapidly during the first period when the filter cake is built up from zero thickness, and an IMPES-type solution will be unstable. We solve this by precomputing the growth of the cake during the timestep, assuming the pressure difference between the well (mud pressure) and the connection block to be constant. The estimated cake properties are used in the computation of the new pressure solution for the total grid, and, finally, the rates from this solution are used to update the real growth of the filter cake.

Filter Cake. The lter-cake model presented here computes cake properties from properties of the individual components present in the mud. The model handles an arbitrary number of polymers and solids.

Static Filtration. The relation between cake thickness hc and filtrate volume Vf is obtained from mass balance (Bourgoyne et al. 1986):

hV C

AC c

c cc

f

c

sm

sc sm

= =

11, , . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where Ac is the filter-cake area and csm and csc are the concentrations of solids in the mud and in the filter cake, respectively. The flow rate through the cake expressed by Darcys law is

q pA kh

c c

c

=

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Setting the rate q = dVf /dt, inserting hc from Eq. 2, and integrat-ing the resulting expression yields the known-filtrate-volume vs. square-root-of-time relation (a spurt-loss volume Vsp is also added to the resulting expression):

V V A C t C pkCf sp c st st

c= + =,

21

. . . . . . . . . . . . . . . . . . . . (4)

There are three unknowns in Eq. 4: the spurt loss Vsp, the cake permeability kc, and the cake porosity c (csc = 1c is part of C1). Two of these variables, kc and c, depend on the mud composition and the pressure drop over the cake, while the spurt loss depends on the mud composition and properties of the rock. The cakes porosity and permeability will decrease with increasing pressure drop because of the compressibility of the cake and thus will reduce the rate increase.

The main effect of temperature is through its influence on the filtrate viscosity and thereby on the filtration parameter Cst. Dif-ferent polymers used in mud will have different upper temperature limits above which they become chemically unstable. Below this limit, only a weak temperature dependency is assumed.

Cake Permeability. The permeability of a porous medium can be estimated with the Carman-Kozeny equation (Lake 1989). Because mud is a multicomponent system containing different kinds of solids of very different sizes, we choose the following variant:

kS

SS c

c

i i

i

=

( ) =

3

20

2 00

2 1, . . . . . . . . . . . . . . . . . . (5)

The tortuosity parameter = (Lt/L)2 accounts for the effective flow-path length Lt, ci is volumetric concentration of species i, and S0 is the specific surface area of the medium defined as the surface-to-volume ratio. In particular for spheres, S0 = 6/Dp, where Dp is the particle diameter. The specific surface area in a mixed system is readily obtained by volume-weighted averaging. The operator used for S0 indicates that the property represents a volume-weighted average.

Another useful expression that relates an effective pore diam-eter to permeability and porosity is

D k

=

32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

The Carman-Kozeny approach treats the porous medium as a bundle of capillary tubes, and D corresponds to the tube diameter when all tubes have the same diameter.

Cake Compressibility. The compression of the cake is a result of the local pressure difference between the cake matrix and the fluid filling the pore space. The cake-matrix pressure is assumed constant and equal to the mud pressure pm. The fluid pressure decreases through the cake as a result of the viscous flow. Because this pressure difference will vary throughout the cake, physical cake properties such as porosity and permeability, which depend on the overburden pressure, will be functions of spatial position. We are interested in some effective average properties, which we will obtain by integrating over the cake thickness.

The local pressure gradient within the filter cake can be described by Darcys equation. At a given time, the flow rate u will be constant through all layers of the cake for an incompressible fluid. Then, according to Darcys equation, the term kcdp/dx must be constant (independent of location in the cake) and equal to u. (assuming is constant).

dd

dd

px

px

u

k xn

c

= =

( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

Let pn represent the local net overburden pressure in the cake (pn = pmp), and let p represent the fluid pressure drop over the cake. The overburden pressure pn will range from zero at the cake inlet to p at the cake outlet. The velocity u can be expressed in


terms of the average cake permeability kca and p. Let kc be a function of pn, and combining kca with Eq. 7 gives

k p p u x u k ph

kk p

c n n

hpca

c

ca

c n

c( ) = =

=

( d d 00

,

)) dpp

n

p

0

. . . . . . . . . . . . . . . . . . . . . (8)

The effective permeability will be a function only of p and will be independent of the cake thickness. To solve the integral of kc(pn), we need a functional relation between kc and pn. In the literature, kc and c are normally treated uncoupled using the same equation type (Outmans 1963):

k k p p n nc on

c o

n= = 1 2 1 2 0and , , . . . . . . . . . . . . . . . . . . (9)

In our case, we want to calculate the permeability using the Carman-Kozeny approach of Eq. 5, include the effect of compress-ibility through the porosity, and integrate kcdpn to obtain the aver-age kca. We use the classic definition for compressibility:

np

e nc on p

cc n

= = 1 0

dd

, . . . . . . . . . . . . . . . . . (10)

Note that the porosity in Eq. 10 converges to the correct limit o when pn approaches zero, which is not the case with Eq. 9. The solution to the average cake permeability, as well as the computa-tion of the average porosity, is given in Appendix A. Here, we show only the final expressions. Defining z(pn) = 1(pn), the average permeability of a compressible filter cake is

kS n

F p Fp

F pz

z

ca

c

k k

k n

=

( ) ( )

( ) = +

12

0

1 2

02

,

ln zz . . . . . . . . . . . . . . . . . . . . . . . (11)

and the average filter-cake porosity is

=

( ) ( )( ) ( )

( ) = +

F p FF p F

F pz

z

k k

n

00

1 3 3

,

ln zz z+ 122. . . . . . . . . . . . . . . . . . . . . . . . (12)

The average filter-cake permeability and porosity computed by Eqs. 11 and 12, respectively, can be used directly in Eqs. 2 and 4.

The compressibility of the filter cake can be split into a revers-ible (elastic) and an irreversible (compaction) part. A simple way to model this is to apply an irreversibility factor, Firf. Let pmax be the historical maximum p. If the current pressure drop p is less than pmax, then an apparent effective pressure drop pa is used:

pa = p + Firf(pmaxp). . . . . . . . . . . . . . . . . . . . . . . . . (13)

Partially reversible compression of a filter cake is illustrated in Fig. 1.

Solids. The information needed in the filter-cake model includes a size distribution of the solid particles, the specific surface area S0, and the solids contribution to the cakes porosity and compressibility.

The size distribution can be either in the form of a log-normal distribution or entered as a table. If a tabular form is used, the volumetric cumulative distribution function F = [0, 1] is taken to be piecewise linear when plotted against ln(D). S0 is computed by integrating S0(D) over the size distribution. If tabular input with n rows is used,

S DD

Se

c x cD

sh

ish

x

x

x

i sh iii

i

06

6 6 1

1

( ) =

= =

,

d

=

= ( )1

1

1

0

1D

cF Fx x

x D

S

i

ii i

i i

,

, ln ,

== Sin

2. . . . . . . . . . . . . . . (14)

The shape factor sh is defined as the ratio of the particles surface area to the surface area of a sphere with the same volume (i.e., for spheres, sh = 1). Examples for other shapes: sh is 1.24 for a cube and 2.3 for a flat particle with dimensions 0.111 in any units.

For each solid type present in the mud, a reference porosity o corresponding to zero overburden pressure and a compressibility parameter nc are required.

Polymer. For polymers, we need to know their effect on fluid viscosity, which is important in dynamic filtration and for flow inside the porous medium. Second, we need the size of the poly-mer molecules to evaluate entrapment in the filter cake and in the formation. Finally, we must estimate the resistance to flow caused by the polymer in the filter cake.

Polymer Viscosity. The effect of polymer concentration on the fluid viscosity is typically expressed at low shear rate with a modi-fied Huggins equation (Sorbie 1991). We use the expression

02 2 3 31= + + + ( )s c k c k c , . . . . . . . . . . . . . . . . . . . . (15)

where c is the polymer concentration (volume fraction), s is the solvent (e.g., water) viscosity, and is a dimensionless form of the intrinsic viscosity [] in units [cm3/cm3]. The intrinsic viscosity [] is defined as the limit of the reduced viscosity when polymer concentra-tion approaches zero. k is the Huggins constant, which, for a range of polymers in good solvents, is reported to be equal to 0.4 0.1. Eventual changes in temperature and in molecular weight Mw caused by, for example, selective trapping of larger polymer molecules in pore throats are accounted for by the following expression:

= [ ] ( )

( ) p ww

a

vTM x t

MB T T

,

001 , . . . . . . . . . . . . . . (16)

where exponent a is reported to be in the range 0.51 (higher value for good viscosifiers). T is the actual temperature, T0 is a reference temperature at which [] is measured, and BT is an empirical parameter. The units for the density p must be consistent with the units used for [] so that becomes dimensionless. The shear-thin-ning effect of polymers is represented by Meters model (Meter and Bird 1964), with the final polymeric viscosity given by

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 5 10 15 20 25

Cake Overburden Pressure, bar

Poro

sity

1st curveFirf=0.2Firf=0.8

n c=0.04

Fig. 1Compressible filter-cake porosity, calculated with o = 0.65, nc = 0.04 bar1, and irreversibility factor Firf = 0.2 and 0.8.


p ss

hf

P= +

+

01

1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

where exponent P has a value between 1 and 2 (high shear thinning) and hf is the shear rate at which (0s) is reduced to one-half. The main influence of temperature is through the solvent viscosity s. We use the same Arrhenius-type equation for oil and water to compute the viscosity as a function of pressure and temperature:

=

+ ( )

o

BT T

B p p

eT

op o

1 1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

where o is the reference viscosity at temperature To and pressure po. BT and Bp are empirical constants, and temperature is in absolute units (K). We use the fluid viscosity in Eq. 18 to describe flow through the filter cake (Eqs. 3 and 4). Yet another correction for the content of solids is made before the effect of polymer is computed:

so

smc=

( )1 2 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

Polymer Size. We assume a log-normal distribution of Mw in the computations. The size of the polymer molecule in solution can be estimated from the intrinsic viscosity [] using Florys empirical equation for the mean end-to-end distance:

D Mhp w= [ ]( )0 00017 1 3. . . . . . . . . . . . . . . . . . . . . . . . . . . (20)The molecular weight Mw is in g/mol, [] in mL/g, and the com-

puted polymer diameter in m. We will use the hydraulic polymer diameter Dhp for computing polymer entrapment in the filter cake and pore-throat trapping of polymer inside the formation.

Flow Resistance. The use of the Carman-Kozeny approach of Eq. 5 requires the polymers contribution to the specific surface area and the cake porosity. The surface of the polymer is estimated by assuming the molecule to be a long rod with diameter Dch. The specific surface area is then

SDp ch

04

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

Note that the specific surface area for a given polymer type depends on the chain diameter Dch of the repetitive unit but not on the molecular weight.

For each polymer type i, we specify a oi and nci according to Eq. 10 that will go into the computations of the effective param-eters for the cake.

Cake Buildup. The cake-buildup calculations involve two peri-ods, the spurt-loss period and the filtration period after a filter cake has been formed.

Spurt-Loss Period. In the spurt-loss period, all polymers and solid particles smaller than a critical size Dcrs are assumed to enter the formation. Solid particles larger than Dcrs will be held back on the rock surface. When sufficient coverage of the rock surface is reached, all solids will be retained. The critical particle size for solids is computed from

D Dcrs s= , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

where D is obtained by Eq. 6 and parameter s represents the lower size ratio of particle size to pore size for particles that will be retained at the rock surface. A common rule of thumb is that particles larger than 1/3 of the average pore size will form an external filter cake (Pautz et al. 1989; van Oort et al. 1993). This rule indicates a critical size where there is a high probability for blockage at the rock surface over an extended time period, during which several particles will pass before the blockage occurs. The

meaning of s is slightly different; it represents the average low-end size of particles held back on the rock surface during a short spurt-loss period. Note also that D is an average property representing both pore-body and pore-throat distributions. Some particles larger than Dcrs will enter the rock, and some smaller particles will be held back. Knowing the particle distribution and assuming an initial cake thickness (see next paragraph), the value of s can be derived from experimental spurt-loss values. Our experience so far indicates that a value for s in the range of 0.30.5 seems to be appropriate. The effect of previous internal permeability reduction is included in the computed pore diameter D. This means also that particle distribu-tions smaller than initial Dcrs will eventually form a cake when the internal permeability is sufficiently reduced.

The average height of the layer needed to form a covering layer on the rock surface is computed by

h n h hS D Dsp sp crs crs crs

= =

>( ),6

0

, . . . . . . . . . . . . . . . . . . . . . (23)

where hcrs represents a characteristic height of the retained particles computed from the specific surface area for the fraction having D > Dcrs. nsp is the number of layers needed to form a completely covering cake. Note that the proposed way of computing hsp place more weight on the smaller particles and reduces the spurt loss for flattened particles. The spurt loss volume is then

V AhC

C cc c

sp csp

spsp

sm

sc sm

= =

11,

*

* *

, . . . . . . . . . . . . . . . . . . . . . . (24)

where csm* is the concentration of solids with D > Dcrs in the mud and csc* is equal to (1o) computed for solids alone. In the special case of spherical and monosized particles, nsp = 1 will correspond exactly to a single monolayer with porosity o. For nsp, a value in the range 12 is suggested.

Filtration Period. In this period, all solids are retained in the cake. Material deposited during the spurt-loss period is ignored (i.e., cake thickness starts from zero). The first mechanism to evaluate is retention of polymer in the cake. The part of the poly-mer molecules with Dhp smaller than a critical size Dcrp will pass through the cake. Dcrp is estimated as a fraction p of the pore size Dc (Eq. 6) for a filter cake made up of solids alone:

D Dcrp p c= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

The effective polymer concentration that will be retained in the cake is then obtained from the Mw size distribution. The criti-cal size ratio p represents the average probability for entrapment over a long filtration period and should be in the low range of standard trapping rules (1/71/3). A value of 0.1 has been used in the simulations presented here.

Finally, the cake properties can be computed. First, effective cake porosity, compressibility, and specific surface area are esti-mated as the volume-weighted average of all involved components, solids, and polymers (nt components):

o c c i o ii

n

c c c i c ii

n

c

c

n c n

S c

t

t

=

=

=

=

=

, ,

, ,

,

,

1

1

0 cc i ii

n

c ii

n

S

c

t

t

, ,

,

,01

11

=

=

= , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)

where cc,i represents the relative concentration of Component i in the filter-cake matrix. The average cake permeability and porosity are obtained readily from Eqs. 11 and 12, respectively, and, finally, the static filtration parameter Cst in Eq. 4 can be computed.


Dynamic Filtration. The filter-cake model outlined so far applies for static conditions at the filter-cake surface. In real situations, the filter-cake surface is exposed to shear forces from mud circulat-ing through the well. Two main mechanisms are considered in the dynamic filtration model, namely, reduced attachment and erosion.

The flow along the cake surface will cause part of the particles to roll off the surface and bounce back into the mud. Only a fraction of the particles will stick to the cake surface. This will reduce the growth rate of the cake. The shear stress at the cake surface may also detach particles from the filter cake and, thus, decrease the cake thickness. This is called the erosion rate Y. The cake growth in terms of and Y can be obtained from mass balance:

ddht

C u Yc

C cc c

cd x

sm

dsm

sc sm

=

=

1

1

,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)

The flow through the cake ux is given by Darcys law if the cake permeability and current thickness are known. This leaves two unknowns to be determined, and Y.

Consider a nonmoving spherical particle just in contact with the filter cake surface with a filter loss rate ux and a perpendicular mud flow along the surface uz. For simplicity, we assume a constant shear rate near the surface, so that the average velocity uz in a layer with the same thickness as the particle will be

dd

,

d

u

x

u x x

uD

x x D

z

z

z

p

D

p

p

=

( ) == =

1

0

12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (28)

The force acting on a spherical particle with constant relative velocity in a fluid is given by

F uDp= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

Then, for the particle close to the surface, the forces in the x and z directions, respectively, are

F u D

F Dx o x p

z p

=

= ( )332

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (30)Note that, in Fz, the viscosity will be a function of the

shear rate while the viscosity acting toward the surface (in Fx) is computed at zero shear rate (o). The dominating mud flow will be along the surface. The reduced viscosity at higher rates (shear thinning) is caused by polymer molecules orienting themselves along the main flow direction. Particles bouncing back into the mud will not see this shear thinning because they will be moving perpendicular to the flow. The ratio of forces acting on the particle along and toward the surface is then

FFF

Du

Rz

x

p

o x

= =

( )

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31)

The fraction of solids (in the mud) that will attach to the surface () is modeled by

=+ ( )

11 F fR n

nF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (32)

The exponent nF determines how fast the transition of a is when the filtration rate ux decreases, and its value should reflect the width of the particle distribution. Values in the range 0.51 are used in the simulations here. Finally, fn is a friction factor.

The shear stress acting on the filter cake surface is given as

z z

z

=

= ( ),

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (33)

Assume a multilayered cake, with layer thickness Dp (a char-acteristic particle diameter size). The forces holding the layer in place are (1) cohesive attraction between the particles (solids and polymers) that form the cake matrix and (2) the viscous pressure gradient from the flow through the cake. We assume cohesive forces to be represented by the filter cake yield strength y, which is measured by Cerasi et al. (2001) to range from a few hundreds to a few thousands of Pascals. The pressure drop over the first layer will be equal to the net overburden pressure at distance Dp into the cake, pn(x = Dp), which we obtain from Eq. A-6. The x-directed forces considered are then

x n p yp x D= =( ) + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (34)We assume the erosion rate to be proportional to the ratio of forces

acting in the z and x directions and to Dp. An empirical rate constant e (in units s1) is introduced. The erosion rate is given by

Y Dp De p

z

n p y

= ( ) +

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (35)

The particle diameter used in Eqs. 28 through 35 is estimated from the size distribution of both solids and polymer using vol-ume-weighted D1.

A limitation of the filter-cake model is that we are using average cake properties. This prevents us from properly including physical processes such as size-selective attachment to the cake and pressure-dependent yield strength y. A reasonable implementation of such effects requires some kind of spatial discretization of the filter cake.

Particle Retention in the Formation. Retention of particles ow-ing through a porous medium can be divided into two main parts, pore-throat plugging and pore-lining retention. The retention kinet-ics for the two parts is very different and so is the damaging effect. In the case of external particles (injection of suspended particles), pore-throat trapping will be most important close to the inlet, while pore-lining retention will dominate deeper in the core. Larger par-ticles will be ltered out from the solution as they are trapped in narrow pore throats, while smaller particles can penetrate deeper. In laboratory experiments, normally only the combined effects of the two processes are obtained, resulting in a large variety of reten-tion models/parameters. The observed results will strongly depend on the experimental setup, and there is no good method for scaling the experimental results to other conditions.

Here, we will try to separate the two retention types. Our main focus is on the pore-throat trapping because it has the higher dam-age potential.

Pore-Throat Trapping. In pore-throat trapping, only particles large enough to be held back in the narrow parts of the pore space (throats) will be involved. To model this, we need to keep track of particle sizes. Solid and polymer components are split into a number of size fractions, each represented as a monosized subcomponent in the simulation. The basic equation used for both solids and polymers (k = s, p) allows for both trapping and re-entrance of particles back to solution. The trapping rate for Subcomponent i is computed by

dd

i w

w

k t i k it

u

Sc= ( )1 2, , , , k = s, p. . . . . . . . . . . . . . . . . . (36)

The units for trapped material and total mobile concentration of Component i are pore volume (PV) units, so that, over a timestep t, we have i = ct,i. The unit for parameters 1 and 2 is 1/length. For solids, the detrapping term 2 will be (approximately) zero. If 2 is zero, Eq. 36 says that a fraction of all passing particles equal to 1 will be retained per length unit and the expected traveling length


E[x] is 1/1. To make the equation more general, we allow 1 to be a function of particle size and 2 to depend on fractional flow. For solids, if the subcomponent size Di > tsD, 1 and 2 are given by

1 10

2 20 21

11

1

s si

s s s o

DD

f

s

=

= +( ),

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (37)

If the particle size is smaller than this limit, trapping is not considered. Finally, D is the pore size estimated by Eq. 6.

Contrary to rigid solid particles, polymer particles will be quite deformable, and their retention in pore throats may be highly dependent on the viscous drag from fluid passing through. For polymers, 1 and 2 are computed by

1 10

2 20 21

11

1

p php i

p p p o

dD

f

p

=

= +(

,

,

)) +

1 22

22

p

p

p

. . . . . . . . . . . . . . . . . . (38)

The size of the polymer Subcomponent i is represented by the hydraulic diameter from Eq. 20, and a lower size limit for pore-throat trapping can be specified by tpD. Particles that are smaller than this limit will not be trapped in the simulations. The main dif-ference from solids is that the re-entrance term will increase with increasing pressure gradient p and cannot be ignored.

Two methods for computing the permeability reduction caused by the pore-throat trapping are implemented. In the first, an inter-nal-cake permeability kic is computed similarly to kc for the filter cake, except that a zero overburden pressure pn is used. Then, the effective absolute permeability is computed by harmonic averaging between kic and the original rock permeability k:

k ak

a

ka

ic

s p

iceff = +

=

+

11

1

,

. . . . . . . . . . . . . . . . . . . . (39)

The internal-cake porosity is a weighted average of all trapped material. This method seemed to overpredict the perme-ability reduction, in particular when polymer is involved. The likely explanation is that small amounts of trapped material will not be packed close together as a cake, but rather will be spread over a larger area within the narrow parts of the pore space. One way to compensate for this is to introduce a modification factor for the specific surface area of the polymer fpm.

In the second method, we compute a pore-throat permeability kpt within the narrow pore space defined as a fraction fpt of the total pore space. A specific surface S0pt for this pore-throat volume is computed by combining the surface area of the original rock, S0r(, k) obtained from Eq. 5, with the surface area of the deposited solids and polymers. The rock porosity is corrected for deposited material pt. Then, kpt(pt, S0pt) is calculated by Eq. 5, and finally the new effective permeability is obtained by harmonic averaging with k:

ptp s

ptr s s

a

SS a S S

=

+

=

( ) + +1

10

0 0 0

,

pp p

s p

pt pt pt

a

k ak S

( )( ) + +( )

= ( ) +1

0

,

,eff

11

1

1

=

+

a

k

a fpt s pic

,

max ,

. . . . . . . . . . . . . . . . (40)

The proposed methods have some interesting differences from common phenomenological models that relate the permeability reduc-tion only to the amount of deposited material or the reduction in poros-ity [e.g., keff/ki = (/i)m]. With the indicated model, any dependency on the nature of the deposited particles and on the initial permeability must be set explicitly by changing the model exponent m. By using the Carman-Kozeny approach, properties of the trapped particles are automatically included through their specific surface area. As a result, the same amount of trapped material will result in larger permeabil-ity reduction when the particles are smaller. The trapping rate will increase at lower permeability because of a smaller pore diameter D, but the relative reduction in permeability per amount of trapped mate-rial will be reduced. This is a result of decreased difference between the core permeability and the internal-cake permeability.

A presumption in the permeability-reduction model is that all streamlines see the same amount of trapped material. This require-ment applies to the computational volume enclosed in a single grid cell. With significant heterogeneities on a scale smaller than the grid cell (i.e., if grid cells are populated with average properties), then the presented model will be less reliable. Furthermore, the accuracy in computed permeability reduction will not be better than the trapping model used (Eq. 36). One obvious simplification is that we disregard the pore-size distribution, which may result in underestimation of initial deposition rate when the first narrow pathways are plugged. However, significant impairment will not occur until the main flow paths start to plug, and, presumably, this part of the process is suf-ficiently represented by using average properties.

Pore-Lining Retention. The pore-lining retention concerns material attached to the pore surfaces. The attachment of particles to the rock surface is governed by a balance between attractive and repulsive forces. This is modeled with a Langmuir-type kinetic equation for adsorption A:

ddAt

c Q A Ak k k m k k k k= ( ) 3 4, , , , k = s, p. . . . . . . . . . . . . . . (41)Qm is the adsorption capacity, c the volumetric concentration, and 3 and 4 are rate parameters representing adsorption and desorp-tion, respectively. The same equation is used for both polymers and solids. For a mixture with several solid and polymer components, the following apply:

One set of polymer-adsorption parameters is used for all poly-mers. The maximum adsorption capacity for individual polymer components is according to their relative concentration and sums up to Qm,p. Adsorption between polymers is competitive.

One set of solid-adsorption parameters is used for all solids. The maximum adsorption capacity for individual solid components is according to their relative concentration and sums up to Qm,s. Adsorption between solids is competitive.

The model assumes that polymer and solid adsorption can be decoupled (i.e., that no competitive adsorption between solid and polymer exists).

A main reason for repulsive forces is the electrical charge on rock and particle surfaces. An increasing electrolyte concentration will reduce electrostatic repulsion. Also, a particle may depart from the rock surface as a result of a high viscous drag. These mechanisms are included in the desorption parameter 4 through the pressure gradient p and the effective salinity Cse terms:

4 4041

611

10

41

, ,

,

,

k kk

pk

= +

++ Cse k42,

, k = s, p. . . . . . . . . (42)

The last salinity term is optional and requires the presence of at least one cation in the simulations. Cse is defined as the equivalent concentration of a reference cation, normally sodium. p in Eq. 42 is corrected for the effect of pore-throat plugging (i.e., this is subtracted from the actual pressure gradient).

Effective Salinity. Both the state of swelling clays and the attach-ment of nes to the pore surface are strongly dependent on the type


of exchangeable cations, such as K+, Na+, Mg2+, and Ca2+, and on the total salinity. The equilibrium between cations in solution and those attached to the clay surface is approximated by the mass-action equations (Hirasaki 1982; Hirasaki and Lawson 1986).

ff K

XX

Naz

XXC

Naz

i

i

i

i

i

= , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (43)

where Xi is the bulk phase concentration of Cation i in N (normality or mEq/mL), zi is the valence, and fXi is the fraction of Cation i at the clay surface. KXCi is an ion-exchange constant for each cation. XNa and fNa represent the bulk phase concentration and the surface fraction, respectively, of the reference sodium (Na+) cation.

To evaluate the salinity effect in mixed brines on, for example, clay stability, we adapt the effective salinity model from UTCHEM (Pope and Nelson 1978; Hirasaki 1982) and extend it to any num-ber of cations. Na+ is used as a reference cation, and the effective strength of other cations is transformed to an equivalent amount of Na+. The effective salinity, Cse, is calculated from the fractional concentrations at the clay surface (for n cations):

C X

fse

Cl

ci Xii

n=

=

11

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (44)

The total salinity is here represented by XCl (aqueous chlorine concentration), which is equal to the sum of all cations in solution. ci is the effective salinity parameter for Cation i.

Clay Swelling. The net transport of water into or out of a swelling clay is driven by osmotic forces, a gradient in the effective salinity. The swelling is modeled as a kinetic water-adsorption model with Cse as the driving force in both terms:

dd

sw sw

se cse

sw sw sw se swt

kC

Q k C=+

( ) 1 2

max , . . . . . . . . . . . . (45)

where sw is the swelling state defined as the volume ratio of adsorbed water to clay; ksw1 and ksw2 are adsorption and desorption rate param-eters, respectively, and Qswmax is the maximum swelling state.

The effect of the swelling is that the pore volume available for flow will be reduced. We model this by reducing the effective water-phase saturation by an amount equal to the trapped water. Consequently, the relative permeability to water will decrease during swelling. The oil relative permeability will be affected indirectly through a decrease in maximum oil saturation equal to trapped water. The amount of trapped water is corrected for the initial content of water in the clay.

ResultsThis section presents simulations of various static and dynamic mud-filtration experiments and a dilute particle-transport experi-ment, and demonstrates simulations of fines migration and clay swelling. Finally, laboratory results are used in simulation of a North Sea well. Mud compositions are given in Table 1.

Input parameters used in the simulations (if not mentioned in the text) are given in Table 2 (polymers), Table 3 (solids), Table 4 (various model parameters), Table 5 (pore-throat trapping), Table 6 (pore-lining adsorption), Table 7 (relative permeability), and Table 8 (dynamic filtration).

TABLE 1MUD COMPOSITIONS (g/L)

Component Mud 1 Mud 2 Mud 3a Mud 3b Polymer (P1, P2) 25 25 Xanthan 4 4 Starch 20 20 20 CaCO3 100 100 50 50 Drilling

solids (OCMA) 40

Barite 155

TABLE 2POLYMER PROPERTIES*

P1 P2 Xanthan Starch

Viscosity, Low Shear

[ ] (mL/g) 1500 3200 5500 600 k' 0.4 0.4 0.4 0.4 k'' 0.01 0.01 0.07 0.07 Mw exponent, a 0.8 0.8 0.8 0.5 BvT [1/C] 0 0 0.006 0.001 T0 (C) 20 20 20 20

Shear Thinning P 1.7 1.7 1.8 1.5 hf (s1) 2 2 1.5 5

Size Parameters Mw (g/mol) 107 107 3 106 5 106

ln(Mw) 1 1 2 2 Dch ( m) 0.0008 0.0008 0.001 0.0012

(g/cm ) 1 1 1 1

Computed Properties S0 (m /cm ) 5000 5000 4000 3333 Mean Dhp ( m) 0.414 0.498 0.428 0.242

Filter Cake Properties

o 0.85 0.85 0.85 0.85 nc (bar1) 0.095 0.095 0.08 0.08

* Numbers in italic do not influence static fluid loss computation.


Static Filtration. The lter-cake model is demonstrated by computing static high-pressure/high-temperature (HP/HT) ltration experiments with two muds performed at 85C and 850 psi on 20- and 60-m ceramic disks. The solids size distributions for the two mud systems are given in Fig. 2. The manufacturer permeability of the two disks is 5 and 20 darcy, respectively. These permeabilities and a porosity of 0.37 were used in the calculations. Filtration volumes are computed for a 45-cm ltration area [American Petroleum Institute (API)].

The 0.0722 volumetric concentration of solids in Mud 1 (S1 in Table 3) includes a mixture of sized CaCO3 and barite. Average poly-mer properties are used arbitrarily to match the filtration volume; see P1 in Table 2. The polymer concentration was set to 0.025.

Mud 2 contained 0.0377 volume-fraction-sized CaCO3 (S2 in Table 3). The only other parameter changed in the Mud 2 calcula-tions was the intrinsic polymer viscosity [], which was increased from 1500 to 3200 mL/g (P2 in Table 2). All other parameters remained unchanged.

Details from computation of the spurt-loss and filtration volumes for Mud 1 on the 5-darcy filter are given in Table 9. The main steps include

1. Determine critical size of solids that can enter the formation (sD = 0.435.8 = 14.3 m) (Eqs. 6 and 22).

2. Use size distribution to find concentration and specific sur-face area of particles trapped at the disk surface. Compute average deposition thickness needed to form the initial cake, hsp = nsp6/S0sp = 21.1 m (Eq. 23).

3. Compute spurt-loss volume, Vsp = 1.25 mL (Eq. 24).4. Determine average filter-cake permeability kca = 9.3106 md

and = 0.416 from average cake properties, = 0.515, = 0.0244 bar1, and p = 58.6 bar (Eqs. 11, 12, and 26).

5. Finally, compute the static filtration parameter Cst = 0.0309 mL/cmmin0.5 and fluid loss Vf (30 minutes) = 8.86 mL (Eq. 4).

Experimental and simulated filtrate volumes are plotted against the square root of time in Fig. 3 (Mud 1) and Fig. 4 (Mud 2). We

TABLE 3SOLIDS PROPERTIES (S0 COMPUTED)

S1* S2*

CaCO3* Mud 3

OCMA Mud 3b Fe2O3*

Dp ( m)** 18.44 29.4 7.05 1.5 0.856 St Dev 1.4 1.34 0.8 1 0.95 sh 1.8 1.35 1.25 1.5 1.8 S0 (m /cm ) 1.5 0.67 1.5 9.91 19.6

(g/cm ) 2.6 2.6 2.7 2.7 5.4 Filtercake

o 0.4 0.4 0.4 0.4 0.4 nc (bar1) 0.0001 0.0001 0.0001 0.001 0.0005

* Tabular distribution, mean Dp, and standard deviation are computed. ** Dp corresponding to mean ln(Dp). Standard deviation in ln(Dp).

TABLE 4VARIOUS MODEL PARAMETERS, VALUES USED IN SIMULATIONS

Variable Value Remarks

s 0.4 size factor in spurt loss

p 0.1 size factor for polymers nsp 1 spurt-loss factor

3 tortuosity factor

TABLE 5PORE-THROAT-TRAPPING PARAMETERS

Solids Polymers

t (fraction) 0.035 0.05

10 (cm1) 40 1.5

11 2 2.5

20 (cm1) 0.0001 0.0015

22 (bar/cm) 0.01

22 0.5 fpm 1 fpt 0.05

fpt2 0.1

TABLE 6PORE-LINING-ADSORPTION PARAMETERS

Solids Polymers Fe2O3 Fines

Qm (pv) 0.05 0.001 0.05 0.2

30 (min1) 0.1 0.1 0.1 0.03

31 0 0 0 0

40 0.0001 0.0002 0.0001 0.0001

41 (bar/cm) 10 0.1 10 10

41 3 1 3 0.25

42 1.5 1.5 1.5 1.5 RPRmax 2 2 2 2

200 1000 200 1000

TABLE 7RELATIVE PERMEABILITY, COREY PARAMETERS

Simulation krwe Ew Srw kroe Eo Sor

FD1-FD3 0.15 3 0.2 0.77 1.5 0.2 FD15 0.15 3 0.2 0.65 1.5 0.2 B14 0.1 4 0.185 0.75 3 0.185 Field 0.4 3 0.14 0.65 4 0.1

TABLE 8DYNAMIC-FILTRATION PARAMETERS

Simulation fn nF e (s1) y (Pa)

DS2 0.1 1.25 0 5000 DS3 0.1 0.5 67 5000 DS4 0.05 0.8 1 1000 Mud98067 300 0.5 1 10

0

0.2

0.4

0.6

0.8

1

0.1 1 10 100 1000

Particle Diameter, m

Cum

ulat

ive

Dis

trib

utio

n, F

, vol

. fra

ctio

n

Mud 1Mud 2Mud 3

Fig. 2Particle-size distribution.


observe that increasing the permeability (pore size) of the disks results in a higher Vsp. This is because a smaller fraction of the solids will be held back at the disk surface and, in addition, the average size of retained particles will be larger so that a larger volume is required to form the initial cake. Vsp is more than doubled for Mud 2 compared to Mud 1. This is mainly because of the lower

solid concentration. All these differences are well reproduced in the calculated curves, as clearly illustrated in Figs. 3 and 4.

The second observation is that the filtration curves are parallel to each other (i.e., the filtration parameter Cst does not depend on the properties of the formation but only on the mud properties and the differential pressure over the cake).

Fig. 5 shows computed filtrate volumes with Mud 1 for a wide range of permeabilities. We show both the analytically computed Vf and Vsp, ignoring the pressure drop over the disk, and the simulated Vf (30 minutes) using a 10-cm core. We observe that Vsp starts to increase above 10 md, while the net filtration volume after the spurt loss (VfVsp) is constant in the analytical calculations. The simulated Vf deviates from the computed value at low permeability (0.01 md) when the resistance over the core becomes significant compared to the filter-cake resistance. Also, at very high perme-ability (100 darcies), the simulated Vf is lower than the computed Vf because of internal blockage of pores that reduces the effective pore size of the formation.

Blaxter Sandstone. The results from three static ltration experi-ments using Blaxter sandstone are given in Table 10. The core dimensions were 8-cm length and 2-in. diameter, with an effective ltration area of 16.8 cm2. The permeability was approximately 150 md, and the porosity was 20%. The mud system used in Core Experiment FD1 contained 50 g/L sized CaCO3 with size distri-bution given in Fig. 2 (Mud 3a), 4 g/L xanthan, and 20 g/L low reticulated starch. 40 g/L drilling solids [Oil Company Materials Association (OCMA) clay] was added in Experiments FD2 and FD3 (Mud 3). All the core oods were conducted at 90C. An overbalance pressure of 15 bar was used in FD1 and FD2, and 35

TABLE 9COMPUTATION OF STATIC FILTRATION, MUD 1, Ac = 45 cm

Model parameters s = 0.4 p = 0.1 nsp = 1 [ ] = 1,500 mL/g s = 0.339 cp Disc properties K (md) = 5000 = 0.37 Ac = 45 cm2 D = 35.8 m sD = 14.3 m Spurt loss Fraction > sD = 0.589 S0sp = 0.284 m2/cm3 hsp = 21.2 m C1sp = 0.0763 Vsp = 1.25 mL

Mud

Concentration

(volume fraction) Effective Concentration

(volume fraction) S0 (m /mL) o nc (bar1) Polymer 0.0250 0.0249 5000 0.850 0.0950 Solids 0.0722 0.0722 1.5 0.400 0.0001 Total 0.0972 0.0971 1283 0.515 0.0244

Filtration p = 58.6 bar kca (md) C1 Cst (mL/cm min0.5) Vf 30 minutes (mL) 0.416 9.27E-06 0.1993 0.0309 8.86

0

2

4

6

8

10

12

0 1 2 3 4 5

Square Root of Time, minutes

V f ,

mL

6

20 m 1 60 m 120 m sim 60 m sim

Fig. 3HP/HT filtration at 850 psi and 85C, experimental and simulated for 20- and 60-m filter disks, Mud 1. (Ac = 45 cm).

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6

20 m 3 60 m 320 m sim 60 m sim

Square Root of Time, minutes

V f ,

mL

Fig. 4HP/HT filtration at 850 psi and 85C, experimental and simulated for 20- and 60-m filter disks, Mud 2. (Ac = 45 cm).

0.1

1

10

100

0.001 0.1 10 1000 100000K, md

Filtr

ate

Volu

me,

mL

Vsp Vf Vsp

Vf Vf sim

Fig. 5Effect of the porous medium on filtrate volumes. Ana-lytical spurt loss (Vsp) and filtrate volume (Vf) and simulated Vf (30 minutes) with a 10-cm core (Vf-sim). (Ac = 45 cm), Mud 1.


bar in FD3. API ltration (100 psi and 25C) and HP/HT (500 psi and 90C) are also reported in Table 10.

Properties of the polymers and solids are given in Tables 2 and 3, respectively. Fig. 6 shows a good agreement between measured and computed shear-thinning mud viscosity. Other model param-eters used in the simulations are given in Tables 4, 5, and 6.

Filtration. All the experimental fluid-loss volumes in Table 10 are well reproduced in the simulations. This indicates that the effects of pressure, temperature, and composition are all well taken care of in the simulations. The pressure sensitivity (735 bar) is handled by the compressible-filter-cake model. The simulated effect of temperature on filtrate volume is only through the change in the brine viscosity, from 0.926 cp in the API test (25C) down to 0.318 cp in the core tests (90C). The effect of composition is more complex. In the core experiments, we observe a reduction in Vf when OCMA clay is added (FD1 vs. FD2). In the calcula-tions, the filter cake in FD2 will be thicker but with a higher kca because of the reduced relative polymer content. However, adding OCMA clay decreases the average solids particle size, which may potentially trap more polymer in the cake. In the simulations, approximately 9% of the xanthan and 30% of the starch is passing through the filter cake in FD1. In FD2, these numbers are reduced to zero for xanthan and 1% starch. The increased entrapment of polymer in the filter cake will reduce kca, and this is the reason, at least in the simulations, for the observed decrease in Vf between FD1 (12 mL) and FD2 (10 mL).

The relative viscosity of the filtrate produced from Mud 3a was measured to be approximately 6 and showed shear-thinning proper-ties. This verifies that a significant amount of polymer was passing through the filter cake in FD1. The lower level of simulated poly-

mer migration through the cake in FD2 agrees well with measure-ments taken by Snchez et al. (2004), where 12% of the polymer was measured (as total organic carbon) to pass through the cake for a mud system similar to Mud 3b. However, the actual values of the model parameters used to simulate this phenomenon are highly uncertain. In the simulation, we identify the degree of polymer entrapment in the cake as a potentially important variable for fil-tration rates; however, the properties used in these calculations are poorly known. The polymer molecular weight Mw is well known as a physical property, but its actual value and its size distribution are highly uncertain. The computation of molecular hydraulic size from [] and the trapping parameter p are also uncertain. Finally, the OCMA properties used are also approximate.

Return Permeability. The oil return permeabilities in Table 10 are obtained after backflooding the core with oil at increasing rates: 1, 2.5, 5, and 8 mL/min, for 30 minutes at each rate. Simulated and experimental return permeabilities are in the same range, 8090% of initial oil permeability koi. The simulated differential pressure over the core and the development of the return permeability RKO during the backflood of FD1 are shown in Fig. 7. We see that RKO is highly dependent on the length of the backflooding period. The profile is governed by relative permeabilities, which are unknown, and viscosity ratio of oil to water, which may be reduced because of polymer invasion. The relative permeabilities used (see Table 7) are typical for water-wet medium, with a final tuning to the results in FD1, except for the oil endpoint permeability.

The average Sw shown in Fig. 7 is still well beyond its initial value of 0.2. Also shown is RKOE, an estimated return permeabil-ity at maximum oil saturation. If the flood were continued, RKO (0.824) should finally reach RKOE (0.916). Note that capillary pressure is ignored (zero) in these simulations and that additional uncertainty in return permeabilities may be caused by capillary end effects.

Damage Profile. Another issue much discussed is what the damage profile looks like in the invaded zones. The simulated damage profile (RKO) for FD1 is shown in Fig. 8 together with profiles of trapped CaCO3 and polymers. CaCO3 particles are trapped in the first millimeters of the core, while polymer shows a deep penetration. It seems that trapped polymer has reached a plateau extending approximately half the filtrate-invasion volume. The damage profile (RKO) follows the trapped-polymer profile. The amount of external CaCO3 particles introduced during the spurt-loss period is obviously very small, and there is no realistic way to simulate any serious damage from trapped CaCO3 particles alone. The observed damage, therefore, is caused most likely by the polymer.

Another experiment, FD8, shows very serious plugging when only the polymer mixture is injected (4 g/L xanthan plus 20 g/L starch), shown in Fig. 9. When simulating FD8 with polymer-trap-ping parameters given in Table 5, only the first part of the pressure-drop curve was matched. The pressure increase during long-term

TABLE 10EXPERIMENTAL AND SIMULATED FLUID-LOSS VOLUMES WITH MUD 3

FD1 FD2 FD3

p (bar) 15 15 35 Drill solids (g/L) 0 40 40 API (100 psi, 25C) exp 6 5 Vf 30 minutes (mL) sim 5.9 4.7 HP/HT* (500 psi, 90C) exp 12 12 Vf 30 minutes (mL) sim 12.4 10.6 Core** (90C) exp 12 10 11 Vf 240 minutes (mL) sim 12.1 10.3 11.1 Oil return exp 83 82 86 Permeability (%) sim 82.9 86.8 85.4

* HP/HT filtration volumes converted to API filtration area. ** Core filtration area is 16.8 cm2.

1

10

100

1000

10000

1 10 100 1,000 10,000

Shear Rate, 1/sec

Visc

osity

, mPa

s

CalcExp.

Mud 3a

Fig. 6Mud 3a, experimental (converted Fann viscosities) and calculated viscosity, 25C.

0

0.15

0.3

0.45

0.6

0.75

0 20 40 60 80 100 120

Time, minutes

Pre

ssur

e D

rop,

bar

0

0.2

0.4

0.6

0.8

1

Ko

/Koi

and

dp RKORKOE

FD1

1 mL/min

8 mL/min

5 mL/min

2.5 mL/min

Fig. 7Simulated back production, FD1. Differential pressure (dp), current (RKO) and maximum (RKOE) oil return perme-ability, and average Sw.


injection shown in Fig. 9 was approximately simulated using a polymer modification factor fpm = 0.15 (see comment after Eq. 39). The amount of polymer that will invade the core in typical mud-fil-tration experiments is well within the part that is matched without modification of the polymer effect (fpm = 1, as in Table 5).

In Fig. 10, the backflooding of Core B14 is extended with an extra step where the final oil in place is displaced with a highly viscous oil (M82) at a reduced rate. Using the pressure derivative with respect to the front position of M82, the permeability profile along the core can be estimated. We approximate the front position by assuming constant So throughout the core. In Fig. 11, we show the estimated ko profile (Experiment VF) together with koi. We also show profiles from simulation of the experiment and an estimated ko profile from the simulated pressure history (Simulation VF). Note that the estimated increase in ko toward the outlet end in the

oilflood, which is to the left in Fig. 11 and the mud side, is a result of dispersion of the viscous-oil front. The same behavior was seen when applying the method on simulated data (Simulation VF).

Core B14 has a length of 4.7 cm and was subjected to static mud exposure for 50 hours (Mud 3b), during which close to 2 PV of filtrate had entered the core. The results in Fig. 9 strongly indicate that ko is reduced throughout the whole core, which supports the previous results (Fig. 8), indicating that polymers may cause deep and rather evenly distributed damage.

Dynamic Filtration. Two examples of simulating dynamic experi-ments are given in Figs. 12 through 14. The rst example is Core Test FD15 (Blaxter sandstone) with Mud 3b (described earlier). The experiment is simulated with different sets of parameters (fn, nF, , y), given in Table 8. The plot of Vf vs. time in Fig. 12 has

0

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0.0002

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0.0006

0.0008

0.001

Trap

ped

CaC

O3

1 an

d po

lym

er1

00, P

V

RKOE

RKO

CaCO3Xanthan

Starch

Ko

/Koi

Fig. 8Simulated oil permeability reduction, current (RKO) and estimated final (RKOE), and pore-throat-trapped solids (CaCO3) and polymers.

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35

0 1 2 3 4

PV Injected

Pres

sure

Dro

p, b

ar

5

Exp. FD8Sim. fpm=1Sim. fpm=0.15

Fig. 9Xanthan+starch injection, 150-md Blaxter sandstone.

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PV Injected

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sure

Dro

p, b

ar

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Oil

Ret

urn

Perm

eabi

lity,

RK

O

Sim.Exp. CLS370Exp. M82RKO

B14: back production

Fig. 10Back production with low-viscosity oil (CLS370) and highly viscous oil (M82) in Experiment B14.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Core Position, x/L

Ko

Sim. Sim VFExp. VF KoiKo max

B14

Fig. 11Interpretation of oil permeability profile (Experiment VF) from viscous-oil flood, B14.


a nonuniform development of its slope, which may suggest some experimental problems. This is seen more clearly when plotting the slowness (inverse velocity) vs. Vf, as in Fig. 13. Simulation DS3 puts more weight on the last part and assumes that a steady-state limit of approximately 6000 s/cm is reached. DS4 puts more weight on the rst part, while DS2 is in between. DS2 and DS4 have not reached steady state within the 4-hour time frame. In DS2, no erosion at all is assumed, only a reduced attachment rate. This illustrates the problem of upscaling a short dynamic experiment to the eld situation, which will be shown in a later section.

The second dynamic case (in Fig. 14) is a filtration experiment taken from Dewan and Chenevert (2001). A bentonite mud with solid content csm = 0.18 is used. Appropriate rheologic mud prop-

erties were simulated using a polymer component. The solid-size distribution was taken from one of the other muds (Duratherm) used in their paper. The simulated slowness curve in Fig. 14 matches the experimental curve very well. The experiment starts with dynamic filtration (100 s1) at 500 psi, switches to static con-ditions, switches back to dynamic conditions again, and ends with increasing the pressure to 1,500 psi. Each period was 30 minutes, which was insufficient to reach steady-state conditions. After the pressure increase, we first see a short transient period where the compressible cake adapts to the new pressure. Then, the curve will slowly progress toward the same steady-state limit as before the pressure increase, although at a slower rate because more mass must accumulate in the cake at the higher pressure.

The uncertainties in the particle-size distribution and the rheologic properties make the value of the model parameters in Table 8 less gen-eral. The main purpose, however, was to demonstrate the capability of the dynamic model to reproduce typical experimental behavior.

Particle Retention. An experiment (Experiment 1) taken from Al-Abduwani et al. (2003) is used to demonstrate the pore-throat-trap-ping model. In this experiment, a dilute 20-ppm hematite (Fe2O3) particle suspension was injected into a Bentheimer sandstone core. The particle size was stated to be in the range of 0.1 to 5 m, with 65% less than 1 m. The size distribution used in our simulations is given in Fig. 15, and additional information is given in Table 3. Core data used in the simulations are L = 12.2 cm, k = 1300 md, = 0.22, and Sw = 1. The trapping parameters for solids tuned to this experiment (Table 5) are also used in the other simulations presented in this paper.

By using dx/dt = uw/(Sw) in Eq. 36, setting 2 to zero, and integrat-ing, we find the effluent concentration for a given particle size to be

c c L= ( )0 1exp , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (46)

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45

50

0 50 100 150 200 250

Time, minutes

Filtr

ate

Volu

me,

mL

exp.DS2DS3DS4

FD15

Fig. 12Dynamic filtration with Blaxter sandstone, Mud 3b. Simulation of Experiment FD15.

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0 10 20 30 40 5

Vf , mL

Slow

ness

, s/c

m

0

exp.DS2DS3DS4

FD15

Fig. 13Slowness vs. Vf in dynamic-filtration experiment FD15.

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10000

20000

30000

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50000

60000

0 30 60 90 120

Time, minutes

Slow

ness

, s/c

m

Exp. Mud 98067Sim

500 psi 100 seconds1

500 psi 0 seconds1



Fig. 14 Experimental and simulated dynamic filtration. Experi-mental data are replotted from Dewan and Chenevert (2001).

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0.1 1 10Particle Diameter, m

Cum

. Vol

. Fra

ctio

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1

Rel

ativ

e Ef

fluen

t Con

c.

F Hematite

CeffCeff (Dlim)

Fig. 15Hematite particle distribution used in simulation. Relative effluent concentration is for the Bentheimer core described in the text, computed with (Dlim) and without a lower size limit for trapping (ts).


where 1(Dp) is given by Eq. 37. The computed size-dependent relative effluent concentration for the present core is shown in the size-distribution plot (Fig. 15). The average effluent concentration is estimated from the size distribution to be 13.1 ppm when 20.7 ppm is injected.

Simulated and experimental results are compared in Figs. 16 through 18. If pore-throat trapping is considered as the only retention mechanism, the simulated effluent concentration seen in Fig. 16 starts at the 13 ppm analytical value and then declines slowly as an effect of previously trapped material. The simulation was extended beyond the experimental time period, and, at some point, when an external filter cake has been formed, the effluent concentration dropped rapidly to zero. The experimental curve is different; it breaks through at a low concentration and increases rather slowly toward a plateau after approximately 300 PV. This curve shape may be explained in part by a gradual change in the trapping pattern caused by a distribution in pore-throat size, but it

is also influenced by other retention mechanisms. A closer match to the experimental profile is obtained by including pore-lining adsorption in the experiment.

A total of 662 PV was injected in Experiment 1. The final distribution of hematite and permeability along the core is shown in Figs. 17 and 18, respectively. In both cases, the simulated and experimental profiles match. The experimental data represent aver-ages over core segments. A corresponding segmental permeability distribution computed from the simulation results almost overlaps the experimental curve shown in Fig. 18.

Fig. 19 shows the specific surface area of trapped hematite particles throughout the core from the simulation. A corresponding effective average particle diameter, computed as Dp = 6sh/S0, is also shown. Obviously, the average size of trapped particles is larger at the inlet end because smaller particles will penetrate deeper.

This case demonstrates the capability of the pore-throat model to reproduce experimental data not only in terms of overall

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0 200 400 600 800 1,000 1,200 1,400 1,600 1,800

PV InjectedEf

fluen

t Hem

atite

, ppm

SimExp.1Exp.2Exp.4Sim A=0

Fig. 16Experimental and simulated effluent profile for hematite. Pore-lining adsorption is turned off in Sim A = 0. Experimental data are replotted from Al-Abduwani et al. (2003).

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FeO

3 C

once

ntra

tion,

PV

Sim

Exp.

Fig. 17Experimental (Al-Abduwani et al. 2003) and simulated hematite distribution in core after 662-PV injection.

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ized

Per

mea

bilit

y

SimSim. segment

Exp. segment

Fig. 18Experimental (Al-Abduwani et al. 2003) and simulated permeability profile in core after 662-PV injection.


permeability reduction but also by matching internal-damage and particle-deposition profiles. Although promising, it is still uncer-tain how well the parameters in Table 5 will adapt to different rock types, other permeability levels, and with other fluid/solid systems. The test was performed with hematite particles in distilled water flooded through a highly permeable homogeneous Bentheimer core and with only approximate knowledge of the particle dis-tribution. One might expect increased trapping rate and more damage in less-uniform rocks with poorly sorted sands and higher tortuosity. One benefit of the applied Carman-Kozeny approach is that core properties such as permeability, porosity, and tortuosity are included in the computations; however, additional testing is required before one can tell if this is done adequately.

Fines Migration and Clay Swelling. Simulations of nes migra-tion and clay swelling upon injection of low-salinity brine are dem-onstrated in Fig. 20. In the nes simulation, 1% of the rock matrix is de ned as nes, held in place using the pore-lining-adsorption model. In the clay-swelling simulations, 2% of the rock matrix is de ned as swelling clay with a maximum swelling capacity of 10 (Qswmax in Eq. 45). Other parameters in Eq. 45 are ksw1 = 0.0002 mEq/(mLmin) and ksw2 = 0.001 mEq/(mLmin). The effective salinity Cse is essentially equal to the NaCl concentration indicated in Fig. 20, and is 0.7 mEq/mL with the initial brine.

Field-Scale Simulations. To test the upscaling to eld conditions, we use Mud 3b on a horizontal well from the North Sea. Perme-ability and porosity distribution along the wellbore is given in Fig. 21. We set the wellbore diameter to 0.24 m and use a constant pres-sure boundary at 100 m in our 2D radial model. The temperature is 90C. Filtrate invasion is simulated for a period of 250 hours with an overbalance pressure of 86 bar. Then, we simulate cleanup by

back producing the well using a 30-bar drawdown for 24 hours. Four cases are run, FS1 with static ltration and FS2 through FS4 with dynamic conditions at shear rate of 170 s1. Cases FS2 through FS4 correspond to various matches to the dynamic core experiment FD15 and dynamic- ltration simulations DS2 through DS4 shown in Table 8.

The simulations are summarized by main numbers in Table 11. Typical results from the simulations are demonstrated by Simula-tion FS4 in Figs. 22 through 25. Fig. 22 shows the saturation distribution around the well after mud exposure for 200 hours and after the 24-hour back-production period. The variation in penetration depth is mainly a result of the porosity distribution along the wellbore because filtration rate through the filter cake is independent of the formation properties. After the back production, we see that there is still a significant amount of water around the well, indicating that substantial production is needed to re-establish original fluids saturation. This is a result of bypassing caused by the heterogeneous permeability distribution.

Figs. 23 and 24 show the computed permeability impairment along the well. The normalized oil permeability for each zone is computed by two methods in Fig. 23: (1) as a productivity index

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S0 T

rapp

ed F

eO3

(m/c

m)

0

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1.5

2

2.5

3

Trap

ped

Part

icle

D,

m

S0

Dp (m).

Fig. 19Profiles of specific surface area and corresponding particle size of hematite in core after 662-PV injection (simulated).

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0 5 10 15PV Injected

Pres

sure

Dro

p, b

ar

FinesSwelling clay

Brine0.5 N NaCl

0.1 N NaCl

0.01 N NaCl

0.001 N NaCl Brine

Fig. 20Simulation of fines mobilization and clay swelling.

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Permeability

Wel

lbor

e, m

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fraction

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Radial Direction, m

Porosity

Wel

lbor

e, m

Fig. 21Field case, permeability (logarithmic) and porosity distributions.


computed from the rate in each zone and (2) by analytically ignoring crossflow between zones (RKO). The error introduced by ignoring crossflow is very small in this case and almost zero for the total well with relative PIoil and RKO equal to 0.794 and 0.792, respectively, in FS4. The same results are plotted in terms of skin distribution in Fig. 24 where S (current) is computed from the oil rate at the end of the 24-hour back-production period and corresponds to RKO. S (kabs) is computed from the reduction in absolute permeability, corresponding approximately to RKOE. The difference between the two skin values is because of the incom-plete back production of water. The radial distribution of RKO and RKOE along the well is given in Fig. 25.

TABLE 11FIELD-CASE SIMULATION RESULT, AFTER 250 HOURS OF MUD EXPOSURE (86 bar, 170 s1)

AND AFTER 24 HOURS OF BACK PRODUCTION (30-bar DRAWDOWN)

Simulation Vf (m3) RKO S (current) S (absolute k)

FS1 55 0.969 0.22 0.011 FS2 504 0.872 1.0 0.005 FS2c* 225 0.924 0.6 0.008 FS3 1717 0.732 2.3 0.003 FS4 702 0.842 1.2 0.004 FS4b** 535 0.833 1.3 0.005 FS4c* 251 0.895 0.8 0.008

* Dynamic plus static periods (50 plus 25 hours); static for the last 50 hours. ** Included rate of penetration 10m/h.

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0.860

0.691

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So, 200 hours #FS4

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0.860

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So, 24 hours Back production. #FS4

Radial Direction, mW

ellb

ore,

mRadial Direction, m

Wel

lbor

e, m

Fig. 22Oil distribution around well after 200 hours of mud exposure at 86 bar, 170 s1 (left) and after 24 hours of back production at 30-bar drawdown (right).

00.10.20.30.40.50.60.70.80.9

1

0 200 400 600 800 1000 1200 1400Wellbore, m

PI, r

elat

ive

Oil RKO#FS4

Fig. 23Simulated FS4. Relative productivity index for oil, from simulated well rate and computed ignoring crossflow (RKO). After 24 hours of production at 30-bar drawdown.

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Skin

S (current) S (Kabs)#FS4

Fig. 24Simulated FS4. Skin distribution computed from current productivity and from current absolute permeability. After 24 hours of production at 30-bar drawdown.


From the upscaling of laboratory results with Mud 3b, we conclude that no significant real damage to the formation should be expected from externally introduced particles in the present case. Other internal damage processes are not evaluated in this example. The real damage, as indicated by RKOE in Fig. 25, is in line with the observed polymer damage observed in the laboratory experiments. However, one should expect temporarily reduced productivity because of incomplete displacement of mud filtrate. The size and duration of this decline depend on filtrate penetration depth, relative permeability, and the variance of the permeability distribution.

The fluid-loss volumes in Table 11 are very high; in fact, they are an order of magnitude larger than expected. The worst case is FS3, in which a high erosion rate is assumed. Using a steady-state slowness of 5628 s/cm corresponding to that obtained for DS3 in Fig. 13 but corrected for the different shear rate in FS3 and the filtration area of the well (DwLw = 1094 m), we compute a filtrate volume of 1750 m compared to 1717 m given in Table 11. Similarly, the static fluid loss for the well is estimated to be 55.6 m using the filtration parameter Cst from FD3 (0.0032 m/mh0.5), which compares well to the simulated 55 m in Table 11. The deviations are because of differences in overbalance pressure and low-permeability zones offering flow resistance comparable to the mudcake resistance. These computations indicate that the field-scale simulations are correct in terms of upscaling the labo-ratory results, but our description of the field conditions may be insufficient. Of course, our interpretation of the laboratory results is very important, and, in the case of FD15, the estimated dynamic parameters are very uncertain. But all the interpretations used pre-dicted large fluid-loss volumes in the field case. Other explanations should be sought.

One obvious note is that the formation will be gradually exposed to the mud as the well is drilled. In the simulations, the total well was exposed from time zero. Exposing the formation according to a well penetration rate of 10 m/h is shown in Fig. 26 (FS4b). However, this will result only in a moderate reduction of the filtrate volume from 702 to 535 m. Simulating with periods

of 50 hours dynamic plus 25 hours static and, last, 50 hours of static reduces the loss to 251 m (FS4c). In Simulation FS2, only reduced mudcake growth and no erosion are assumed. Apply-ing the same periodic dynamic/static history on FS2 reduces the fluid loss from 504 to 225 m (FS2c). Variations in shear rate are obviously important, and knowledge about how the mudcake will respond to increased shear stress after a static period is vital for good predictions at the field scale. Other possibilities that may contribute to higher than expected computed fluid loss at the field scale are that the effective shear rate is lower than projected because of off-centered drillstring and the presence of significant surface roughness in the well compared to the flat core ends used in laboratory experiments.

Summary and ConclusionsMaximize is a program for computation of formation damage imposed on the formation during drilling. Among the main physical phenomena modeled in the program are filter-cake buildup under static and dynamic conditions, fluid loss to the formation, transport of solids and polymers in the formation including effects of pore-lining retention and pore-throat plugging, and salinity effects on fines stability and clay swelling. The models handle multicompo-nent water-based-mud systems at both core and field scales.

Our filter-cake model differs from previous models in that we compute the filter cakes porosity and permeability from proper-ties ascribed to the individual solid and polymer components in the mud. The necessary data for each component are porosity and compressibility, if packed as a sandpack, and specific surface area. The specific surface area is obtained from the size distribu-tion for solid particles and from the average chain diameter of polymers. The models capability to reproduce various fluid-loss experiments is demonstrated. For a CaCO3 mud, the effects of varying temperature, differential pressure, and addition of drilling solids are all simulated with a single set of input parameters.

The particle retention is split into two main mechanismspore-throat trapping and pore-lining retention. We focus mainly on the pore-throat trapping, which is the process with the main damage potential. A dilute-particle-injection experiment was selected for testing the model. The simulation provided excellent matching of

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RKO, #FS4, 24 hours back production

Wel

lbor

e, m

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Radial Direction, m

RKOE, #FS4, 24 hours back production

Wel

lbor

e, m

Fig. 25Normalized oil permeability after 24 hours of produc-tion RKO (left) and RKOE computed at maximum So (right), simulated FS4.

Fig. 26Simulated FS4b, with well penetration rate of 10 m/h.

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Radial Direction, m

So, 125 hours, Sim. SF4b

Wel

lbor

e, m


final experimental profiles of retarded particles and permeability along the core.

Upscaling of laboratory results to the field scale is demonstrated for a North Sea well. Only damage from externally introduced mud particles and polymers was considered. The simulations indicated essentially no permanent damage, in line with the moderate poly-mer damage observed in the laboratory experiments. However, a temporary decline in productivity because of slow displacement of invaded fluids was observed. The slow restoration of initial satura-tion around the well is attributed to the heterogeneous permeability distribution and to large fluid-loss volumes in the simulations. The simulated fluid-loss volume is an order of magnitude larger than that normally observed in the studied field cases. The dynamic fluid loss had been tuned to a dynamic laboratory experiment, and analytical calculations verify that the simulated results are correct in terms of upscaling the fluid-loss rate at the laboratory scale.

Nomenclature a = Mw exponent for polymer viscosity A = area, L2 A = adsorbed material (PV), solids or polymer Bp = viscosity parameter for pressure, Lt2/m BT = viscosity parameter for temperature, T BvT = polymer viscosity parameter, T1 c = concentration (volume fraction) C1 = ltration coef cient, L/L Cse = effective salinity, N Cst = static ltration parameter, L3t0.5 D = diameter, L Dch = polymer-chain diameter, L Dhp = hydraulic polymer diameter (polymer size in solution), L fn = friction factor fX = fractional concentration of cation at the rock surface, N/N h = thickness, L k = permeability, L k, k = polymer-viscosity parameters, Eq. 15 ksw1 = clay-swelling parameters, Nt1 ksw2 = clay-deswelling parameters, N1t1 KXC = cation-exchange parameter, Nz1, where z is the valence Mw = molecular weight, g/mol n = number nc = lter-cake compressibility, Lt2/m nF = dynamic- ltration exponent N = normality, meq/mL p = pressure, m/Lt2 P = polymer shear-thinning exponent, Eq. 17 PI = productivity index, rate/drawdown PV = pore volume, used as normalized volume unit q = ow rate, L3/t RKA = absolute permeability normalized on initial k RKO = oil permeability normalized on initial koi RKOE = ko at current maximum So normalized on initial ko (maxi-

mum So) S = skin factor (dimensionless), saturation S0 = speci c surface area (particle surface/particle volume), L1 t = time, t T = temperature, T u = Darcy velocity, u = q/A, L/t V = volume, L3 X = ion concentration, N Y = erosion rate, L/t z = 1, ion valence = effective mud fraction in dynamic ltration = critical size parameter for particle ow in porous medium nk = n = 11, 22: trapping parameters for solids or polymer (k =

s, p). n = 41, 42: adsorption parameters for solids or polymer (k = s, p).

c = effective salinity parameter e = erosion-rate parameter, t1 = shear rate, t1 hf = polymer-viscosity parameter, t1 [] = intrinsic polymer viscosity, L3/m nk = n = 1, 2, 10, 20,

Documents

SPE 122241 PA (Formation Damage and Well Productivity Simulation)