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Prediction of surface subsidence due to oil- or gasfield development

RYSZARD HEJMANOWSKI Department of Mine Surveying and Environmental Engineering, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krakow, Poland

ABSTRACT Compaction phenomenon caused by oil- or gasfield development can induce the considerable surface deformations. Surface deformations above the oil- or gasfields were observed and measured in Wilmington, Bolivar Coast and at last over the Ekofisk field. When rocks, building the oil- or gas-reservoir have significant compaction coefficient, thickness of the reservoir are big and the intensive pore pressure decline are prognosed the compaction phenomenon, surface and ground deformations may occur. The new method for prediction of surface subsidence, caused by planned oil- or gasfield development is presented in the paper. The following factors were considered in the model: characteristic of the reservoir rocks, pressure of overburden, relationship between compaction and pressure gradient in the reservoir, pressure decline dynamic and its influence on the compaction rate and the surface subsidence. In the presented method the stochastic model of the ground as well as the relation between compaction and the pressure gradient in reservoir was applied. Therefore the predicted characteristic of pore pressure decline is possible to consider. Influence of the retardation of reservoir rocks and overburden on the propagation of ground deformations can also be taken to account. Changes of geomechanic parameters in reservoir are possible to consider in the model thanks reservoir discretisation. It enables to compaction / subsidence modeling above any reservoir with sophistical space forms. The presented method is initially tested for Gronningen reservoir using the accessible from literature data. The method, presented in the paper, was a subject of author's doctor thesis in Clausthal Technical University, Germany.

INTRODUCTION The porosity of reservoir rocks decreases under the conditions of a permanent pressure of overburden and the internal reservoir pressure varying during exploitation. The increasing compaction may sometimes lead to destruction of porous structure in case of weak reservoir rocks. This compaction causes land subsidence, reaching even several meters down (Wilmington). The subsidence troughs forming above the exploited oil or gas beds are in general big in horizontal dimensions. This is due to such factors as: Considerable exploitation depths (to 200 m in Wilmington, to 2900 m in Groningen and to 2900 m in Ekofisk), 2 2 2 Wide exploited areas (90 km in Wilmington, 900 km in Groningen, 200 km in Ekofisk). The prevailing location of exploitation equipment in central areas of fields causes that the most important factor - from the point of view of the land surface protection - are the vertical shifts of the land (subsidence). The remaining components of the shift vector as well as deformations will

rather affect the objects located close to edges of exploited fields. Many works stressing the harmful effect of land subsidence on the exploitation equipment have been published over the last years. The hazards and costs caused by land subsidence and preventive means in the coastal regions were also described (Van Kesteren, 1973; Sulak, 1991 among the others). Both the qualitative and quantitative prediction of possible deformations of the land surface above the oil and gas reservoirs already at the design of the reservoir exploitation is important in prevention. Equally important is a permanent control (updating of predictions) during exploitation. The known methods of predicting land subsidence above oil and gas beds are based mainly on mechanics of continuous medium (Geertsma, 1973; Kosloff et al., 1980; De Waal & Smits, 1985). In this paper another solution, based on the stochastic model of the ground, is presented. The method presented is an extension and complement of the attempt to apply the stochastic model to the oil-field (Sroka, 1988).

THEORETICAL MODEL The movement of ground particles forced by compaction of the rocks of reservoir and by gravitation is extremely difficult to describe. The main difficulty results from the lack of knowledge on mechanical, mutual behavior of ground particles and the spatial variation of mechanical properties of rocks. For these reasons the Author decided to apply the stochastic model of the ground. Such model enables the statistical treatment of the ground movements and has already been applied in Poland for description of the ground deformations in hard coal mines (Litwiniszyn, 1956). Applications of this type of the model involve the assumption that the ground will attain the most probable state of geomechanical equilibrium when the exploitation ends. In consequence, the normal distribution function (the Gauss distribution function) is taken for the function transforming the cause of the deformation - compaction due to pressure in the form of subsidence trough (Fig.1). This function is also called the influence function. elementary subsidence trough

S

F

KE reservoir element

Fig.1 Elementary relationship between compaction and subsidence

The basic relationship describing the subsidence of any point may be written as follows: S(x, y, z) = ∫ ∫ ∫Q( xQ , y Q , zQ ) ⋅ F(x, y, z,x Q , y Q , z Q )dΩ , Ω

(1)

where Ω is the exploited area of a field, Q the distribution of reservoir compaction, F the transforming function (the influence function), x, y, z are coordinates of the point, xQ, yQ, zQ are coordinates of so called "reservoir element". Not only the final subsidence but also the stage subsidence will be interesting for the long lasting land subsidence courses, characteristic for oil and gas reservoirs. For this reason two time relationships were taken into account in the method presented: - Delay in the reservoir compaction, - Retarding action of the ground. Further considerations are devoted to so called “reservoir element”. The whole exploited reservoir is divided into the fragments elementary, from the point of view of calculations (Fig.2). Z X

land surface offshore platform

Y

drilling wells

gas reservoir reservoir element Fig.2. Threedimensional picture of the reservoir (thickness) after its discretisation

Each reservoir element is characterized by its coordinates xQ, yQ, zQ, its local thickness Mi, its primary pore pressure p0 and the change in pressure to the moment t. In this way all reservoir elements are determined both geometrically and physically.

Course of compaction The final axial compaction in the reservoir element may be written as the change in the thickness of the element, caused by total change of pressure: ∆ M E = K E ⋅ M = cm ⋅ ( p0 - p E ) ⋅ M (2) M - primary reservoir thickness in the reservoir element, E K - relative final compaction, cm - uni-axial compaction coefficient, p0 - primary pressure pE - final pressure (at the end of exploitation). The value of the predicted compaction at the time t of exploitation is of some interest: ∆ M E (t) = K E (t) ⋅ M = c m ⋅ ( p0 - p(t)) ⋅ M

(3)

Because of some delay in reaction of the reservoir rocks on the drop in pore pressure the actual

value of compaction at the time t will be somewhat lower than the value resulting from eq. (3). As this value is responsible for the movements in the ground its determination is necessary. For this reason the following differential equation was used: δ K(t) = ξ ⋅ [K E (t) - K(t)] , δt

(4)

where ξ is the time coefficient for the reservoir - relative velocity of the rocks compaction [1/year]. It is the model parameter, dependent on mechanical properties of the reservoir rocks, on the overburden pressure (depth) and on exploitation rate. Eq.(4) may be solved only when the pressure gradient in reservoir is known. The pressure gradient may be estimated from the exploitation design and known exploitation conditions. This value will be varying and will dependent on the velocity of exploitation for this particular part of reservoir. In further presentation the example of linear changes in pore pressure was taken into account (Fig.3). p(t) = -b t + p0 pore press. p(t) = p E

p

p

t ≤ T t>T p -p b= 0 E T-t 0

0

E

exploitation t =0 0

time

T

Fig.3. The assumed model of linear changes in reservoir pressure

In this model it was assumed that porosity changes may be approximated with a line section over small intervals of pressure changes. Eq.(4) was solved for this linear model, giving the volume compaction V(t). The volume compaction is the product of axial compaction, K(t) and volume of the reservoir element V. The following formulae were obtained for both sections from Fig.3: for the moment t: t0 < t≤ T, t0=0:  E  1 E ⋅ [1 - exp(-ξ ⋅ t )] , V K (t) = K(t) ⋅ V = V ⋅  K (t) - K ⋅ ξ ⋅T  

(5)

for the moment t: t> T:

E V K (t) = V K (T) + [V ⋅ K - V K (T)] ⋅ [1 - exp(-ξ ⋅ (t - T ))]

(6)

Then, the volume of the elementary land subsidence, forming at the surface due to the volume compaction of the reservoir, was determined as: δ V M (t) = c ⋅ [a ⋅ V K (t) - V M (t)] δt

(7)

a - volume coefficient; 0.9 < a < 1.0 c - time coefficient for overburden. Solution of eq. (7) lead to the equations enabling calculation of the volume for the elementary land

subsidence, at any stages of exploitation: for the moment t: t0 < t≤ T, t0=0:

  1 1  1 [exp(-ξ ⋅ t ) - exp(- c ⋅ t )]  ⋅  ⋅ 1 - exp(- c ⋅ t ) V M (t) = a ⋅ V K (t) - V ⋅ K E ⋅ T c  c -ξ   

(8)

for the moment t: t> T: V

M

(t) = V

M

(T)

[a

+

[

+ a V ⋅ K

E

⋅ V

K

- V

K

(T) (T) -

- V

]⋅

M

(T)

{1 + c

] ⋅ {1 ξ c - ξ

⋅ exp

c - ξ

- exp

[- c (t

- T

)]} +

⋅ exp

[- c (t

- T

)]

[- ξ (t

- T

-

(9)

)]}

The graphical interpretation of the presented solution of eq. (9) facilitates understanding the meaning of both time parameters ξ - for reservoir and c - for overburden (Fig.4). E

K (t) V E K V

V ( t)

VK(t)

M

1

1

1 c

0

T

Time

Fig.4. The meaning of time coefficients assumed course of compaction

The influence function As it was mentioned in the introduction, the influence function (see eq.(1)) is the appropriate parametric function of the normal distribution (Knothe, 1953):  d2 1 F = 2 ⋅ exp -π 2  (10) R  R  R - so called radius of scattering of main influences, R=H ctg β, d - the horizontal distance between the calculated point and the reservoir element, H - exploitation depth. The radius of scattering of main influences is functionally related to so called angle of the range of influences (β), which is the ground parameter, characterizing the rigidity of rocks of the overburden.

CALCULATIONS OF SUBSIDENCE The way of computing the subsidence of any point results from eq. (1) and the notes concerning the reservoir discretization. It is the sum of subsidence values for this point, caused by compaction of particular reservoir elements: N

S(x, y, z) = ∑ V M i ( xQ , yQ , zQ ) ⋅ F(x, y, z,xQ , yQ , zQ )

(11)

i=1

N - number of reservoir elements. EXAMPLE OF COMPUTATIONS The computational method presented in this paper was pre-tested for the example of the natural gas field. The necessary data were taken from the literature on the subject. To some approximation they correspond to the data on the field in Groningen (Teeuw, 1973; Van Kesteren, 1973 and the others). The prediction of land subsidence was computed for three periods: 1965-1975, 1975-1987 and 1987-2030. As the Author knew only the total drop in the reservoir pressure, reaching 30 MPa at maximum, he assumed the following simulation of pressure drops during exploitation: 1965 - the initial state (pore pressure in the whole reservoir equal to about 35 MPa), 1965-1975 - the maximal change of pressure equal to 11 MPa (exploitation shifts from the South to the North), 1975-1987 - the maximal change of pressure equal to 6 MPa, 1987-2030 - the maximal change of pressure equal to 13 MPa, 2030 - the final state (pore pressure equal throughout the field, of about 5 Mpa). Next discretization of the exploited field into 132 elements was carried out. The following parameters of the computational model were assumed: -1 cm = 1.45 E-04 MPa , a = 1.0, -1 = 0.1 year , ξ -1 c = 5.5 year . The predicted values of land subsidence above the exploited natural gas reservoir are presented in Figs. 5 - 8.

Fig.5. Estimated surface subsidence in year 1975.




SUMMARY In opposite to the models applied up to now, the presented method of calculations is based on the stochastic model of ground. It makes some use of experience gained in the hard coal and salt mining. The parameters of the theoretical model are relatively easy to determine. The course of compaction and movements in overburden were treated as time dependent, which made possible predicting the land subsidence at any time of exploitation. The algorithm of calculations enables prediction of land subsidence for exploitation fields of any shape. The computations were carried out for the spatial model. The method presented may be applied also for computing the horizontal shifts and deformations (Hejmanowski, 1993). The results of computations may be used as a basis for design of necessary protections (emergency stands or additional drilling platforms).

REFERENCES De Waal, J.A. & Smits, R.M.M. (1985) Prediction of reservoir compaction and surface subsidence: Field application of new model. SPE 14214, 1-11. Geertsma, J (1973) A basic theory of subsidence due to reservoir compaction: the homogeneous case. Verhandelingen van het Koninklijk Nederlands geologisch mijnbouwkundig Genootschap. DEEL Vol.28 Hejmanowski, R. (1993) Zur Vorausberechnung förderbedingter Bodensenkungen über Erdölund Erdgaslagerstätten (Prediction of land subsidence induced by oil and gas exploitation). PhD Thesis Clausthal Technical Univ., Clausthal-Zellerfeld, Germany. Knothe, S. (1953) Equation of finally subsidence trough. (in Polish) Archiwum Górnictwa i Hutnictwa, Vol.1, Zeszyt 1, Warsaw 1953, Poland. Kosloff, D., Scott, R.F. & Scraton, J. (1980) Finite element simulation of Wilmington oil field subsidence: linear and nonlinear modeling. Tectonophysics, 65 and 70.

Litwiniszyn, J. (1956) Application of the equation of stochastic process to mechanics of loose bodies. Archiwum Mechaniki Stosowanej, Vol.8. Krakow, Poland. Sulak, R.M. (1991) Ekofisk Field: The first 20 Years. Journal of Petroleum Technology, October, 1265-1271 Sroka, A. (1988) Selected problems in predicting influence of mining-induced ground subsidence and rock deformations. Proc. 5th Int. Symp. on deformation measurements and Canadian Symp. on mining surveying and rock deformation measurements, Fredericton, Canada. Teeuw, D. (1973) Laboratory measurement of compaction properties of Groningen reservoir rock. Verhandelingen van het Koninklijk Nederlands geologisch mijnbouwkundig Genootschap. DEEL Vol.28 Van Kesteren, J. (1973) The analysis of future subsidence resulting from gas production in the Groningen field. Verhandelingen van het Koninklijk Nederlands geologisch mijnbouwkundig Genootschap. DEEL Vol.28