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2006 ABAQUS Users Conference 125

Numerical Simulation of Compactionand Subsidence Using ABAQUS

Gaia Capasso and Stefano Mantica

Eni S.p.A., Exploration and Production Division, Milan (Italy)

Abstract: The evaluation of reservoir compaction and surface subsidence induced by hydrocarbon

production represents a critical concern to oil companies and government environmental

agencies. In this paper, we present an approach to evaluate strain and deformations at reservoirscale by using ABAQUS, linked to the flow simulator used in Eni E&P.

Keywords: Reservoir Compaction, Surface Subsidence, Environment, Cam-Clay, Geomechanics,

Hydrocarbon Production.

1. Introduction

The study of hydrocarbon production from an underground reservoir involves two basic elements:the rock and the fluid contained in its pore space. Fluid flow and fluid pressure evolution together

with stress-strain variation in the solid skeleton are the processes associated with the exploitationof the field. Fluid flow analysis is essential in any reservoir study in order to forecast the

production and manage the development of the field; nonetheless, the geomechanical processes

associated with the reservoir exploitation are also of primary interest, since they can affect the

behaviour of the reservoir itself and cause environmental impact as a consequence of productive

layer compaction and land subsidence. Then, the evaluation of reservoir compaction and surfacesubsidence induced by hydrocarbon production represents a critical concern to oil companies and

government environmental agencies.

Reservoir compaction may alter the rock permeability over time with a consequent reduction inwell rates, a delay in reserves recovery and a decrease in ultimate recovery from compaction-drive

reservoirs (Ostermeier, 1995). In this case, a reliable forecast of the compaction is required for an

optimized management of reservoir production. Moreover, the accurate prediction of soil

deformation is crucial for the design of casing and completion to avoid well failures induced byhigh levels of compaction (da Silva, 1990; Bruno, 1992). In addition, surface subsidence that may

be induced by hydrocarbon production must be considered in the design of surface facilities or to

prevent adverse environmental impact when the reservoir is located close to an area of ecological,historical or social significance.


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126 2006 ABAQUS Users Conference

Surface subsidence and reservoir compaction due to hydrocarbon production have beendocumented for fields located in many different areas in the world, from the North Sea to the

United States and South America. Subsidence at the Ekofisk oil field (North Sea) is widely known

because of its absolute magnitude, of the order of meters (Zaman, 1995); in the Netherlands,subsidence at the Groningen gas field, though only of the order of tens of centimeters, is of great

significance since large areas of the Netherlands are below sea level (Boot, 1973).

In Italy, interest has increased during recent years because of the exploitation of the gas fields

located off-shore in the Adriatic Sea; as a consequence, advanced methodologies have beendeveloped internally in Eni E&P in order to study the problem of reservoir compaction, to fully

understand the mechanisms involved in surface subsidence and to forecast and prevent adverse

environmental impacts.

In this paper, we present a procedure to evaluate stress and strain at reservoir scale by usingABAQUS, linked to the flow simulator Eclipse (Schlumberger, 2004) used in Eni E&P. First, we

give a description of the problem and of the general methodology by providing an overview of the

workflow that has been developed. Next, we illustrate the application of this approach to a realistictest case. Finally, we conclude with a discussion on the obtained results and the perspective forfuture work.

2. Problem description

Rock compaction and the associated subsidence, i.e. the sinking or settlement of the land surface,

may occur due to hydrocarbon withdrawal from the subsurface.

Considering the reservoir as a porous medium, the basic mechanism controlling compaction and

subsidence phenomena can be explained referring to Terzaghis principle of effective stress, which

governs the interaction between solid skeleton and fluid, stating that:

pijijij ='

where ijis the effective stress, ij is the total stress, andp is the pore pressure. The effectivestress governs the mechanical behaviour of the porous medium since all measurable effects of achange of stress, such as compression, distortion and a change of shearing resistance, are

exclusively due to changes in the effective stress (Terzaghi, 1936).

With reference to a reservoir, the above principle can be applied by considering that the weight ofthe overburden is supported partially by the rock matrix and partially by the pressurized fluid in its

pore space; the reduction of the pore pressure due to reservoir exploitation will induce an increase

of the effective stress with a consequent compaction effect on the formation (Figure 1).


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start production end production

pp11pp11 pp22pp22

CompactionCompaction

SubsidenceSubsidence

Reservoir pore-pressure

Effective stress

reservoir p2 < p1

Figure 1 Compaction and subsidence due to reservoir exploitation.

In order to study the phenomenon, analytical and semi-analytical methods have been developed in

the past. One of the simplest approaches that allows for the quantitative determination of surfacesubsidence due to the depletion of a reservoir, as well as the deformation of the whole half-space

surrounding the reservoir itself, is Geertsmas analyticalsolution (Geertsma, 1973), based on the

concept of the nucleus of strain (Mindlin, 1950). This approach, however, is valid under anumber of simplifying hypotheses, which are usually not respected in a real case: cylindrical shape

of the reservoir, uniform depletion, linear elastic behaviour and homogeneity of the porous

medium (Figure 2, left). Some of these hypotheses can be overcome by using asemi-analytical

approach based on the same concept (Geertsma, 1973a); in this case the discretization of the

depleting volume allows for a more realistic description of the reservoir geometry and depletiondistribution (Figure 2, right).

Land surfaceLand surface Land surfaceLand surfaceLand surfaceLand surface

Figure 2 Analytical (left) and semi-analytical Geertsma approaches.


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However, the semi-analytical approach is still limited because of the hypotheses of linear elastichomogeneous material. The homogeneity of the material means that the model is forced to

describe the whole medium, i.e. reservoir, side-, over- and under-burden, with the same

mechanical properties. Decreasing compressibility usually occurring with increasing depth, as wellas different characterization of various lithologies cannot be taken into account. The hypothesis of

linear elastic behaviour imposes the same linear relationship between stress and strain, both during

loading (depletion) and unloading (repressurization); on the contrary, it is known that the soil can

have a highly non linear behaviour, with a strong influence of previous stress paths.

In many cases, the use of a finite element (FE) model is than much more suitable in order to fullydescribe the geomechanical behaviour of a depleting reservoir and of the surrounding material. It

is possible, in this case, to build an FE model with a detailed description of the geometry, taking

into account the geological structure of the producing layers and of the over- and under-burdenlayers; regions with different mechanical properties and complex constitutive laws can be defined

in order to correctly consider the behaviour of the materials; the system can be loaded with the

measured drawdown which is function of space and time. A workflow has been developed

internally in Eni E&P by using ABAQUS as the main numerical tool for geomechanicalsimulations.

3. Reservoir modeling

The results of the standard reservoir studies carried out for the management of the field productionprovide part of the inputs necessary for a geomechanical finite element analysis. The typical

workflow of a reservoir study consists of a static study and a dynamic study.

The static model includes the detailed reconstruction of the geological structure of the reservoir(e.g. the shape of the layers and the trend of the faults), the definition of the mineralized volumes

and the attribution of the petrophysical parameters (initial porosity and permeability) as a function

of the location. The result of a static study is a 3D model of the reservoir and of the surrounding

region, describing all its geological, lithological, stratigraphical and petrophysical aspects. Figure3 shows, as an example, the geometry of the top of a gas field layer, where all the faults present inthe area are reported.


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Figure 3 3D static model: representation of a geological horizon and faults.

The dynamic model is built with the flow simulator currently used in Eni, which is a fullyimplicit, three phases, 3D finite difference code. The dynamic model takes as input all the

information of the static model and, by introducing a series of additional parameters regarding the

characteristics of the fluids, the rock and the well system, provides the information required for thefield management, such as the dynamic reserve evaluation and the production profiles as a

function of the development scenarios. As an example, in Figure 4 we show the finite differencediscretization of a dynamic model for a real gas field.

Figure 4 Flow model: grid discretization and initial pressure distribution.

The dynamic model provides as output sets of data that are used in the geomechanical finite

element simulation: the grid discretization of the reservoir and of the surrounding aquifer areas;

the initial values of porosity and permeability; the evolution of the fluid pressure as a function ofspace and time. As explained in detail in the following section, all these information are converted

with an interface code and used to build the ABAQUS FE model.


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4. Subsidence studies workflow

A workflow has been developed in Eni E&P in order to carry out geomechanical simulations withthe FE code ABAQUS; the aim is to assess the stress/strain evolution of the field and of the

surrounding rock during the productive life of the reservoir and after the abandonment. Thecompaction of the depleting layers as well as the evolution of surface subsidence can then be

quantitatively evaluated.

The workflow includes the following steps:

1. Model construction: a FE grid for the reservoir region is built starting from the FD(Eclipse) discretization; the FE grid is then enlarged in order to include over-, under and

side-burden; the FE model is created and populated with the reservoir data derived by the

flow model;

2. Linear elastic simulations: FE simulations under linear elastic hypotheses are carried out

and compared to a reference semi-analytical solution in order to validate the gridding, theboundary conditions and the pressure attribution at the finite element nodes;

3. Elasto-plastic simulations: FE simulations using realistic constitutive behaviour (elasto-plasticity) and appropriate description of the heterogeneity of the materials are performed

including all the information available for the field.

4.1 Model construction

A Fortran90 interface code that provides an automated link between the flow model and

ABAQUS has been developed in-house: files with the results of the flow simulation are processed

and the needed information is re-written as input files to run ABAQUS.

The geometrical information of the Eclipse 3D corner point grid are directly extracted from the

relevant output files of the flow model and processed to build the FE mesh in the reservoir region.This approach allows for the definition of a FE model which is fully consistent with the reservoir

FD model.

The typical FD and FE grid structures are shown in Figure 5 for a 2D mesh.


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F.E.M.F.E.M. GRIDGRID

Node #

1 x, y

12 x, y

Coords

Element # Node #

1 1 2 6 5

6 7 8 12 11

F.E.M.F.E.M. GRIDGRID

Node #

1 x, y

12 x, y

Coords

Element # Node #

1 1 2 6 5

6 7 8 12 11

Element # Node #

1 1 2 6 5

6 7 8 12 11

F.D.F.D. GRIDGRID

Cell

(1,1) (x1,y1)(x4,y4)

(2,1) (x1,y1)(x4,y4)

(2,3) (x1,y1)(x4,y4)

Node coords

F.D.F.D. GRIDGRID

Cell

(1,1) (x1,y1)(x4,y4)

(2,1) (x1,y1)(x4,y4)

(2,3) (x1,y1)(x4,y4)

Node coords

o o o o

o o o o

o o o o

j=1 j=2 j=3

i=1

i=2

1 2 3 4

5 6 7 8

9 10 11 12

1 2 3

4 5 6

o o o o

o o o o

o o o o

j=1 j=2 j=3

i=1

i=2

j=1 j=2 j=3

i=1

i=2

1 2 3 4

5 6 7 8

9 10 11 12

1 2 3

4 5 6

Figure 5 - From FD to FE grid.

It has to be noted that the FD mesh is built according to the geological structure of the reservoir,

i.e. stratigraphy and fault geometries are respected as much as possible. This may produce a series

of geometrical irregularities that are allowed in FD simulations but cannot be straightly

maintained in a FE grid. In order to describe the real geometry of soil levels with vanishingthickness (pinch-outs), some cells in the FD grid may have nodes collapsing in a single point;

the presence of faults, in addition, is often described in a FD grid with its real dislocation, causing

the shift between nodes of two neighboring cells. Small irregularities are removed during the grid

processing by slightly adjusting the node position in order to get a suitable FE grid without losingthe correct geological structure (see Figure 6).


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F.D. grid: pinch out F.E. grid: pinch outF.D. grid: pinch out F.E. grid: pinch out

F.D. grid: faultF.E. grid: fault

F.D. grid: faultF.E. grid: fault

Figure 6 - FE grid construction: removal of irregularities.

Another important point is related to the number of cells originally describing the flow model. In

some cases the huge number of FD grid cells, and the fact that the FE final grid must also include

over-, under-, and side-burden, makes it impossible to maintain the same discretization. Theinterface code that generates the FE grid allows for a merging of grid cells in the vertical and in

the horizontal direction to reduce the number of dof. The FD grid, however, is the result of a

detailed study of the field that considers information coming from seismic surveys, well data,

geological knowledge of the area and reservoir development program. The vertical discretization

is of topmost importance to correctly describe the material properties of the porous medium andthe pressure drawdown. Therefore, when a vertical merging is needed, a comprehensive analysis

must be executed in order to lump layers with similar properties and production histories.

The result of this process is a FE grid which discretizes the field region, including all themineralized area and the surrounding aquifers if any. This model is then extended in the horizontal

direction to describe the side-burden, up to the surface to describe the over-burden and down to a

fixed basement to describe the under-burden. This external part of the grid, needed to correctly

simulate the geomechanical behaviour of the system, is automatically built by the interface code,provided that the final model size is given. The element type attributed to the depleting regions

(hydrocarbon + aquifers) is 8(20)-node brick stress/displacement/pore pressure (C3D8P/C3D20P),

while 8(20)-node brick stress/displacement (C3D8/C3D20) are assigned to the external regions.

4.2 Linear elastic simulations

According to our workflow, ABAQUS calculations are first performed under the hypotheses of alinear elastic behaviour of the material with homogeneous and isotropic mechanical properties. In

this way the FE results must match the semi-analytical solution (Geertsma, 1973a). This approach

can be considered as a general methodology for the validation of the gridding, upscaling andattribution of pressure at the nodes, as well as for the correct setting of the boundary conditions.

The following properties have to be assigned to construct the full model in ABAQUS:


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bulk density(elements C3D8) and dry density d=-.f/g(elements C3D8P); elasticitymodulusEand Poisson ratio ;

specific weight of the pore fluid f(elements C3D8P); initial void ratio e=/(1-) (elements C3D8P);

tabulated values of permeability vs. void ratio (elements C3D8P).Note that density, initial void ratio and permeability do not affect the result of elastic simulations.

A sensitivity study is then carried out on different grids by possibly varying the number ofelements modeling the over-, under- and side-burden; the extent of the model itself; and, finally,

the element merging procedure.

The results of the ABAQUS models are compared with the semi-analytical solution: the final gridstructure is chosen to be the one that best reproduces the semi-analytical solution with the

minimum computational effort.

4.3 Elasto-plastic simulations

Once the geometry of the model has been defined in terms of size and discretization, the final FEmodel is built with constitutive laws and material properties according to the available data.

In this section we introduce the criteria used for the definition of the hydro-mechanical regionsand the evaluation of the material properties associated to them. Next, the definition of the initial

state is described in terms of stress condition, pore pressure equilibrium and initial void ratio.

4.3.1 Region definition

The FE geomechanical model, as already discussed in 4.1, is built by considering two differentclasses of elements. It comprises porous elements corresponding to the fluid saturated zones of the

flow model and non-porous elements corresponding to over-, under- and side-burden.

The region definition in the non-porous zone depends on the distribution of the mechanicalproperties and, in particular, on its heterogeneity. In the fluid saturated zones, as a general rule,

each layer constitutes a region: the layering of the flow model, which is preserved in the FE

model, is in fact generated by taking into account the real stratigraphy of the reservoir.

For each porous layer a further region subdivision is necessary if a fluid-fluid contact is present.This permits to account for different fluids with different properties and behaviour into the single

phase ABAQUS simulator. Thence, each porous layer is split into up to three regions (gas, oil and

water) defined by the relevant contact depth.

4.3.2 Material properties

Density


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The density is given in the FE model as a function of depth. Since this option is not directlyavailable in ABAQUS, the following simple approach has been adopted: the variable temperature

is introduced in the model as an initial condition with values corresponding to the node depth, so

that the density can be assigned as a function of temperature. This is of course a strong restrictionbecause it makes impossible the use of the temperature as a physical variable.

The bulk density profile(z) is usually obtained from sonic and density logs measured along thewells; in the non-porous regions it is provided in a tabular form as a function of depth, while in the

porous regions the dry densitydof the material has to be entered, which is defined as:

g

f

d

=

being the region porosity,gthe gravity acceleration and f the specific weight of the saturatingfluid. For each porous region the dry density is then provided as function of depth.

Fluid specific weight

In the porous regions, the specific weight of the saturating fluid has to be provided. For each

hydro-mechanical region the value off(gas, oil or water) is taken as constant; it is determined asan average value by using the fluid contact depths and the initial pressure distribution of the region

as calculated by the flow model.

Constitutive law

Depending on the material characterizing the reservoir and the surrounding zones, the most

appropriate behaviour is used in the model, by choosing among the elasto-plastic constitutive laws

implemented in ABAQUS for geomaterials. Usually, the Modified Cam Clay model is used in thecase of sandy/shaly materials, while models based on strength criterions (Mohr-Coulomb,

Drucker-Prager) are preferred when dealing with limestone or dolomite. The property values

needed for the law definition are derived by laboratory or in situ tests.

4.3.3 Initialization

Stress

The stress initialization is a critical point when using elasto-plastic constitutive laws, since thematerial behaviour is controlled by the current value of the stress and by its path, not only by the

stress variations.

The stress initialization is performed through the following steps:

an initial in situ stress field is computed using the material density(z) and aK0 value(the ratio between horizontal and vertical stress) obtained from in situ measurements of

horizontal stress available for the field;

the model is then equilibrated with this stress field as initial condition, assuming linearelastic behaviour of the material and using the *GEOSTATIC option to verify that the


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geostatic stress field is in equilibrium with the applied loads and boundary conditions onthe model and to iterate, if needed, to obtain equilibrium;

the resulting stress field is then used to initialize the elasto-plastic simulation;

a further *GEOSTATIC step is performed at the start of the elasto-plastic simulation inorder to verify that negligible displacements occur.

The correct attribution of the initial stress is a crucial step since the initial geometry should not be

changed by the initialization process, being known as it is at the beginning of production.

Void ratio

The initial void ratios e, assigned to the porous regions of the ABAQUS model, are assumed

constant region by region and are obtained by averaging the initial porosity values of the flowsimulation.

Pore pressure

The initial pore pressure field is given as calculated by the flow model initialization, then

consistently with the fluid contacts and with the specific weight of the saturating fluids.

4.3.4 Boundary conditions

The boundary conditions assigned to the model consist of null displacement at the bottom of thegrid and no horizontal displacement at the sides. Step 2 of the workflow assures that the mesh is

sufficiently extended so that the boundary conditions imposed do not affect the simulation results.

4.3.5 Load history

The pressure time evolution is automatically extracted from the flow model output files at the time

steps of interest and re-written in order to be directly available as ABAQUS input.

It must be noted that pressure is cell centered in the FD flow simulation, while it is node centered

in the ABAQUS simulator. The pore pressurePi at each node i of the FE grid is then obtained as a

weighted average of the cell centered pressurespj of the FD simulation as follows:

( ) ( )

( )

=

=

=

8,1

8,1

j

j

j

jj

iiporv

iporvip

P

where thej sum runs over the 8 neighboring cells sharing node i, characterized by pressurepj(i)and pore volumeporvj(i).

The pressure values obtained with this processing are then applied as boundary conditions at eachstep of the FE ABAQUS calculation, providing the load history of the model.


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5. Application to a realistic test case

In this paragraph we present the application of the proposed procedure to the study of a realistictest case. The PUNQ problem (Floris, 1999) has become quite popular, in the oil industry

community, as a sort of benchmark for history matching and risk analysis methodologies. It is adynamical reservoir model inspired to a real West Africa field. Water support comes from north

and west aquifers, while two faults close the reservoir at east and south, with a small gas cap.

Figure 7 - PUNQ model: map of the top layer.

An history period, simulating 8 years of production from six wells located close to the gas-oilcontact (GOC), was generated by The Netherlands Organization for Scientific Research (TNO)

using geostatistical distributions of porosities and permeabilities. Then, 8 years of forecast with 5

additional infilling wells have been simulated. The data-set, consisting of noisy well-data, gridstructure, permeability and porosity distributions, is available at TNO web site (The Punq project,

see reference).

Following the workflow that has been outlined in the previous sections, the geomechanical

modelling for the PUNQ field has been performed in three different steps. First, a finite elementmodel has been built starting from the Eclipse finite differences grid and extending the

computational domain. Next, the validity of the FE model has been checked by ensuring that the

results obtained with ABAQUS, under the hypotheses of a homogeneous material with linear-


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elastic behaviour, and the semi-analytical solution be equivalent. Finally, one-way coupled poro-elasto-plastic simulations based on the Cam-Clay material behaviour have been performed.

The finite difference grid of the PUNQ field used for the Eclipse flow simulations is a corner pointmesh made of 19x28x5 cells in the I, J and K directions respectively. The original Eclipse model

covers an area of about 3.5x5.0 km2, but includes only a small portion of the hydraulicallyconnected aquifers. It has to be noted that, in this case, the aquifers are considered as point sources

on the boundary of the reservoir and no depletion is considered into these zones. Being this a

small model, no upscaling has been performed. The reservoir model has been further extendedhorizontally into the FE model, obtaining a final areal extent of about ~13.5x15 km2. The original

grid has been extended vertically up to the surface and down to a fixed basement 5000 m deep.

The global FE grid results in 29x38x25 (27550) elements and 30420 nodes for a total of 93690

degrees of freedom, using 8-node linear brick stress/displacement elements (C3D8) outside the

reservoir and 8-node linear brick stress/displacement/pore pressure elements (C3D8P) in the activecells of the reservoir and aquifer regions. The global FE model is shown in Figure 8.

Figure 8 Global finite element grid.

As it can be seen from Figure 8, 10 layers have been used for the over-burden and for the under-burden.

In Figure 9 we show a top view of the FE grid for the first reservoir layer. The colour zone

corresponds to the Eclipse grid while the white elements have been added into the FE grid toproperly model the side-burden. The boundary conditions assigned to the model consist of nulldisplacement at the bottom of the grid and no horizontal displacement at the sides. It has been


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verified numerically that the mesh is sufficiently extended so that the boundary conditionsimposed do not affect the simulation results (see discussion below).

Figure 9 - Top view of the FE grid for the first reservoir layer.

The first FE calculations have been performed under the hypotheses of a linear elastic behaviourof the material with homogeneous and isotropic mechanical properties. The grid shown in Figure 8

that has been used for the elasto-plastic geomechanical study is the result of a sensitivity

evaluation carried out by running a number of elastic simulations on different grids, with a rangeof size and number of elements, in order to choose a reliable model that gives a good agreement

with the semi-analytical solution. The results obtained in terms of iso-subsidence curves are

shown in Figure 10, together with the semi-analytical results (and with the analytical results, for

the sake of completeness). It is evident that a very good agreement is obtained; the correspondinggrid has then been chosen for the further elasto-plastic simulation.


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Figure 10 Iso-subsidence lines (in cm) for the elastic runs: analytical (red), semi-analytical (blue) and FE (black-dashed). The hydrocarbon saturated area of the fifth

layer (green) and the surrounding aquifer (blue) are also shown.

In the reservoir zone each layer of the FE model represents a region with specified mechanicalproperties. The same mechanical parameters have been assigned to the side-burden in the

corresponding layers. The over-burden and under-burden are discretized by 10 layers each, every

one of them corresponding to a mechanical region.

The porous layers are then split into different hydro-mechanical regions, according to the gas-oil

and water-oil contact (WOC). This subdivision scheme results in a total of 38 hydro-mechanical

regions: 13 porous regions in the gas (3), oil (5) and water (5) saturated areas, 5 in thecorresponding side-burden, 10 for the over-burden and 10 for the under-burden.

In Figure 11 we show the hydro-mechanical regions for the first reservoir layer: the green area is

the gas saturated region (above the GOC, located at 2355 m), the red area is the oil saturated zone

(below the GOC and above the WOC, located at 2395 m), the blue area is the water saturatedregion and the white area is the non-porous region.


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gas

oil

water

non porous

gas

oil

water

non porous

Figure 11 Hydro-mechanical regions for the first reservoir layer.

The bulk density profile adopted in this kind of studies is usually obtained from sonic and density

well logs. In this case, we used the following expression: ( ) 01615.06.1453 zz = , whereis inexpressed kg/m3 andzis in m.

The constitutive law used in the geomechanical elasto-plastic simulations is the Modified Cam

Clay model. Note that, for the PUNQ model, we have considered elasto-plastic behaviour only forthe fluid saturated regions where the pressure drawdown induces a large variation of effectivestress, while all the other zones are assumed to behave elastically. In the porous regions, the

following properties are given:

- slope of the normal compression line in e:lnpplane;

- Ne intercept of the normal compression line in e:lnpplane;

- slope of the unloading-reloading (swelling) line in e:lnpplane;- Poisson ratio;

- M slope of the critical state line in the q:pplane.

In the above definitions q is the deviator stress andpis the mean effective stress.

For each porous region, the slope has been obtained by using the corresponding initial void ratioeini and the initial uniaxial compressibility coefficient cm: ( ) inivinim ec ,'1 += . The value ofdescribing the behaviour of the porous material under unloading-reloading conditions has been


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assumed as 1/3 of the corresponding . The Poisson ratio is 0.3 and, finally, the slope Mof thecritical state line has been taken as 1.33.

The mechanical behaviour of the non-porous regions of the model is described by a linearisotropic elastic constitutive law by giving, as input to ABAQUS, the elasticity modulusEand the

Poissons ratio .

The uniaxial compressibility cm, as a function of the effective stress, has been assumed to be:

)1347.1('01367.0 = vmc

where v is the vertical effective stress in bar. The vertical effective stress has been evaluated as

pvv =' , being v the total vertical stress, calculated using the bulk density given above,

andp the pore pressure. In the overburden and underburden layers,p has been obtained through

the following relationship (zin m andp in bar):p=0.102 z; while, for each region of the reservoiran average weighted on pore volume has been adopted.

After the initialization of the model, the pressure evolution has been imposed as a boundary

condition at each node of the FE grid using 17 loading steps, approximately one per year of

production of the field.

The FE model predicts a maximum forecasted subsidence of 0.81 cm, at the end of the production,

compared to 0.65 cm given by the semi-analytical approximation. The maximum extent of the

predicted iso-subsidence 0.05 cm line is around 6 km for the semi-analytical and 4.5 km for the

elasto-plastic FE model. From Figure 12, it can be seen that the subsidence bowl presents the samenearly circular shape in all cases, and that it is much smaller for the elasto-plastic model than for

the semi-analytical one: the subsidence bowl deepens in the case of elasto-plastic FE modelling

but its effects are restricted to a smaller area.

The whole time evolution of the 0.05 cm iso-subsidence contour line is shown in Figure 13. It has

to be noted, however, that the results in terms of subsidence are absolutely negligible since nomeasuring instrument has accuracy comparable to the results of our simulations (0.81 cm in 16years). In fact, the iso-subsidence line of 2 cm is usually assumed to be the limit of subsidence.

Finally, in Figure 14 the vertical strain at the top of 4 reservoir layers is shown at years 1983.


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Figure 12 Subsidence results: analytical (red), semi-analytical (blue),Elasto-plastic FE (black).

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Figure 13 Time evolution of 0.05-cm iso-subsidence contour line.


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Top of layer 1 Top of layer 2

Top of layer 3 Top of layer 4

Figure 14 Vertical strain maps at the top of 4 reservoir layers at year 1983.

6. Conclusions

The workflow presented in this paper has been applied in Eni E&P to real cases with excellentresults in terms of compaction and subsidence evaluation associated with hydrocarbon production.


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An open issue is related to the hydro-mechanical coupling between the flow and geomechanicalsolvers. Our plan for future work includes an iterative coupling between Eclipse and ABAQUS

that will take into account the influence of porosity and permeability variations as computed by

ABAQUS into Eclipse flow simulations.

7. References

1. Boot, R., Level Control Surveys in the Groningen Gas Field, Verhandelingen Kon. Ned.Geol. Mijnbouwk. Gen., Vol. 28, pp. 105-109, 1973.

2. Bruno, M. S., Subsidence-Induced Well Failure, SPE Drilling Engineering, 1992.

3. da Silva, F. W., Debande G. F., Pereira C. A., and Plischke B., Casing Collapse AnalysisAssociated With Reservoir Compaction and Overburden Subsidence, SPE 20953, 1990.

4. Floris, F.J., Bush, M.D., Cuypers, M., Roggero, F., and Syversveen, A.R., Comparison ofProduction Forecast Uncertainty Quantification Methods - An Integrated Study, paper

presented at 1st Conference on Petroleum Geostatistics, Toulouse 1999.

5. Geertsma, J., A Basic Theory of Subsidence due to Reservoir Compaction: theHomogeneous Case, Verhandelingen Kon. Ned. Geol. Mijnbouwk. Gen., Vol. 28, pp. 43-62,1973.

6. Geertsma J. and Van Opstal G., A Numerical Technique for Predicting Subsidence AboveCompacting Reservoirs Based on the Nucleus of Strain Concept, Verhandelingen Kon. Ned.

Geol. Mijnbouwk. Gen., Vol. 28, pp. 63-78, 1973a.

7. Mindlin R.D. and Cheng D.H., Thermoelastic Stress in the Semi-Infinite Solid, J. ofApplied Physics, Vol. 21, p. 931-933, 1950.

8. Ostermeier, R. M., Deepwater Gulf of Mexico Turbidites Compaction effects on Porosityand Permeability, SPE 26468, 1995.

9. Schlumberger, Eclipse Reference Manual 2005A, 2005.

10. The PUNQ Project, URL http://www.nitg.tno.nl/punq/.

11. Terzaghi, K., The Shearing Resistance of Saturated Soils and the Angle Between the Planeof Shear, Proc. of 1st Int. SMFE Conference, Harvard Mass., Vol.1, pp. 54-56, 1936.

12. Zaman, M. M., Abdulrahheem, A. and Roegiers, J. C., Reservoir Compaction and SurfaceSubsidence in the North Sea Ekofisk field, Subsidence due to fluid withdrawal, Elsevier

Scince, pp. 373-419, 1995.

8. Acknowledgement

The authors acknowledge Eni S.p.A. for the permission to publish this paper. We are also gratefulto R. Vitali and to E. Sguanci of ABAQUS Italia s.r.l. for their help.

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