A Coupled Reservoir-Geomechanics Model and Applications to Sand Production Simulation

A Coupled Reservoir-Geomechanics Model and Applications to
Sand Production Simulation
SUN Feng, XUE Shifeng
China University of Petroleum, Dongying, China, 257061
[email protected]
Abstract: Sand production is a serious problem in petroleum industry, it’s becomes more critical as
operators follow more aggressive production strategies. In this paper, focusing on the coupling between
fluid flow and solid deformation/collapse, a consistent geometrical model for sand production prediction
is established. Finite element method with explicit and sequential iterative algorithm is used to solve the
coupled governing equation system. Simulation results of perforation cavity stability and sand
production rate are presented. The calculation indicates that the critical pressure drawdown and the
pressure gradient variable take as the important roles in sand production process. The study suggests that
the proposed model may be used to generate quantitative information for predicting sand production and
optimizing pressure drawdown.
Keywords: sand production, fluid-solid coupled effect, finite element method, pressure drawdown.
1.
Introduction
Sand production occurs when the well fluid under high pumping rate dislodges a portion of the
formation solids leading to a continuous flux of formation solids. Sanding process occurs in formation
failure leading to the well bore and perforation cavity instabilities. A high stress distribution, especially
near a well and perforation tips, often induces local formation collapse. Such collapse region may spread
with fluid flow and sand production process. These sanding effects become more critical these days as
operators are following aggressive production schedules.
In recent years, a great deal of work has been done in the general area of sand production, and various
sand prediction models have been published in the literature[1],[2],[3]. These models have the capability to
indicate whether initial sand production may take place somewhere during the lifetime of an oil fields.
However, these models are unable to predict whether the sand production will be problematic. Reviews
of sand rate prediction models and models describing the erosion of sand grains in a rigid oil sand
matrix were recently presented [4],[5],[6].
The main objective of this paper is to develop a model that quantifies the production caused by
formation impairment and induced by the transient pressure gradients from the drawdown operations.
Focusing on the coupling influence between hydro-mechanical aspects of sanding developing process, a
consistent geometrical model for sand prediction is established. This model captures both the
geomechanical aspects (rock deformation and failure) and the fluid flow aspect (role of drawdown),
which includes the capacity to predict wellbore/perforation cavity instabilities, sand quantities and sand
production rates. Galerkin finite element method with explicit and sequential iterative algorithm is
adopted to solve the coupled governing equation system.
2.
Coupled Reservoir-geomechanics Model for Sanding Process
To evaluate the sand production and wellbore stress in continuum mechanics frame, a series of basic
equations and boundary conditions should be derived. For simplicity, the basic equations in this paper
are developed for a single-component single-phase flow system. We classify the complicated sanding
process equations into three parts: the first mainly contains porous solid matrix deforming, the
single-phase fluid flow equations are included in the second part, and the third part is the sand
production model equations.
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2.1 Solid Matrix Deformation Equations
The poro-mechanical behavior of sandstone is described by the theory of poro-elastoplasticity. For
steady state conditions, poro-elastoplastic processes are described by the following equations.
The Equilibrium equation is as follows:
σ ij' , j + f i − αP , j δ ij = 0
(1)
A general stress-strain constitutive equation for the porous solid matrix is formulated in an incremental
formula as follows:
d σ ij = Cijkl d ε kl − αdPδ ij
(2)
The boundary condition equation is as follows:
(3)
(σ ij' , j − αP, j δij ) ⋅ n j − Ti = 0
Where in the above equations, σ ij' denotes the effective stress tensor, P is the pore pressure,
δ ij is the
Kronecker tensor, f i is the body force component, α is the Biot coefficient, Cijkl is the tangent
elastoplastic stiffness matrix, ε kl is the stains, Ti is the boundary force, and n j is the cosine of exterior
normal direction.
According to the Galerkin finite element method, equation (1) and (3) can be converted into the weak
integral form:
(4)
σ ij′ δε ij d Ω = α Pδε ij d Ω + Tiδ ui d Γ + f iδ ui d Ω
∫Ω
∫Ω
∫Γ
∫Ω
Where δui is the displacement components variation, δε ij is the strain components variation, Ω is the
solved area and Γ is the boundary.
Thus a Galerkin FEM formulation for solid skeleton deformation can be represented as:
Where
{ui }
 K ij  {ui } = Fp + Fs + Fg
(5)
is solid displacement component,  K ij  is stiffness matrix for skeleton deformation,
Fp is fluid pressure load, Fs is the load of stress along boundary, and Fg is body force load.
2.2 Fluid Phase Flowing Equations
The conservation of fluid in a deformable porous medium can be expressed as [2]:
∂ (φρf )
∂ε
k

(6)
φρf v +
− ∇ ⋅  ρf ∇ P  = 0
∂t
∂t
µ

Where ρ f is the fluid density, k is the permeability, µ is the fluid viscosity, φ is the reservoir porosity,
and
εV
is the solid volumetric strain
Similar to the solid matrix FEM formulation, if a virtual “displacement” δP is introduced into equation
(6), a weak integrate formulation for pore fluid has the form of:

k

∂P
 ∂ε v 
(7)
∫Ω δP  Cl φ ∂t − ∇ ⋅  µ ∇P   d Ω = ∫Ω δP  − ∂t d Ω
Where Cl =
1 ∂ρ f
ρ f ∂P
is the coefficient of fluid compressibility.
A simplified Galerkin FEM formulation for fluid flow can be taken as:
p
&
Cij  {P} −  K ij  {P} = Fε v + FQ
(8)
p
Where Cij   K ij  are the mass matrix and stiffness matrix for fluid seepage respectively, Fε v is
the solid volumetric strain load, and FQ is boundary flux load.
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Fluid flux is expressed using Darcy’s law, which establishes its relation with the pressure gradient
k
q = − ∇P
µ


φ3
k = k0 
2 
(1
−
)
φ


▽P.
(9)
(10)
Where k is the reservoir permeability, which can be related to porosity via Carmen-Kozeny equation
(8), subscript “0” denotes the initial state.
2.3 Sand Production Model Equations
The stress concentration and failure near wellbore and perforation tips are treated based on the solid
matrix strength theory and failure element concept.
Corresponding to strength classifications, two types of rock failure are mainly expected in sand
production scenarios: shear failure and tensile failure. Shear failure refers to the condition when the
effective tangential stresses near the cavity wall exceed the shear strength of the rock and cracks develop.
Tensile failure in sand production arises when the radial hydrodynamic drag force, i.e. radial effective
stress, exceeds the rock tensile strength [7].
Two classical rock strength criterions are presented as the following.
(a) Shear failure —Drucker-Prager criterion
f = aI1 + J 2 − xk = 0
2sin ψ
a=
3(3 + sin ψ)
xk =
6c0 cos ψ
(11)
3(3 + sin ψ)
Where I1 = σ 1 + σ 2 + σ 3 , J2 = σ1σ2 + σ2σ3 + σ3σ1 , c0 is cohesion, ψ is internal friction angle,
and σ 1 , σ 2 , σ 3 are the three principal stress components.
(b) Tensional failure criterion
Because of the rock media’s low resistance to tensile stress, it is always ruptured perpendicular to the
direction of maximum tensional stress.
(12)
f = σ max − Tcut = 0
Where Tcut is the cut-off strength, σ max is the maximum tensional stress.
3.
Solution Methods
The system of equations (5), (8) can be solved numerically by imposing time stepping, a grid and
solving for discrete unknowns. We use explicit coupled approach, which the pressure is computed first
from equation (8), given a set of initial values. Then the displacement u , strains and stress are computed
from equation (5) from updated P and boundary conditions.
With pressure gradient, porosity, and load history changing, the stress concentration region may develop
continual failure or collapse. Simulating this dynamic collapse developing process is especially
important to understand wellbore/perforation cavity instabilities’ mechanism in sand formation.
σ =
nodes
∑σ
i
(13)
i =1
When the stress components are solved by numerical methods, the rock failure criterion (11) and (12)
can recognize the failure point in the scale of elements level. The three principal stresses are averaged in
element nodes, as equation (13). If the failure conditions are reached, this means the failure elements
haven’t any bearing load capacity. Thus the failure elements should be cut off in next load step. The
failure elements integrated in the time and space fields of simulation model, and we get the quantitative
information of failure element. As the foregoing statement, sand grains come from the formation
structure failure. So the cumulative sand mass is as the cumulative failure elements volume.
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In consideration of results precision and computing time, we determine to use the scheme of elements’
stiffness weakens technology, which means weak the stiffness of failure elements and modify the
stiffness of integrated matrix. The failure elements information, failure type, time, and position, is
recorded in a dataset at each time step.
4.
Numerical Examples and Discussion
A practical simulation model is established based on the geological formations data of Shengli oilfield.
Its numerical simulating parameters are listed in Table 1. The numerical geometry model of 2D-RZ
coordinate systems is shown in Figure 1.The phase of perforation is 72°, shot density is 20 shots/m, and
penetration is 0.5m. Computed data are the result of average equivalent unit-thickness layer.
Table1 Numerical simulating parameters
E
(GPa)
ν
σV
σh
8.0
0.26
(MPa)
22.0
(MPa)
18.0
D=0.18m
k
(m2)
50×10-15
φ
0.32
P
(MPa)
12.0
α
0.8
c0
Tcut
(MPa)
3.0
(MPa)
0.7
σV
σh
10m
(a) Schematic diagram of 2D-RZ coordinate
(b) local enlargement of the perforation and grid
Figure 1 Schematic diagram of the FEM model and grid
Numerical simulation results of reservoir stabilities in near wellbore and perforation cavity at different
drawdown are shown as figure 2, which are following rock failure criterions.
(a)
△P=1.0MPa
△
(b) P=3.5MPa
Figure 2 Effect of drawdown on cavity stability
(c)
△P=5.0MPa
Perforation cavity stability is determined by the balance of contact and drag forces, which are directly
the pressure drawdown and reservoir strength. Local collapse occurs when the drawdown is 1.0MPa,
and followed by a transient sand burst, as figure 2a. As sanding continues, effective stresses are
redistributed around the cavity as the failed material is removed. This stress redistribution, in turn, may
fail additional material that provides more grains for production. Perforation cavity propagation with
drawdown increases, as figure 2b. It’s accompanied by massive failure zone around perforation tunnel
and catastrophic failure of the sand pack when drawdown is 5.0MPa, as figure 2c. Sand control
measures should be taken as drawdown increase. The critical drawdown of this reservoir strength
condition is 5.0MPa
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9
Sand product ion rat e
Bott om-hole pressure
0.03
8
0.02
7
0.01
6
0.00
5
1
2
3
4
5
6
7
8
9
10
0.04
Bott om-hole pressure
0.03
8
0.02
7
0.01
6
0.00
10
5
0
1
2
Production time(days)
3
8
0.02
7
0.01
6
0.00
3
9
0.03
Sand production rate(m /day)
Bott om-hole pressure
Bottom-hole pressure(MPa)
3
Sand production rate(m /day)
10
Sand product ion rat e
5
1
2
3
4
5
6
5
6
7
8
9
10
(b) ∆t=2.0 day
0.05
0
4
Production time(days)
(a) ∆t=1.0 day
0.04
9
Sand product ion rat e
7
8
9
0.05
10
Sand production rat e
Bot tom-hole pressure
0.04
0.03
8
0.02
7
0.01
6
0.00
10
5
0
Production time(days)
1
2
3
4
5
6
7
Production time(days)
(c) ∆t=4.0 day
(d) ∆t=6.0 day
Figure 3 Effect of drawdown operation on sand production rate
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9
8
9
10
Bottom-hole pressure(MPa)
0
3
0.04
0.05
Bottom-hole pressure(MPa)
10
Sand production rate(m /day)
0.05
Bottom-hole pressure(MPa)
3
Sand production rate(m /day)
Pressure build-up process decreases the reservoir bottom-hole pressure and increases the drawdown.
These transient gradients of pressure in reservoir can generate substantial fluid drag forces. They impose
additional loads on the formation leading to fines mobilization, formation impairment, and formation
failure. Practical drawdown guidelines mitigate the influence of high transient pressure gradients and
avoid unnecessary impairment to the formation.
Figure 3 compares the simulation results for effect of drawdown operation on sand production rate.
Numerical simulations show the fact that the pressure gradient variable takes as the important roles in
sand production process. Sand production in initial period (t≤3.0 days) is the effect of in-situ stress.
Rapid pressure drawdown operation creates the high transient gradient of pressure, its effect is
cumulative, and become problematic in the later stages of a well’s life. As can be seen from figure 3 (a)
to (d), the peak value of element damage rate decreases with operation time increase. Each individual
pressure-step magnitude may not necessarily have appreciable effect, but the cumulative effect will
gradually destroy light cementation and reservoir structure. Figure (a), (b) show that transient damage
phenomenon appear during the steady pressure drawdown periods.
Optimized build-up strategy varies with the type of sand in terms of cementation, particle-size
distribution and rock strength, as well as the type of sand control. In general, the larger the pressure
drawdown step, the greater the potential for solid movement and fines generation that may either create
sanding or increase skin. The best practice is: keep pressure drawdown steps, small with long time
interval, and minimize frequency of shutdowns and avoid rapid shutdowns.
Discussion of pressure drawdown operation on cumulative sand production volumes is shown as figure
4. Build-up time has a strong effect on cumulative sand production volumes.
Cumulative sand producrion volume(m 3)
0.05
∆t=1 day
∆t=2 day
∆t=4 day
∆t=6 day
0.04
0.03
0.02
0.01
0.00
0
1
2
3
4
5
6
7
8
9
10
Production time(days)
Figure 4 Effect of drawdown operation on cumulative sand production volume
5.
Conclusion
Based on the continuum mechanics theory, a coupled reservoir-geomechanics model was proposed and
used to sand production simulation. Galerkin finite element method with explicit and sequential iterative
algorithm was used to solve the coupled governing equation system.
Numerical simulations show that this model can be used to evaluate the onset of sand production and
critical pressure drawdown. During sanding process, high permeability channels can be generate around
the wellbore region and perforation hole as a result of rock failure.
Drawdown pressure plays a key role in sand management. The inner surface of the perforation cavity
was stable to a certain critical flow rate. In general, the larger the pressure step, the greater potential for
solids movement and fines generation.
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