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Theoretical

and

xperimental Studies

o

Sandstone cidizing

A.D. Hill,

* SPE

Texas Petroleum Research Committee

D.M. Lindsay, Texas

Petroleum

Research Committee

I.H. Silberberg,

SPE

Texas Petroleum Research Committee

R.S. Schechter,

SPE

U. of Texas

bstract

The matrix acidization of sandstone by a

hydrochloric/hydrofluoric acid mixture is described

through use of a capillary model. The model was

solved first in linear coordinates so that it could be

compared with the results

of

coreflood experiments

performed on Berea sandstone. The model

predictions showed reasonable agreement with the

experimental data and yielded specific information

about the reaction characteristics of the sand

stone/HCIIHF system.

The acidization model then was applied in radial

coordinates to generate design curves for a matrix

acidization treatment. While these curves strictly

apply only to those sandstones having similar

mineral compositions, the approach

is

general.

I t is

based on matching the location of the HF reaction

front to the depth of a damaged zone. This method

introduces the concept of an optimum injection rate

and, in this regard, differs from other design

methods reported in the literature.

Introduction

The matrix acidization of sandstone by an

HCIIHF

acid mixture is an often-employed oilwell stimulation

technique designed to increase permeability in a zone

around the wellbore. The acid mixture flowing into

the porous medium reacts with the various mineral

species present, thus effecting an increase in the

matrix porosity and, it is hoped, the permeability.

Clearly, one of the factors controlling the depth of

acid penetration

is

the chemical composition of the

minerals which the acid contacts. Smith and Hen

drickson,l Gatewood,

2

and Lund et

al.

3

5

have

shown that the reaction with calcite is more rapid

Now

with Marathon Oil Co.

0197-7520/81/0002-6607 00.25

Copyright 98

Society

of

Petroleum Engineers

of

AIME

30

than with silicate minerals clay or feldspar), which

is, in turn, more rapid than the reaction with silica.

Several papers describing the distance of

penetration have been published. Smith and Hen

drickson 1 and Smith et

al.

6

first suggested the use of

linear core tests to predict radial penetration. Farley

et

al.

7

reported tests similar to those conducted by

Smith and Hendrickson but measured many ad

ditional parameters including the effluent acid

concentration, which is quite useful since the effluent

concentrations may yield information about reaction

characteristics. Experiments conducted in linear

systems are difficult to translate in terms

of

penetration in a radial system, since the fluid velocity

varies inversely with radial distance. The obvious

approach has been to develop a mathematical model

that can be calibrated based on linear flow data and

then applied to a radial system.

Gatewood

2

proposed that the acid penetration

distance be determined by assuming that the reaction

of

HF with the silicate minerals is much faster than

with the silica. The distance of penetration is

determined in this model by the formation com

position and by the stoichiometry of the reactions.

Lund et

al.

5

8 and Fogler and McCune

9

developed a

model which neglects the reaction

of

HF with silica

but does consider the reactions with the silicate

minerals. The advantage of these approaches is that

the penetration depth can be predicted based on the

formation composition. However, the reaction with

silica cannot be neglected in determining the depth of

penetration, as will be seen.

Williams and Whiteley 1 used a somewhat dif

ferent approach which includes an empirical

determination of the reaction rate based on linear

core flood experiments. The analysis assumes that a

quasistationary state exists. Williams 11 used these

empirical reaction rates to ascertain the total acid

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volume and injection rate needed to obtain a given

penetration distance. This approach includes the

silica reaction in the kinetic model, since the model

is

determined empirically, and requires a separate test

at reservoir temperature

of

each new formation to be

acidized. Further, the analysis method that assumes a

quasistationary state is

not

valid for short times.

The conclusion provided by the analyses of

both

Lund and Fogler

8

and Williams II

is

that the depth of

penetration increases with acid velocity. However,

some field experience indicates that injecting at the

maximum rate may not be the best strategy when the

volume of acid

is

limited.

12

This question

is

ad

dressed in this paper.

Matrix acidizing

is

generally effective only when

applied to remove near-well bore damage.

13

Otherwise, the stimulation resulting from a treatment

will be quite small and is almost independent

of

the

permeability increase in the acidized zone. A design

to implement this strategy can be developed by

focusing primarily on the depth

of

acid penetration

and ensuring sufficient volumes to at least restore the

original permeabilities. This feature is a fortunate

one, because permeability changes caused by

acidization are complex, as evidenced by the

data of

Smith and Hendrickson. I Their results indicate that

the permeability first declines and then increases.

There are two possible mechanisms for this

phenomenon, both

of

which probably contributed to

the observed permeability decline. Labrid 14

discussed the mechanism

of

permeability reduction

by precipitation of reaction products. However, the

movement of fines also may be responsible for the

observed permeability reduction. Evaluating the

relative importance of these two mechanisms is

difficult.

cid Balances

Schechter and Gidl

ey

l5

and

Guin

et

al

6

•

7

have

developed a capillary model which predicts the

change in pore-size distribution resulting from acid

attack. The model approximates the porous medium

as a collection of cylindrical pores of varied sizes that

become enlarged as a result of the acid reaction

at

the

pore walls and allows for coll isions between

pores. The pore structure

is

characterized by a pore

size distribution

1 /

A,x,

t),

where 1 /

A,x,

t) dA

is

defined as the number of pores per unit volume

having a cross-sectional area between

A

and

A

dA.

The change in 1 / as a function of time

is

described

by an integrodifferential equation given in Appendix

A. Also shown in Appendix A are the equations

relating the permeability and porosity

to

moments

of

the pore-size distribution.

The rate of acid reaction

is

an important feature

and is characterized as follows.

dA

- = 1/; A,x,t). . 1)

dt

The pore growth function 1/; depends on A because,

in general, acid must diffuse to the mineral surface to

react, and it depends on both time and position

because the local acid concentration must depend on

these factors.

FEBRUARY

1981

In sandstone acidization, many reactions take

place; the most important of them are the reactions

of

HF and HCI with carbonates, the reaction

of HF

with silicates such as clays and feldspars, and the

reaction of

HF

with quartz. To model the acid

concentrations

as

the acid mixture flows through and

reacts with the porous medium, acid balances for

HF

and HCI must be written as follows.

a ¢C

HF

)

aC

HF

u

at ax

L XH F

r

O

1/;HF 1 / A,x,t)dA 2)

o

and

a ¢C

HCl

)

ac

HCl

u

at ax

= - Lx

HCl

I

oo

1/;HCl

1 /

A,x,

t) dA , (3)

o

where

u is

the flux,

i is

the moles

of

acid

i

expended

per volume of rock dissolved, and C

i

is

the con

centration of acid

i.

The function

1/;

is

related directly

to

the overall

reaction rate as shown by Eq. 4

2V7rRaA

V2

1/;

= ,

4)

P

s

where

ex is

a stoichiometric coefficient, Ps

is

the

density of the solid, and R is the average overall

reaction rate over the entire reactive surface of one

pore. This overall rate depends on a series of in

dividual processes including diffusion of the reac

tants to the solid surfaces, reaction with the surface,

and

diffusion of the products from the surface. In

extreme cases, the overall reaction rate may be

diffusion controlled

or

in others, reaction limited.

Guin

16

has developed overall reaction rates for a

number of cases, and these general results need not

be repeated here. There are two limiting cases that

apply to this study.

In

pores of the size range

characteristic

of

sandstones, the reaction rate for the

dissolution of carbonates

is

much larger than the rate

of diffusion,

and

this permits use of the diffusion

controlled approximations given in Appendix B.

The second limiting case

that

applies

to

the

reactions

of HF

with silicates (feldspars and clays)

and with silica also

is

given in Appendix B in a form

developed assuming that these reactions are first

order in

HF

concentration. This approximation has

been shown

to

be a good one for the minerals

of

interest in this work.

4

•

I1

The reaction rate constant of

HF

with quartz

differs from that with silicates so that the propor

tions of these minerals present must be taken into

account. They may be combined to yield

rHF =

[ksilfclay + k

qtz

1 fclay ]C

HF

,

(5)

where

fclay is

the fraction

of

the surface area oc

cupied by clay minerals.

In this kinetic model all the clays

and

feldspars

have been lumped into one average parameter. This

31

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Mineral

Quartz

T BLE

1 -

MINER L CONTENT Y

PETROGR PHIC N LYSIS

Dolomite and siderite

Chlorite and

illite

Feldspar

Percent

of

Total Rock

75 ± 5

10

±

3

10

±

3

5±3

combining

is

not necessary, but, on the basis of the

system studied, the approximation appears to be

valid.

A dynamic description of the parameter

fcJay is

now necessary to track the changing average reaction

rate constant and permit the correct evaluation

of

the

lfHF function. The two rate laws for the individual

reactions of HF with silicates and quartz may be

written as

rHF

=

kSi C

HF

6)

and

- rHF = kqtzCHF

7)

Recalling

that

the reaction rates are expressed as

mass of acid reacted per unit area per unit time, then

dVsi

dt

=

CisiI ksiI

CHFSsiI

Psi

8)

and

_

dv

qtz

= CiqtzkqtzCHFSqtz

, 9)

dt

Pqtz

where vsiI and V

qtz

are the volumes

of

silicate and

quartz per unit volume and

SsiI

and Sqtz are the

surface areas per unit volume

of

the two minerals,

respectively.

I f

it

is

assumed

that

the fraction of the exposed

surface area occupied by each mineral

is

propor

tional to the volume per unit volume, there results

d v tfcJay)

Cisi ksi

-

- - f c J a y S tCHF

10)

dt PsiI

and

d[ v

t

1 - fcJay)]

dt

Pqtz

. 1- fcJay)StCHF , (11)

where St

=

2hL ( A Y 1/

A,x,t)dA.

o

Using the chain rule

of

differentiation and solving

each equation for

dVt ldt

results in the following

equations.

dV

t

dt

_ dfcIay _

qsiI ksiI

S C 12)

t

HF·· · ·

fcJay dt PsiI

and

dV

t

v

t

dfcJay _ Ciqtzkqtz S C

t HF· 13)

dt 1 - fcJay dt Pqtz

Equating Eqs.

12

and

13

and recognizing that

v

t

total volume

of

solid per unit volume) is 1 - C/> the

differential equation describingfcJay

is

obtained.

32

dfcIay = Ciqtzkqtz _ CisiI ksiI )

dt Pqtz Psi

. fcJay (1 - fcJay) S C

c/> t HF·

14)

An examination

of

Eq.

14

reveals it to have the

correct qualitative behavior - i.e., for

ksi

ap

preciably greater than

k

qtz

, fcJay

will decline but at a

progressively slower rate as the silicate minerals

dissolve.

This equation, coupled with the acid balances,

predicts the acid concentrations throughout the

linear flow regime. t also

is

coupled with the

evolution equation, from which is obtained 1/ the

pore-size distribution function) at all space positions

through time. A solution

of

this coupled system.

of

integrodifferential, partial differential, and ordinary

differential equations will yield the desired predic

tions

of

acid concentrations plus matrix porosity and

permeability values. The numerical techniques used

to solve this set

of

equations are described briefly in

Appendix C and in full detail elsewhere.

19

Similar approaches have been used by Lund

et al

5

and by Williams and Whiteley.

10

These studies both

employ empirically determined kinetic data to design

an acid treatment and do not consider the pore

geometry. Lund et al also have neglected the

reaction with silica, whereas Williams and Whiteley

have analyzed their results assuming that a

quasistationary acid front is achieved. This latter

assumption

is

not valid during the initial stages of the

treatment, and neglecting the reaction of HF with

quartz is never valid, as is discussed in a subsequent

section.

The model presented here uses surface reaction

rates, which should apply independently

of

the

composition of the formation only

fcJay

must be

adjusted to allow for compositional variations), and

does not make the simplifying assumptions imposed

by Lund

et al or

Williams and Whiteley with regard

to reaction rates. In addition to considering the

reactions with

HF

it was found necessary to account

for the reactions with HCI by including an HCI

balance. Previous studies have neglected this latter

reaction.

xperiments

To test the predictions of the capillary model, a series

of

coreflood acidization experiments were per

formed.

23

These experiments all were conducted

using Berea sandstone cores with initial composition

shown in Table

1

and an acid mixture

of

3

wtOJo HF

and 12 wtOJo HCr.

The core 0.1 m long and 0.0254 m in diameter)

was placed in a Hassler holder, which then was

mounted into a nitrogen porosimeter, and the pore

volume was measured. Suitable corrections for the

void volume

of

the holder were obtained by

calibrating with a solid plug having the same external

dimensions as the Berea core.

After the initial porosity was determined, the

Hassler core holder with core) was mounted into a

Ruska

™

constant-temperature oven and allowed to

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reach thermal equilibrium at the desired reaction

temperature. The core was evacuated and filled with

a 3 wtOJo NaCl solution, and the initial permeability

was measured. The flow rate was determined in all

cases by employing a Ruska proportioning pump. A

backpressure regulator was used to maintain a system

pressure of 5 to 7 MPa. This pressure level

is

necessary to prevent CO

2

bubbles from forming in

the core.

Each experiment consisted of the injection of at

least

90

PV

of

acid. The effluent was collected in

either 1 or 2-PV samples using a sample collector.

Pressure drops across the cores were measured

during the course of the experiment.

After the core holder had been removed from the

flow system, it was mounted into the porosimeter

apparatus without having to remove the core from

the holder. The final porosity was determined.

The experimental studies required a reliable

technique for determining the amount

of

hydrofluoric acid that reacted. The procedure used

by Williams and Whiteley was adopted. The

method involves an ionometric technique which is

rapid and effective if carefully calibrated by

measurement

of

the amount

of

silica dissolved in the

effluent acid solutions. This two-step approach is

made necessary by the presence

of

fluosilicic acid and

other fluosilicates in the residual acid solutions. The

residual acid solution must be buffered before

analysis to a pH of 4.2

to

4.5 to ensure complete

dissociation of all hydrofluoric acid. At this pH level

the fluosilicates partially dissociate and release

fluoride ions. These then are included in the con

centration sensed by the ion specific electrode. To

compensate, a glass slide etch procedure which

allows for direct determination of reactive HF was

used. The details of this procedure are available. 23

I t was found that a new calibration curve

is

required if either the proportions of HF and HCl or

the mineral content of the core are changed.

I f

sufficient care

is

exercised, the ionometric analysis

is

rapid and reliable.

Discussion of Results

Effluent Concentrations

Typical acid (HF) effluent curves are shown in Figs. 1

through 4. Theoretical calculations are complex, and

the ·numerical techniques are described in Appendix

C. These results show that initial acid concentrations

are small but increase as more acid is injected until a

plateau value less than the injected acid con

centration

is

reached. Qualitatively, this profile can

be understood in terms of the differences in the

reaction rates between the

HF

and the minerals

present. The carbonates are expected to react first,

with the reactions with silicates and quartz following

in that order. The importance of the HF/quartz

reaction

is

shown by the plateau region that develops

in the latter stages of the experiment. This plateau

provides a measure

of

the HF/quartz reaction, as

most of the other minerals that are accessible should

have dissolved by this stage of the experiment. The

level

of

this plateau indicates that the HF/quartz

FEBRUARY

1981

T=

25.0C a ,

0.342

MLlSEe

EXPER\MENTRL DATA. RUN NO.

4

- MODEl PRED1CllDN

t.aa

20.00

30.00

40.00

50.00 60.00 70.00 80.00

PORE VOLUMES

OF

RCID INJECTED

Fig. 1 - Comparison

of

predicted and experimental ef

fluent acid concentrations (Run 4) for initial

concentrations of 3 wt% HF and 12 wt% HC .

T ,

2S.0C 0=

0.250

tiLiSEC

( )

E'(PER1MENTAL OqHL

RUN

N O 5

- MODEL PREGICT ION

20·00

30·00 40.00

50.00 50.00

CjO.oo

P ~ f vOLUMES OF ACID iNJECTED

Fig. 2 - Comparison

of

predicted and experimental ef

fluent acid concentrations (Run 5) for

initial

concentrations

of

3

wt%

HF and 12

wt%

HC .

>

< u ~ 0

T"

2S.0C Q", 0.130 r L/5EC

( )

[XPERIMENHll DATA. RUN

NO 6

- MODEL PREDICT

JON

~ o o

10.00 20.0:)

J:J.O l (:J.OO

so.::m

60.00 70.00 "10.00

PORf "OuJl1ES ACID iNJECTED

Fig. 3 - Comparison of predicted and experimental ef

fluent acid

concentrations

(Run 6) for

initial

concentrations of 3

wt%

HF and 12

wt%

HC .

33

8/10/2019 SPE-6607-PA.pdf


q T 2S .OC Q= 0.063 MLlSEe

( ) EXPER IMENTAL DATA.

RUN

NO. 7

- MODEL PREDICT ION

. . .

. . . . .

\,:+,.,,,.-,--,,:,,- .,,:===::;,,:--:.,::-, ---=, ,.,::-,---,. -,.=,,-s ',.-:c,,c---:<.'-:c. =-- 70,-;;.';O-, ---;1,,:--:.,::-, ---;;)90.00

PORE VOLUMES

OF ACID INJECTED

Fig. 4 - Comparison of predicted and experimental ef·


concentrations of 3

wt%

HF and 12

wt%

HCI.

u

«

T=

25

.OC

ao:

0 .250

MLISEC

( )

EXPER I MENTAL DATA. RUN

NO.

5

{ )

FRACTION OF CLAY=.D5

X

FRACTION

OF

CLAY=.166

.

W ~ ~ _ ~ ~ ~ ~ ~ ~ = ; ~ ~ ~ ~ - ; ; : ; ; - ~ : - ; ; ; ; - - - - - ; 1 ~ - - - - ; ;

J .00 10.00

20.00 30.00

.. 0.00

50.00 60.00 10.00 80.00

fO.OO

PORE

VOLUMES OF

RCID

INJECTED

Fig. 5 - E ffec t of parameter Ac1ay(O) on effluent acid con·

centration predicted by the model.

52.0C

Q=

0.128

t lL Sf [

(') f X P f R l ~ F N T L DqTR. RuN N O . 9

- MODfl

P ~ E D I C T

ION

'0

: : : ~

. .

0

0

'

0

. ~ , . ( , . , : : - , ......... ,,,-"'.

,=- ;:zo-:.

,:-- ,,:-:.-=-, -:1,,:--:.

=-,

--- ,,:--:.::-,----:,, .

::-,

----:,';-,. ::-,

----:,"-,

= --;:90.00

P O ~ E

~ O L U M f S

R[ID

INJECTED



concentrations of 3

wt%

HF and

12 wt%

HCI.

34

reaction is significant and cannot be neglected if the

depth

of

acid penetration

is

a concern.

To

clarify this point further, it may be noted

that

at

52°e

the stabilized acid effluent concentration

ratio is about 0.6 as shown in Figs. 6 and 7. Thus,

40070 of

the HF is consumed within 0.1 m, the length

of the core. Thus, even

though

the reaction

of

HF

with quartz is relatively slow, it is still an important

feature

of

sandstone acidizing and cannot be

neglected.

By

using the first -order reaction rate constant

reported for feldspar

4

as

ksil

and by empirically

adjusting

k

tz and

fela

(0) (the fraction

of

the

reactive surface occupied by the silicates after the

carbonates have been removed), the effluent com

positions obtained by integrating the acid balance

equations could be brought into reasonable,. but

certainly not perfect, agreement with the ex

perimental data as shown in Figs. 1 through 4. The

values of the parameters found to give reasonable

representation

of

the

data at 25°e

were as follows.

=

7.6

x 10

- 6 _g_H_F

2 (kmOI

HF

m ·s

m

3

(from Fogler

4

) .

kg HF

k

qtz

=5.0x 10-

8

2 (kmOI

H F

m

's

m

3

felay O)

= 0.05 .

The empirically determined k

qtz

at first appeared to

be

too

small when compareo with the results

of

Blumberg,18 Born,24 Mowrey,25 and Guin.

16

These

investigators all report reaction rate constants of

apf,roximatel

y

5

x

10-

7

kg HF m

2

·

(kmol HFI

m ) for the reaction

of

HF

with

quartz

at

25°e.

However, those investigations all were conducted

using amorphous silica, whereas most if not all

of

the

silica found in sandstones

is

a-quartz. I t

is

well

known that the solubility of

amorphous

silica in

water

is

generally (dependinf.

on

the temperature)

larger than

that of a-quartz.

2

,26

Only one paper was

found reporting the reaction

of a-quartz

with HF

at

the concentration levels

of

interest in this study.

Analysis

of

the data presented by Bergman

28

yielded

a first-order rate constant at 25°e r a n g i n ~ between

4.2 x 10-

8

and 6.8 x 10-

8

kg HF m ·s (kmol

HF m

3

.

These values compare favorably with the

rate constant which was found to fit the experimental

data and

are consistent with the reported differences

in solubility

of

a-quartz as compared with amor

phous silica.

The second parameter that was adjusted to obtain

a reasonable fit of the effluent acid concentration

profiles was

felay O),

the fraction

of

the reactive

surface occupied by silicates after the carbonates

have been reacted. Assuming fclay O) to be equal to

the volume fraction

of

silicates present (0.16) gave

unrealistic predictions of the acid effluent con

centration; a value

of

0.05 yielded a much better fit

of

the

data,

as shown in Fig. 5. Thus, it appears

that

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all the silicates did not react at the high rate found for

feldspars.

To shed light on the significance

of

the parameter

fclay(O), a petrographic analysis was made

of

a

sandstone core that had been acidized with 90 PV of

acid mixture. This analysis consisted of examining a

thin section of the acidized rock with a petrographic

microscope. The study indicated that most

of

the

feldspars had been removed by the acid treatment,

whereas the clays present appeared to have reacted

only slightly. Thus, it appears that the parameter

fclay(O) is related to the fraction of feldspar present

initially (see Table 1). Furthermore, it appears for the

sandstone studied that most

of

the clay reacted at a

rate close

to

that

of

quartz.

An analysis of total aluminum present before and

after acidization showed that, in the front

of

the

acidized core, approximately 7 OJo of the original

aluminum present had been removed. This result

indicates that the clay minerals reacted with the acid

mixture

to

some extent and that the aluminum layers

in the clays possibly were leached preferentially. Such

a preferential attack of acid on the octahedral layers

of

a clay has been found previously for the reaction

ofHCI

with clay.29

Note that the clay minerals in the Berea sandstone

were in the form of rock fragments such as pieces of

mica and shale. The reactivity to clay in this state

cannot be generalized to other types of clay com

monly found in sandstones. In particular, the specific

surface area of clay in this state may be much less

than that

found for clay in the form

of

small

platelets, thus greatly reducing the reactivity.

Using a similar fitting procedure and fclay(O) =

0.05, values for k

qtz

and

ks l

were obtained that gave

reasonable fits ot the experimental data at 52°C

(Figs. 6 and 7). The values used

at

this temperature

were

and

Permeability Response

The matrix permeabilities during acidization

predicted by the model did not agree well with those

found experimentally (Fig. 8). As the capillary model

used in this study had given excellent predictions

of

permeability changes in homogeneous matrix

materials in previous studies, 16,30 it

is

presumed that

the failure of the model when applied to sandstone

is

due primarily

to

the heterogeneity of the material.

Experimental permeability

profiles during

acidization usually show an initial decline in per

meability, due most likely to the downstream

migration of fine particles as cementing substances

are dissolved. Such a permeability decline cannot be

predicted by this capillary model, since particle

migration

is

not accounted for. A further difficulty

FEBRUARY

1981

1:: 52. DC Q ,

0.250

MLiSEC

( )

EXPERIMENTAL DATA.

RUN NO 9

- MODEL PREDICTION

~ . o o

10.00

20.00 JO.oo ~ o o o

50.00 6::1.00 10.00

80.00 90.0?

PORE

VOLUMES OF RC 0 N

JEUEO


fluent acid concentrations (Run 9 for initial

concentrations of 3

wt

HF and

12 wt

HCI.

o

~

8 ~ ~ r ~ T ~

I I

7 -

6

5

2 f-

1 ,..

Acid

Rate:

O 250

ml/sec

Temperature: 23 5°C

D Experimental Data, Run N o . 5

- Model Prediction

-

•

-

-

-

-

-

-

-

~

~

I

~ _ _ _ _ J

o

20 40 60 80

100

PORE VOLUMES

OF

ACID

INJECTED

Fig. 8 - Comparison of permeability prediction with ex·

perimental results for initial concentrations of 3

wt

HF and 12wt HCI.

35

8/10/2019 SPE-6607-PA.pdf


1.5

4>

0.15

0

-e-

Vr

0.153

-e-

2 1 T r ~ h

-

1200

ct

r.

a:

1.25

>-

q

1.15

:

qo

J)

0

a:

0

Q

1.0

1.0

Fig. 9 - The optimal strategy

for

an undamaged formation.

encountered in predicting permeability changes arose

from the necessity to use the approximate growth

function shown in Appendix B

to

model the dif

fusion-controlled reaction of

HF and

HCI with

carbonates, because the exact function created a

numerical instability in the evolution equation.

6

Design of an Acid Treatment

Optimal Strategy

In the final analysis, economic considerations govern

the design of an acid treatment. However, even given

the same technical information , the

best

treatment

may vary from company to company because

economic criteria vary. Instead of determining that

most profitable action, it

is

possible

to

calculate the

greatest stimulation that can be produced using a

given volume of acid. Given the solution

to

this

problem, which

is

called the optimal strategy, then

the best course

of

action defined in some economic

sense can be selected.

To

obtain the optimal strategy, it is assumed that

the acid reaction rate

is

so well controlled that it

is

possible

to

increase the porosity in the formation

surrounding the well bore in any desirable way. Thus,

4> r) (the porosity following treatment)

is

the control

variable. The volume of rock dissolved by

an

acid

treatment

is

rd

r

= 27rh

r 4)-4>o)dr .

............... 15)

Here, 4>

is

the porosity before treatment, h

is

the

thickness of the formation

to

be treated, and r

wand

r

d

are the radii

of

the well bore and drainage area,

respectively. The volume of rock dissolved by a

volume

of

acid

is

fixed by stoichiometry; thus,

specifying the treatment volume fixes Vr- Using

Darcy's law for radial flow, the new production

resulting from the acid treatment

is

given by

(16)

36

Zone

Initially

4>

0.15

Damaged

3.0

I

Vr

0.153

0

2 1 r r ~ h a

-e-

I

-e-

I

rw

1200

0

I

~

k DAMAGED

0.01

a:

2.0

k FORMATION

>-

:::

J)

q

7.5

0

a:

1.5

qo

0

Q

1.0

1.0

3.0 4.0

r / r

Fig. 10 - The optimal strategy for a damaged

f o r m a t i o ~

Here, ko

r) is

the permeability before treatment,

while k r)

is

the resulting one. The problem

is to

maximize qlqo for a fixed

V

r

Since the numerator

is

determined by the initial state

of

the reservoir, the

problem reduces

to

min

[ d

dr ]

¢ r) J rk r)

..................... 17)

rw

subject

to

r being constant.

To

proceed, the per

meability must be related to the change in porosity.

Lund and Fogler

8

have proposed

~ ; ; ) = ~

f·

18)

This expression

is

used here, although more complex

relationships which better define the response of a

given formation may be substituted easily.)

Pontryagin's maximum principle defines the

solution

of

variational problems such as that posed

here. Using this principle, it can be shown that

4> [ n ]lI(n+ )

=

2 for

4> ;::: 4>

•

4>

{ r

4>oko

r)

(19)

Given the volume of acid

to

be used, the constant

(3

can be evaluated by substituting Eq. 19 into Eq.

15

and 4> r) then determined. The function 4> r)

represents the new porosity at each point r which the

acid

is to

create

if

the maximum stimulation is

to

be

obtained. Obviously, it

is

not possible

to

tailor the

acid reaction rate so that this optimal porosity

distribution will result; however, the desired porosity

distribution should be approached as nearly as

possible.

The optimal porosity distributions for two dif

ferent cases are shown in Figs. 9 and 10.

For

an

undamaged reservoir, the maximum stimulation

ratio

is

1.15 when about 1.8 m

3

of acid per meter of

formation thickness

of

3.0

wtO o HF

and

12

wt% HCI

mixture are used in a 0.3-m-diameter wellbore. Large

stimulation ratios can be achieved if there

is

near

well bore damage. The optimal porosity distribution

for the case in which the damaged zone extends a

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distance 1.5 w is shown in Fig. 10. Note the

discontinuity in the porosity that occurs at the outer

limit

of

the damaged zone, indicating that the

maximum stimulation is achieved when the greatest

portion

of

the acid

is

expended in removing damage.

The stimulation ratio for this case

is

7.5, indicating

that large increases in production can be achieved by

damage removal.

These calculations provide a measure of the best

that can be achieved given a carefully defined

situation. However, they

do

not

provide a practical

means

of

designing an acid treatment.

Application of the Model

A sandstone acidization model will be useful if it can

yield information needed for the design

of

an actual

acidization treatment. Such information may be

obtained from the model used in this study by ap

plying it

in

radial coordinates.

It

was found from the

model (expressed in radial coordinates) that a

quasistationary HF concentration profile is

established after the

carbonate

material has reacted

and that the HF front moves slowly, as can be seen in

Fig. 11. Furthermore, the position

of

the

HF

front

depends on the injection rate - the higher the flow

rate, the farther into the matrix the front will extend.

Williams 11 constructed acidization design curves

based on a definition

of

radius

of

permeability in

crease as that radius at which the formation porosity

had reached a certain level.

For

a given desired radius

of

permeability increase, such a definition

automatically requires that operation at the highest

injection rate possible will use the least total volume

of

acid and, thus, be the most desirable treatment

scheme. Since the optimal strategy developed in the

previous section revealed that the most effective use

of

a given volume

of

acid

is

to penetrate a damaged

zone, the Williams criterion, which requires the

highest injection rate, may not always be the

preferred one. Since the depth of acid penetration

is

a

sensitive function

of

the injection rate (a

quasistationary front

is

established), an alternative

design strategy is proposed as follows.

1.

Choose an injection rate such

that

the HF acid

front reaches the desired radius

of

permeability

improvement. This radius should correspond

to

the

radius of damage.

2. Inject enough total volume

of

acid at the chosen

rate so that the total

amount of

solid reacted in the

penetrated region reaches a certain level. This

amount is

chosen from experimental permeability

date

to

ensure a permeability increase

throughout

the

region.

Based on this design strategy, illustrative design

curves have been generated

by

the model for sand

stone with three different feldspar contents and a

well radius

of

3 in. These curves are presented in

Figs.

12 through

14. The reaction rate constants at

the various temperatures needed to generate these

design curves were obtained from Arrhenius

relationships based on the rate constants known at

25

and 52°C. Such an extrapolation

of

reaction rates to

higher temperatures

is

not always valid

but

was

FEBRUARY 1981

°

l . ~ ~ - - - - - - - - ~ - - - - - - - - - - - r - - - - - - - - - - -

)

i:i o

S .

5

)

0

.

..

o

i l

:l

Injection

ra te

0.1 bbl/min/n

of

reservoir thickness

Temperature:

200°F

v =

66

gal/It

of reservoir thickness

S

· ~ o ~ ~ ~ ~

RADIAL mSTAN CE FROM WELLBOR E (INCHES)

Fig. - HF concentration profiles in a radial flow system

for initial concentrations of 3 wt HF and 2

wt

HCI.

w

E-o

0

p::

E=:

t.J

w

....

0.7

0.6

~

w

t.J 0.5

53

E-o

p::

5

> 0.4

p::

w

00

w

p::

0.3

0

E-o

.......

;:;:: 0.2

.......

l

ill

~

0.1

0.0

t-......; ... ....

I

500F

200°F

150°F

lOO°F

~

____ ____

______

____

o

3 6 9 12

PENETRATION DISTANCE (INCHES)

Fig. 2 - Design curves for 3 wt HF and 2 wt HCI in

2 -feldspar formation.

37

8/10/2019 SPE-6607-PA.pdf


necessitated by the lack of reaction rate information

above 52°e.

An inspection of Figs. 12 through

14

shows that, to

a reasonable approximation, the total acid volume

required to increase substantially the permeability

throughout a region about the wellbore

is

sensitive to

the formation composition but not to temperature.

Thus, the recommended acid volume 3 wtO o HF and

12

wt% HCl) depends on the depth of the damaged

zone and the formation composition. The optimal

injection rate depends

on

all

of

the variables but is

most sensitive to temperature. At temperatures in

excess

of

93.3°C, it is not possible to obtain deep

penetration because the injection rates required

cannot be achieved in most cases without fracturing

the formation.

The most important conclusion is that the highest

0 . 7 r r ~ ~ ~ ~ ~ ~ ~

o. 1

o. 0

~ - - - - - - - - - - ' - - - ' - - - - - f

300

200

100

o

~ - - ~ ~ - - - - - - ~ - - - - ~ ~ - - - - ~

o 3 8 2

PENETRATION

DISTANCE (INCHES)

Fig. 13 - Design curves

for

3 wt HF and 2

wt

HCI in

5 ·feldspar

formation.

38

rate is not always the optimum. Damage near the

wellbore may be removed best by relatively small

treatments applied at modest rates.

The results can be viewed in another way. Once the

volume of acid is selected, the best rate then is given.

The depth

of

acid penetration also

is

fixed. Ob-

viously, these design curves can be used best when the

extent of the damaged zone

is

known; however, their

use also is recommended if the acid volume is

prescribed, perhaps on the basis of experience in a

particular formation.

In extending this design procedure to types of

sandstones other than Berea, it

is

anticipated that the

reaction rates of some clay minerals will be quite

different from those found in Berea. Unfortunately,

data on reaction kinetics are not now available. It

will be necessary to establish these reaction rates.and

00

UJ

r.:l

0 . 7 ~ - - - - - - ~ - - ~ ~ ~ - - ~ ~ - - - - - -

0.6

u 0.5

5

E <

0.4

o. 1

o o ~

______ ______ ______ ______

300

200

100

OL ' J

o

3 6 9 12

PENETRATION

DISTANCE

(INCHES)

Fig. 14 - Design curves

for

3

wt

HF and 2

wt

HCI in

10 ·feldspar formation.

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their temperature dependence to develop design

curves similar to those presented here for Berea.

A second possible difficulty is

that

often the

damage

is

a result of drilling fluid invasion. The clays

present in the fluid may react at a different rate,

thereby altering the reaction rate. In turn, both the

penetration distance and volume of acid needed to

achieve good stimulation will be altered.

onclusions

Matrix acidizing often is conducted

at

the highest

rate the formation can accept without fracturing. t

has been argued correctly

that

increased rates will

result in increased depths of acid penetration. The

strategy used here differs from this one. Given a

limited acid volume, it was found best to adjust the

rate so as to increase effectively the permeability of a

near-well bore damage zone.

To

implement this

strategy, a means of predicting acid penetration

depth is needed. A model which uses both the

chemical composition of the formation and its pore

size distribution has been derived. Although the

model did not represent accurately the observed

variations in permeability, it did predict measured

acid effluent concentrations.

In addition to the development of a design strategy

and

illustrating its implementation, the following

conclusions have been drawn from this study.

1 The capillary model predicts effluent acid

concentrations well.

2. Rate constants determined independently were

used for the reactions

of HF

with quartz

and

with

silicates, and the model yielded reasonable fits to the

experimental data. This agreement provided support

for the validity

of

the model.

3

The feldspar in Berea sandstone reacts at a

higher rate than the clays, which are present

primarily in rock fragments. The low reactivity

of

clay observed in this study could be due to the fact

that the surface area available for attack was much

smaller in those rock fragments than that found for

clays with a fine platelet surface.

4. Though the clays in this study appeared to react

at a slow rate similar to that

of

quartz, it is not

known at what rate the fine clay particles, sometimes

responsible for well damage, will react. The reaction

rates of such materials would affect the design

results.

5. Illustrative design curves were obtained from the

model in radial geometry based

on

the location of the

HF front. This criterion shows that the acid injection

rate should be chosen based

on

the desired radius

of

permeability increase

and that

neither the fastest nor

the slowest rate is always optimal.

6. The reaction of quartz with

HF

is not negligible

as has been assumed in other works

9

but is, in fact,

crucial to the effectiveness of sandstone acidization.

Nomenclature

A

= cross-sectional area

of

a pore, m

2

sq ft)

A

p = reference pore cross-sectional area, m

2

sq ft)

FEBRUARY 1981

C

j

=

concentration of acid

i

mol/m

3

mol/cu

ft)

D

j

= molecular diffusivity

of

acid i, m

2

Is

sq ft/s)

fcJay

fraction

of

surface area occupied by

reactive silicates after dissolution of

carbonates

FG

=

geometric factor

h

=

formation thickness, m ft)

k

=

permeability, m

2

md)

k r) = permeability after acid treatment at

radius r m

2

md)

o r) = permeability before acid treatment at

radius r m

2

md)

k

j

= first-order reaction rate constant,

kg i /m2

·s mol HF/m

3

[Ibm ilsq ft·s mol HF/cu ft)]

first-order reaction rate constant for

reaction of

HF

with quartz,

kg HF/m2 ·s mol

HF/m

3

[Ibm HF/sq ft·s mol HF/cu ft)]

first-order reaction rate constant for

reaction

of HF

with silicates,

kg HF/m2 ·s mol HF/m

3

[Ibm HF/sq

ft·s mol

HF/cu ft)]

L

=

average pore length, m ft)

M

j

= the jth moment

of

the pore-size

distribution function

n

=

exponent of porosity Ipermeabili ty

relationship

q

= well production after acid treatment,

m

3

Is

cu ft/s)

qo = well

production

prior

to

acid treatment,

m Is cu ft/s)

r =

radial space dimension, m ft)

r

d

=

drainage radius, m ft)

rHF rate

of

disappearance

of

HF,

kg HF/m

.

s Ibm HF/sq ft·s)

- rHF

-rHF

rate of disappearance of

HF

due to

reaction

of HF

with silicates,

kg HF/m

2

·s

Ibm HF/sq ft·s)

rate of disappearance of

HF

due to

reaction

of

HF with quartz,

kg

HF/m

2

·s Ibm HF/sq ft·s)

rw

=

well radius, m ft)

R

average reaction in a pore,

kg

acid/m

2

·s

Ibm

acid/sq ft·s)

specific surface area of quartz, m

2

m

3

sq

ft/cu

ft)

specific surface area of silicates, m

2

/m

3

sq

ftlcu

ft)

specific surface area

of

all solids, m

2

/m

3

sq ftlcu ft)

t

=

time, seconds

u = volumetric flux, m

3

1m2

·s cu

ftlsq

ft·s)

U

qtz

specific volume of quartz, m

3

/m

3

cu ftlcu ft)

usil specific volume of silicates, m

3

1m

3

cu ftlcu ft)

39

8/10/2019 SPE-6607-PA.pdf


V

t

specific volume

of

all solids, m

3

/m

3

(cu ft/cu ft)

Vr volume of rock dissolved, m

3

(cu ft)

x

space dimension in linear flow system, m

(ft)

Xi

moles of acid expended per cubic meter

(35.3 cu ft) of rock dissolved

a stoichiometric factor (solid dissolved per

acid reacted), kg/kg (Ibm/Ibm)

{

constant

r

gamma function

J pore-size distribution, m - 5 (ft - 5)

Ps density

of

a solid, kg/m

3

(Ibm/cu ft)

7

transformed time variable

¢ r) porosity at radius r

¢c

= porosity after carbonates have been

removed

¢o initial porosity

1 ;_

pore growth function, m

2

/s (sq ft/s)

cknowledgments

We

thank

the Texas Petroleum Research Committee

for sponsoring this work and the U. of Texas for

University Fellowship support of one of the authors.

We also express our appreciation to Halliburton

Services for conducting additional experiments and

to Charles R. Williamson for performing the

petrographic analyses.

References

1. Smith,

C.F. and

Hendrickson, A.R.:

Hydrofluoric

Acid

Stimulation

of

Sandstone Reservoirs," J Pet. Tech. (Feb.

1965) 215-222; Trans. AIME, 234.

2. Gatewood,

J.R.,

Hall, B.E., Roberts,

L.D., and

Lasater,

R.M.: "Predicting Results

of

Sandstone Acidization," J

Pet.

Tech. (June 1970) 28, 693-700.

3. Lund, K., Fogler,

H.S., and

McCune, C.C.: "Acidization I -

On

the Dissolution

of

Dolomite in

HCI, Chem. Eng. Sci.

(March 1973) 28, 691-700.

4. Fogler, H.S.

and

Lund, K.: "Acidization III - The Kinetics

of

the Dissolution

of

Sodium and Potassium Feldspars

in

HF/HCI Mixtures," Chem. Eng. Sci. (Nov.

1975

1325-1332.

5. Lund, K., Fogler,

H.S.,

and McCune, C.C.: "Predicting the

Flow

and

Reaction

of

HCIIHF Acid Mixtures in Porous

Sandstone

Cores,

Soc. Pet. Eng. J (Oct. 1976 248-260;

Trans. AIME, 261.

6. Smith,

C.F.,

Rose, W.M., and Hendrickson, A.R.:

Hydrofluoric Acid Stimulation - Developments for Field

Application," paper

SPE

1284 presented at

SPE

40th Annual

Meeting, Denver, Oct. 3-6,1965.

7. Farley, J.T.,

Miller, B.M., and Schoettle, V.: "Design Criteria

for Matrix Stimulation with Hydrochloric-Hydrofluoric

Acid,

J

Pet. Tech.

(April 1970) 433-440.

8. Lund,

K.

and Fogler, H.S.: "Acidization

V:

The Predictions

of

the Movement

of

Acid

and

Permeability Fronts

in

Sand

stones, Chem. Eng. Sci.

(May 1976) 31, 381-392.

9. Fogler, H.S.

and

McCune, C.C.:

On

the Extension

of

the

Model of Matrix Acid Stimulation to Different Sand stones,"

A IChE J.

(July 1976) 22, 799-805.

10. Williams, B.B.

and

Whiteley, M.E.: "Hydrofluoric Acid

Reaction with a Porous

Sandstone, Soc. Pet. Eng. J.

(Sept.

1971) 306-314;

Trans.

AIME, 251.

11.

Williams, B.B.:

Hydrofluoric

Acid Reaction with Sandstone

Formations,

J Eng. Ind. Trans. ASME (Feb. 1975) 252-

258.

12.

Templeton, C.c., Richardson,

E.A.,

Karnes,

G.T., and

Lybarger, J.H.: "Self-Generating Mud

Acid,

J Pet. Tech.

(Oct. 1975) 1199-1203.

40

13.

Williams, B.B., Gidley,

G.L.,

and Schechter, R.S.: Acidizing

Fundamentals Monograph Series, Society of Petroleum

Engineers, Dallas (1979) 6, Chap. 2.

14.

Labrid, J .C.: "Thermodyn amic and Kinetic Aspects

of

Argillaceous Sandstone Acidizing,"

Soc. Pet. Eng. J

(April

1975) 117-128.

15. Schechter, R.S. and Gidley,

J.L.:

The Change in Pore Size

Distributions from Surface Reactions in Porous

Media,

AIChE J. (May 1969 339-350.

16.

Guin, J .A.: "Chemically Induced Changes in Porous Media,"

Report No. UT 69-2, Texas Petroleum Research Committee,

Austin, TX (Nov. 1969);

PhD

dissertation, U. of Texas,

Austin (1969).

17. Glover, M.C. and Guin,

J.A.:

"Dissolution

of

a

Homogeneous Porous Medium by Surface Reaction,"

AIChE

J (Nov. 1973) 1190-1195.

18. Blumberg, A.A.: "Differential Thermal Analysis and

Heterogeneous Kinetics: The Reaction

of

Vitreous Silica with

Hydrofluoric

Acid, J. Chem. Phys.

(July 1959) 1129-1132.

19. Hill, A.D.:

Flow and

Simultaneous Heterogeneous Reactions

in Porous

Media,

Report No. UT 78-1, Texas Petroleum

Research Committee, Austin,

TX

(1978);

PhD

disseration, U.

of Texas, Austin (1978).

20. Reigle, E.G.:

A

Study

of

the Effect

of

Core Size on Apparent

Pore Size Distribution,"

MS

thesis, U.

of

Texas, Austin (June

1962).

21. Smith, C.S.: A Systems Engineering Approach to the

Simulation

of

Distributed Parameter Processes,"

PhD

dissertation, Heriot-Watt U., Edinburgh, Scotland (Oct.

1975).

22.

Gear, C.W.:

The

Automatic Integration

of

Ordinary Dif

ferential

Equations, Comm. ACM March

1971) 185-190.

23. Lindsay, D.M.: An Experimental Study

of

Sandstone

Acidization," Report No.

UT

76-1, Texas Petroleum Research

Committee, Austin (July 1976).

24. Born, H.K.H.: The Mechanism of the Dissolution of Silica in

Hydrochloric-Hydrofluoric Acid Mixtures,"

PhD

disser

tation, U.

of

Texas, Austin (May 1976).

25. Mowrey, S.L.: The Theory

of

Matrix Acidization and the

Kinetics

of

Quartz-Hydrogen Fluoride Acid Reactions,"

Report No. UT 73-4, Texas Petroleum Research Committee,

Austin (Sept. 1973).

26. Wollast, R.: The Silica

Problem, Sea

M.N. Hill

et

al

(eds.),

John

Wiley and Sons Inc., New York City (1974) 359-

392.

27. Owen, L.B.: "Precipitation

of

Amorphous Silica from High

Temperature Hypersaline Geothermal Brine," Publication

OCRL-51866, Lawrence Livermore Nat . Labora tory,

Livermore,

CA

(June 1975).

28. Bergman, 1.: "Silica Powders of Respirable Sizes IV. The

Long-Term Dissolution

of

Silica Powders in Dilute

Hydrofluoric Acid:

An

Anisotropic Mechanism

of

Dissolution

for the Coarser Quartz Powders,"

J Appl. Chem.

(Aug. 1963)

356-361.

29. Osthaus, B.B.: "Chemical Determination

of

Tetrahedral Ions

in Nontronite and Montmorillonite,"

Clays and Clay

Minerals A. Swineford and N. P lummer (eds.) Nat . Research

Council Publication 327, Washington, DC (1954).

30. Sinex, W.E.

Jr.,

Schechter, R.S., and Silberberg,

LH.:

"Dissolution of a Porous Matrix by Slowly Reacting Flowing

Acids,"

Ind. Eng. Chem. Fundam.

(May 1972) 205-209.

31. Champion, L.S.: The Ultimate Yield from Oil Well

Stimulation with Acids,"

MS

thesis, U.

of

Texas, Austin

(1970).

PPENDIX

A

Capillary Model

In this model, the porous medium is approximated as

a collection

of

cylindrical pores all having the same

length

L.15-17

The distribution

of

pore areas

is

defined by a pore-size distribution

J A,x, t),

where

1J A,x,t)dA is the number of pores per unit volume

having a cross-sectional area between A and A +

dA.

It

can be shown 15 that the two importan t matrix

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properties - porosity and permeability - are given

by

cf>(x,t) = L ~

1](A,x,t)dA

A-I)

and

k(x,t)

o

ko r A

2

1](A,X,t)dA

o

~ O O

21]

(A,x,O)dA

o

, A-2)

where

ko is

the initial permeability.

It is

evident that

knowledge

of

the pore-size distribution during the

acidization process would yield predictions

of

porosity and permeability throughout the reaction

region. An equation, called the evolution equation,

describing the behavior of the pore-size distribution

has been derived based on the concepts that the pores

continuously are being enlarged due to acid attack

and that, as the pores become larger, the walls

separating adjacent pores may dissolve, leaving a

single pore having an area equal to the sum of the

original areas. These concepts can be expressed in

mathematical terms as

a ] + /;1])

= L

[ ; (A,X,

t)

1] (A - A,X,

t)

at vA 0

1] (A,X,t)dA - ext1 ; (A,X,

t)

o

+1/;(A,x,t) ]1](A,x,t)1](A,x,t)dA } A-3)

1/;(A,x,t) is the pore-growth function defined by

dA

dt

= 1/;(A,x,t). .

A-4)

Note that 1 ; depends on the many factors governing

the overall reaction rate in a pore. These factors

include acid concentration, pore geometry, dif

fusion, and surface reaction rate. Knowledge

of

what

takes place in a single pore

is

required to determine

the evolution

of

the pore sizes. However, the solution

of

the integrodifferential equation Eq. A-4)

is

complex, and a numerical approach generally is

required.

PPENDIXB

Overall Reaction Rates

Guin 16 has developed overall reaction rates for a

number

of

cases, and these general results are not

discussed here. There are, however, two limiting

cases that apply to the present study. If the reaction

rate

is

much larger than the rate

of

diffusion, which

is the case for the dissolution of carbonates, the

overall rate

is

diffusion controlled and

u

C·a

.f. - _1_ A2

for

A < A (B

la)

f HCI

- L2M2

P

s

P •• , -

and

~

36)2/3 aC

i

~ )

1/3

D 7rA

)

213

1/;HCI r 1I3)p

s

M2 2L

for

A

> Ap , B-Ib)

FEBRUARY

98

where

-2

18.IDuiL

M2) V2

Ap =

and

Di is

the molecular diffusivity

of

the acid. The

second limiting case is surface reaction controlled,

and

_

2hakC

i

V2

1/;(A,t) - A , B-2)

Ps

where

k is

a surface-reaction rate constant.

PPENDIXC

Numerical Techniques

As long as the 1 ; function is a separable function

of

C

i

and A having the form

1/;i = fe(Ci) fA

(A)

, C-l)

the evolution equation may be transformed so that it

may be solved independently from the acid balance

equations.

a ]

a[fA (A)1]] _

- [

r

(A- '

)

-

LJ1]

I \,T

aT aA 0

1]

(A,T)f

A

A)dA- VA (A) +fA

A)]

o

1](A,T)17(A,T)dA} ,

C-2)

where

t

T(X,t) =

fe(Ci)dt

.

C-3)

o

The 1 ; function used

to

model the reaction

of HF

and HCI with sandstone was

1/;=

1/;slow C

HF

)

for cf

>

cf>e, C-4)

where

°

or

A <

Ap

36

2

/

3

aC

i ~ ) 1/3

(Di

) 213

r 1I3)ps

M2

2L

for

A > Ap

C-5)

2hakC

HF

Y;

1/;slow C

HF

)

= A

2 , C-6)

Ps

and cf>e

is

the porosity after carbonates have been

removed. This 1 ; function assumes an initial dif

fusion-controlled reaction

of

acid with carbonates

followed by the slower reaction-rate controlled

reaction with silicates and quartz.

I t

should be noted

4

8/10/2019 SPE-6607-PA.pdf


that the

1 ;

fast function used in the model for A < A

is an approximation to the derived function (Eq. i

1). This approximation was necessary because the 1 ;

function proportional to A

2

causes a numerical

instability in the evolution equation. 16

As both

1 ;fast

and

1 ;slow

are separable functions of

C and A the evolution equation can be transformed

to Eq. C-2 and solved before the acid balance

equations. This solution was made using a modified

finite difference technique,16,17

and

the pore-size

distribution

1

7)

then was stored as input

to

the acid

balance equations. In this form the appropriate pore

size distribution can be obtained, given a value of

7.

To obtain 7, the time dependence of the acid con

centration at a point must be established and the

integral defined by Eq. C-3 evaluated.

The initial condition needed

to

solve the evolution

equation

is

the initial pore-size distribution 1/ A,x,O).

For

the Berea sandstone studied, the original pore

size distribution was obtained from mercury in

jection data taken by Core Laboratories Inc. A

comparison of these

data

with those obtained by

Reigle

20

in studies

on

several sandstones indicated

that

the pore-size distributions differed little for the

various sandstones. Thus, it

is

felt that the initial

pore-size distribution of Berea sandstone

is

an ap

propriate initial condition when studying many other

sandstones as well.

Once the pore-size distribution has been

established, the acid balance equations (Eqs. 2 and

42

3), the A

clay

equation (Eq. 14),

and

the

7

integral

(Eq. 22) all must be solved simultaneously. This

solution was accomplished using a previously

developed program

21

for solving coupled partial and

ordinary differential equations. The program em

ploys orthogonal collocation to generate a system of

ordinary differential equations, which then are

solved using Gear's method

22

for stiff systems. The

complete details

and

computer program are

presented elsewhere. 19

SI Metric onversion Factors

bbl

x

1.589873

E-Ol

= m

3

cu ft

x

2.831 685 E - 02

= m

3

of

F -

32)/1.8

=oC

ft x

3.048*

E-01

m

gal

x

3.785412

E-03

m

3

in. x 2.54*

E+Ol mm

Ibm

X

4.535924

E-01

kg

Ibm mol

x

4.535924

E-01

kmol

mL x 1

cm

3

psi

x

6.894757

E-03

MPa

sq ft

x

9.290304* E-02 m

2

'Conversion factor

is

exact.

SP J

Original manuscripl received in Sociely of Pelroleum Engineers office May

19, 1977. Revised manuscripl received Dec. 27 1979. Paper accepled for

publicalion

Aug.

13

1980. Paper SPE 6607)

firsl

presenled

al

Ihe SPE 1977 Inll

Symposium on Oilfield and Geolhermal Chemislry, held in La Jolla, CA, June

27·28.

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