A numerical method for analyzing fault slip tendency under fluid injection with XFEM

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POLITECNICO DIMILANO

DIPARTIMENTO DI MATEMATICA "F. BRIOSCHI" PH.D. COURSE IN MATHEMATICAL MODELS AND

METHODS FOR ENGINEERING

A NUMERICAL METHOD FOR

ANALYZING FAULT SLIP TENDENCY

UNDER FLUID INJECTION WITH XFEM

Doctoral Dissertation of:

LIU Daqing

Supervisor:

Prof. Luca Formaggia

Tutor:

Prof. Anna Scotti

The Chair of the Doctoral Program:

Prof. Luca Formaggia

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Abstract

I

Nmany industrial operations, we need to inject or extract fluid into/from

a rock reservoir. For instance for petroleum extraction, geologic stor-age of carbon dioxide etc. The fluid injection can alter the stress field in the reservoir and induce the preexisting fault to reactivate. Our research focuses on this problem and investigates the case of eXtended Finite Ele-ment Method to develop the numerical method able to assess the probability of fault reactivation. The eXtended Finite Element Method is sufficiently general to allow also the description of planar faults. Due to the method-ological nature of this work, we will deal only with two dimensional cases, but the methodology can be extended to a three dimensional setting.

In order to study the slipping behavior of the fault in the process of re-activation, our research considers the fault as a zero-thickness interface. Besides, we present a reduced model of the fluid flow in the fault so that we can also account for the variation of the hydraulic aperture and the perme-ability of the fault in the process of reactivation even if the fault is geomet-ically described as a zero-thickness interface.

We particularly focus on the map-view two-dimensional case, while when considering the dip of the fault, we regard the system as a two-and-one-half dimensional system based on Anderson’s fault theory which as-sumes the vertical direction as principal stress component. We use the Rate-and state- dependent friction model as the fault friction model, Rate-and Biot’s theory of poroelasticity to study the coupling between fluid flow and me-chanical deformation in porous media. Since the Fully Coupled Method between fluid flow and poromechanics is computationally expensive, we

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have proposed an Iteratively Coupled Method. In particular, we apply the Fixed-stress Split [32], readapted to our problem. In such scheme, the fluid flow problem is solved firstly by freezing the total means stress field, and then the results are used to solve the mechanical problem. The Fixed-stress Split is unconditionally stable, consistent and more accurate for a given number of iterations compared with the undrained split [32]. In order to verify our method, some test cases are presented. For the coupling between fluid flow and poromechanics, we consider the Terzaghi Problem and the Mandel Problem, the results are similar to those in previously published works. While, for the mechanic problem, we compare the results with those obtained by using the software Pylith.

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Summary

W

E propose a numerical method for analyzing fault slip tendency

under fluid injection using the extended finite element method (XFEM) both for fluid flow and poroelasticity. The fault is mod-eled as a zero thickness interface, and we use a reduced model for the fluid flow in the fault to account for its hydraulic behavior. We use the rate- and state- dependent friction model as the fault friction model, and Biot’s the-ory of poroelasticity to study the coupling between fluid flow and mechan-ical deformation in the surrounding porous media. Since a fully coupled method between fluid flow and poromechanics is computationally expen-sive, we have investigated the use of the so-called fixed-stress split in this context. In such a scheme, the fluid flow problem is solved firstly by freez-ing the total means stress field, and then the results are used to solve the mechanical problem. The fixed-stress split is unconditionally stable, con-sistent and more accurate for a given number of iterations compared with other type of splitting strategies. In order to verify our method, some test cases are presented. For the coupling between fluid flow and poromechan-ics, we consider the Terzaghi Problem and the Mandel Problem, compar-ing our results with those of previously published works. While, for the mechanic problem, we compare the results with those obtained using the software Pylith.

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CHAPTER

1

Introduction

Figure 1.1: Stress Field affected by Fluid Injection

In several applications related, for instance, to the exploitation of oil and geothermal reservoirs and CO2sequestration activities, the injection or

ex-traction of fluids from the subsurface can alter the stress field in the porous medium, which could induce preexisting faults to reactivate. Induced

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seis-micity has been documented since the 1920s and it is usually of modeate magnitude. However, these events deserve attention: they usually occur close to industrial areas, possibly with shallow hypocenter, and can cause economic damages. Nowadays as a promising technology to reduce an-thropogenic CO2 emissions into the atmosphere (e.g., [39], [56], [9], [28],

[47], [8]), geologic storage of carbon dioxide CO2 draws more and more

attentions from researchers all around the world. In our research we will focus on the reactivation of preexisting fault under fluid injection, because the reactivation of faults could produce a path for CO2 to leak, playing a

critical role to determine the success of geologic storage of carbon dioxide. Fault reactivation induced by fluid injection is a complex problem in-cluding several aspects such as the crustal deformation, fluid flow, coupling between fluid flow and mechanics, fault reactivation, and so on.

As concerns crustal deformation, boundary integral methods which have been used in dynamic simulations with complex geometry, (e.g., [42], [49], [46], [58]) cannot simulate nonlinear bulk rheologies, and finite differ-ences which have been used to simulate long, repeated rupture problems (e.g., [12], [10], [11]) lack the ability to deal with complex fault system geometry. While for the Finite Element Methods (FEM), although it could incorporate bulk rheology and nonplanar fault geometry (e.g., [13], [7]), it is very difficult to generate a mesh aligned with a nonplanar fault. Curved faults and faults with roughness are much more realistic in nature, espe-cially when the fault geometry is uncertain and the mesh needs to change with the fault geometry to study different possible scenario. So the eX-tended Finite Element Method (XFEM) which was originally developed by Belytschko and colleagues (e.g., [57], [31], [50]) and is still being re-searched, see [59], as a mesh-independent finite element method for em-bedded interfaces looks as a promising technique also for this type of prob-lems. XFEM extends the finite dimensional functional space to include discontinuous functions across interfaces ( in our case, across the fault), and it can deal with these discontinuities without remeshing.

For the interface conditions between the rock blocks across the fault, Lagrange multiplier methods and penalty methods are the most common methods. Set Lagrange multiplier methods is extremely mesh dependent and is not obviously extensible to three dimensions, and standard penalty methods typically result in ill-conditioned systems of equations [23]. So we have resorted to Nitsche’s method, see Hansbo [3], which could be re-garded as a hybrid of penalty methods and Lagrange multiplier methods, as our choice, because it provides a stabilized linear system of equations and is better conditioned than standard penalty methods. Moreover, we avoid

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saddlepoint problem complications inherent to Lagrange multiplier meth-ods [23].

For the simulation of fluid flow, in [15] the authors proposed an unfit-ted, mixed finite element method which viewes the fluid pressure and the fluid velocity as two variables of the system of equations for Darcy’s flows in fractured porous media. In their work, fractures were regarded as in-terfaces by using suitable reduced models (e.g., [14]; [60]). A study of the well posedness of XFEM method for flow in fractured porous media is found in [43].

For the coupling between fluid flow and geomechanics, various strate-gies have been investigated in the literature: fully coupled, iteratively cou-pled, explicitly coucou-pled, and loosely coupled methods. In a fully coupled strategy, the governing equations are solved simultaneously at each time step, and a solution is obtained through iteration. Although the fully cou-pled strategy is unconditionally stable, it is computationally expensive [32]. In iteratively coupled schemes, one of the fluid flow problem and the me-chanical problem will be solved firstly, and then the solution will be used to solve the other problem. The sequential procedure is iterated at each time step until the solution converges. At convergence the solution of an iterative scheme is the same as that obtained by the fully coupled method. Besides, the scheme can use different domains for the flow problem and the mechanical problem, which is useful because usually the domain of the mechanical problem is larger than that of reservoir simulation in order to deal with the reservoir boundary conditions (e.g., [5], [40]). The explicitly coupled scheme is a special case of iteratively coupled scheme in which only one iteration is taken (e.g., [38], [52], [26], [25]). In loosely coupled scheme, the coupling is resolved only after a certain number of flow time steps (e.g., [18], [54], [53]). Although this latter scheme can save compu-tational cost, it is less accurate and requires reliable estimates of when to update the mechanical response [32]. Among the four strategies, the itera-tively coupled scheme is the better choice for this research.

For the iteratively coupled methods, there are four operator-split strate-gies: drained split, undrained split, fixed-strain split, and fixed-stress split. The drained split method solves the mechanical problem with frozen pres-sure field firstly, and then solves the fluid flow problem with frozen dis-placement field. Compared with the drained split method, the undrained split method allows pressure to change but keeps the fluid mass in each grid block remaining constant during the mechanical step. Fixed-strain split scheme solves the flow problem with frozen strain field, and then uses the solution to solve the mechanical problem. While, the fixed-stress split

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scheme solves the flow problem with frozen total means stress field firstly. From practical point of view, the framework for reservoir simulation could benefit from solving fluid flow problem firstly [32]. Besides, by analyz-ing these four operator-split strategies, J. Kim [32] found that the drained and the fixed-strain splits are only conditionally stable and are inconsistent if performed with a fixed number of iterations, while the undrained and the fixed-stress splits are unconditionally stable regardless of the coupling strength and are consistent even in the case of a single iteration per time step. Moreover, the fixed-stress split is more accurate than the undrained split under the condition of a given number of iterations. Under his recom-mendation, we prefer to use the fixed-stress split as our time discretization strategy.

For the fault reactivation, in the latest research (e.g., [27], [34], [8]), there are two methods used to represent the fault: one is employing solid elements in the fault aperture, and another one is using zero-thickness el-ements which act as an interface in the media. The first one enables to describe accurately the hydraulic behavior of the fluid flow in the fault, but when we mainly study the slipping behavior of the fault in the process of reactivation, the latter one is much more effective. Fault reactivation may change the hydraulic aperture and the permeability of the fault, which will affect fluid flow in the fault and in the surrounding media. In order to account for the hydraulic behavior in the fault and its effect on the sur-rounding media, in (e.g., [60]; [43]), a reduced model of the fluid flow in the fault has been deduced, which we employ in our research. As concerns the mechanical behavior of the fault, there are three main friction models in literature: static friction model, slip-weakening model, and the rate- and state-dependent friction model. Among these three models, the rate- and state- dependent friction model is much more realistic since it can account for, so we will use it as our fault friction model.

Compared with the previous research (e.g., [27], [34], [8]) of the fault reactivation induced by fluid injection, we have chosen XFEM to deal with the discontinuities of the rock deformation and the fluid flow across the fault. We regard the fault as a zero-thickness interface and use a reduced model to account for the hydraulic behavior in the fault. We start consider-ing the two-dimensional map-view case, but in order to consider the dip of the fault, we will regard the system as a two-and-one-half dimensional sys-tem based on Anderson’s fault theory which assumes the vertical direction as principal stress component. This thesis is structured as follows: in Chap-ter 2 we discuss the mathematical model for the geomechanics of faulted rocks, focusing on the two dimensional case. In Chapter 3 we present the

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model for fluid flow in the presence of faults. In Chapter 4 we introduce the coupled poroelastic problem and discuss the detail of its numerical dis-cretization. In Chapter 5 we present some test cases with the aim of veri-fying our method against benchmarks in literature, and to test the effect of some physical parameters on the risk of fault reactivation. Finally, Chapter 6 is dedicated to conclusions.

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CHAPTER

2

Geomechanic problem

In this chapter we present the mathematical model that describes rock de-formation.

The mechanical problem is governed by the equations of momentum balance. Although the exact rheology of the rock underground is unknown, the linear, constant coefficient elasticity is considered to be enough to de-scribe many of the features of crustal deformation [23]. So in our research, linear elasticity will be used to supply the equations for the momentum bal-ance that govern earthquake physics.

Besides, in this work, we will consider poroelasticity, i.e. the effect of the pressure of a fluid, so we introduce the effective stress tensor σ0 defined by

σ0 = σ + bpI, (2.1)

where σ is the total (Cauchy) stress tensor, b is the Biot coefficient, p is the fluid pressure, and I is the identity tensor [44].

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2.1

Balance of Linear Momentum

For a fully three-dimensional system, the momentum balance equation gov-erning displacement under isotropic linear elasticity is given by

ρ∂

2_u

∂t2 = ∇ · σ (u) ,

in the computational three-dimensional domain and for t > 0, supple-mented by appropriate boundary and initial conditions. Here, u is the dis-placement, ρ is the density of the material, while the constitutive equation reads

σ0(u) = λtr ( (u)) I + 2µ (u) , (u) = 1

2

∇u + (∇u)T, where is the strain, λ and µ are the Lam´e’s parameters.

In this work, we firstly focus on the map-view two-dimensional case which means that we consider a two-dimensional domain Ω that represents a top view of three-dimensional domain, as shown in Fig. 2.1.

In order to research on the map-view two-dimensional case, we

sim-Figure 2.1: Map-view Case

plify the equations of the three-dimensional system to the so-called Mode III (anti-plane) deformation and Mode I/II (within-plane) deformation for a two-dimensional slice of the three-dimensional system.

In Mode III case, the deformation is only allowed to occur in the anti-plane direction, so the equations can be reduced to the two-dimensional wave equation:

∂2u

∂t2 = ∇ · σ (u) ,

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2.2. Boundary Conditions

While in Mode I/II (Mixed Mode) case, deformation is only allowed to occur within the two-dimensional map-view plane Ω. Then the equations can be reduced to two coupled wave equations for a compressional (p-) wave and a shear (s-) wave:

ρ∂

2_u

∂t2 = ∇ · σ (u) , (2.2)

σ0(u) = λtr ( (u)) I + 2µ (u) , (u) = 1

2

∇u + (∇u)T. (2.3)

2.2

Boundary Conditions

Figure 2.2: The domain and the boundaries

We assume that the domain Ω is a convex polygon or has a C1boundary. On the boundary, we will consider a Dirichlet boundary ΓD where condi-tions on the displacement are imposed, a Neumann boundary ΓN where traction is imposed, and ”Mixed” boundary conditions where displacement is only allowed to be imposed on the direction parallel to the boundaries.

We use a far-field loading rate

u (t) = g (t) on ΓD, t > 0,

to set Dirichlet boundary conditions on ΓD, where g is a given function of time. For a constant velocity, assuming zero displacement at time 0, we

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have g (t) = vpt in which vp is the given velocity on ΓD. While, on the

Neumann boundary, we set

σ0(t) · n = F(t) on ΓN, t > 0,

where n is the unit vector normal to the boundary, and F is a given traction. In view of the approximation of the problem with finite elements, we need to formulate the equations in weak form. To this aim, we introduce the sobolev space H1(Ω), i.e. the space of square integrable vector functions whose gradient is also square integrable. The weak form then reads: find u ∈ Vg = {w ∈ H1(Ω) : w|ΓD = g(t)} such that ∀v ∈ V0, Z Ω (v) : σ0(u) = Z ∂Ω v · σ0(u) · n = Z ΓN v · F + Z ΓD v · σ0(u) · n, where V0 = {w ∈ H1(Ω) : w|ΓD = 0}.

However, it may be convenient for practical reasons, to seek the solution on the whole H1 space, avoiding the need to apply the constraint due to the Dirichlet condition. This will be done by implementing Dirichlet boundary condition numerically by the Nitsche approach ( [33], [19]). Details will be given later on.

2.3

Mathematical Model of the Fault

Even if the fault is a complex three-dimensional region, in this work, we regard it as a zero-thickness interface ( [27], [23]), then in the map-view two-dimensional case, the fault is simply represented by a line γ.

2.3.1 Fault Division

To model fault reactivation we need to consider the possibility that part of the fault be active (slipping). So, at any time t, the fault γ can be subdivided into two parts: the slipping region (that we denote as the Neumann part γN) where the two sides are allowed to have different displacements, and the stuck region, which we denote as γD, where we have a given value of displacement jump.

In Mode III case, the condition is given by: σ0(u) · ˆn = f on γN,

J ∂u

∂tK = 0 on γ

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2.3. Mathematical Model of the Fault

where f is the friction force per unit area exherted by the material inside the fault, {·} and _{J·K indicate, correspondingly, the average and the jump} of a quantity which may be discontinuous across the fault. More precisely, JuK = u

+ _{− u}−

, the two sides of the fault being identified by + and − accordingly to Fig. 2.2.

In Mode I/II (Mixed Mode) case, the tangential component of slip is constrained by friction and failure, while normal components are set by no-interpenetration constraint: ˆ t ·σ0(u) · ˆn = f on γN, J ∂u ∂tK ·ˆt = 0 on γ D_.

where here ˆt is a unitary vector tangent to the fault, i.e. ˆt · ˆn = 0. 2.3.2 No Interpenetration Constraint

Because no interpenetration of material is enforced, we also have that JuK · ˆn ≥ 0 on γ,

In Mode I/II (Mixed Mode) case, since we mainly consider compressed faults, we can require:

JuK · ˆn = 0 on γ. 2.3.3 Friction and Weakening

The extent of γN and γD depends on the status of the fault. In order to find the partition of γ into γN and γD, we have to define the friction term f and failure/resticking conditions.

When a region is stuck, the magnitude of friction is exactly the right amount to keep the region stuck [23]. Once the applied load reaches a critical value, the fault begins to slip. If it is in the rate-hardening situation (the friction increases with the increasing slip rate), slipping is stable and the fault creeps at small, steady slip rate. While if fricton decreases with increasing slip rate, slipping is unstable and earthquake occurs.

In frictional models, friction is taken equal to f = µfσ

0

nf , where µˆ f is

the friction coefficient, σ0_n = ˆn · σ0 · ˆn is the effective normal stress, and ˆf is the friction direction, which opposes the applied internal forces when the fault is stuck, while it opposes slip rate when the fault is slipping.

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pressure. Slip’s direction is always ±z, so ˆf = −sign J ∂u ∂tK .

In Mixed Mode case, the effective normal stress is σ_n0 = ˆn · σ0 · ˆ

n. Because slip direction is taken in tangential direction, we have ˆf = −sign J ∂u ∂tK ·ˆt .

In reality, the fault has a dip angle. We recall that the dip of the fault is the angle between the horizontal surface and the fault, i.e. the dip is 90ofor vertical faults. Although in the two-dimensional map-view case the dip of fault could not be defined, it is very important because it affects the normal stress, so we have to find a way to consider the dip of fault in the two-dimensional map-view framework.

2.3.4 Dipping Faults

Figure 2.3: Normal and tangential vectors on dipping faults

In order to consider a dipping fault in a two-dimensional framework, we regard the system as a two-and-one-half dimensional system based on Anderson’s fault theory ( [21], [22]) which assumes the vertical direction as principal stress component. We view the vertical stress as a constant, lithostatic value and assume that the vertical gradients can be neglected, and there is no variation in the fault along the dip [23].

So here the given displacement u is a vector in the two-dimensional map-view layer, while the stress is a three-dimensional tensor in the two-and-one-half dimensional system. By comparing the three-dimensional tractions on the fault plane to the fault strength, it can be determined whether

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the fault slips or not. If slip occurs, the three-dimensional traction is spec-ified by friction, then the equivalent dimensional stresses in the two-dimensional plane of the layer can be computed [23].

Under the assumption of this theory, the three-dimensional stress tensor on the fault is:

σ0 =    σ_x˜0_˜_x+ pls σ 0 ˜ x˜y 0 σ_x˜0_˜_y σ_{y ˜}0_˜_y + pls 0 0 0 pls   

where pls= ρgdΩis the lithostatic pressure on the layer and dΩis the depth

of domain Ω. In this three-dimensional frame of reference, the effective normal stress is σ0(3)_n = ˆn(3)·σ0(3)·ˆn(3), where the three-dimensional normal vector ˆn(3) =

sin(θ) · ˆn(2)₁ , sin(θ) · ˆn(2)₂ , cos(θ) T

for a fault dip angle θ. Besides, the shear stress can be decomposed along two directions identified by two unit vectors ˆt⊥and ˆtk: ˆt⊥=

−cos(θ) · ˆn(2)₁ , −cos(θ) · ˆn(2)₂ , sin(θ)T and ˆtk =

−ˆn(2)₂ , ˆn(2)₁ , 0T. In the three-dimensional case, slip in the ˆt⊥

direction corresponds to slip in the cross-view plane, and slip in the ˆtk

di-rection corresponds to slip in the map-view plane [23]. 2.3.5 Friction Model

For the friction coefficient µf, there are mainly three fault friction models

that could be used [8]:

1. Static friction model (µf = constant).

2. Slip-weakening model: µf is a function of the slip magnitude |d|,

where d =_{JuK ·}ˆt is the tangential slip, according to the following law

µf =    µs− (µs− µd) |d| dc , |d| ≤ dc, µd, |d| > dc,

with dca given critical value, µs the constant static friction coefficient, and

µdthe dynamic friction coefficient.

In this model, as shown in Fig. 2.4, when the rupture front approaches the point which we focus on, the shear stress begins to increase from its initial value τ0. When the shear stress reaches the peak value, or the static

yield stress τs = µsσ

0

n, the fault begins to slip. With the fault slipping,

the shear stress decreases until the amount of slip reaches the critical slip-weakening distance dc. Then the fault continues slipping with a constant

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Figure 2.4: Slip weakening model, where τsis the static yielding stress,τ0 is the initial

shear stress,τdis the dynamic yielding stress,dc is the critical slip distance, and the

dashed areaGcrepresents the energy released by the fracture.

dynamic friction stress τd = µdσ

0

n. The difference between the static yield

stress and the dynamic friction stress is known as the breakdown stress drop ∆τb = (µs− µd) σ

0

n. The dashed area in Fig. 2.4 equals the fracture

en-ergy Gc = ∆τbdc/2. In this model, seismic rupture occurs when the strain

energy release by rupture extension becomes sufficiently large to overcome Gc.

3. Rate- and state-dependent friction model. In this model µf is a more

complex function of the fault history: µf = µ0+ Aln V V0 + Bln V0θ dc , dθ dt = 1 − θV dc ,

where V = |_dtdd| is the slip rate magnitude, µ0is the steady state friction

coefficient at the reference slip rate V0, A and B are empirical

dimension-less constants, and θ is the macroscopic variable characterizing the state of the fault surface.

In this model, the evolution of the friction coefficient is determined by the combined effects of the evolution of the state variable θ, and the slip rate or velocity V . Here, the state variable θ maybe could be understood as the frictional contact time [35], or the average maturity of contact as-perities between the sliding surfaces [4]. The evolution of θ is assumed to

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Figure 2.5: Rate- and state-dependent friction model. The friction coefficient µf on the

fault evolves with the slip rate or velocityV and the state variable θ. With the slip velocity increasing fromV1toV2suddenly, the friction coefficient increases sharply

due to a sudden increase in resistance from contact asperities firstly, and then declines slowly due to slip-weakening. The final steady state value of the friction coefficient can be lower than the initial steady state value ifA − B < 0, as shown in the figure. [8]

be independent of changes in the effective normal traction σ_n0 which can accompany the fault slip due to changes in fluid pressure. This model ac-counts for the decrease in friction (slip-weakening) as the slip increases, and the increase in friction (healing) as the time of contact or slip velocity increases. The two effects act together such that A > B leads to strength-ening of the fault, stable sliding, and creeping motion, and A < B leads to weakening of the fault, frictional instability, and accelerating slip. By this way, the model could capture repetitive stick-slip behavior of faults and the resulting seismic cycle ( [36], [17]).

In this work, we use the Dieterich-Ruina rate and state friction model (a variational form of the rate- and state-dependent friction model) [6] as our fault friction model:

µf =        µ0+ Aln V V0 + Bln V0θ dc V ≥ Vlinear, µ0+ Aln Vlinear V0 + Bln V0θ dc − A 1 − V Vlinear V < Vlinear,

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θ (t + ∆t) =        θ (t) exp −V (t) ∆t dc + ∆t −1 2 V (t) ∆t2 dc V (t) ∆t dc < ε, θ (t) exp −V (t) ∆t dc + dc V (t) 1 − exp −V (t) ∆t dc _{V (t) ∆t} dc ≥ ε,

where Vlinearis a cutoff for a linear slip rate dependence, and it is used

to prevent failure due to null slip velocity, V = 0. Here, ε is set to 10−5. 2.3.6 Failure Criteria

We use a history-dependent criterion to check whether the fault will slip. A stuck point x ∈ γD begins to fail when the effective shear stress ex-ceeds a critical friction value. In practice, at each time step of our scheme, we perform the following test:

∀x ∈ γD that satisfies |τ 0 (u (x)) | |σ0 n(u (x)) | ≥ µf(x) ⇒ x ∈ γN,

where µf is given by the friction model, and γN is the portion of the fault

where the failure criteria is met and the fault can slip. Here we define |τ0(u (x)) |

|σ0

n(u (x)) |

as the slip tendency of the fault.

Conversely a slipping point x x ∈ γN re-sticks when slip rate returns to zero: if ∀x ∈ γN satisfies _J∂u ∂t(x)K = 0 ⇒ x ∈ γ D . 2.3.7 Earthquake Magnitude

Once the fault slips, the widely used expression of Kanamori [29] could be used to calculate the magnitude of the earthquake:

Mw =

2

3log10M0− 6.1,

where the seismic moment defined by Kanamori and Brodsky [30] is M0 = µAd.

Here, A is the area of the rupture [m2], d is the magnitude of the final slip vector at the end of earthquake [m], and µ is the shear modulus [Pa].

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2.4. Quasi-static Problem

2.4

Quasi-static Problem

To analyze the slip tendency of the fault, we need to focus on the phenom-ena that occur before fault reactivation. This process before fault slipping could be described using a quasistatic assumption which neglects the ac-celeration term in the momentum balance equation, since the time scale of this process is of the order of years. So for the map-view two-dimensional case, the linear elasto-static problem becomes:

∇ · σ (u) = 0, (2.4) σ = σ0− bpI, (2.5) JuK · ˆn = 0 on γ, (2.6) ˆ_{t ·}n_σ0 (u)o· ˆn = f on γN, (2.7) JuK ·ˆt = g on γ D , (2.8)

for the displacement u, where f is the friction traction, g is the given dis-placement jump on the tangential direction across the fault, ˆn is the unit vector normal to the fault, and ˆt is the unit vector tangential to the fault.

Here, we mainly focus on the geomechanical problem, so we neglect the fluid pressure p. To extend the weak formulation to the case of faulted me-dia we consider the domain Ω split into the two subdomains Ω+and Ω−so that ¯Ω+_{∩ ¯}_Ω− _{= γ and Ω = in( ¯}_Ω+_{∪ ¯}_Ω−_{). We indicate with Ω}

γ = Ω+∪ Ω−

and with H1_(Ω

γ) the broken space defined as H1(Ωγ) = {w : Ωγ → R :

w|Ω± ∈ H1(Ω±)} with standard definition of inner product

(u, v)H1_(Ω

γ) = (u|Ω+, v|Ω+)H1(Ω+)+ (u|Ω−, v|Ω−)H1(Ω−)

where H1_{(Ω) = {f ∈ L}2_{(Ω) : D}α_{f ∈ L}2_(Ω) _{∀α : |α| ≤ 1}.}

We can then extend naturally the definition to two dimensional vector function: H1(Ωγ) = H1(Ωγ) × H1(Ωγ) or in general n dimensional vector

functions.

To write the weak form we will impose both the interface conditions on γ and the boundary conditions via the Nitsche approach which enables essential boundary conditions to be implemented within the weak formula-tion by adding a consistent penalizaformula-tion term. Especially for the interface conditions, γD and γN are functions of time, so by this way, we will not need to change the discrete space at every time step. In the following, since we are still dealing with the continuous formulation, h has to be interpreted as a positive coefficient, in the FEM discretization, it will indicate the mesh

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size.

We choose a set of test functions v ∈ V ⊂ H1(Ωγ). Then the weak

form reads: for any time t > 0, find the displacement vector u ∈ H1(Ωγ)

such that B (u, v) = `t(v) ∀v ∈ V, (2.9) where B (u, v) = Z Ω (v) : σ0(u) (∗) − Z γ (_JvKγ· ˆn) σ 0 n(u) + σ 0 n(v) (JuKγ· ˆn) (a1) +βI h Z γ (_JvKγ· ˆn) (JuKγ· ˆn) (a2) +βD h Z γD JvK γD · ˆt JuKγD · ˆt (b) − Z ΓD v · σ0(u) · ˆn + σ0(v) · ˆn · u (c1) +β∂Ω h Z ΓD (v · u + (v · ˆn) (u · ˆn)) (c2) , ` (v) = Z γN JvK γN · ˆt f (∗) +βD h Z γD JvK γD · ˆt g (b) + Z ΓN v · F (∗) − Z ΓD σ0(v) · ˆn · g(t) (c1) +β∂Ω h Z ΓD (v · g(t) + (v · ˆn) (g(t) · ˆn)) (c2) .

for all test functions v ∈ H1(Ωγ). The ( ∗ ) terms are the typical weak

for-mulation of the momentum balance where the divergence of σ has moved to test function v, and the natural boundary terms from Eq. (2.7) arise in linear form. The (a), (b), (c) terms apply three separate boundary condi-tions respectively: term (a2) applies the no interpenetration constraint Eq. (2.6) with constant penalty coefficient βI, (b) applies the essential boundary

conditions Eq. (2.8) with penalty coefficient βD, and (c2) applies Dirichlet

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2.5. Dynamic Problem

β∂Ω.

2.5

Dynamic Problem

In order to research on the slipping behavior of the fault after the fault slip and calculate the corresponding earthquake magnitude, we need to focus on the phenomenon of fault reactivation. This process normally only lasts less than some seconds, and could be described as the dynamic problem. For the dynamic problem, we need to solve:

ρ∂

2_u

∂t2 − ∇ · σ (u) = 0, (2.10)

σ = σ0 − bpI, (2.11)

with the following conditions on the fault:

JuK · ˆn = 0, (2.12) ˆ t ·nσ0(u)o· ˆn = f on γN, (2.13) J ∂u ∂tK ·ˆt = 0 on γ D_. (2.14)

As the static problem of the geomechanics, we multiply (2.11) by a test function v, integrate over the domain, apply the divergence theorem, ex-ploit symmetry of the stress tensor, and use Nitsche’s method to deal with the Dirichlet boundary conditions. We get the weak form for the mechanic problem, reading

Find displacement vector u ∈ H1(Ωγ) such that

v, ρ∂ 2_u ∂t2 + B (u, v) = ` (v) ∀v ∈ V, (2.15)

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where v, ρ∂ 2_u ∂t2 = Z Ω ρv · ∂ 2_u ∂t2 B (u, v) = Z Ω (v) : σ0(u) (∗) − Z γ (_JvKγ· ˆn) σ 0 n(u) + σ 0 n(v) (JuKγ· ˆn) (a1) +βI h Z γ (_JvKγ· ˆn) (JuKγ· ˆn) (a2) +βD h Z γD JvKγ D · ˆt J ∂u ∂tKγD · ˆt (b) − Z ΓD v · σ0(u) · ˆn + σ0(v) · ˆn · u (c1) +β∂Ω h Z ΓD (v · u + (v · ˆn) (u · ˆn)) (c2) , ` (v) = Z γN JvKγ N · ˆt f (∗) + Z ΓN v · F (∗) − Z ΓD σ0(v) · ˆn · vpt (c1) +β∂Ω h Z ΓD (v · vpt + (v · ˆn) (vpt · ˆn)) (c2) .

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CHAPTER

3

Fluid Flow Problem

In this chapter we focus on the fluid flow problem in the porous medium. The porous medium consists of the solid and the pores. We assume that the fluid fills the porous medium completely which means that there is no void space left. We first start by recalling same basic definitions and results.

Let us consider a domain Ω formed by a porous material. Let Br(x)

be the ball contained in Ω centred in x0 with radius r (as in Fig. 3.1). We

define the void space indicator function at the microscopic level as i(x, t) :=

(

1 x ∈ non-solid matrix at time t, 0 x ∈ solid matrix at time t. then we define the porosity φ at time t as

φ (r, x, t) := 1 |Ω0|

Z

Br(x0)

i (x, t) dx.

The value of φ at position x0 and time t changes with different values of the

radius r as shown in Fig. 3.2. If, like in Fig. 3.2, we can find a value of the radius r such that φ stabilizes for all x0 and t ∈ (0, T ), then Br takes the

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Figure 3.1: Averaging volume Ω0in a porous mediumΩ [1]

Figure 3.2: Porosity φ (x0, t) changes with different value of radius r of Ω0[1]

l being the value for which φ reaches the plateau as in Fig. 3.2. The exis-tance of the REV indicates that there is a clear separation of scales between the ”pore scale” and the scale where the material can be considered as a homogenized continuum.

In this work, we focus on the two-dimensional map-view case and as-sume that the motion of the incompressible fluid in the porous media Ω ⊂ R2 is governed by Darcy’s equation. Besides, we also assume that ∂Ω = ΣD_{S Σ}N _{with Σ}D_{T Σ}N _{= ∅, and that the boundary flux ¯}_{u : Σ}D _{× R}+ _→

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3.1. Governing Equations

3.1

Governing Equations

3.1.1 Balance of Fluid Mass

For the isothermal single-phase flow of Newtonian fluid which occupies the entire void space in a porous medium, the fluid mass conservation equation is

∂ (ρfφ)

∂t + ∇ · (ρfw) = ρffq in Ω, t > 0, (3.1) where w is the fluid flux velocity, ρf is the density of the fluid, φ is the

porosity of the porous medium, and fq is a volumetric source term. In this

work we will consider a porous medium saturated with fluid.

In next chapter, when we research on the coupling between the geome-chanics and the fluid flow problem, we will focus on the variation of the porosity in the porous medium with the time, but in this chapter we focus on the fluid flow problem, so we neglect it. Then, the balance of fluid mass becomes

∇ · w = fq.

3.1.2 Darcy’s Law

We assume that fluid flow can be modeled by Darcy’s Law. Then, the rela-tionship between the Darcy velocity w and the fluid pressure p is expressed by

w = −k

µ(∇p − fv) , (3.2)

where k is the absolute permeability tensor, µ is the fluid viscosity, and f_v is a generic forcing term, for instance due to gravity.

We do not consider the variation of the fluid viscosity here, and we de-fine K = k

µ, then Eq. (3.2) becomes

w = −K (∇p − f_v) .

3.2

Boundary Conditions

For the fluid flow problem, since we are considering the mixed formulation of the problem, the normal component of velocity is imposed on the Dirich-let boundary ΣD_{, while the boundary conditions on pressure are natural.}

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More precisely, we impose (

w · n = ¯u on ΣD_,

p = ¯p on ΣN, with ΣD _{∪ Σ}N _{= ∂Ω and Σ}D _{∩ Σ}N _{= ∅.}

3.3

A Reduced Model for the Fluid Flow in the Fault

Figure 3.3: The domain division

In several applications, especially in the geologic storage of carbon diox-ide, the fluid flow problem in the fault is very important, because it is crit-ical to check whether the carbon dioxide could leak from the fault, so the fluid flow problem in the fault needs to be resolved accurately. Since the fault is very thin compared to the surrounding media, the grid size of the surrounding domain is usually too coarse to capture the feature. Therefore, in our research, we will consider the fault as a geometric interface with zero-thickness also for the fluid problem. The presence of fluid flow in the fault and the exchanges with the surrounding medium are accounted for by a suitable reduced model.

We divide the whole domain into three parts: Ωi, i = 1, 2, f . As shown

in Fig. 3.3, Ω1 and Ω2 are the two parts of the original domain divided by

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3.3. A Reduced Model for the Fluid Flow in the Fault

The Darcy equations are given by:            ∇ · w = fq in Ω, K−1w + ∇p = f_v in Ω, p = ¯p on ΣN_, w · n = ¯w on ΣD.

Defining the Darcy flux w and the pressure p in each sub-domain Ωi as

wi := w|Ωi and pi := p|Ωi for i = 1, 2, f , then the coupled problem for the

single-phase Darcy system is:

Find (w, p) ∈ Hdiv(Ω) × H1(Ω) which satisfy for i = 1, 2, f

( ∇ · wi = fqi

K−1_i wi+ ∇pi = fvi

in Ωi,

with the coupling conditions for j = 1, 2 (

wj· nj = wf · nj

pj = pf

on γj,

and the boundary conditions for i = 1, 2, f (

pi = ¯pi on ΣNi ,

wi· n = ¯wi on ΣDi .

where Hdiv(Ω) = {w ∈ L2(Ω) : ∇ · w ∈ L2(Ω)}, γj := ∂ΩjT ∂Ωf for j =

1, 2, and Σi := ∂ΩiT ∂Ω for i = 1, 2, f .

3.3.1 Reduced Model of the Fluid Flow in the Fault

As already mentioned we want to regard the fault as a lower dimensional interface, so we replace Ωf with its centerline γ and use a reduced model

for the flow following [1], [2].

• The conservation equation in the fault is

∇τ · ˆw = ˆfq+Jw · nKγ on γ,

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Figure 3.4: Example of Ωf

• The reduced Darcy equation in the fault is ˆ

η ˆw + ∇τp = ˆfˆ v on γ,

where ˆη (s) := 1

a (s) Kf,τ(s)

.

• The first coupling condition for the global problem is ηγ{w · n}γ =JpKγ+ a {fv· n}γ,

where {·} indicates the mean quantity across fault γ, and ηγ(s) :=

a (s) Kf,n(s)

.

• The effects of the variation of pressure and velocity across the fault is ξ0ηγJw · nKγ+

a

4Jfv· nKγ = {p}γ− ˆp on γ,

where ξ0is a closure coefficient derived from the assumption made on

the variation of the pressure across the fault when deriving the reduced model, see [1]. In this work, we have taken the value of ξ0 equivalent

to assuming a parabolic pressure variation across the fault.

3.3.2 Reduced Model

The reduced model for the flow with a generic momentum forcing term is: find the Darcy velocity wi and the pressure pi associated to the porous

matrix, i = 1, 2, and the reduced Darcy velocity ˆw and reduced pressure ˆp associated to the fault, to satisfy the following equations:

           ∇ · wi = fqi in Ωi, K−1_i wi+ ∇pi = fvi in Ωi, ∇τ · ˆw = ˆfq+Jw · nKγ on γ, ˆ η ˆw + ∇τp = ˆfˆ v on γ, (3.3)

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3.4. Weak Formulation

coupled with the interface conditions on γ ( ξ0ηγJw · nKγ+ a 4Jfv · nKγ = {p}γ− ˆp ηγ{w · n}_γ =JpKγ+ a {fv· n}γ on γ, (3.4)

and with the boundary conditions on ∂Ω and ∂γ:            pi = ¯pi on ΣNi , wi· n = ¯ui on ΣDi , ˆ p = ¯pˆ on ∂Nγ, ˆ w · n = ¯w · nˆ on ∂Dγ, (3.5)

for i = 1, 2, where ∂Nγ is the Neumann part of the boundary of the fault

region while ∂Dγ is the Dirichlet part of the boundary of the fault region.

3.4

Weak Formulation As shown in [1], we define D : 2 [ i=1 Ωi = Ω \ γ.

We introduce the functional spaces for i = 1, 2

Vi := {vi ∈ Hdiv(Ωi) :< vi· nΓ, v >= 0 ∀v ∈ H_0,Σ1 N i (Ωi) and vi· ni ∈ L2(γ ∩ ∂Ωi)} and Qi := H1(Ωi), where < vi · nΓ, v >= R Ωvi · nΓ · vdΩ, H 1 0,ΣN(Ω) = {f ∈ H1(Ω) : f |∂Ω\ΣN = 0} and L2(Ω) = {f : Ω → R s.t. R Ω(f (x)) 2_{dΩ < +∞}.}

By putting Vi, i = 1, 2 together, we get the broken functional space

V : D → R

V := {v : D → Rn: vi := v|Ωi ∈ Vi for i = 1, 2},

and by putting Qi, i = 1, 2 together, we get the broken space Q : D → R

Q := {q : D → R : qi := q|Ωi ∈ Qi for i = 1, 2}.

Then we introduce the functional spaces for the problem in the fault: the spaces of functions living in the tangent space, that is

ˆ

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and < ˆv · nΓ, v >= 0 ∀v ∈ H_0,∂γ1 N(γ)},

ˆ

Q := {ˆ_{q : γ → R, ˆ}q ∈ L2(γ) : ∇nq = 0 and ∇ˆ τq ∈ [Lˆ 2(γ)]n}

Analogously, we define the global space for the Darcy velocity and the global space for the pressure

V := {v = (v, ˆv) ∈ V × ˆV} and Q := {q = (q, ˆq) ∈ Q × ˆQ}

which are Hilbert spaces endowed with inner products (·, ·)V : V × V → R

and (·, ·)Q : Q × Q → R, defined as

(w, v)V := (w, v)V+ ( ˆw, ˆv)ˆ_V and (p, q) := (p, q)Q+ (ˆp, ˆq)_Qˆ,

and norms || · ||V : V → R and || · ||Q : Q → R, defined as

||w||2

V := (w, w)V and ||p||2Q := (p, p)Q,

with w, v ∈ V such that w = (w, ˆw), v = (v, ˆv) and p, q ∈ Q such that p = (p, ˆp), q = (q, ˆq).

Finally, we define the global space for the whole problem D := {z = (v, q) ∈ V × Q},

which is an Hilbert space endowed with the inner product (·, ·)D : D ×D →

R and associated norm || · ||2D : D → R, defined as

(z, t)D := (v, w)V+ (q, p)Q and ||z||2D := (z, z)D,

with z = (v, q) ∈ D and t = (w, p) ∈ D.

Then by multiplying by the test functions v and q, which act as weight-ing functions in the integral form of the differential equations, integratweight-ing over the domain, applying the divergence theorem, and inserting the essen-tial and natural boundary conditions, we could get the weak form of the governing equations for the fluid flow problem:

Find (w := (w, ˆw) , p := (p, ˆp)) which are the Darcy velocity and the pressure in the medium and in the fault respectively belonging to appro-priate functional spaces satisfying the corresponding boundary conditions such that

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3.4. Weak Formulation where A [(w, p) , (v, q)] = Z Ω K−1w · v + Z ΓD i γ0h−1(w · n) (v · n) + Z γ ηγ{w · n}γ{v · n}γ+ ξ0 Z γ ηγJw · nKγJv · nKγ − Z Ω p (∇ · v) + Z ΓD i p (v · n) + Z Ω q (∇ · w) − Z ΓD i q (w · n) + Z γ ˆ η ˆw · ˆv + Z ∂Dγ γ0h−1(ˆw · n) (ˆv · n) − Z γ ˆ p (∇τ · ˆv) + Z γ ˆ p_{Jv · nK}γ+ Z ∂Dγ ˆ p (ˆv · n) + Z γ ˆ q (∇τ · ˆw) − Z γ ˆ q_{Jw · nK}γ− Z ∂Dγ ˆ q (ˆw · n) , and F (v, q) = Z Ω f_v_i · v + Z Ω fqiq − Z ΓN i ¯ pi(v · n) + Z ΓD i γ0h−1w¯i(v · n) − Z ΓD i ¯ wiq + Z γ ˆf_v· ˆv + Z γ ˆ fqq −ˆ Z ∂Nγ ¯ ˆ pi(ˆv · n) + Z ∂Dγ γ0h−1w (ˆ¯ˆ v · n) − Z ∂Dγ ¯ ˆ w ˆq + Z γ af_v_i · n γ{v · n}γ− Z γ a 4Jfvi· nKγJv · nKγ, for all test functions v ∈ V and q ∈ Q.

Note that we are imposing the essential boundary conditions on the nor-mal flux on the Dirichlet boundary by means of a consistent penalization. Here, γ0 and h are positive coefficients, while in the finite element context

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CHAPTER

4

Numerical Formulation

In this section we discuss the numerical approximation of the coupled flow and geomechanic problem. Firstly, the coupled modeling of single-phase flow and poromechanics will be illustrated. Secondly, we will address the problem of discretization in time of the coupled problem and discuss the iterative splitting method, and then we will describe the discretization in space of the fluid flow and mechanics problem by means of the eXtended Finite Element Method (XFEM). The numerical method described in this section has been implemented in an original C++ code based on the fi-nite element library Getfem++, getfem.org, which provides excellent support for the implementation of XFEM and non-matching methods in general thanks to ad hoc tools for geometrical operations and quadrature.

4.1

Coupled Modeling of Fluid Flow and Poromechanics

We here recall the system of equations introduced in the previous chapters in view of their coupling.

The domain Ω is split into two parts Ω1 and Ω2, separated by a fault

described by the line γ = int ¯Ω1TΩ¯2.

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by setting Ωγ = Ω1S Ω2, we have that at any given time t > 0,              ∇ · σ (u, p) = ∇σ0(u) − b∇p = ρf , ∂(ρfφ) ∂t + ∇ρfw = ρffq in Ωγ, K−1w + ∇p = ρfg,

σ0(u) = λtr ( (u)) I + 2µ (u) .

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Porosity and permeability depend on the stress state. We assume that porosity variation is governed by

∂φ ∂t = b ( ˙u) + 1 M ∂p ∂t = b∇ · ˙u + 1 M ∂p ∂t u =˙ ∂u ∂t, where M is the Biot modulus.

We choose to ignore the effects of water compressibility, so the second equation in Eq. (4.1) may be rewritten as

1 M

∂p

∂t + ∇ (w + b ˙u) = fq in Ωγ,

while for the permeability in the bulk, we use the model developed by Chin et al [41]: K = K0

φ φ0

α

where α ≥ 1 is a power-law exponent experi-mentally determined, and it depends on the type of rock considered, while K0 is the permeability at a reference porosity φ0. This model is justified

by some experimental observations and allows us to describe the fact that a reduction of porosity induces a reduction of rock permeability. The system is supplemented by boundary conditions on ∂Ωγ, which correspond on

fix-ing either the normal stress or the displacement on different portions of the boundary.

In the fault, the evolution of the tangential velocity flux ˆw obeys the model described in Eq. (3.3), since we assume that variations of porosity in the fault for the phenomena under study are negligible. We recall it for completeness,

(∇τ · ˆw = ˆfq+Jw · nK on γ, ˆ

η ˆw + ∇τp = ˆˆ fv = ρg · ˆt on γ,

with the boundary conditions on ∂γ: (

ˆ

p = ˆpN on ∂Nγ,

ˆ

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4.1. Coupled Modeling of Fluid Flow and Poromechanics

where ˆwD and ˆpN are given data.

The effective permeability of the fault in the reduced model is given by ˆ

η = (Kf,τa) −1

. However the material permeability Kf,τ in the fault may

vary with the state. In particular, even if we are assuming a compressive state and we do not have variation in the fault aperture due to normal dis-placements, we may have an effect related to shear dilation [20]. To account for this effect we define an effective aperture aeff = a + bhs where bhs is

the aperture increase due to shear displacement, and it based determined according to [27]. Then, we set Kf,τ = Kf 0

aeff

a0

2

, where Kf 0is a

refer-ence value at zero shear displacement. I Coupling Conditions

We are left with the coupling conditions, which we write as follows                          ξ0ηγJw · nKγ = {p} − ˆp, ηγ{w · n}γ =JpKγ+ a{fv· n}, ˆ t · {σ0(u)} · ˆn = f on γN_(t), J ∂u ∂tK ·ˆt = 0 on γ D_(t), Ju · ˆnK = 0 on γ, Jσ · ˆnK · ˆn = 0 on γ. (4.2)

γN_{(t) is defined as the partion of γ where the ratio of tangential and normal}

stress exceeds a given value, more precisely, at any time t > 0,

γN(t) = {x ∈ γ : |τ 0_(x)| |σ0 n(x)| > µf or J ∂u ∂tK · t (x) < f}, (4.3)

where τ0 = {σ · n} · t and σ0_n = {σ · n} · n are the tangential and normal component of the average effective Cauchy stress on γ, respectively. We then set γD(t) = γ \ γN_{(t). The first condition in Eq. (4.3) is the classic}

failure condition, the second condition allows a point in the slipping region γN _{to remain in that region until its tangential velocity goes below a}

thresh-old value f. Here, µf is the friction coefficient, as described in a previous

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4.2

Time Discretization

Let us consider the time interval (0, T ) and define a partition 0 = t0 ≤

t1 ≤ ...tN = T , with N ∈ N. The time step ∆tk = tk+1− tk is assumed

to be constant and will be denoted simply as ∆t. We introduce the semi-discretization in time of problem (Eq. (4.1)) with the Implicit Euler method, reading        ∇ · σ0(un+1_{) − b∇p}n+1 _{= 0} _{in Ω} γ 1 M pn+1_{− p}n ∆t + b ∇ · un+1_{− ∇ · u}n ∆t + ∇ · w n+1 _{= f} q in Ωγ ∇τ · ˆwn+1 = ˆfq+Jw n+1_{· n} K on γ

with the constitutive equations        σ0(un+1) = λtr(un+1)I + 2µ(un+1) K(un+1_{, p}n+1_)wn+1_{+ ∇p}n+1 _{= f} v Kτ(un+1) ˆwn+1+ ∇τpˆn+1 = ˆfv

and the coupling conditions (Eq. (4.2)) evaluated at time tn+1. Note that the fluid and mechanics problems are coupled by some linear and nonlin-ear terms: the pressure gradient in the equilibrium equation, the divergence of the displacement in the mass conservation equation are linear coupling terms, while the dependence of permeability on porosity and effective aper-ture introduces nonlinearities in the system. Moreover, the interface condi-tions associated with the frictional contact are strongly nonlinear.

4.2.1 Solution Strategies for the Coupled Problem

Nowadays, the most common coupling methods used for modeling the in-teractions between flow and geomechanics could be classified into four types: fully coupled, iteratively coupled, explicitly coupled, and loosely coupled [32].

• Fully Coupled

Coupled governing equations are solved simultaneously at every time step, and a converged solution is obtained through iteration, typically using the Newton-Raphson method. This method is unconditionally stable, but is computationally expensive [32].

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4.2. Time Discretization

• Iteratively Coupled

One of the flow and the mechanical problems is solved firstly, and then the solution is used to solve the other problem. The sequential proce-dure is iterated at each time step until the solution converges to within an acceptable tolerance. By this method, the solution at convergence is the same as that obtained by the fully coupled method. Besides, this partitioned methods also have some other advantages such as the possibility to use different domains for the flow and the mechanical problems which is useful because sometimes the computational do-main of the mechanical problem is taken larger than that of reservoir simulation in order to deal with the far-field reservoir boundary con-ditions, see [5, 40]. Moreover, different pre-existing software for the two sub-problems can be adapted and used in an iterative framework. • Explicitly Coupled

It is also called the non-iterative sequential method. It is a special case of the iteratively coupled method in which only one iteration is taken, see [38], [52], [26], [25].

• Loosely Coupled

The coupling between the problems is resolved only after a certain number of flow time steps ( [18], [54], [53]). Compared with other strategies, it can save computational cost, but it is less accurate and re-quires reliable estimates of when to update the mechanical response [32]. In this work we will adopt an Iteratively Coupled method to deal with the coupled fluid flow and reservoir geomechanics in our research, for reasons of computational cost, in particular in view of a future extension fo the code to three dimensional cases.

4.2.2 The Fixed Stress Splitting

For the iteratively coupled solution methods there are four different pos-sible strategies: drained and undrained splits solve the mechanical prob-lem firstly, fixed-strain and fixed-stress splits solve the flow probprob-lem firstly. Nowadays, most of the methods developed solve the mechanical subprob-lem firstly, but from a practical point of view, the framework for reservoir simulation would benefit from solving the flow problem firstly within each time step [32].

• Drained Split

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and then solves the fluid flow problem with a frozen displacement field. Let i denote the iterations, then the coupled problem is split, at each time step, as follows:

ui pi Au dr −→ p=const ui+1 pi Ap_dr −→ =const ui+1 pi+1

where the operators Au_dr, Ap_drformally denote the (drained) mechanics problem and the fluid problem respectively.

• Undrained Split

Compared with the drained method, the undrained split allows the pressure to change but keeps the fluid mass in each grid block re-maining constant during the mechanical step (δm = 0). The splitting consists of the following steps:

ui pi Au ud −→ m=const " ui+1 pi+12 # Ap_ud −→ =const ui+1 pi+1

Here, the operators Au_ud, Ap_udformally denote the (undrained) mechan-ics problem and the fluid problem respectively.

• Fixed-strain Split

This scheme solves the flow problem with a frozen strain field firstly, and then uses the solution to solve the mechanical problem. If we formally denote with Ap_sn the solution of the modified flow problem and with Au_snthe mechanics problem the splitting reads

ui pi Apsn −→ =const " ui+12 pi+1 # Au sn −→ p=const ui+1 pi+1 • Fixed-stress Split

This case solves the flow problem with a frozen total means stress field firstly, then solves the mechanics problem with a given pressure field,

ui pi Apss −→ σv=const " ui+12 pi+1 # Au ss −→ p=const ui+1 pi+1

Here we have denoted with Ap_ss the modified flow problem under the assumption of fixed stress, and with Au

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4.2. Time Discretization

In all the aforementioned methods the two subproblems are solved it-eratively for every time step until convergence is achieved. It has been proven ( [32, 45]), that the fixed-strain splits are only conditionally stable, their stability limits are independent of time step size and only depend on the strength of the coupling between flow and mechanics, which means that they could not be used to solve problems with strong coupling. Be-sides, even when the stability limit is honored, oscillations may occur in the solution. In contrast to them, the undrained and the fixed-stress splitting methods are unconditionally stable regardless of the coupling strength, and they do not suffer from oscillations. It is also shown that the drained and the fixed-strain split methods with a fixed number of iterations are inconsistent, while the fully coupled method, the undrained split and the fixed-stress split are consistent even in the case of a single iteration per time step. Among the unconditionally stable schemes, the fixed-stress method is more accu-rate for a given number of iterations than the undrained method. Based on these considerations we employ the fixed-stress split as our solution strat-egy.

As already mentioned, in this scheme the flow problem is solved first assuming a constant (”frozen”) mean stress field, and thus, a constant total volumetric stress. This allows, as we will show, to derive an expression for ∇ · un+1_{in the mass conservation equation.}

Let the index i denote the i-th iteration of the fixed stress scheme. At time step n+1 we impose, for each iteration, that the total volumetric stress at iterations i and i − 1 are the same. Since σv = Kdr∇ · u − bp, where Kdr

is the drained bulk modulus, equal to λ + µ in the two-dimensional case, we obtain

Kdr∇ · un+1,i− bpn+1,i= Kdr∇ · un+1,i−1− bpn+1,i−1. (4.4)

Substituting Eq. (4.4) in the mass balance equation we obtain

1 ∆t 1 M + b2 Kdr pn+1,i+∇·wn+1,i= fq+ 1 ∆tMp n₊ b ∆t∇·(u n_−∇·un+1,i−1₎₊ b2 Kdr pn+1,i−1 (4.5)

This modified fluid problem does not depend on un+1,i_{and can be solved}

with the displacement fields at the previous time and at the previous itera-tions. The fixed stress splitting algorithm can be summarized as follows: Initialization At each time step n + 1 set un+1,0_{= u}n_{and p}n+1,0 _{= p}n_.

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Fluid Subproblem Solve Eq. (4.5) coupled with the flow problem along the fault to obtain pn+1,i:

                       1 ∆t 1 M + b2 Kdr pn+1,i+ ∇ · wn+1,i= fq+ 1 ∆tMp n₊ b ∆t∇ · (u n_{− ∇ · u}n+1,i−1_{) +} b 2 Kdr pn+1,i−1 _{in Ω} γ wn+1,i+ K∇pn+1,i = fv ∇τ · ˆwn+1,i = ˆfq+ 1 ∆tJw n+1,i−1_{· n} K in γ ˆ wn+1,i_{+ K} τ∇τpˆn+1,i = ˆfv (4.6) with coupling conditions on γ

( ξ0ηγJw n+1,i_{· n} K = {p n+1,i_{} − ˆ}_pn+1,i_, ηγ{wn+1,i· n} =Jp n+1,i K + a{fv· n},

as well as the given boundary conditions for the fluid problem. Identification of γN _{and γ}D _{With the computed value of fluid pressure we}

compute an intermediate total stress σ = 2µ(un,i_{) + λ∇ · u}n,i ₋

b∇pn+1,i_{in order to estimate, with the help of (4.3), the extension of}

γD_{and γ}N _{at the current iteration, as well as the values of the friction}

coefficient.

Mechanic Subproblem Solve the equilibrium equation on Ωγ,

2µ(un+1,i) + λ∇ · un+1,i= b∇pn+1,i, (4.7) together with the coupling interface conditions given by the last four equations in (4.2) and the given boundary conditions for the mechan-ical problem, to obtain un+1,i.

Convergence Test Test for convergence, then repeat steps 2 and 3 or pro-ceed to the next time step. For the convergence we monitor kpn+1,i+1₋

pn+1,i_k

L2_(Ω)and kun+1,i+1− un+1,ik_H1_(Ω γ).

Remark 1. We point out that, as suggested in [37], the coefficient β = b2

Kdr

multiplying pressure in the fluid sub-problem can be regarded as a stabilization parameter, thus,β could take different values with an impact on the number of iterations needed to achieve convergence. In this work we have adopted the classical, physically based value.

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4.3. Space Discretization

4.3

Space Discretization

We can now introduce the space discretization strategy for the two prob-lems: the fluid flow and the mechanics. In this work we employ a finite element discretization, in particular, to achieve high flexibility in the pres-ence of complex geometries, we resort to XFEM to represent interfaces across mesh elements.

4.3.1 The Grids

In this work we use the same mesh for the fluid flow problem and the ge-omechanical problem. This choice has the advantage of avoiding interpo-lation of material properties and variables, such as for instance the fluid pressure, between different grids. However, this is only possible because we are considering relatively simple synthetic cases: in real life applica-tions the geomechanical model is usually larger than the fluid reservoir of interest and different grids are often used.

Both at the geomechanical and the reservoir scale the construction of a computational grid can be a challenging task in realistic configurations: indeed, grids should account for

• heterogeneous layers

• multiple and intersecting fractures and faults.

Moreover, the fault surface is usually non planar and has a roughness that should be accounted for to compute reliable predictions of the slip tendency. In this work we choose to employ non-conforming grids, i.e. we build the ”bulk” mesh of the porous medium irrespective of the fault geometry and position. Thus, the domain Ω and the fault γ are meshed independently.

We discretize the domain Ω into an approximation Ωhwhich is a convex

polygon, and consider a family of triangulations Th. Let h be the maximal

diameter of the elements of Th. The approximation Ωh is such that if h

goes to 0, Ωh → Ω. In our examples, Ω is itself a polygon, so Ωh = Ω.

Analogously we define for γ its piecewise linear approximation γh (details

will be given later on) and consider a family of triangulations Lˆ_h.

Remark 2. The mesh sizes h and ˆh can be chosen independently. However it has been observed, [48], that oscillations can arise in the fault if ˆh << h. For this reason we will use independent grid for the bulk and the fault, but with comparable mesh sizes.

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Figure 4.1: δ(s) is the distance between the curve fault γ and the approximation γhin the

convex

Since This not conformal with the fault γh, for the triangles K ∈ Th cut

by the fault γ, we define Ki = K ∩ Ωi, where Ωi, i = 1, 2, are the two parts

of the original domain divided by the fault γ, and use Gh = {K ∈ Th :

γ ∩ K 6= ∅} to denote the collection of the elements cut by the fault. In the following we will assume, following [15], that:

A-1 the triangulation is shape-regular, i.e., cρK ≤ hK ≤ CρK where hK

is the diameter of element K, ρK is the diameter of the largest circle

contained in K and c, C are positive constants;

A-2 if K ∈ Gh, then γ intersects the boundary ∂K twice exactly, and for

each edge of K, it intersects at most one time;

A-3 γh is ”a good approximation” of γ, i.e. if γK = γ ∩ K and γK,h =

γh∩ K then γK = {(t, s) : 0 < s < |γK,h|, t = δ(s)} while γK,h =

{(t, s) : 0 < s < |γK,h|, t = 0}, where δ(s) is the distance between

the fault γ and the approximation γh.

I The Use of Level Sets to Represent the Fault

The geometry of the fault γ is assumed to be given as a parametrized curve as γ : x = g(s) with s1 ≤ s ≤ s2. Let us define the signed distance of a

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4.3. Space Discretization

Figure 4.2: The fault as the zero of a level set function equal to the signed distance from the interface (left) and a zoom of a detail (right). γ is represented by the white line, and its approximationγhby the red line.

ψ(x) = min

x∗_∈γ|x − x

∗_{|sign(x − x}∗

) · n

where n is the normal to the surface. Then γ can be identified as the zero-level of ψ. We then replace ψ with its finite element approximation (with linear finite elements) ψh and obtain γh as its piecewise linear

zero-level set as shown in Fig. 4.2.

Thus, the difference between γ and γh depends on the size of the bulk

triangulation Th. This guarantees that if γ has bounded curvature

Assump-tion A-3 is always satisfied for a sufficiently refined grid.

The definition of the interface by means of a level set function is moti-vated by the capabilities of the finite element library Getfem++, used ad the basis of our implementation, which provides the tool to define quadrature formulas on subdomains defined by the positive, negative or zero values of level sets.

4.3.2 Space Discretization with the XFEM

As we mentioned in the previous sections the geometrical complexity of realistic cases motivates our choice of using non-matching grids, i.e. to al-low the fault to cut the elements of the bulk mesh. However, the material properties can change abruptly across the fault surface, and the solutions (pressure, fluid velocity and even displacement) can be discontinuous. For this reason we employ the eXtended Finite Element Method (XFEM) to account for the discontinuities across the fault by means of suitable enrich-ments of some standard finite element spaces ( [31], [50], [57])

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The XFEM are currently one of the most popular methods to discretize mechanical problems in linear elastic material in the presence of fractures, for industrial and geophysical applications. They have been applied to the problem of fault equilibrium and failure by E. Coon in [24] and [23]. In these works however the effect of fluid pressure and flow along and across the fault are not taken into account. The use of XFEM for the approxi-mation of flow in porous media with fractures and faults is more recent but quite established, see for instance [15], [51] and [55], where a single-phase Darcy flow is solved in undeformable fixed domains. In this work we will discretize both the mechanical and fluid subproblems by means of the XFEM using the same non-matching grid for the two problems to obtain a natural coupling and avoid interpolations.

I Flow Problem

For the numerical approximation of the flow problem in the bulk and along the fault we proceed as proposed in [15] and [1]: starting from a stable mixed finite element pair for fluid velocity and pressure we enrich the fi-nite element spaces to represent possible discontinuities across the fault. In particular we start from the lowest order Raviart Thomas finite elements (RT0) for the Darcy velocity and piecewise constant (P0) finite elements

for pressure in the two dimensional domain Ωh; analogously on the fault

γh we approximate the flux with P1 finite elements and the pressure with

P0 elements. The choice of mixed finite elements is motivated by their

property of local mass conservation and the robustness in the presence of permeability contrasts.

For each element K ∈ Th we define the restriction of the standard RT0

and P0 functions to the cut elements as follows:

RT0(Ki) = {vh|Ki : vh ∈ RT0(K)}

P0(Ki) = {qh|Ki : q ∈ P0(K)}.

Note that if K /∈ Gh then Ki either coincides with K or it is empty. On

each side of the fault, i.e. for i = 1, 2 we define the discrete spaces Wi,h,

Qi,has

Vi,h = {vh ∈ Hdiv(Ωi) : vh|Ki ∈ RT0(Ki) ∀K ∈ Th},