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SUBSIDENCE PREDICTION CAUSED BY THE OIL AND GAS

DEVELOPMENT

Anton Sroka

Institute of Mining Surveying and Geodesy

Technische Universität Bergakademie Freiberg/ Germany

Email: anton.sroka@tu-freiberg.de

Ryszard Hejmanowski

Dept. of Mining Damage and Geoinformatics

Faculty of Mining Surveying and Environmental Engineering

AGH University of Science and Technology/Cracow/ Poland

Email: hejman@agh.edu.pl

Abstract: Oil and gas development from underground reservoirs disturbs original rock mass

balance. The tending of the rock mass to achieve a new, even only temporary balance is

manifested in the movements of the ground surface. Movements can affect ground

infrastructure like offshore platforms, pipelines and buildings. For increasing the efficiency of

the preventive actions a’priori precisely prediction of the subsidence is necessary. By the

prediction of surface subsidence changes of pore pressure in time due to exploitation and

geometry of the reservoir have to be taken into account. The prediction method based on the

influence function of Knothe will be presented in the paper. Some applications according to

the oil and natural gas developments will be discussed

1. Introduction

Surface subsidence of areas where oil, gas and water are exploited are a serious problem in

various parts of the World. In the coastal regions, vertical movements of the surface may

result in flooding or generate extra costs for securing the banks. Such problems were

encountered, e.g. in the area of Maracaibo Lake in Venezuela (ca. 3.5 m – maximum

subsidence), Mexican Gulf, in California (ca. 10 m – maximum subsidence) and in Japan. The

subsidence troughs may be huge in size and the damage to the objects standing on them is

comparable to those in the mining areas. Considerable deformations of surface usually can be

found in the places where thick fluid reservoirs occur and the host rocks have compaction

qualities.

Surface deformations in the areas of oil and gas exploitation can be efficiently predicted on

the basis of methods employing the influence functions. These methods make use of relatively

uncomplicated mathematical models and are simple in use. Regarding their high efficiency,

these methods are not frequently met in the World’s literature, therefore the Authors decided

to discuss the prediction model based on an influence function applicable to fluid mineral

reservoirs.

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

2. Rudiments of the proposed method

2.1. Discretization of reservoir

All the considerations made in this paper will refer to an elementary part of a reservoir. The

whole reservoir will be divided into elementary cubicoids having a square base of side L and

height M

0

, corresponding to the original thickness of the fluid reservoir [7](fig.1).

0

L

L

Fig.1. Element of reservoir

This approach enables a description of a varying thickness of a reservoir and simplification of

numerical operations. Once the whole gas reservoir field has been divided into elements, the

following situation is obtained, fig. 2.

X,Y,Z,M ,p ,p(t)

00

Fig. 2. Gas field divided into elements

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

2.2. The main cause-and-effect relation

Geometrical-integral influence exploitation model bases on the following cause-and-effect

relation (1). Accordingly, the result, i.e. surface subsidence can be determined as a loop of

two functions: influence function or transformation function

),,( tzr

ϕ

, and a function

describing the cause of rock movement, i.e. compaction

)(tK

∆

.

),,()(),,( tzrtKtzrS

ϕ

∗∆=∆ (1)

where:

- increase of compaction volume of reservoir element in time t. )(tK∆

φ

(H,t)

φ

(z ,t )

1

z= z

0

z= z

1

z= H

r = 0

cause

effect z

1

effect z

Η

∆

M(t)

M

0

Fig.3. Causal nexus for exploitation of one reservoir element

It should be noted that both functions in eq. (1) are functions of time.

3. Calculation model

Being a cause of rock mass movements, in the case of oil and gas exploitation, compaction

depends on changes of pore pressure in the gas or oil field. This relation may be described

linearly to some extent:

[]

00

)()( MtppctM

m

⋅−⋅=∆

(2)

where:

∆

M(t) – change of thickness of fluid reservoir,

c

m

– compaction coefficient,

p

0

– original/primary? pore pressure,

p(t) – pore pressure in a moment t,

M

0

– original/primary? thickness of the analysed element of reservoir.

The compaction coefficient c

m

depends on the properties of reservoir rocks, depth of

reservoirion and original/primary? pressure p

0

[e.g., 15, 16]. The compaction volume for thus

defined element of reservoir (fig.1) is:

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

2

)()( LtMtK ⋅∆=∆ (3)

where:

∆

K(t) – bulk/volume compaction in a time t

The influence functions have been known to be used for predicting surface deformations since

Bals published his work in 1931 [1]. In other works employing influence functions written by

Litwiniszyn [12] and Knothe [9], the influence functions were justified in the form of a

validated Gauss function. The form of the function for predicting the surface may assume the

following form:

⋅−=

2

2

2

exp

1

),(

KK

R

r

R

Hr

πϕ

(4)

where: H – depth of exploitation,

r – horizontal distance of calculated point from the reservoir element,

R

K

– radius of scattering of exploitation influences after Knothe for a surface:

β

ctgHR

Hz

K

⋅=

= )(

(5)

β

- angle of the main influence range, after Knothe.

Predicting the subsidence within a rock mass, the R

K

value should be calculated from the

following relation:

β

ctgzH

H

z

RzR

nnn

H

KK

⋅⋅==

⋅−1

)(

)()( (6)

where: n – coefficient of boundary surface [11, 2, 3, 4, 10, 13].

For calculating this coefficient, n = 0.5 can be assumed.

Basing on elasticity theory, in 1970s Geertsma proposed an influence function in the

following form [5, 6]:

3

1

)(

d

H

d ⋅

−

−=

π

υ

ϕ

(7)

where: υ - Poisson coefficient,

d – distance of the calculated point from the reservoir element.

Other examples of influence functions to be potentially used for predicting deformations in

the exploitation conditions can be found in literature, e.g. Kochmański, Perz. The application

of the function in the form presented in (4) seems to enable a more precise validation of the

model. Owing to the fact the process of rock mass deformation over the oil or gas exploitation

area develops in time, it is crucial to introduce a function of time f(t). It describes the delaying

impact of the rock mass:

)(),(),,( tfzrtzr ⋅=

ϕ

ϕ

(8)

According to Knothe, the function has the form:

)exp(1)( tctf ⋅

−

−= (9)

where: c – time coefficient, which gives a full description of the delaying influence of a

porous reservoir rock on the lowering pore pressure and the delayed influence of the onlying

rocks.

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

Predicting the surface subsidence, they can be calculated by superpositioning the elementary

subsidence’s. Elementary subsidence caused by exploitation of a fluid from one or any

element of reservoir will be written in compliance with (1) as:

),,()()(

,,,

tzrtKtS

ijijiij

ϕ

⋅∆=∆ (10)

where: j – number of calculation point,

i – number of reservoir element,

()()

22

, ijijij

yyxxr −+−= ,

.

ijij

zzz −=

,

Using the linear superpositioning, a subsidence of any point in a time t can be calculated

from:

∑

=

=

∆=

Ni

i

ijj

tStS

1

,

)()( (11)

where: N – number of reservoir elements (

Ni

≤

≤

1

).

The subsidence distribution over a large fluid reservoir with the on-going exploitation can be

calculated on the basis of known reservoir pressure distributions. Having divided of the whole

reservoir into constituent elements it was possible to precisely describe the reservoir pressure

distribution, its changeability in time and space, and hence to precisely model the surface

subsidence in time. If the element of a reservoir were treated as an object initiating rock mass

movements, it can be described values of such attributes as, e.g. location (x, y, z), thickness

M

0

, primary pore pressure p

0

, pressure caused by exploitation in the successive moments of

time p

1

(t

1

), p

2

(t

2

), etc. Contemporary numerical techniques enable operation of attributes for

each reservoir element not only at the stage of calculation and prediction of subsidence’s.

Having had a description of attributes, spatial analyses can be made. Their objective can be,

e.g. determining efficiency of a given exploitation method or prevention against too big

deformations of the sea bed on shelves or on the surface.

4. Practical application

4.1. Prediction of surface subsidence

The Authors verified the presented model on some real-life examples. These were oil

reservoirs under the North Sea and gas reservoirs in one of European countries.

The modeling in the case of the gas field was made in stages. The results of calculations were

compared with the levelling measurement and presented in figure 5. Parameters of the

calculation model, i.e. R

K

, c, c

m

, were determined on the basis of data measured as subsidence

points on surface by a moment t and as compaction properties of the reservoir. The prediction

of subsidence for the successive period (t

i+1

) was calculated with the use of these parameters

and expected decreases of pore pressure value in time t

i+1

. Therefore, it was possible to obtain

an exceptionally high accuracy of predicted subsidence’s, which was not attainable with other

methods [7, 8]. In view of the cause-and-effect relation mentioned in section 2, the parameters

of the model can be also identified. This was indicated by, among others, results of analyses

carried out for a gas field, where the inverse analysis of measured surface subsidences enabled

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

determining the compaction coefficient value c

m

[14]. The obtained values were convergent

with the results of laboratory tests by Teeuw [15].

0 102030405060708090

Observed points [km]

observed predicted

Fig. 4. Prognosed surface subsidences against measured values

4.2. Geodesy surveying measurements of surface subsidence

While presenting the applicability of the model, attention was drawn to the fact that

knowledge of the actual course of surface deformation was extremely important.

Measurements of deformations are usually made with the use of such geodesy surveying

methods as levelling of measuring points. For the last 10 years GPS and remote sensing

methods (e.g., InSAR) have been more and more commonly applied. By determining the

increase of subsidence between individual moments of time, a whole description of changes

in the trough over the exploited reservoir is being made. Designing the parameters or

elaborating remote sensing data, the measurement points should be so densely distributed as

to make their maximum distance equal to:

K

Rl ⋅= 1,0 (12)

Taking into account (5), this can be written as:

)(1,0

β

ctgHl ⋅⋅= (13)

It should be noticed that this distance in the average conditions of fluids exploitation is

considerable.

The time elapsed between measurements is also important for the specific fluid reservoirs. It

should be less than the time of occurrence of the assumed boundary difference of subsidence

∆S

G

. With such a criterion, it is possible to determine an average admissible time span

between the successive measurements of surface subsidence:

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

β

2

2

_

__

0

_

ctg

F

H

pMc

S

t

tm

G

⋅

∆⋅⋅

∆

=∆

∆

(14)

where:

- low thickness of reservoir,

0

_

M

- average decrease of reservoir pressure in time ∆t,

t

p

∆

∆

_

_

H

- average depth of deposition,

F – total surface of the reservoir.

As the decrease of pressure can be written as:

tpt

t

p

p

t

∆⋅=∆⋅

∂

∂

=∆

•

∆

_

, (15)

where: - gradient of pore pressure, then eq. (14) will finally take the form:

•

p

β

ctgH

FpMc

S

t

m

G

_

_

0

_

⋅

⋅⋅⋅

∆

=∆

•

(16)

It follows from eq. (14) that the time span between individual observations depends on a

parameter typical of mineral exploitation rate. In this case it is described by a gradient of pore

pressure decrease.

References:

[1] Bals R.: Beitrag zur Frage der Vorausberechnung bergbaulicher Senkungen. Mitt. Aus

dem Markscheidewesen, 1931/32.

[2] Bartosik-Sroka T., Sroka A.: Zmienność wartości parametrów teorii T.Kochmańskiego

i S.Knothego w górotworze. Rudy i Metale Nieżelazne, nr 7., 1974.

[3] Drzęźla B.: Badania teoretyczne i modelowe ruchów górotworu przy eksploatacji

górniczej. Politechnika Śląska, Gliwice, 1971 (niepublikowane).

[4] Drzęźla B.: Zmienność zasięgu wpływów eksploatacji w górotworze. Przegląd Górniczy,

nr 10. Katowice 1979.

[5] Geertsma J.: Land subsidence above compaction oil and gas reservoirs. SPE-European

Spring Meeting 1972; SPE AIME Nr 3730, pp 17.

[6] Geertsma J., Van Opstal, G.: A numerical technique for predicting subsidence above

compaction reservoirs, based on the nucleus of strain concept. Verhandlingen Koniklijk

Nederlands Geologisch Mijnbouwkundig Genootschap., Vol. 28, pp.63-78, 1973.

[7] Hejmanowski R.: Zur Vorausberechnung förderbedingten Bodensenkungen über Erdöl-

und Erdgaslagerstätten. PhD. Thesis. Technical University Clausthal, Germany, 1993.

[8] Hejmanowski.R., Sroka A.: Time-space ground subsidence prediction determined by

volume extraction from the rock mass: Land subsidence : proceedings of the sixth

international symposium on Land subsidence : Ravenna 24–29 September 2000. Vol. 2:

Measuring and monitoring theory and modeling, 2000. pp. 367–375.

[9] Knothe. S.: Równanie profilu ostatecznie wykształconej niecki osiadania. Archiwum

Górnictwa i Hutnictwa, t.1, z.1, 1953.

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

[10] Kowalski A.: Określenie zmienności parametru promienia zasięgu wpływów głównych

w górotworze teorii Budryka-Knothego na podstawie badań geodezyjnych

przemieszczeń pionowych górotworu. Praca doktorska. GIG, Katowice, 1984

(niepublikowane).

[11] Krzysztoń D.: Parametr zasięgu niecek osiadania w ośrodku sypkim. Archiwum

Górnictwa, t.10, z.1., 1965.

[12] Litwiniszyn J.: Przemieszczenia górotworu w świetle teorii prawdopodobieństwa.

Archiwum Górnictwa i Hutnictwa, t.2, z.4., 1954.

[13] Mikołajczak J.: Uwagi do równania promienia zasięgu wpływów głównych. Ochrona

Terenów Górniczych, nr 69/3. Katowice 1984.

[14] Sroka A., Schober F.: Studie zur Analyse und Vorhersage der Bodensenkungen und

Kompaktionsverhaltens eines Erdgasfeldes. Abschlussbericht, 1989.(niepublikowane)

[15] Teew.D.: Laboratory measurement of compaction properties of Groningen reservoir

rock, Verhandlingen Koniklijk Nederlands Geologisch Mijnbouwkundig Genootschap.,

Vol. 28, pp.19-32, 1973.

[16] Van Kesteren. J.: Estimate of compaction properties of Groningen reservoir rock.

Verhandlingen Koniklijk Nederlands Geologisch Mijnbouwkundig Genootschap., Vol.

28, pp.33-40, 1973

3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006