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14 PETROPHYSICS February 2005

The Influence of Water-Base Mud Properties and Petrophysical

Parameters on Mudcake Growth, Filtrate

Invasion, and Formation Pressure

Jianghui Wu1, Carlos Torres-Verdín

2, Kamy Sepehrnoori

2, and Mark A. Proett

3

INTRODUCTION

Mud filtrate invasion takes place in permeable rock for-

mations penetrated by a well that is hydraulically overbal-

anced by mud circulation. The invasion of mud filtrate into

permeable rock formations is responsible for the develop-

ment of a mudcake on the borehole wall (solids deposition),

as well as for the lateral displacement of existing in-situ flu-

ids from the borehole. Drilling variables such as mud den-

sity and chemistry, mud circulation pressure, and time of fil-

tration may significantly affect the spatial extent of mud-fil-

trate invasion. In-situ rock formation properties such as

porosity, absolute permeability, relative permeability, pore

pressure, shale chemistry, capillary pressure, and residual

fluid saturations, also play important roles in controlling

both the dynamic formation of mudcake and the time

evolution of the invasion process.

One of the technical problems often considered in

mud-filtrate invasion studies is the description of mudcake

buildup and invasion rates. Over the years, many laboratory

investigations have undertaken the phenomenological

description and quantification of this problem (e.g., Fergu-

PETROPHYSICS, VOL. 46, NO. 1 (FEBRUARY 2005); P. 14–32; 30 FIGURES, 3 TABLES

ABSTRACT

The work described in this paper models the complete

invasion process quantitatively with a finite-difference

invasion simulator that includes the dynamically coupled

effects of mudcake growth and multiphase, immiscible

filtrate invasion. A fully coupled mudcake growth model

is assumed and the flow rate of filtrate invasion is deter-

mined from both mud parameters and rock formation

properties.

Specific parametric representations of the assumed

invasion model are based on previously published labora-

tory experiments on mudcake buildup. As part of the

numerical validation of the simulator, we reproduced

available experimental data and obtained very good

agreements.

The influence of several mud and petrophysical param-

eters on both mudcake growth and filtrate invasion is

quantified with a sensitivity analysis. These parameters

include mudcake permeability, mudcake porosity, mud

solid content, relative permeability, capillary pressure,

formation permeability, cross flow between adjacent lay-

ers, and gravity segregation. Our simulations reveal the

physical character of invasion profiles taking place under

realistic petrophysical conditions. Results also character-

ize formation pressure changes and pressure supercharg-

ing observed during wireline formation testing.

Keywords: mud filtrate, mudcake, invasion, super-

charge

Manuscript received by the Editor July 30, 2004; revised manuscript received December 17, 2004.1Baker Atlas, 2001 Rankin Road, Houston, Texas 77073; e-mail: [email protected] of Petroleum and Geosystems Engineering, 1 University Station C0300, The University of Texas at Austin, Austin, Texas

78712; e-mail: [email protected], [email protected] Energy Services, 3000 N. Sam Houston Parkway E., P. O. Box 60070 (77204), Houston, Texas 77032; e-mail:

[email protected]

©2005 Society of Petrophysicists and Well Log Analysts. All rights reserved.

son and Klotz, 1954; Bezemer and Havenaar, 1966;

Fordham et al., 1988 and 1991; Dewan and Chenevert,

2001). Based on laboratory experiments of mud circulation,

Dewan and Chenevert (2001) reported a methodology to

predict the time evolution of mudcake buildup, as well as

the effective petrophysical properties of mudcake. Dewan

et al.’s description is entirely based on six mud filtrate

parameters, all of which can be determined from a standard

on-site mud filtrate test.

A theoretical basis for laboratory and field observations

was first presented by Outmans (1963). He described a

method applicable to high permeability formations assum-

ing that, even at the onset of the process of mud-filtrate

invasion, the full overbalance pressure was absorbed across

the mudcake. In his single-phase fluid flow study, Outmans

derived the well-known t law. However, the latter descrip-

tion is not accurate when the net flow resistance offered by

the formation is comparable to that of the mudcake.

Semmelbeck et al. (1995) introduced a two-phase fluid

flow simulator that assumed several of the mud properties

described in Dewan and Chenevert (2001). Gravity force

was ignored in Semmelbeck et al.’s work because only low

vertical permeability cases were considered in the analysis.

Dewan and Chenevert (2001) presented a single-phase flow

mathematical model to reproduce laboratory measure-

ments. The simulation results matched laboratory measure-

ments using a 0.25-inch core to represent the rock forma-

tion. However, this model cannot be applied to low perme-

ability formations, where the net flow resistance offered by

the formation is comparable to that of the mudcake. Proett

et al. (2001) developed an immiscible invasion simulator

using a fully coupled mudcake growth model. Wu et al.

(2004) proposed a methodology to simulate mud-filtrate

invasion in deviated and horizontal wells.

In this paper, a new invasion simulator, termed INVADE,

is developed that accurately reproduces the process of

mud-filtrate invasion in multi-layer formations, including

the effect of salt mixing.

FINITE-DIFFERENCE SIMULATOR-“INVADE”

The finite difference simulator INVADE was developed

based on the solution of fluid-flow differential equations

and boundary conditions for immiscible radial flow and

coupled mudcake growth. INVADE was built upon the

existing multi-phase, multi-component, and multi-chemical

species fluid-flow simulator UTCHEM, developed by The

University of Texas at Austin (Center for Petroleum and

Geosystems Engineering, 2000). UTCHEM can simulate

the advection, dispersion, diffusion, and transformation of

different species (oil, surfactant, water, salt, polymer, etc.)

in porous media under various production and injection

conditions. Because of these features, INVADE can be used

to numerically simulate the transport of salt due to invasion

of mud filtrate when there is a difference in salinity between

mud filtrate and connate water.

It is generally acknowledged that static filtration governs

the initial growth of mudcake and that the fundamental role

of dynamic filtration is to limit this growth (e.g., Chin,

1995; Dewan and Chenevert, 2001). In this paper, we

assume that mudcake stops growing after reaching its limit-

ing thickness. By using a static filtration model other than

the dynamic filtration model, mudcake growth is acceler-

ated because the dynamic filtration tends to decrease the

rate of mudcake growth. Well conditioning can also affect

the mudcake and influence the process of invasion. Fre-

quently, the well is reconditioned prior to a logging run or in

the process of drilling. To account for reconditioning, we

can assume that the mudcake is either partially or com-

pletely removed, and that it is henceforth allowed to reform.

In the worst-case scenario, the mudcake is completely

removed from the borehole wall. Therefore, the limiting

mudcake thickness is a parameter input to INVADE, and so

is the option to remove the mudcake in order to simulate

reconditioning of the well.

The INVADE software has one extra input file in addi-

tion to the input files used by UTCHEM. This file, named

MUD contains parameters such as mud properties, mud

pressure, and “rub-off ” time. The flow rate of mud filtrate

calculated with equation (A.29) will be treated as a standard

rate of injection for the well.

In the Appendix, the mudcake growth model is coupled

with the two-phase immiscible Darcy flow boundary value

problem developed for radial invasion. It is noted that equa-

tion (A.29) used for the calculation of filtrate flow rate is

derived from a one-dimensional model. For multi-layer

cases, cross-flow between layers is simulated with

UTCHEM, while equation (A.29) is used for the calculation

of the rate of flow of mud filtrate.

Relative permeability and capillary pressure curves

In the development of the immiscible flow model, no

assumptions were made concerning the behavior of relative

permeability and capillary pressure curves. Therefore,

these curves are completely arbitrary and can be a power

function adjusted to match core measurements. The curves

shown in Figure 1 correspond to a typical behavior of rela-

tive permeability for a water-wet sandstone (� = 0.25, K =

300 md). In this case, the relative permeability curves can

be characterized by the saturation-dependent Brooks-

Corey-type equations (Brooks and Corey, 1966), given by

k k Srw rw wtew� �0 ( ) ,* (1)

February 2005 PETROPHYSICS 15

The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure

k k Sro ro wteo� � �0 1[ ( )] ,* (2)

and

SS S

S Swt

w wi

wi or

* ,��

� �1(3)

where S wt* is normalized water saturation, Sw is water satura-

tion, Swi is irreducible water saturation, Sor is residual oil sat-

uration, krw is water relative permeability, kro is oil relative

permeability, krw0 is water relative permeability endpoint,

kro0 is oil relative permeability endpoint, ew is the exponent

for krw, and eo is the exponent for kro.

The capillary pressure curve shown in Figure 2 is charac-

terized by a relationship of the form

P PK

Sc c wte p� � �0 1

�[ ( )] ,* (4)

where Pc is capillary pressure between oil and water, � is

formation porosity, K is formation permeability, Pc0 is the

coefficient for capillary pressure, and ep is the exponent for

capillary pressure.

The exponents in equations (1) through (4) (i.e., ew, eo,

ep) control the shape of the curves, whereas the coefficients

in the same equations (i.e., krw0 , kro

0 , Pc0) control the location

of the endpoints. The coefficient for capillary pressure, Pc0,

is set to 2 for the case of a 300 md formation, while for the

case of low-permeability formations, Pc0 is set to 0.2. These

saturation-dependent functions can approximate most prac-

tical cases of relative permeability and capillary pressure

curves and are used for the INVADE examples described in

this paper.

COMPARISON OF INVADE PREDICTIONS WITH

LABORATORY MEASUREMENTS

To validate the numerical simulator, we attempted to

reproduce the experimental data acquired during a static fil-

tration test performed with field Mud 97074 (Dewan and

Chenevert, 2001). In this particular case, mud properties are

as follows: mudcake reference permeability Kmc0 = 0.003

md, mudcake reference porosity �mc0 = 0.59, solid fraction

fs = 0.231, mudcake thickness = 0.25 cm, compressibility

exponent for mudcake permeability v = 0.63, and exponent

multiplier for mudcake porosity � = 0.1. In step 1, a pressure

of 300 psi is applied during 30 minutes and the filtrate vol-

ume and slowness (inverse of flow rate) are recorded after

the onset of invasion. In step 2, the pressure is raised to 1000

psi and the recording continues for another 30 minutes.

A radial invasion model is constructed such that the fil-

tration area is the same as that of filter paper (45.8 cm2). We

assume a wellbore radius equal to 10 cm. The distance

between the wellbore and the formation’s outer-boundary is

0.635 cm (0.25 inch) to match the thickness of the filtration

medium used in the experiment. INVADE enforces a con-

stant-pressure condition at the outer boundary, whereas flu-

ids can move freely out of the outer boundary. The influ-


Wu et al.

FIG. 1 Water-oil relative permeability curves assumed in thenumerical simulations described in this paper. The solid anddashed curves describe relative permeabilities as a function ofwater saturation for water and oil fractions, respectively.

FIG. 2 Capillary pressure curve used in the sensitivity analysisof mud-filtrate invasion. This curve represents two formations: a300 md formation with Pc

0 = 2, and a 3 md formation with Pc0 =

0.2.

ence of the radial geometry of the simulation is negligible in

this case. Only single-phase fluid-flow (water) was consid-

ered in the experiment, whereupon the initial water satura-

tion for the model was set to 1.0.

In Figure 3, S1 represents the slowness at the end point of

step 1, while S2 represents the slowness at the start point of

step 2. Figure 3 indicates that INVADE results exhibit a

good agreement with slowness measurements of step 1 and

S1, but S2 calculated with INVADE is about 30% higher

than the measured S2. This discrepancy causes a transient

pressure kick during the measurement sequence when pres-

sure increases from 300 psi to 1000 psi.

The compressibility index v is calculated from the mea-

sured slowness, S1 to S2, when pressure is increased from P1

to P2 in the two-step recording shown in Figure 3, i.e.,

vS S

P P� �1

1 2

2 1

log( / )

log( / ). (5)

Conversely, S2 could be predicted if other parameters

were known. In the example shown in Figure 3, we find S1 =

37500 sec/cm, v = 0.63, P1 = 300 psi and P2 = 1000 psi,

resulting in S2 = 24000 sec/cm. This value agrees well with

the calculation performed with INVADE.

To assess mudcake growth in low-permeability forma-

tions, we make use of the following constants applicable to

UT mud: kmc0 = 0.03 md, �mc0 = 0.8, fs = 0.06, v = 0.9 and � =

0. The wellbore radius of the radial model is 10 cm and the

distance between the wellbore and the formation’s

outer-boundary is set to 0.635 cm (0.25 in.) in order to

match the thickness of the filtration medium used in the

experiment. Again, only single-phase fluid-flow (water)

was assumed in the experiment, whereupon the initial water

saturation for the model was set to 1.0.

Figure 4 (solid lines) compares the calculated volume of

filtrate as a function of t against the experimental mea-

surements performed at UT Austin. Aside from a 1.2 cc off-

set at t = 0 for the experimental data, (commonly referred to

as “spurt loss”) the agreement is good. By shifting the time

axis, simulation results will properly match experimental

measurements. To match this experiment in the presence of

“spurt loss,” it is expected to take 4.6 minutes less than the

calculated time (400 minutes) for the mudcake to reach the

thickness of 0.25 cm.

Also shown in Figure 4 (dashed line) is the calculated

mudcake thickness, which grows linearly with t , reaching

a value of 0.25 cm in approximately 6.7 hours. Figure 5

shows the pressure buildup across the mudcake as a func-

tion of time after enforcing a 50 psi overbalance pressure.

The 3 md curve in Figure 5 shows that even though it takes

hours to fully build, sufficient mudcake deposits within 2

seconds to absorb 90% of the full overbalance pressure.



FIG. 3 Comparison of the time evolution of the measured andsimulated slowness of static filtration for Mud 97074. Slownessis the inverse of invasion flow rate (the filtration area of filterpaper is 45.8 cm2).

FIG. 4 Comparison between measurements and numericalsimulations. Time evolution of volume of filtrate and mudcakethickness during a static filtration test performed through a 3 mdrock core sample. For convenience, time-dependence isdescribed with the square root of the actual time of invasion(adapted from Dewan et al., 1993). The square-dashed andsolid lines describe measured and simulated volumes of filtrate,respectively, whereas the dashed line describes the simulatedmudcake thickness.

INVADE SIMULATION RESULTS

Base Case

A “Base Case” was chosen as a reference example for the

simulation of mud filtrate invasion. Input data for this base

case include the relative permeability and capillary pressure

curves shown in Figures 1 and 2, respectively, with the

remaining variables described in Table 1. Results from the

INVADE simulation are shown in Figure 6 in the form of

profiles of water saturation as a function of radial distance

away from the borehole wall. The first curve shows the

most significant invasion advance, with the front advancing

to 0.2 m within six hours. This advance is due to the fact

that initially there is no mudcake and it takes about 19 hours

for it to reach a thickness of one centimeter (see also Figure

12). The water saturation curves do not exhibit the pis-

ton-like behavior that would be expected with immiscible

invasion that does not consider capillary pressure. To mini-

mize the numerical dispersion effects, a third-order spatial

discretization option is selected in the simulation input file

for INVADE.

Radial profiles of formation water salinity are shown in

Figure 7. Salt concentrations are converted into equivalent

values of connate water resistivity, Rw, using the formula

(Dresser Atlas Inc., 1982)

R rC r T

w

w

( )( )

,�

��

�

�� 0.0123

3647.5 82

1.8 390.955(6)

where T is temperature measured in degrees centigrade, Cw

is salt concentration measured in ppm, and�

r is the location

of the observation point. In turn, electrical resistivities are

calculated via Archie’s law from the corresponding spatial

distribution of water saturation shown in Figure 6. The cal-

culated radial profiles of electrical resistivity are plotted in

Figure 8.

Mud filtrate invasion flow rate, volume of filtrate,

buildup of pressure across the mudcake, and mudcake

thickness are shown as solid lines (Kmc0 = 0.03 md) in Fig-

ures 9, 10, 11 and 12, respectively. The coupled mudcake

model allows the thickness to increase to a maximum of 1

cm, where it is assumed that dynamic filtration limits its

growth. It takes about eight seconds (0.0001 day) before a

noticeable increase of mudcake properties is observed. At

this time, the invasion flow rate is reduced by nearly 95%

from its initial rate. The pressure drop across the mudcake

has also increased to 450 psi of the 500 psi overbalance


Wu et al.

FIG. 5 Pressure at the sandface during static filtration. Timeevolution of pressure at the sandface for six different values ofrock-core permeability. The simulation of static filtration wasperformed through a 0.25-in. core sample maintained at a con-fining pressure of 50 psi.

TABLE 1 Summary of mudcake, petrophysical, fluid, andinvasion parameters used in the numerical simulations ofmud-filtrate invasion considered in this paper.

Variable Units Value

Mudcake reference permeability md 0.03

Mudcake reference porosity fraction 0.30

Mud solid fraction fraction 0.06

Mudcake maximum thickness cm 1.00

Mudcake compressibility exponent v fraction 0.40

Mudcake exponent multiplier � fraction 0.10

Water viscosity (filtrate) cp 1.00

Oil viscosity cp 3.00

Rock compressibility 1/psi 0.0E-6

Water compressibility 1/psi 0.0E-6

Initial formation pressure psi 5000.00

Mud hydrostatic pressure psi 5500.00

Formation permeability md 300.00

Formation porosity fraction 0.25

Permeability anisotropy fraction 1.00

Coefficient for capillary pressure Pc0 psi 2.00

Exponent for capillary pressure ep n/a 6.00

Total invasion time hours 48.00

Mudcake rub-off time hours N/A

Wellbore radius cm 10.00

Formation outer-boundary cm 610.00

Mud filtrate salinity ppm 43,900.00

Formation water salinity ppm 102,500.00

Archie’s tortuosity/cementation factor a n/a 1.00

Archie’s cementation exponent m n/a 2.00

Archie’s saturation exponent n n/a 2.00

Temperature °C 24.00



FIG. 9 Sensitivity analysis of mudcake reference permeabilityand flow rate across the mudcake. Time evolution of the flowrate of mud filtrate for three different values of mudcake refer-ence permeability.

FIG. 7 Time-lapse simulation: radial profiles of salt concentra-tion. The curves describe salt concentration in the radial direc-tion from the wellbore and across the center of the permeablelayer at 6-hour increments after the onset of mud-filtrate inva-sion. The corresponding radial distributions of water saturationare shown in Figure 6.

FIG. 6 Time-lapse simulation: radial profiles of water satura-tion. The curves describe water saturation in the radial directionfrom the wellbore and across the center of the permeable layerat 6-hour increments after the onset of mud-filtrate invasion.

FIG. 8 Time-lapse simulation: radial profiles of electrical resis-tivity. The curves describe electrical resistivity in the radial direc-tion from the wellbore and across the center of the permeablelayer at 6-hour increments after the onset of mud-filtrate inva-sion. The corresponding radial distributions of water saturationand salt concentration are shown in Figures 6 and 7, respec-tively. Electrical resistivity was calculated assuming Archie’sequations with tortuosity/cementation factor a = 1, cementationexponent m = 2, and saturation exponent n = 2.

pressure. The mudcake continues to thicken until a 1 cm

thickness is reached at about 19 hours after the onset of

invasion. At this point, the invasion flow rate has been

reduced to a small fraction of its initial value and the

sandface pressure has reached its steady state.

Mudcake permeability decreases with increasing pres-

sure across the mudcake. The solid line (Kmc0 = 0.03 md) in

Figure 13 shows that mudcake permeability is stabilized at

2.5 10–3 md after eight seconds of invasion.

Case of mudcake removal

In this case, mudcake is removed after one day of inva-


Wu et al.

FIG. 12 Sensitivity analysis of mudcake reference permeabilityand mudcake thickness. Time evolution of mudcake thicknessfor three different values of mudcake reference permeability.

FIG. 10 Sensitivity analysis of mudcake reference permeabilityand accumulated volume of mud filtrate. Time evolution of theaccumulated volume of mud-filtrate for three different values ofmudcake reference permeability.

FIG. 11 Sensitivity analysis of mudcake reference permeabilityand pressure across the mudcake. Time evolution of the pres-sure across the mudcake for three different values of mudcakereference permeability.

FIG. 13 Sensitivity analysis of mudcake reference permeabilityand mudcake permeability. Time evolution of mudcake perme-ability for three different values of mudcake reference perme-ability.

sion. As shown in Figure 14(a), during the next day of inva-

sion the thickness increases. Mudcake thickness increases

at nearly the same rate as before the removal of mudcake,

taking about 19 hours to reach the 1 cm maximum thick-

ness. Figure 14(b) shows that the invasion flow rate

increases nearly instantaneously to about 30% of the initial

rate and then is slowed down to a fraction of its original rate as

the mudcake reforms. Comparison of the water saturation pro-

files described in Figure 15 against those shown in Figure 6

indicates that there is a sudden increase of invasion length

after mudcake is removed at the end of one day, and the final

invasion front is 8 cm deeper than in the base case (46 cm).

SENSITIVITY ANALYSIS

A sensitivity analysis was performed using the base case

example as reference by varying the selected parameters

shown in Table 2. In the base case, mudcake reference per-

meability (0.03 md) is very small compared to that of the

formation permeability (300 md). Therefore, the mudcake

will control the flow rate of filtrate invasion. Three parame-

ters related to mud properties are selected for the sensitivity

analysis, i.e., mudcake reference permeability, porosity,

and filtrate solid fraction.

Mudcake Permeability

Mudcake properties, particularly permeability, have a

significant influence on the invasion process. The assertion

that mudcake permeability is significant is clearly true for

high permeability zones since the mudcake is the primary

regulator of the flow rate of filtrate invasion.

Figure 10 shows that the total filtrate volume after two

days of invasion increases with increasing mudcake perme-

ability. As expected, pressure across the mudcake decreases

with increasing values of mudcake permeability. The latter



FIG. 15 Time-lapse simulation: radial profiles of water satura-tion. The curves describe water saturation in the radial directionfrom the wellbore and across the center of the permeable layerat 6-hour increments after the onset of mud-filtrate invasion.Mudcake is removed after one day of invasion and is allowed tore-grow during the subsequent one-day interval.

FIG. 14 Time evolution of (a) mudcake thickness, and (b) flowrate of mud filtrate during the process of invasion. Mudcake isremoved after one day of invasion and is allowed to re-grow dur-ing the subsequent one-day interval.

TABLE 2 Summary of the test cases considered in the sen-sitivity analysis of mud-filtrate invasion reported in thispaper. The table shows input variables and results obtainedfrom the simulations.

Mudcake properties Invasion Results

Filtrate Mudcake

volume buildup

Case Kmc0 after 2 days time

no. (md) �mc0 fs (m3/m) (hours)

Base 0.030 0.3 0.06 0.124 19.0

1 0.010 0.3 0.06 0.066 56.9

2 0.003 0.3 0.06 0.037 189.5

3 0.030 0.5 0.06 0.117 15.1

4 0.030 0.8 0.06 0.106 9.3

5 0.030 0.3 0.20 0.097 4.8

6 0.030 0.3 0.40 0.091 1.8

observation is confirmed by the family of curves shown in

Figure 11.

The mudcake buildup time is reduced with increasing

mudcake permeability because the deposition of solids

depends on the flow rate of filtrate, which is higher for

high-permeability mudcake. It takes 190 hours for a

0.003-md mudcake to thicken to 1 cm, while it takes 19

hours for a 0.03 md mudcake to reach the same thickness.

Mudcake porosity

Mudcake porosity has an influence on mudcake growth.

The mudcake buildup time decreases with increasing

mudcake porosity because the total amount of solid deposi-

tion decreases with an increase of porosity.

It takes 9.3 hours for a 0.8 porosity mudcake to build

while it takes 19 hours for a 0.3 porosity mudcake to reach

the same thickness (Table 2).

In Figure 16, the flow rate for a high-porosity mudcake

decreases because of relatively thicker mudcake. The total

filtrate volume decreases with increasing mudcake porosity

(Table 2). In Figure 17, the pressure across mudcake

increases with increasing mudcake porosity because of a

thicker mudcake.

Solid fraction

Filtrate solid fraction also has a considerable influence

on mudcake growth. The time of mudcake buildup

decreases with increasing solid fraction because the speed

of solid deposition increases.

Figure 18 shows that the flow rate for a high solid-con-

tent mud decreases because of relatively thicker mudcake.

The total filtrate volume decreases with an increase of mud

solid fraction (Table 2). In addition, Figure 19 shows that

the pressure across the mudcake increases with increasing

mud solid fraction because of thicker mudcake.


Wu et al.

FIG. 17 Sensitivity analysis of mudcake porosity and pressureacross mudcake. Time evolution of the pressure acrossmudcake for three different values of mudcake reference poros-ity.

FIG. 16 Sensitivity analysis of mudcake porosity and flow rateacross mudcake. Time evolution of the flow rate of mud-filtratefor three different values of mudcake reference porosity.

FIG. 18 Sensitivity analysis of filtrate solid fraction and flowrate across mudcake. Time evolution of the flow rate acrossmudcake for three different values of filtrate solid fraction.

As shown in Table 2, it takes 1.8 hours for a mud with a

solid fraction of 0.4 to build while it takes 19 hours for a

0.06 solid fraction mud to reach the same thickness.

Invasion in low-permeability formations

When formation permeability exceeds a few milli-

darcies, sufficient mudcake forms in a matter of seconds,

and virtually the entire overbalance pressure driving the

invasion is absorbed across the mudcake. Therefore, the

rate of invasion is entirely controlled by mudcake proper-

ties rather than by formation properties. However, Dewan

and Chenevert (1993) showed that conditions are different

when permeability is lower than a few millidarcies. The ini-

tial invasion rate, limited by formation permeability, is suf-

ficiently low so that the pressure drop across the mudcake

increases very slowly.

To study how formation properties will affect invasion,

we make use of the set of formation and mudcake variables

given in Table 3. The mud properties are the same as those

of UT Austin Mud and the distance between the wellbore

and the formation’s outer-boundary is set to 610 cm (20 ft).

Two-phase immiscible flow is simulated with the

water-oil relative permeability curves shown in Figure 1.

The coefficient for the capillary pressure equation, Pc0, is

set to 0.2 and the exponent for the capillary pressure equa-

tion, ep, is set to 6.

Two formation properties, i.e., formation permeability

and oil relative permeability endpoint, are selected for this

sensitivity analysis. Capillary pressure influences the rate

of change of water saturation in the invaded zone but the

changes are small. Simulation tests indicate that other for-

mation properties will have only a slight influence on the

process of mudcake buildup.

Figure 20 shows the pressure across the mudcake as a

function of time after the enforcement of a 50 psi overbal-

ance pressure. It takes the 3 md formation 6.7 hours to build

a 45 psi pressure across the mudcake rather than the several

seconds indicated in Figure 5. The maximum pressure

across the mudcake takes place when the mudcake reaches

its maximum thickness. When the mudcake stops growing,

mud filtrate continues to invade into even deeper regions in

the formation, and therefore the resistance resulting from

the formation will increase. As a result, the pressure across

the mudcake will decrease and the pressure at the sandface

will increase. Figure 21(a) shows that the flow rate is higher

for a high permeability formation and will converge after

3.8 hours of invasion. After that, the flow resistance from

the 0.3 md formation continues to increase, resulting in a

gradually decreasing flow rate. As expected, Figure 21(b)

shows that the volume of filtrate is higher for a high perme-

ability formation.



FIG. 19 Sensitivity analysis of filtrate solid fraction and pres-sure across mudcake. Time evolution of the pressure acrossmudcake for three different values of filtrate solid fraction.

TABLE 3 Case of a low-permeability formation. Summaryof mudcake, petrophysical, fluid, and invasion parametersused in the numerical simulations of mud-filtrate invasionconsidered in this paper.

Variable Units Value

Mudcake reference permeability md 0.03

Mudcake reference porosity fraction 0.80

Mud solid fraction fraction 0.06

Mudcake maximum thickness cm 0.25

Mudcake compressibility exponent v fraction 0.90

Mudcake exponent multiplier � fraction 0.00

Water viscosity (filtrate) cp 1.00

Oil viscosity cp 1.00

Rock compressibility 1/psi 0.0E-6

Water compressibility 1/psi 0.0E-6

Initial formation pressure psi 5000.00

Mud hydrostatic pressure psi 5050.00

Formation permeability md 3.0/1.0/0.3

Formation porosity fraction 0.25

Permeability anisotropy fraction 1.00

Coefficient for capillary pressure Pc0 psi 0.20

Exponent for capillary pressure ep n/a 6.00

Total invasion time hours 48.00

Mudcake rub-off time hours N/A

Wellbore radius cm 10.00

Formation outer-boundary cm 610.00

The oil relative permeability endpoint affects invasion

through the resistance offered by the formation to the flow

of filtrate. As shown in Figure 22, increasing the value of

the endpoint translates to a decreasing resistance and, con-

sequently, the overbalance pressure lost in the formation

decreases, and pressure across the mudcake increases.

Figure 23(a) shows that higher values of kro0 cause higher

flow rates of invasion. Therefore, as shown in Figure 23(b),

mudcake buildup is faster and, consequently, the volume of

filtrate is slightly higher.


Wu et al.

FIG. 20 Time evolution of (a) pressure across mudcake, and(b) pressure at the sandface during a static filtration test for sixdifferent values of formation permeability. The static filtrationtest was performed assuming a 20-ft formation maintained at aconfining pressure of 50 psi.

FIG. 21 (a) Time evolution of filtrate flow rate for three differentvalues of formation permeability; (b) Time evolution of total fil-trate volume per formation thickness for three different values offormation permeability.

FIG. 22 Time evolution of pressure at the sandface during astatic filtration test performed assuming a 20-ft formation at aconfining pressure of 50 psi for three different values of forma-tion kro

0 .

FIG. 23 (a) Time evolution of filtrate flow rate for three differentvalues of formation kro

0 ; (b) Time evolution of the total filtrate vol-ume per formation thickness for three different values of forma-tion kro

0 .

FLOW RESISTANCE RATIO OF MUDCAKE

Formation properties will affect invasion only when the

formation flow resistance is comparable to that of the

mudcake. We denote the ratio of mudcake flow resistance

over total flow resistance by the variable �. The relation-

ship between � and the remaining parameters is summa-

rized by the equation

�

�

��

��

��

1

10K

K k P

R R

R R

mc

rwv

f well

well mc

o

( )

ln( / )

ln( / )�

K

K k P

R R

R R

mc

w rov

out f

well mc

0

� �� ( )

ln( / )

ln( / )

,

�

(7)

This last expression is derived from Darcy’s law by making

four assumptions: (1) there is no capillary pressure, (2) the

invasion profile is piston-like, (3) water is the only flowing

fluid in the invaded zone, and (4) oil is the only flowing

fluid in the virgin zone.

Supercharging

The flow resistance parameter � can also be used to

assess supercharging effects. Supercharging is defined as

the increased pressure observed at the wellbore sandface

caused by invasion. This is an important factor in wireline

formation testing since the pressures recorded are influ-

enced by supercharging. The supercharge pressure can now

be estimated by subtracting the pressure across the

mudcake from the overbalance pressure. Consequently, the

degree of supercharging is proportional to 1 – �.

Figures 24 and 25 show the relationship borne by the

supercharging index (1 – �) with K/Kmc0 and uo /uw. The

value of (1 – �) will decrease with increasing values of

K/Kmc0 and the value of (1 – �) will increase with increasing

values of uo /uw.

Now consider the time when the mudcake reaches its

maximum thickness. For the three cases (K = 3, 1, and 0.3

md) shown in Figure 20, the corresponding flow resistance

ratios are 0.92, 0.77, and 0.32, respectively. This ratio mul-

tiplied by the overbalance pressure will yield the pressure

across the mudcake when the mudcake reaches its maxi-

mum thickness. Pressures across the mudcake for these

cases are 46.0, 38.5, and 16.0 psi, respectively. The calcu-

lated pressures are in good agreement with the values

shown in Figure 20. As a rule of thumb, when � is smaller

than 0.9, the flow resistance offered by the formation

becomes comparable to that of the mudcake.

Figure 26 shows that (1 – �) will increase with increas-

ing invasion front radii. This agrees well with the fact

observed in Figure 20 that after the mudcake reaches its

maximum thickness, the pressure at the sandface increases

while the radius of the invasion front increases.

Figure 27 shows that (1 – �) will decrease with increas-

ing values of oil relative permeability. This agrees well with

the results shown in Figure 22.

In summary, the value of � calculated from equation (7)



FIG. 24 Graphical description of the relationship between thesupercharge index (1 – �) and K/Kmc0 and �o /�w. The plot wasconstructed assuming Rw = 10 cm, Rf = 18.3 cm, Rout = 610 cm,Rmc = 9.75 cm, overbalance pressure = 50 psi, kro = 1.0, krw =0.2, and compressibility exponent for mudcake permeability v =0.4 to 0.9.

FIG. 25 Family of curves that graphically describe the relation-ship between the supercharge index (1 – �) and K/Kmc0 and�o /�w. The curves were constructed assuming Rw = 10 cm, Rf =18.3 cm, Rout = 610 cm, Rmc = 9.75 cm, overbalance pressure =50 psi, kro = 1.0, krw = 0.2, and compressibility exponent formudcake permeability v = 0.4 to 0.9.

is a close approximation of the simulation results. A tem-

plate of � can be constructed to guide the assessment of

whether the resistance to flow offered by the formation is

comparable to that of mudcake flow resistance.

The influence of supercharging on wireline formation

tester measurements was described by Stewart and

Whittman (1979) and Proett et al. (2001).

INVASION IN MULTIPLE-LAYER FORMATIONS

All the invasion processes discussed above take place in

a single layer formation. In practice, cross flow will exist

when mud filtrate invades non-isolated multiple-layer for-

mations.

A three-layer formation model was constructed to illus-

trate the effect of cross flow between adjacent layers. The

mud properties are the same as those listed in Table 1. Layer

permeabilities are 30 md, 300 md, and 1000 md. Each layer

has a thickness of 61 cm (2 ft) and the formation’s

outer-boundary is set to 610 cm (20 ft). Densities for water

and oil are 1.0 and 0.85 g/cm3, respectively.

Figure 28 shows radial profiles of filtrate saturation after

two days of invasion. It can be observed that the low perme-

ability formation lost some filtrate to the higher permeabil-

ity formation due to cross flow between layers. The profile

of filtrate saturation in the 1000 md layer suggests the influ-

ence of gravity effects.

In addition to supercharging, gravity force can influence

the pressure gradient. In the three-layer case, the pressure

gradients for filtrate and oil were 0.433 and 0.368 psi/ft,

respectively. Figure 29 shows the time evolution of pres-

sures at the sandface for the 1000 md layer. At the initial

condition, the oil-bearing layer exhibits a pressure gradient

of 0.368 psi/ft. After one hour of mud-filtrate invasion, the

pressure gradient decreases to 0.328 psi/ft because of the

influence of the upper layer (K = 30 md), which sustains a

higher pressure due to supercharging. The pressure gradient

gradually increases to 0.378 psi/ft at 10 hours after the onset

of invasion. At the end of two days, the gradient becomes

0.398 psi/ft, i.e. 8% higher than the initial condition. As

suggested by Figure 29, the oil-water-contact location cal-

culated based on pressure tests after invasion may be raised

because of pressure-curve shifts, assuming that no changes

occur in the pressure gradient of the underlying water-bear-

ing zone.

SUMMARY AND CONCLUSIONS

1. The simulator described in this paper, referred to as

“INVADE”, can be used to match laboratory measure-

ments as well as to calculate invasion into multi-layer

formations. INVADE can also be used to simulate the

process of salt mixing between mud filtrate and connate

water.

2. For high permeability zones, both mudcake growth rate

and mud filtrate invasion rate are controlled primarily by


Wu et al.

FIG. 26 Graphical description of the relationship between thesupercharge index (1 – �) and K/Kmc0 for three different value ofRf. The curves were constructed assuming RW = 10 cm, Rout =610 cm, Rmc = 9.75 cm, overbalance pressure = 50 psi, kro = 1.0,krw = 0.2, compressibility exponent for mudcake permeability v =0.9, and �o /�w = 1.

FIG. 27 Graphical description of the relationship between thesupercharge index (1 – �) and K/Kmc0 for three different value ofkro. The curves were constructed assuming Rw = 10 cm, Rf =18.3 cm, Rout = 610 cm, Rmc = 9.75 cm, overbalance pressure =50 psi, krw = 0.2, compressibility exponent for mudcake perme-ability v = 0.9, and �o /�w = 1.

mud properties (i.e., mudcake permeability, mudcake

porosity and mud solid fraction).

3. For low permeability zones, both mudcake growth rate

and mud filtrate invasion rate will be influenced by for-

mation properties (i.e., formation permeability, oil rela-

tive permeability endpoint) in addition to mud proper-

ties.

4. A mudcake flow resistance parameter was introduced in

this paper to characterize the potential for invasion. This

parameter can be used to closely approximate the pres-

sure differential supported by the mudcake and potential

supercharging sensed by wireline formation testers.

5. In a multi-layer formation zone, low permeability forma-

tions can lose filtrate to higher permeability formations

due to cross flow between layers. The initial pressure

gradient can be affected by mud filtrate invasion, which

may cause errors in the estimation of the location of

oil-water contacts.

NOMENCLATURE

C constant

Cw salt concentration, ppm

eo exponent for kro equation

ep exponent for capillary pressure equation

ew exponent for krw equation

fs solid fraction

H height of the zone

i grid number index

K formation permeability

Kc core permeability

Kmc mudcake permeability

Kmc0 mudcake reference permeability

knw non-wetting phase permeability

krw water relative permeability

krw0 water relative permeability endpoint

kro oil relative permeability

kro0 oil relative permeability endpoint

kw wetting phase permeability

Pc capillary pressure

Pc0 coefficient for capillary pressure equation

Pnw pressure for non-wetting phase fluid

Pw pressure for wetting phase fluid

q mud filtrate flow rate

Q total sandface flow rate

Rf invasion front radius

Rmc mudcake radius

Rout formation outer boundary radius

Rw connate water resistivity, ohm-m

Rwell borehole radius

r radius in cylindrical grid system�

r location of the observation point

S1 slowness at pressure P1

S2 slowness at pressure P2

Sor residual oil saturation

Sw wetting phase or water saturation

Swi residual wetting phase saturation

S wt* normalized wetting phase saturation

t invasion time



FIG. 28 Spatial cross-section (radial and vertical directions) ofwater saturation simulated in a three-layer formation. The inva-sion time is two days. Individual layer permeabilities and porosi-ties are indicated on the figure.

FIG. 29 Time evolution of the pressures at the sandface for thelower layer (K = 1000 md) of the three-layer formation describedin Figure 28.

T temperature measured in degrees centigrade

v compressibility exponent of cake permeability

v(t) total production Darcy velocity

|vn| Darcy velocity of the filtrate through the cake

vnw Darcy velocity for non-wetting phase fluid

vw Darcy velocity for wetting phase fluid

Vl volume of liquids in the mud suspension

Vs volume of solids in the mud suspension

xmc mudcake thickness

Greek Symbols

�P overbalance pressure

�p pressure drop across the mudcake

� exponent multiplier for mudcake porosity

� formation porosity

�mc mudcake porosity

�mc0 mudcake reference porosity

� ratio of mudcake flow resistance over total

flow resistance

�f filtrate viscosity

�nw non-wetting phase fluid viscosity

�o oil viscosity

�w wetting phase fluid or water viscosity

ACKNOWLEDGMENTS

We would like to express our gratitude to Anadarko

Petroleum Corporation, Baker Atlas, Conoco-Phillips,

ExxonMobil, Halliburton Energy Services, the Mexican

Institute for Petroleum, Schlumberger, Shell International

E&P, and TOTAL, for funding of this work through UT

Austin’s Joint Industry Research Consortium on Formation

Evaluation. A special note of gratitude goes to Dick Wood-

house and David Kennedy for their constructive technical

criticism and editorial comments on the first version of this

paper.

REFERENCES

Bezemer, C., and Havenaar, I., 1966, Filtration behavior of circu-

lating drilling fluids: Society of Petroleum Engineers Journal,

vol. 6, no. 4, p. 292–298.

Brooks, R. H., and Corey A. T., 1966, Properties of porous media

affecting fluid flow, Journal of the Irrigation and Drainage

Division, Proceedings of the American Society of Civil Engi-

neers 92, no. IR2, p. 61–88.

Center for Petroleum and Geosystems Engineering, 2000,

UTCHEM Technical Documentation, The University of Texas

at Austin.

Chin, W. C., 1995, Formation invasion with applications to mea-

surement-while-drilling, time-lapse analysis, and formation

damage, Gulf Publishing Company, Houston, Texas.

Dewan, J. T., and Chenevert, M.E., 1993, Mudcake buildup and

invasion in low permeability formations; application to perme-

ability determination by measurement while drilling, paper

NN, in SPWLA/CWLS 34th Annual Logging Symposium

Transactions: Society of Professional Well Log Analysts, Cal-

gary, Alberta.

Dewan, J. T., and Chenevert, M. E., 2001, A model for filtration of

water-base mud during drilling: determination of mudcake

parameters: Petrophysics, vol. 42, no. 3, p. 237–250.

Dresser Atlas Inc., 1982, Well Logging and Interpretation Tech-

niques, Dresser Industries, Houston, Texas.

Ferguson, C. K., and Klotz, J. A., 1954, Filtration of mud during

drilling: Petroleum Transactions of AIME, vol. 201, p. 29–42.

Fordham, E. J., Ladva, H. K. J., and Hall, C., 1988, Dynamic filtra-

tion of bentonite muds under different flow conditions, SPE

18038, in SPE Annual Conference Proceedings: Society of

Petroleum Engineers, Houston, TX.

Fordham, E. J., Allen, D. F., and Ladva, H. K. J., 1991, The princi-

ple of a critical invasion rate and its implications for log inter-

pretation, SPE 22539, in SPE Annual Technical Conference

Proceedings: Society of Petroleum Engineers, Dallas, TX.

Holditch, S. A., and Dewan, J. T., 1991, The evaluation of forma-

tion permeability using time lapse logging measurements dur-

ing and after drilling: Annual Report, Contract No.

5089-260-1861, Gas Research Institute, December.

Outmans, H. D., 1963, Mechanics of static and dynamic filtration

in the borehole: Society of Petroleum Engineering Journal,

Sept., p. 236–244.

Proett, M. A., Chin, W. C., Manohar, M., Sigal, R. and Wu, J.,

2001, Multiple factors that influence wireline formation tester

pressure measurements and fluid contacts estimates, SPE

71566, in SPE Annual Technical Conference Proceedings:

Society of Petroleum Engineers, New Orleans, LA.

Semmelbeck, M. E., Dewan, J. T., and Holditch, S. A., 1995, Inva-

sion-based method for estimating permeability from logs, SPE

30581, in SPE Annual Technical Conference Proceedings:

Society of Petroleum Engineers, Dallas, TX.

Stewart, G. and Whittman M., 1979, Interpretation of the pressure

response of the repeat formation tester, SPE 8362, in SPE

Annual Conference Proceedings: Society of Petroleum Engi-

neers, Dallas, TX.

Wu, J., Torres-Verdín, C., Sepehrnoori, K., and Delshad, M., 2004,

Numerical simulation of mud-filtrate invasion in deviated

wells: SPE Reservoir Evaluation and Engineering, vol. 7, no.

2, p. 143–154.

APPENDIX

Consider a simple constitutive model for incompressible

mudcake buildup. The filtration of a fluid suspension of

solid particles by a porous but rigid mudcake can be con-

structed from first principles. First let xmc(t) > 0 represent

cake thickness as a function of time, where xmc(0) = 0 indi-

cates zero initial thickness. Also, let Vs and Vl denote the

volumes of solids and liquids in the mud suspension,

respectively, and let fs denote the solid fraction defined as


Wu et al.

fV

V Vs

s

s l

��( )

. (A.1)

If the solid particles do not enter the formation, Chin

(1995) shows that the time evolution of mudcake thickness,

xmc(t), satisfies the ordinary differential equation

dx

dt

f

fv

mc s

s mc

n�� ( ) ( )

,1 1 �

(A.2)

where fs is the solid fraction of mud, fmc is the mudcake

porosity, and |vn| is the Darcy velocity of the filtrate through

the mudcake and through the filter paper.

Now consider a one-dimensional, constant density, sin-

gle liquid flow. The corresponding Darcy velocity is given

by

vK p

xn

mc

f mc

��

�, (A.3)

where Kmc is mudcakepermeability, �p is the pressure drop

across the mudcake, and �f is filtrate viscosity.

Substitution of equation (A.3) into equation (A.2) leads

to

dx

dt

f

f

K p

x

mc s

s mc

mc

f mc

�� ( ) ( )

.1 1 � �

�(A.4)

If the mudcake thickness is infinitesimally thin at t = 0, with

xmc(0) = 0, equation (A.3) can be integrated to yield

x tt pf

f

Kmc

s

s mc

mc

f

( )( ) ( )

.��

2

1 1

�

� �(A.5)

This latter result demonstrates that mudcakethickness in a

linear flow grows with time in proportion to t . Equation

(A.5) is valid only when Kmc, �p, and �mc are constant. If

Kmc, �p, and �mc are functions of time, equation (A.4) can

also be integrated numerically. To obtain the filtrate produc-

tion volume, we combine the relation dVl = |vn|dAdt and

equation (A.3) to obtain

dVK p

xdAdtl

mc

f mc

��

�, (A.6)

where dA is an elemental area of filter paper. Further substi-

tution of xmc from equation (A.5) yields

dVpK

t

f

fdAdtl

mc

f

s mc

s

��

2

1 1

�

�( ) ( ). (A.7)

Direct integration of this last expression under the assump-

tion of initial zero filtrate gives

V t pK f

fdAl

mc

f

s mc

s

��

21 1

��

�( ) ( ). (A.8)

From equation (A.8) it follows that filtrate production vol-

ume in a linear flow also grows with time in proportion to

t .

By taking the derivative of equation (A.8) with respect to

time we obtain the mud filtrate invasion flow rate, q(t),

given by

q t pK f

t fdA

mc

f

s mc

s

( )( ) ( )

.��

��

�1 1

2(A.9)

In reality, mudcake may be compressible, that is, its

mechanical properties may vary with the applied pressure

differential. Compressibility effects can be determined by

performing the filtration experiment at increasing pressure

differentials. Experiments on mudcake properties by

Dewan and Chenevert (2001) indicate that the permeability

during the initial mudcake buildup can be estimated with

the expression

K tK

p tmc

mc

mcv

( )( )

,� 0(A.10)

where pmc is the mudcake pressure differential (psi), Kmc0 is a

reference permeability defined at 1 psi differential pressure

and v is a “compressibility” exponent. Typically, v is in the

range of 0.4 to 0.9. A value of zero for v corresponds to com-

pletely incompressible mudcake. Dewan and Chenevert’s

(2001) work also indicates a similar relationship for mudcake

porosity during the initial mudcake buildup, given by

��

�mc

mc

mcv

tp t

( )( )

.� 0(A.11)

In the above expression, the new exponent multiplier � was

found to vary from 0.1 to 0.2 based on a porosity-permeabil-

ity cross plot. Further experiments indicated that mudcake is

not completely elastic and that it exhibits a hysteresis on the

first compression and decompression cycle. While addi-

tional relationships can be determined for hysteresis effects,

in most cases mudcake builds to a stable thickness and the

hydrostatic pressure does not change appreciably.

Up to this point, we have discussed static mudcake

growth wherein the borehole fluids are not moving. In a

dynamic condition, annular flow can limit the growth of

mudcake by continuously shearing the mudcake surface.

Also, the mechanical action of the rotating drill pipe can



directly remove the mudcake and can increase the surface

shear of the annular fluids. In essence, dynamic filtration

limits the growth of mudcake. Static filtration left

unchecked would continue to grow until the mudcake

would literally plug off the wellbore. Normally, static filtra-

tion is sufficiently slow such that total plugging does not

pose operational problems.

The dynamic filtration process has been studied exten-

sively in the industry. Fordham, et al. (1991), for example,

introduced the concept of a “critical invasion rate” beyond

which mudcake will not form, while Holditch and Dewan

(1991) have attributed it to an “adhesion fraction” which

controls the buildup process. Most authors agree on the

physical principle that erosion occurs when the hydraulic

shear stress on the surface of the mud closest to the center of

the borehole exceeds the mudcake shear strength (e.g.,

Chin, 1995; Dewan and Chenevert, 2001). Only Chin

(1995) has fully developed the theoretical basis for

dynamic mudcake erosion for Newtonian and non-Newto-

nian annular flow and has offered solutions for a variety of

wellbore conditions. Factors that influence dynamic filtra-

tion include: mud rheology, pumping rates, annular flows

(concentric and eccentric) and borehole shape (round ver-

sus elliptical).

It is generally acknowledged that static filtration governs

the initial mudcake growth and that the role of dynamic fil-

tration is to limit this growth (e.g., Chin, 1995; Dewan and

Chenevert, 2001). In this paper, we will assume that

mudcake reaches such a limiting thickness but will not pres-

ent the analysis for the corresponding prediction. Mudcake

thickness is normally estimated using caliper logs from

openhole wireline or logging-while-drilling logs shortly

before a formation tester is run. Mudcake thickness esti-

mated using the caliper log can be assumed to be the limit-

ing filtrate thickness.

Well conditioning can also affect mudcake and influence

invasion. Frequently, the well is reconditioned prior to a

logging run or in the process of drilling. To account for well

reconditioning, we can assume that the mudcake is either

partially or completely removed, and henceforth allowed to

reform. In the worst-case scenario, the mudcake is com-

pletely removed during well conditioning. In practice, this is

unlikely because inaccessible mudcake forms within the

near-surface rock grain structure, and it is difficult to remove.

The flow equations for one-dimension immiscible, con-

stant density radial flow are obtained by combining Darcy’s

law and the equation of mass conservation. The Darcy

velocities are

vk P

w

w

w

w��

�

dr, (A.12)

and

vk P

drnw

nw

nw

nw��

�, (A.13)

where �w and �nw are viscosities, kw and knw are

permeabilities, and Pw and Pnw are pressures. The subscripts

w and nw here are used to denote wetting and non-wetting

phases.

The mass continuity equations in cylindrical radial coor-

dinates take the form

�

��

�v

r

v

r

S

dt

w w w� �� , (A.14)

and

�

��

�v

r

v

r

S

dt

nw nw nw� �� , (A.15)

where the signs of vw and vnw are taken positive for invasion

(i.e., injection). By combining equations (A.12) through

(A.15) we obtain the production Darcy velocities, given by

��

� �

�S

dt r

k P

dr

v

r

w w

w

w w��

��

�

�� , (A.16)

and

��

� �

�S

dt r

k P

dr

v

r

nw nw

nw

nw nw��

��

�

�� . (A.17)

Furthermore, by adding equations (A.14) and (A.15),

together with substitution from

S Sw nw� �1, (A.18)

one obtains

rv v

rv v

w nw

w nw

�

�

( )( ) ,

�� 0 (A.19)

or, equivalently, {r(vw + vnw)}r = 0. It then follows that

( ) ( ) ,v v v tw nw� � (A.20)

where v(t) is the total production Darcy velocity. For incom-

pressible flow, v(t) is the mud filtrate velocity at the

wellbore.

At this point, it is convenient to introduce the capillary

pressure function Pc and write it as a function of the wetting

phase saturation Sw, namely,

P S P Pc w nw w( ) .� � (A.21)


Wu et al.

Then the non-wetting velocity in equation (A.13) can be

written as

vk P S

dr

P

drnw

nw

nw

c w w��

��

�

��

�

� �( ). (A.22)

We now combine equations (A.17), (A.18), and (A.22) to

obtain an expression for the non-wetting production per unit

volume per unit time in terms of the wetting saturation and

pressure, given by

��

� �

� �( ) ( )1��

�

��

�

��

�

�

��

S

dt r

k P S

dr

P

dr

vw nw

nw

c w w nw

r.

(A.23)

By making use of equation (A.20), the wetting and non-wet-

ting production per unit volume per unit time equations

(A.16 and A.23, respectively) can be combined to yield an

alternate form of equation (A.23), i.e.,

��

��

�

��

�

��

�

��

�

� �

�

� �

�

r

k P S

dr

k k P

dr

nw

nw

c w nw

nw

w

w

w( )�

�

��

v t

r

( ).

(A.24)

This last equation can be used to solve for Pw and subse-

quently equation (A.16) can be used to solve for Sw. Such a

procedure is known as the “implicit pressure-explicit satura-

tion” or IMPES method and is widely used in reservoir sim-

ulation.

Having the immiscible two-phase flow formulations and

the properties of mudcake growth, we can now approach

the problem where mudcake forms and grows, thereby cre-

ating a Darcy flow that appears at the inlet of our radial

geometry. This flow satisfies its own pressure differential

equation and is characterized by the moving mud-to-

mudcake boundary and a fixed mudcake-to-rock interface

shown in Figure A.1.

At the wellbore surface, the total sandface flow rate is

defined as

Q t HR v twell( ) ( ) ,�2� (A.25)

where Rwell is the radius of borehole, H is the height of the

zone and v(t) is the velocity through the mudcake obtained

from equation (A.24). Using finite differencing, let i be the

grid block index where i = 1 at the mud-to-mudcake bound-

ary, i = 2 at the fixed mudcake-to-rock interface, and i = n at

the outer boundary. For incompressible flow, the mass flow

rate between each grid block should be equal and this condi-

tion leads to

rk P S

rr

k k P

r

nw

nw

c w nw

nw

w

w

w

� � �

�

�

�

�

( )�

��

�

��

�

��

�

��

�

�

��

i

C , (A.26)

where C is a constant and 2 � i � n. The latter equation can

be expressed for each i-th grid block as

k P P

P P

k knw

nw

c i c i

w i w i

nw

nw

w

w� � �, ,

, ,

�

�

�

��

�

��

�

��

�

1

1

��

��

�

�

��

�

�

�

��

�

��

�

P P

r rC

w i w i

i i

, ,

ln( ) ln( ).

1

1

(A.27)

For a mudcake sustaining single-phase flow, the mass

balance condition for equation (A.27) dictates that C be

equal to the mud filtrate flow, i.e.,

Kr

P

rC

mc

f

w

�

�

�� , (A.28)

where �f is filtrate viscosity and Kmc is mudcake permeabil-

ity.

Mass balance can also be enforced for the total flow rate

through the entire radial model (i.e., i = 2 to n). This can be

done by combining equations (A.25), (A.27), and (A.28).

Accordingly, it can be shown that the inlet filtrate flow rate

is coupled to the formation flow by the expression

Q tH P t P t

r r

k

w w n

r i

nw

nw

( )( ( ) ( ))

ln( ) ln( )

, ,��

��

��

��

2 1

1

�

� ��

�

�

��

�

��

�i

c i c i

w i w i

nwP t P t

P t P t

k, ,

, ,

( ) ( )

( ) ( )

1

1 �nw

w

w i

f

mc

well

mci

n

k K t

R

R t�

�

��

�

��

� ��

��

�

��

�

�

�

�

( )ln

( )

.

2

(A.29)

As the mudcake grows, the total inlet flow rate can be calcu-

lated directly from the above equation. The immiscible

radial Darcy flow equations are solved using equations

(A.24) and (A.16) for each grid block, i, for each time step.

In summary, the linkage between the mudcake model and

the immiscible invasion model is governed by the flow rate

using the coupled Darcy flow described by equation (A.29).

From equation (A.29), we notice that at the start of inva-

sion Rmc = Rwell and hence the flow rate reaches its maxi-

mum. As the mudcake builds and Rmc becomes smaller than



FIG. A-1 Diagram of a one-dimensional mudcake-rock model.The mudcake is located on the left and the rock formationextends to the right of the diagram.

Rwell the mudcake flow rate decreases. The flow rate is also

influenced by the saturation of the fluid in the formation, Sw.

As the mudcake builds and reaches a maximum, the

value of Rmc can remain unchanged in the numerical simula-

tion. In this case, the invasion is governed by the invasion

front represented by the fluid saturations. In the case of

mudcake removal or “rub-off,” mudcake thickness is

reduced to 0 and allowed to rebuild thereafter. Again, the

mudcake growth is dynamically coupled to the invasion

model and grows at a time rate that is linked to the invasion

front that continues to penetrate the formation.

ABOUT THE AUTHORS

Jianghui Wu is a research scientist with Baker Atlas. His

research areas include wireline formation testing, reservoir simu-

lation, and integration of logging data. He holds BS and MS

degrees from the University of Petroleum in China and a PhD

degree from The University of Texas at Austin, all in petroleum

engineering. Formerly, he was a reservoir engineer with CNPC.

Carlos Torres-Verdín received a PhD in Engineering

Geoscience from the University of California, Berkeley, in 1991.

During 1991–1997 he held the position of Research Scientist with

Schlumberger-Doll Research. From 1997–1999, he was Reservoir

Specialist and Technology Champion with YPF (Buenos Aires,

Argentina). Since 1999, he has been with the Department of Petro-

leum and Geosystems Engineering of The University of Texas at

Austin, where he currently holds the position of Associate Profes-

sor. He conducts research on borehole geophysics, formation eval-

uation, and integrated reservoir characterization. Torres-Verdín

has served as Guest Editor for Radio Science, and is currently a

member of the Editorial Board of the Journal of Electromagnetic

Waves and Applications, and an associate editor for Petrophysics

(SPWLA) and the SPE Journal.

Kamy Sepehrnoori is the Bank of America Centennial Profes-

sor in the Department of Petroleum and Geosystems Engineering

of The University of Texas at Austin. His teaching and research

interests include computational methods, reservoir simulation,

parallel computations, applied mathematics, and enhanced oil

recovery. Sepehrnoori holds a PhD degree in petroleum engineer-

ing from The University of Texas at Austin.

Mark A. Proett received a BSME degree from the University

of Maryland and a MS degree from Johns Hopkins. He has been

involved with the development of formation testing systems since

the early 1980s, and has published extensively. Proett holds 16 pat-

ents, 14 of which deal with well testing and fluid flow analysis

methods. He has served on the SPWLA and SPE technical com-

mittees and served as the Chairman for the SPE Pressure Transient

Testing Committee. He is currently a Senior Scientific Advisor for

Halliburton Energy Services in the Strategic Research group.


Wu et al.

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