Analysis of Experimental Analysis of Experimental Data for Flow Thorough Data for Flow Thorough Fractures using GeostatisticsFractures using Geostatistics

DICMAN ALFREDDICMAN ALFREDDr. ERWIN PUTRADr. ERWIN PUTRA

Dr. DAVID SCHECHTERDr. DAVID SCHECHTER

L

ppbq L

03

12

Fracture model

W = 2b

Cubic law of fractures

From the experiments knowing pressure drop and flow rate , the aperture b can be calculated.

Actual core surface

Actual fracture surface

2be

Louis (1974) proposed that when e/D < 0.033, then f = 1e/D >0.033, then f = (1 + 8.8(e/D)^1.5)

L

pp

f

bq L

03

12

e/D is defined relative roughness,

where D is the hydraulic diameter = 2*2b

Modified cubic law

Work by researchers, such as Neuzil and Tracy (1981), Brown (1987), Tsang and Tsang (1987), Tsang et al. (1988) and Moreno et al. (1988), have shown that the flow through a fracture follows preferred paths or flow channels due to the variation in fracture aperture.

Previous Research

Detailed measurements of fracture apertures have been obtained by joint surface profiling (Bandis et al. 1981, Brown and Scholz 1985, Gentier 1986), low melting point metal injection (Pyrak-Nole et al. 1987, Gale 1987), and resin casting technique (Hakami 1988, Gentier et al. 1989). BUT THEY ARE EXPENSIVE AND THE DATA MAY NOT BE A TRUE REPRESENTATIVE OF THE FRACTURE.

Tsang (1990) chose a statistical description of a fracture with variable apertures by means of three parameters , performed numerical flow and transport experiments with them with particular emphasis of correlate the fracture geometry parameters. But concluded that the correspondence between observations and the hydrological properties is STILL AMBIGUOUS.

Our Approach

Experimental data-DP,K,Q,Kavg

Expermental data analysis b,Kf Qf, Qm

Fracture surface generated randomly through geostatistics

Simulation model with varying permeability distribution

Study the effect of variance and friction factor on flow

Simulate and match the pressure drop from experimental data

•Multiphase flow•Upscaling to outcrop studies

Include the friction factor to derive fracture permeabilty

2ln

2

1exp

2

1)(

x

xxf

2

22 1ln

2ln

2

x

zln

Probability Density Function for Log Normal Distribution

If is the mean and2 is the variance

To standardize this ,

Similar to the normal distribution

How do we apply this to our research ??????

Variogram : summarises the relationship between the variance of the difference between measurements and the distance of the corresponding points from each other.

Kriging : uses the information from a variogram to find an optimal set of weights that are used in estimating a surface at unsampled locations.

Variogram and Kriging

Lag distance

Co-

var

ianc

e

Sill : describes where the variogram develops a flat region, i.e. where the variance no longer increases. Range : the distance between locations beyond which observations appear independent i.e. the variance no longer increases.

Nugget variance : when the variogram appears not to go through the origin.

Kriging

We can use the variogram to estimate values at points other than where measurements were taken. This process is termed kriging.

Log Normal Distribution

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 50 100 150 200 250 300

Aperture

Rel

ativ

e Fr

eque

ncy

variance 50

variance 300

variance1000

Variance 1800

Variance 2320

Variance 2200

What is the effect of changing variance on permeability ????

So lets get started !!!!

500500 1.21.2 2.72.7 4.154.15

10001000 2.262.26 4.84.8 7.67.6

15001500 4.64.6 9.39.3 15.115.1

5 cc/min 10 cc/min 15 cc/minPressure Drop

Experimental Data

500500 3.753.75 6.756.75 10.2410.24

10001000 2.582.58 4.414.41 6.316.31

15001500 0.640.64 0.820.82 0.690.69

Flow through fracture

Log Normal Distribution

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 20 40 60 80 100 120 140

Aperture

Rel

ativ

e Fr

eque

ncy

core56.4 var300

core40 var100

core20 var 30

Core56 var 200 Core40 var 100

Core20 var 30

VARIOGRAM MODELING

CORE 56.4 VAR 200

CORE 40 VAR 100

CORE 20 VAR 30

PERMEABILITY DISTRIBUTION CORE 56.4 VAR 200

PERMEABILITY DISTRIBUTION CORE 40 VAR 100

PERMEABILITY DISTRIBUTION CORE 20 VAR 30

The volume of the core is maintained constant

Grid definition 31*15*15

RESULTS

• Sensitivity studies

• Pressure Drop match

• Rate comparisons between theoretical and

simulated flow

• Permeability comparison

• Variance vs Overburden pressure

• Comparison between cubic law and modified

cubic law

Variance vs Flow rate

3.7

3.75

3.8

3.85

3.9

3.95

4

4.05

4.1

0 200 400 600 800 1000 1200

Variance

Flow

rate

, cc/

min

Variance vs Dp

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 200 400 600 800 1000 1200

Variance

Dp,

psi

a

Pressure Drop Match

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 200 400 600 800 1000 1200 1400 1600

Overburden Pressure, psia

Pre

ssur

e D

rop,

psi

a observed

simulated

Matrix Flow Match

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 200 400 600 800 1000 1200 1400 1600

overburden pressure psia

flow

rate

cc/

min

observed

simulated

Fracture Flow Match

0

0.5

1

1.5

2

2.5

3

3.5

4

0 200 400 600 800 1000 1200 1400 1600

Overburden Pressure, psia

Flow

Rat

e, c

c/m

in

observed

simulated

Aperture Comparison

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

0 200 400 600 800 1000 1200 1400 1600

Overburden Pressure, psia

Aper

ture

, mic

ron

Cubic Law

Modified Cubic Law

Variance vs Overburden pressure

0

50

100

150

200

250

0 200 400 600 800 1000 1200 1400 1600

Overburden presure, psia

Varia

nce

Future Considerations

• Extending it to outcrop studies

• Modeling 2-phase flow.