MF04

MMINIINI F FRACRAC S SPREADSHEETPREADSHEET

April 2000

Dr Peter P. Valkóvisiting associate professor

Harold Vance Department Petroleum Engineering

Texas A&M University

TABLE OF CONTENTS

1 EXECUTIVE SUMMARY.....................................................................................................................5

2 DATA REQUIREMENT....................................................................................................................... 6

3 RESULTS............................................................................................................................................ 8

4 THEORY.............................................................................................................................................. 9

5 SAMPLE RUN................................................................................................................................... 14

A NOMENCLATURE............................................................................................................................ 17

11 EEXECUTIVEXECUTIVE

SSUMMARYUMMARY

The MF Excel spreadsheet is a minifrac (calibration test) evaluation package. Its main purpose is to ex-tract the leakoff coefficient from pressure fall-off data.

Currently it contains the following worksheets:

Analysis with PKN model, Nolte-Shlyapobersky method

Analysis with KGD model, Nolte-Shlyapobersky method

Analysis with Radial model, Nolte-Shlyapobersky method

Analysis with PKN model including the estimate of the fracture propagation retardation (in form of the Continuum Damage Parameter, Cl2). This option uses the "overpressure" experienced during the minifrac treatment to estimate the deviation of the fracture dimensions from the traditional PKN model. (There is no CDM option for the KGD and Radial models.)

The basic result of the analysis is the apparent leakoff coefficient. The apparent leakoff coefficient de-scribes the leakoff with respect to the total created fracture area.

From this apparent leakoff coefficient then a "true" leakoff coefficient is calculated, which value is valid only for the permeable layer, assuming that there is no leakoff outside the permeable layer.

In case of the CDM analysis, an additional parameter is determined from the overpressure. Overpressure is defined as the additional net pressure arising during the minifrac, which cannot be explained by the tra -ditional PKN model. The additional parameter is the CDM parameter: Cl2, measured in ft/(psi-sec). The obtained value may be used as an input parameter in the CDM version of the FD design spreadsheet.

Chapter 5 - Sample RunMF manualMF manual

22 DATADATA

REQUIREMENTREQUIREMENT

The following table contains the description of the input parameters.

Input Parameter Remark

Permeable (leakoff) thickness, ft It is assumed there is no leakoff outside the permeable thickness

Fracture height, ft Needed for the traditional PKN and KGD analysis and for the PKN-CDM analy-

sis. It is not needed for the radial analysis.

Plane strain modulus, E' (psi) Defined as Young modulus divided by one minus squared Poisson ratio.

It is almost the same as Young modulus, and it is about twice as much as the

shear modulus, because the Poisson ratio has little effect on it.

Closure pressure, psi It affects the leakoff coefficient in case of the KGD and Radial models.

It has no effect on the leakoff coefficient for the PKN case.

Also it affects the CDM parameter.

Rheology, K' (lbf/ft^2)*s^n' Power law consistency

Needed only for the CDM analysis.

Rheology, n' Power law flow behavior index

Needed only for the CDM analysis

Chapter 4 - Theory MF ManualMF Manual

The following tabular data are needed:

t, min time elapsed from start of pumping

qi_liq, bpm liquid injection rate (for two wings, valid at the bottom at the given time)

Bottomhole pressure, psi Needed only for the shut-in period, but can be given for the injection period, too

The last value during the injection period is used for overpressure in the CDM analysis.

Indicator whether to in-

clude the injected volume

Should be 1 if the point is in the injection period (the injected volume will be accounted for)

Should be zero for the shut-in period

Indicator, whether to in-

clude the point into the

straight line fit

Should be 1 if the point is to be included into the straight line fit (into the least squares objective

function)

Notice: points after closure are not included automatically if there is a 1 in here.


33

RESULTSRESULTS

The most important result is the leakoff coefficient.

The apparent leakoff coefficient is what obtained without distinguishing between permeable and not per -meable layer.

The "true" leakoff coefficient is obtained by attributing all the leakoff to the permeable layer only.

As a by-product, we obtain the dimensions of the created fracture: xf, (or Rf) and w , that is fracture extent and average width.

The fluid efficiency is also obtained. The efficiency is valid only for the minifrac. We do not encourage to use it as an input to the design, because the efficiency at the real tretment will be significantly different.

The methods involved assumes that the spurt loss can be neglected.

Results:

Apparent leakoff coefficient, ft/min^0.5 Can be considered as the "average" of the leakoff coefficient in the permeable

layer and the zero coefficient outside.

Leakoff coefficient in permeable layer, ft/min^0.5 The leakoff coefficient in the permeable layer. Outside the permeable layer the

leakoff is considered zero.

Half length, ft For PKN and KGD models

Radius, ft For the radial model

Efficiency (fraction) For all models

CDM Cl^2, ft^2/(psi-sec) This combined CDM parameter (together with the closure pressure) will influ-

ence the fracture propagation velocity. If this value is "large" ( for instance is on

the order of 1) there is no retardation of the fracture propagation and essentially

the model behaves as a traditional PKN model. If this value is less (e.g. 0.01

ft^2/(psi-sec) then the fracture propagation is retarded. It takes more time to

reach a certain length and the width and net pressure are higher than calculated


with the traditional PKN model. This parameter is obtained from the maximum

pressure experienced during the minifrac.


44

TTHEORHEORYY

The key to the material balance is fluid leakoff. Fluid leakoff is controlled by a continuous build-up of a thin layer (filter cake) which manifests an ever increasing resistance to flow through the fracture face. In reality, the actual leakoff is determined by a coupled system, of which the filter cake is only one element. A fruitful approximation dating back to Carter, 1957 (Appendix to Howard and Fast, 1957), is to consider the combined effect of the different phenomena as a material property. According to this concept, the leakoff velocity, v L , is given by the Carter equation:

vC

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

where CL is the leakoff coefficient (length/time0.5) and t is the time elapsed since the start of the leakoff process. The idea behind Carter's leakoff coefficient are that: 1) if a filter-cake wall is building up it will allow less fluid to pass through a unit area in unit time; and, 2) the reservoir itself can take less and less fluid if it has been exposed to inflow. Both of these phenomena can be roughly approximated as "square-root time behavior". The integrated form of the Carter equation is:

V

A= C 2 t + SLost

LL p ............................................(2)

where VLost is the fluid volume that passes through the surface AL during the time period from time zero to time t. The first term, 2C tL can be considered as width of the fluid passing through the surface during

the main part of the leakoff process. (The factor 2 appears because the integral of 1/ t is 2 t ).

Fluid efficiency is defines the fraction of the fluid remaining in the fracture: V Vi/ . The fracture surface, A, is the area of one face of one wing and the average width, w , is defined by the relation: V V Awi .

It is often assumed that the created fracture remains in a well defined lithological layer (mostly the pro-ducing formation), and the fracture is therefore characterized by a constant height, hf.

A hydraulic fracturing operation may last from tens of minutes up to several hours. Points of the fracture face near to the well are opened at the beginning of pumping while the points at the fracture tip are “younger”. Application of Eq. 1.25 necessitates the tracking of the opening-time of the individual fracture face elements. If we divide the injected volume by the surface area of one face of one wing, A, we obtain the so called "would-be" width. The would-be width can be decomposed into average fracture width, leakoff width and spurt loss width:


V

A= w 2 C t Si

L e p 2 .........................................(3)

where the factor 2 is introduced because the fluid leaks off through both faces of one wing. The dimen-sionless factor, , is the opening-time distribution factor. It reflects the effect of the distribution of the opening-time. If all the surface is opened at the beginning of the injection, then reaches its absolute maximum, = 2.

In order to obtain an analytical solution for constant injection rate Carter considered a hypothetical case when the fracture width remains constant during the fracture propagation (the width "jumps" to its final value in the first instant of pumping.) Nolte (1986) postulated a basically similar, but mathematically sim-pler assumption. He assumed that the fracture surface evolves according to a power law,

A tD D .................................................(4)

where A A AD e / , t t tD e / and the exponent remains constant during the whole injection period.

If this assumption is accepted the opening time distribution factor is easily obtained from the exponent . Selected values are given in Table 1.2.

Table 1 Opening time distribution factor

Model PKN KGD Radial

4/5 2/3 8/9

1.415 1.478 1.377

The information obtained from a minifrac calibration treatment includes the closure pressure, pc, the leakoff coefficient and possibly perforations and near-wellbore conditions.

The fall off part of the pressure curve is used to obtain the leakoff coefficient for a given fracture geome-try. The original concept of pressure decline analysis due to Nolte (1979) is based on the observation that during the closure process the rate of pressure fall-off provides useful information on the intensity of the leakoff process. (During the pumping period the pressure is affected by many other factors, and hence the influence of leakoff is masked.)

If we assume that the fracture area has evolved with a constant exponent and remains constant after the pumps are stopped, at time (te+t) the volume of the fracture is given by

V =V 2A S 2A g t C tt t i e p e D L ee , ..............................(5)

where the dimensionless shut-in time is defined as

t t tD e / ,................................................(6)

and the two-variable function: g tD , introduced by Nolte is the generalization of the opening time distribution factor.


For computational purposes the approximations in Table 2 can be used. Notice that the above two vari -able functions give back the values shown in Table 1 for zero shut-in time.

Table 2 Approximation of the g-function for various exponents (d = tD)

g d ,

2

3

1.47835 + 81.9445 d + 635.354 d + 1251.53 d + 717.71 d + 86.843 d

1. + 54.2865 d + 372.4 d + 512.374 d + 156.031 d + 5.95955 d - 0.0696905 d

2 3 4 5

2 3 4 5 6

g d ,

8

9

1.37689 + 77.8604 d + 630.24 d + 1317.36 d + 790.7 d + 98.4497 d

1. + 55.1925 d + 389.537 d + 557.22 d + 174.89 d + 6.8188 d - 0.0808317 d

2 3 4 5

2 3 4 5 6

Dividing Eq. 6 by the surface area of one face, the fracture width at time t after the end of pumping is given by

- e

wV

AS C t g tt t

ip L e De 2 2 , . ................................(7)

The first term on the right-hand-side is the "would-be" width. To obtain the actual width the spurt width and the leakoff width are subtracted from the would-be width. The leakoff width increases even after the pumps are stopped, and the g function is the mathematical description of this process. As seen, the time variation of the width is determined by the g(t D,) function, the length of the injection period and the leak-off coefficient, but is not affected by the fracture area.

Unfortunately the decrease of average width cannot be observed directly, but according to linear elastic-ity theory the net pressure during closure is directly proportional to the average width:

p S wnet f ..................................................(8)

The coefficient Sf is the fracture stiffness, expressed in Pa/m (psi/ft). Its inverse, 1/Sf, is sometimes called the fracture compliance. For the basic fracture geometries, expressions of the fracture stiffness are given in Table 3.

Table 3 Proportionality constant, Sf and suggested for basic fracture geometries

PKN KGD Radial

4/5 2/3 8/9

S f 2E

hf

'

E

x f

'

3

16

E

R f

'


The combination of Eqs. 7 and 8 yields:

- -p pS V

AS S S C t g tC

f i

ef p f L e D

2 2 , ..........................(9)

Equation 9, first derived by Nolte, shows that the pressure fall-off in the shut-in period will follow a straight line trend:

- Np b m g tN D , ........................................(10)

if plotted against the g-function (i.e., transformed time). The g-function values should be generated with the exponent, , considered valid for the given model. The slope of the straight line, mN , is related to the unknown leak-off coefficient by:

Cm

t SLN

e f

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

Substituting the relevant expression for the fracture stiffness the leakoff coefficient can be estimated as given in Table 4.

Table 4 Minifrac Analysis by the Nolte-Shlyapobersky method

PKN KGD Radial

Leakoff coef-ficient,

CL

Fracture Ex-tent

Fracture Width

Fluid Effi-ciency

This table shows that for the PKN geometry the estimated leak-off coefficient does not depend on un-known quantities since the pumping time, fracture height and plane strain modulus are assumed to be


known. For the other two geometries considered, the procedure results in an estimate of the leak-off coef -ficient which is strongly dependent on the fracture extent (xf or Rf).

The effect of the spurt loss is concentrated in the intercept of the straight-line with the g = 0 axis, there-fore:

SV

A

b p

Spi

e

N C

f

1

2 .........................................(12)

Unfortunately we do not know the would-be width (Vi/Ae) because the fracture extent is not known. As suggested by Shlyapobersky (1987), Eq. 12 can be used in a reverse manner to obtain the unknown frac-ture extent, if we assume that the spurt loss is negligible. The second row of Table 4 shows the estimated fracture extent for the three basic models using this concept. Note that the "no-spurt loss" assumption also results in an estimate of the fracture length for the PKN geometry, but this value is not used for obtaining the leakoff coefficient. For the other two models, the fracture extent is obtained first and then the value is used in interpreting the slope.

Once the fracture extent and the leakoff coefficient are known, the average width and the fluid efficiency is easily calculated as shown in the subsequent rows of Table 4.

It is not acceptable to take the fluid efficiency from a minifrac treatment and use it as an input variable to the design of the main treatment because the fluid efficiencies in the minifrac and main treatment are dif-ferent. The transferable parameter is the leakoff coefficient itself, but some caution is needed in its inter -pretation. The apparent leakoff coefficient determined from the above method is obtained with respect to the whole fracture area. If we have information on the permeable height hp, and it indicates that only part of the fracture area falls into the permeable layer, the apparent leakoff coefficient should be converted into a "true value" with respect to the permeable area only. This is done simply dividing the apparent value by the ratio of active area to total area.

In some cases there is a considerable discrepancy between the net pressure predicted by the traditional models (such as the PKN model) and the value experienced during the minifrac treatment. The overpres-sure" can be used to determine an additional parameter. The PKN-CDM analysis worksheet determines the so called CDM parameter, which can be used in the fracture design as an additional parameter, con -trolling the lateral fracture propagation. The smaller this parameter is, the fracture propagation is more "retarded" that is the shorter and wider fracture will be predicted during the design.


55 SSAMAM

PLEPLE R RUNUN

Traditional Radial Minifrac Analysis

Input

Permeable (leakoff) thickness, ft 42

Plane strain modulus, E' (psi) 2.00E+06

Closure Pressure, psi 5850

Tabular Input

Time, min BH Injection rate, bpm BH Pressure, psi Include into inj volume Include into g-func fit

0.0 9.9 0.0 1 0

1.0 9.9 0.0 1 0

2.0 9.9 0.0 1 0

3.0 9.9 0.0 1 0

4.0 9.9 0.0 1 0

5.0 9.9 0.0 1 0

6.0 9.9 0.0 1 0

7.0 9.9 0.0 1 0

8.0 9.9 0.0 1 0

9.0 9.9 0.0 1 0

10.0 9.9 0.0 1 0

12.0 9.9 0.0 1 0

14.0 9.9 0.0 1 0

16.0 9.9 0.0 1 0

18.0 9.9 0.0 1 0


20.0 9.9 0.0 1 0

21.0 9.9 0.0 1 0

21.5 9.9 0.0 1 0

21.8 9.9 0.0 1 0

21.95 0.0 7550.62 0 0

22.15 0.0 7330.59 0 0

22.35 0.0 7122.36 0 0

22.55 0.0 6963.21 0 1

22.75 0.0 6833.39 0 1

22.95 0.0 6711.23 0 1

23.15 0.0 6595.02 0 1

23.35 0.0 6493.47 0 1

23.55 0.0 6411.85 0 1

23.75 0.0 6347.12 0 1

23.95 0.0 6291.51 0 1

24.15 0.0 6238.43 0 1

24.35 0.0 6185.85 0 1

24.55 0.0 6135.61 0 1

24.75 0.0 6090.61 0 1

24.95 0.0 6052.06 0 1

25.15 0.0 6018.61 0 1

25.35 0.0 5987.45 0 1

25.55 0.0 5956.42 0 1

25.75 0.0 5925.45 0 1

25.95 0.0 5896.77 0 1

26.15 0.0 5873.54 0 1

26.35 0.0 5857.85 0 0

26.55 0.0 5849.29 0 0

26.75 0.0 5844.81 0 0

26.95 0.0 5839.97 0 0

27.15 0.0 5830.98 0 0

27.35 0.0 5816.3 0 0

27.55 0.0 5797.01 0 0

27.75 0.0 5775.67 0 0


Output

Slope, psi -4417

Intercept, psi 13151

Injected volume, gallon 9044

Frac radius, ft 39.60

Average width, inch 0.49205

Fluid efficiency 0.16708

Apparent leakoff coefficient (for total area), ft/min^0.5 0.01592

Leakoff coefficient in permeable layer, ft/min^0.5 0.02479

Notice that the leakoff coefficient with respect to the permeable layer is greater than the apparent value, because part of the 39.6 ft radius fracture falls outside the permeable area.

Chapter A - Nomenclature MF ManualMF Manual

AA NNOMOM

ENCLATUREENCLATURE

hp = net pay thickness, ft

hf = fracture height, ft

E' = plain strain modulus, psi

K' = Power law consistency index , lbf/(ft2-sec)

n' = Power law flow behavior index

qi = fluid injection rate, bpm

rp = permeable to total area ratio

Rf = created fracture radius, ft

Sf = fracture stiffness, psi/ft

te = pumping time, min

Vi = injected volume, ft3

xf = fracture half length, ft

w = fracture width, ft

Chapter A - NomenclatureMF manualMF manual

Chapter A - Nomenclature MF ManualMF Manual

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