PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering

Stanford University, Stanford, California, February 11-13, 2013

SGP-TR-198

ESTIMATING NATURAL-FRACTURE PERMEABILITY FROM MUD-LOSS DATA

Serhat Akin

Middle East Technical University

Petroleum & Natural Gas Eng Dept, Dumlupinar Blvd No:1

06800 Cankaya, Ankara, Turkey

e-mail: serhat@metu.edu.tr

ABSTRACT

Knowing locations, distributions and apertures of

fractures crossing a geothermal well is of vital

importance in order to minimize costs and increase

efficiency of drilling. Complete or small losses of

drilling fluid flowing from wellbore to the

surrounding formations have been used to identify

fracture zones in the past. An analytical model based

on transient radial mud-loss invasion from a borehole

into a fracture plane coupled with an artificial neural

network approach is developed to estimate natural-

fracture permeability. The developed model is

compared to pressure transient test results obtained

for several wells located in a liquid dominated

geothermal reservoir in west Turkey. It has been

observed that the model fracture permeability values

are in accord with well test derived permeability

values.

INTRODUCTION

The idea of associating mud losses with fracture

permeability in a naturally fractured reservoir was

proposed in sixties (Drummond, 1964). Apart from a

few studies most studies are only qualitative (Dyke et

al, 1995). Most of the mud loss modeling studies

focuses on compressible Newtonian mud propagating

in a non-deformable fracture of constant aperture

with impermeable walls. Coupling the diffusivity

equation with a constant pressure difference

boundary condition and interpolating a tabulated

solution of the problem, Sanfillippo et al (1997)

obtained approximate analytical solutions between

time and mud (Newtonian) volume lost, which was

then converted to fracture hydraulic aperture. They

claimed that eventual mud loss stops were due to

fracture plugging by mud particles although this was

not accounted for in the model. Lietard et al (1999,

2002) developed type curves that describe mud loss

volume vs time. Type curve matching was then used

to obtain natural fracture’s hydraulic width. Majidi et

al (2008a, 2008b) developed mathematical solutions

using yield-power law fluids as opposed to Bingham

plastic fluids used by Lietard et al (1999, 2002).

Huang et al (2011) recently proposed a cubic

equation with input parameters given by the

overpressure ratio, maximum mud loss volume and

the well radius to obtain effective hydraulic fracture

aperture.

DRILLING MUD INVASION INTO

FRACTURES

Lietard et al. (1999, 2002) developed a model based

on the radial flow of a Bingham plastic drilling fluid

through an infinite fracture. The drilling fluid flow

was described by the local pressure drop caused by

the laminar flow inside the fracture with a width of w.

Linear momentum equation relates the local pressure

gradient and average velocity using the following

equation,

(1)

Where and p are the yield stress and plastic

viscosity of the drilling fluid. The local velocity of

the mud in the fractures under radial conditions

around the well is given by vm.

( ) ( )

(2)

Introducing dimensionless mud invasion radius (rd),

time (td) and dimensionless mud-invasion factor (d)

Lietard (2002) obtained the following differential

equation with initial conditions rd=1 when td=0.

( )

( ) (3)

Where

(4)

(5)

. ⁄ / (6)

(7)

Solution of equation 3 is given by Civan and

Rasmussen (2002).

( ) ( ( ) 0

.

/1 ∑

[.

/

] .

/

)

(8)

Where rdmax is the maximum dimensionless mud

invasion radius given by the following equation.

(9)

Lietard et al. (1999, 2002) expressed the cumulative

volume of mud loss Vm by substituting equations 4

and 5 leading to the definition of a parameter X and

equations 5 and 6 leading to the definition of a

parameter Y as shown below.

*, ( )- + (10)

(11)

(12)

A series of type curves relating X and Y can be

constructed by changing w and rdmax which then can

be used to matched field data to obtain w.

More recently Huang et al (2011) proposed a simpler

method based on Lietard et al.’s solution. The

solution is based on the fact that the mud losses will

eventually stop because of the overpressure

eventually reaching the yield stress of the drilling

fluid. The ultimate invasion radius that depends on

the wellbore radius, the yield value of the drilling

fluid and the amount of overpressure can be written

as

(13)

The maximum mud-loss volume is then given by

( ) [

] (14)

Substituting equation 14 into equation 13 gives a

cubic equation (15) in the fracture width (w) with

coefficients dependent on the well radius (rw), the

overpressure ratio (p/y), and the maximum mud-

loss volume (Vmmax). Solution of this equation for the

fracture aperture by discarding physically

meaningless roots is a simple and direct way of

obtaining the fracture aperture when compared to the

curve fitting method proposed by Lietard et al. (1999,

2002).

(

)

(

)

( ) (15)

RESULTS & DISCUSSIONS

A knowledgebase consisting of 162 different data

sets has been developed using the parameters

provided in Table 1. These parameters are then used

to obtain fracture apertures by solving the

aforementioned cubic equation. The model results

are used to train a feed forward artificial neural

network (ANN) model using error back propagation.

Then using this knowledgebase several different

ANN’s were trained. The ANN is supplied with

mud-loss volume, number of events and overpressure

ratio values as input data and the average fracture

aperture is obtained. During the training process for

determining the weights, some data should be

withheld for later verification of network accuracy.

These data are often referred to as test or validation

data. Once the weights have been determined

through back propagation, the test data were used as

network inputs for determining the network’s

accuracy in predicting unprocessed data sets. The

quality or goodness of training was judged based on

the closeness of the prediction of the remaining

“testing” data (i.e. 5% of the total fracture aperture

results using the input data that was not used for

training). This process was repeated for various

networks and the network with the highest accuracy

was used as the model. Rather than randomly

selecting the initial weight matrix, previously

generated successful matrices were used at the start.

This feature decreased the iterations approximately

30% and also guaranteed training of a “good”

network (Yilmaz et al, 2002).

Table 1: Mud-loss volume, number of events and

overpressure ratio values used in

sensitivity analysis. .

Mud-loss Volume, bbl 10 - 80000

Number of Events 1 - 400

Overpressure ratio 364420 - 840000

Since the use of more than a single layer can lead to a

very large number of local minima and make the

training extremely difficult (Haykin; 1994), a single

hidden layer network was used. Several networks

with varying degree of complexity have been trained.

The best result was obtained with an ANN model

composed of 20 hidden nodes, with the momentum

and learning parameters of 0.3 and 0.5 respectively.

Figure 1 shows the average mean square error per

input for training and validation sets.

Figure 1: Average mean square error per input for

training and validation sets.

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0 100 200 300 400 500 600

Me

an s

qu

are

err

or

Epoch

Training set

Validation set

Three wells drilled in Alaşehir Graben in west

Anatolia, Turkey will be used to demonstrate the uses

of the proposed ANN methodology. First observed

mud losses will be used in Huang et al (2011) cubic

equation. Then the same data will be analyzed with

the ANN model. Two of these wells (Well-1 and

Well-2) showed limited mud loss rates at the

reservoir zone before a total loss was observed at the

fault zone that consisted of marble, calcschist and

calcschist-schist (Fig. 1 and Fig. 2) sequence. Well-3

mud loss rates (Fig. 3) were at the same order of

magnitude with the rates observed in the other wells.

However, total loss of mud was not observed in this

well. The mud loss volumes, number of events and

overpressure ratio observed in these wells are given

in Table 2.

Table 2: Mud-loss volume, number of events and

overpressure ratio values observed during

the drilling. Well 1 2 3

Drill bit diameter,

inch

81/2 81/2 81/2

Mud-loss Volume,

bbl

1666 1147 1264.9

Number of Events 36 22 138

Overpressure ratio 641777.8 684000 684000

Calculated

aperture, m

3.71 x10-4 3.7 x10-4 2.07 x10-4

Calculated

permeability, mD

745.12 1100.23 83.37

Well Test Derived

permeability, mD

517.21 968.35 255

ANN aperture, m 3.87x10-4 4.45x10-

4

1.68x10-4

ANN permeability,

mD

846.26 1286.62 44.5

Figure 2: Mud loss rate observed in Well-1.

Figure 3: Mud loss rate observed in Well-2.

Figure 4: Mud loss rate observed in Well-3.

First average mud loss volumes were calculated by

dividing the total mud loss volume to the number of

events monitored during drilling the reservoir section.

This value is used together with the overpressure

ratio in equation 15 and the fracture aperture values

presented in Table 2 are obtained by solving the

cubic equation using MS Excel’s nonlinear solver.

Fracture aperture values are then converted to

permeabilities using the cubic law (Golf Racht,

1982). In doing so several assumptions are made

(Norbeck et al, 2012):

1. All natural fractures are finite with a constant

aperture,

2. The fractures are transverse to the wellbore and

have circular geometry,

3. Cubic law relationship is valid,

4. Matrix permeability is significantly lower than

fracture permeability,

5. Fractures are not charged during the process

Using the Huang et al cubic equation average mud

loss was calculated by dividing the mud loss volume

to the number of events. Substituting these values

together with overpressure ratio the average fracture

aperture was calculated as 3.71x10-4

m. Since

conventional petrophysical logs as well as fracture

detection logs and core plugs are not available for

any of the wells used for this study to quantify and

950

1050

1150

1250

1350

1450

1550

1650

0 100 200 300 400 500

De

pth

, m

Mud Loss Rate, bbl/hr

950

1050

1150

1250

1350

1450

1550

1650

0 100 200 300 400 500 600

De

pth

, m

Mud Loss Rate, bbl/hr

950

1150

1350

1550

1750

1950

2150

0 10 20 30 40 50

De

pth

, m

Mud Loss Rate, bbl/hr

prove the existence of fractures, the results obtained

from cubic equation approach has been compared to

the results of pressure buildup analyses. Using cubic

law (Golf-Racht, 1982) the corresponding

permeability value was calculated as 745.12 mD,

which compared favorably with the well test derived

permeability of 517.21 mD. Typically, pressure

buildup test volume is much larger than the volume

sampled based on the mud losses observed during

drilling. That’s why we expect a similar order of

magnitude in permeability but the average values

may differ. Similar observations were obtained for

the other wells. That is to say the derived

permeability was in the same order of magnitude but

the value was somewhat different.

When ANN results were compared to model and well

test derived permeability values, it was observed that

the permeability values were somewhat higher than

both the model and well test derived permeabilities

for complete mud loss wells. On the other hand the

permeability was somewhat smaller in well #3. One

reason for the mismatch could be the small number

training data. When only a relative small data set is

available for the application of a neural network a

number of drawbacks occur. Firstly, one has to split

the already small data set into a training set and a

testing set. Secondly, overtraining is more likely to

occur with small datasets, because the degrees of

freedom in the network rapidly increase with the

number of neurons. For a good model, the number of

data pairs should exceed the number of weights in the

neural network. One way to tackle this problem

could be the use of orthogonal functions such as

Fourier transforms to transform the input data before

they are used to train the network (Silvert and

Baptist, 1998).

CONCLUSIONS

An analytical model based on transient radial mud-

loss invasion from a borehole into a fracture plane

coupled with an artificial neural network approach is

developed to estimate average natural-fracture

aperture and the corresponding average fracture

permeability. The developed ANN model is

compared to pressure transient test results obtained

for several wells located in a liquid dominated

geothermal reservoir in Alaşehir graben, west

Turkey. It has been observed that the model fracture

permeability values are in accord with well test

derived permeability values that shows that it is

possible to use the model during drilling to predict

permeability of natural fractures.

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