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Numerical Upcaling of Coupled Flow and Geomechanics in Highly Heterogeneous Porous Media Daegil Yang, Texas A&M University; George J. Moridis, Lawrence Berkeley National Laboratory; and Thomas A. Blasingame, Texas A&M University Copyright 2013, Unconventional Resources Technology Conference (URTeC)

This paper was prepared for presentation at the Unconventional Resources Technology Conference held in Denver, Colorado, USA, 12-14 August 2013.

The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s). The contents of this paper have not been reviewed by URTeC and URTeC does not warrant the accuracy, reliability, or timeliness of any information herein. All information is the responsibility of, and, is subject to corrections by the author(s). Any person or entity that relies on any information obtained from this paper does so at their own risk. The information herein does not necessarily reflect any position of URTeC. Any reproduction, distribution, or storage of any part of this paper without the written consent of URTeC is prohibited.

Abstract This paper shows that numerical upscaling of permeability and elastic stiffness tensors can be applied to a very heterogeneous and deformable reservoir system. Fluid flow in deformable porous medium is a multiphysics problem that considers flow physics and rock physics simultaneously. This problem is computationally demanding since we need to solve different types of governing equations such as the mass balance and the equilibrium equations. Numerical upscaling of the transport properties and the mechanical properties using flow and mechanics solvers will provide a coarse reservoir model that represents fine scale contribution of fluid flow and geomechanics. This would help us perform more efficient modeling and simulation of coupled flow and geomechanics in a petroleum reservoir.

Introduction In the reservoir simulation community researchers want to incorporate more realistic physics while modeling and simulating the reservoir performance. At the same time, they have high demands for very efficient computation.

Imagine that we obtained a detailed fine scale geologic description of a petroleum reservoir from geologists. Running flow simulation with this reservoir model is not practical due to an expensive computation cost. We can parallelize the reservoir simulator to achieve faster computation but the number of linearly independent equations and the number of memory to save during the simulation do not change. Furthermore, the number of nodes that can be used for parallel computation in a company or university is limited.

When we are dealing with a reservoir system that needs to have a more accurate estimation of geomechanical impact on the reservoir performance such as an unconsolidated reservoir, we need to couple a geomechanics simulator to the reservoir simulator. This procedure makes the computation more demanding.

To resolve these problems efficiently, we need to define different scales of the reservoir model (fine grid scale and coarse grid scale) and develop a method that effectively captures the fine scale effect on the coarse scale without directly computing all the small features. This process is called upscaling, which assigns equivalent properties to the coarse scale cells, which are determined by solving fine scale boundary value problems. Therefore, the upscaled model can represent the complex physics of the fine scale model using the coarse grid that contains the contribution of the fine scale physics.

Upscaling technique can reduce not only the size of the global matrix but also the number of solutions and parameters to save, allowing an efficient computation to be achieved. The purpose of the numerical simulation is to obtain approximate solutions of the partial differential equations that describe physical phenomena on discretized points, namely, mesh. The upscaling procedure can coarsen the mesh so the number of discrete points is less than the original problem. Therefore, the most important work is to assign the most accurate equivalent properties to each discrete point after coarsening.

Flow based numerical upscaling has been widely used since this can capture the complex flow physics using a pressure solver (Warren and Price 1961; Begg and Carter 1989; Durlofsky et al. 1991; King et al. 1995; King and

SPE 168873 / URTeC 1606914

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Mansfield 1999; Chen et al. 2003; Wen et al. 2006). The pressure solver solves for the governing equation that describes single phase flow, incompressible fluid and an incompressible medium. Darcy’s equation is either implicitly included in the governing equation or explicitly solved by using a mixed framework. By solving local boundary value problems, we can obtain the upscaled permeability or transmissibility using Darcy’s equation. Therefore, the upscaled permeability or transmissibility obtained from the pressure solver is dependent on the choice of boundary conditions.

For the solid mechanics problem, a mechanics solver has been used to obtain the upscaled mechanical properties of the solid materials (Guedes and Kikuchi 1990; Huet 1990; Ghosh et al. 1995; Smit et al. 1998; Kouznetsova et al. 2001; Miehe and Koch 2002; Zysset 2003, Wang 2006; Pahr and Zysset 2008). The mechanics solver solves the quasi-static equilibrium equation. Hooke’s law defines the stress and strain relation and the elastic stiffness tensor is the intrinsic material property to be upscaled. The upscaled elastic stiffness tensor is also dependent on the imposed boundary conditions of the mechanics solver.

Even though upscaling of the permeability and the elastic stiffness tensor is common in reservoir simulation community and solid mechanics community, the application of both methods for solving coupled flow and geomechanics problems has not been employed extensively. Chalon et al. (2004) showed a method of upscaling elastic stiffness tensor that can be used for a large-scale coupled flow and geomechanical simulation. Larsson et al. (2010) applied computational homogenization to model a 2D uncoupled consolidation of asphalt-concrete (as in asphalt-concrete pavement). A classical first-order homogenization was used to upscale the micro-scale heterogeneity of the porous medium on a representative volume element. Later, they extended their work to model a 2D fully coupled consolidation problem (Su et al. 2011) where they obtained more accurate numerical solution than the uncoupled approach. Zhang and Fu (2010) modeled a consolidation problem for a highly heterogeneous porous media with a fully coupled single-phase flow and geomechanics formulation. They used the flow based upscaling approach developed by Wen et al. (2003) and the mechanics upscaling proposed by Huet (1990). In their work, the heterogeneity of the porosity was not considered for the numerical experiments. In addition, they assumed the permeability is a constant value rather than a function that is dependent on pressure and displacement. Recently, Settari et al. (2013) presented a methodology to determine a dynamic equivalent stiffness tensor that can be applied to a heterogeneous compacting reservoir. The analytical upscaling method, based on uniaxial deformation, was used to determine the equivalent stiffness tensor.

In this work, we upscaled the permeability and elastic stiffness tensors using numerical upscaling techniques for the flow and the mechanics problems. Porosity upscaling is done by volume weighted averaging. We used the flow based upscaling method for upscaling the permeability. For the elastic stiffness tensor upscaling, we used the strain energy based homogenization method widely used in solid mechanics community. We used DEAL.II C++ Finite Element Library (Bangerth et al. 2007) for the future expansion of a large scale, dimension independent, and object-oriented code.

The flow solver for the flow based upscaling used a mixed finite element discretization (Chavent and Roberts 1991; Durlofsky 1994; Hoteit and Firoozabadi 2006a, 2006b) that satisfies local mass conservation of the mass balance equation and the mechanics solver used a continuous Galerkin finite element discretization. The upscaled permeability and elastic stiffness tensors will be used to run fully coupled flow and geomechanics simulations and the results will be compared with the fine scale solution. The fully implicit and fully coupled flow and geomechanics simulator with mixed formulation for the flow problem (Jha and Juans 2007) is capable of using both fine scale and coarse scale permeability and elastic stiffness tensors

Numerical experiments with highly heterogeneous model adapted from the SPE10 problem show that there is a good agreement between the coarse scale solution and the fine scale solution. This indicates that numerical upscaling of coupled flow and geomechanics provides good approximation of the numerical solution of highly heterogeneous fine scale models.

Methodology Numerical Upscaling. In flow problem, porosity and permeability are spatially different properties for a heterogeneous reservoir model. In mechanics problem, elastic stiffness tensors are spatially different properties.

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Since porosity is a simple volumetric ratio we can easily upscale it using a volume weighed averaging. However, since the permeability and elastic stiffness tensors are the components of the constitutive relations, which describe different physical problems, an accurate upscaling of these parameters is very important.

One of the phenomenologically derived constitutive equations for flow problem is Darcy’s law that is expressed as (ignoring gravity effect)

........................................................................................................................................... (1)

where , , , are a velocity vector, the second order permeability tensor, viscosity and pressure, respectively. The constitutive equation for the mechanics problem is so called Hooke’s law that is expressed as

.............................................................................................................................................. (2)

where , , are the second order stress tensor, the fourth order elastic stiffness tensor, and the second order strain tensor respectively. The purpose of upscaling a coupled flow and geomechanics problem is to determine an equivalent and that represent closely the fine scale physics on the upscaled domain. The flow solver provides the pressure and velocity solutions of the mass balance and Darcy’s equations. The mass balance equation for the incompressible flow and medium is expressed as

.............................................................................................................................................. (3)

where and indicates velocity and source and sink. In order to solve the Eq. 1 and Eq. 3 we used a mixed finite element formulation. Unlike the control volume finite element method, the mixed finite element method solves for both pressure solution and velocity solution individually and this provides more accurate approximation of fluid velocities (Durlofsky 1994). We used the lowest order Raviart Thomas space for the velocity solution and discontinuous Galerkin element for the pressure solution to overcome a possible saddle-point problem (Fortin and Brezzi 1991). In order to make a finite element formulation we can define spaces of solutions and test functions and as

{ ( ) } ...................................................................................... (4)

{ } ............................................................................................... (5)

............................................................................................................................................ (6)

Then the finite element formulation (weak formulation) is to find and such that

.......................................................................................................... (7)

( (

)

) ................................................................ (8)

where ∫

and ∫

. The velocity and pressure solution are approximated as

∑ .............................................................................................................................. (9)

∑

............................................................................................................................... (10)

where e indicates an edge of the element and i indicates the center of the element. The resulting linear system is

[

] [

] [ ] ...................................................................................................................... (11)

The global matrix is indefinite so we can introduce Schur complement (Diaz and Shenoi 1994) to solve the linear system.

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Following example shows the way to determine the upscaled full tensor permeability of heterogeneous media. The upscaled permeability tensor for a 2D domain has four components. Therefore, we need at least four independent equations to solve. When dealing with the mixed finite element formulation we need two boundary value problems for the mass balance and Darcy’s equations. In this work, the core-flood boundary condition is assumed. For a 2D heterogeneous domain of size Lx and Ly, the boundary condition is expressed as (x-direction flow)

......................................................................................................................................... (12)

........................................................................................................................................ (13)

( ) ............................................................................................................ (14)

where is a velocity vector and is an outward normal vector on the surface where it is located. Fig. 1 shows the size of the domain that we want to upscale and the x-direction core-flood boundary condition.

(a) (b)

Fig. 1—(a) A 2D domain for upscaling and (b) the x-direction core-flood boundary condition on the domain.

Likewise, the core-flood boundary condition of the other direction (y-direction flow) is determined by specifying the constant inlet and outlet pressures and no flow condition to the sides parallel to the flow direction. In order to use Darcy’s law on the coarse grid blocks, we need to obtain the volume weighted average of velocities and pressure gradients on the fine scale domain that we want to upscale as

⟨ ⟩

∫

................................................................................................................................. (15)

⟨ ⟩

∫

.......................................................................................................................... (16)

where i = 1, 2 indicates x- and y-direction core-floods respectively. For example, when the index i equals to one the x-direction core-flood boundary condition is imposed on the boundary. We can rewrite Darcy’s equation as four independent equations as (assuming that the viscosity is one)

⟨ ⟩ (

⟨

⟩

⟨

⟩ ) ....................................................................................................... (17)

⟨ ⟩ (

⟨

⟩

⟨

⟩ ) ....................................................................................................... (18)

⟨ ⟩ (

⟨

⟩

⟨

⟩ ) ....................................................................................................... (19)

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⟨ ⟩ (

⟨

⟩

⟨

⟩ ) ....................................................................................................... (20)

where ,

, , and

are the components of the upscaled permeability tensor in a coarse cell. We can manipulate the above equations to make linearly independent equations with respect to the permeability tensors that can be written as a matrix form as

[ ⟨

⟩ ⟨

⟩

⟨

⟩ ⟨

⟩

⟨

⟩ ⟨

⟩

⟨

⟩ ⟨

⟩

]

[

]

[ ⟨ ⟩

⟨ ⟩

⟨ ⟩

⟨ ⟩

]

..................................................................................... (21)

The above matrix and vector forms of linearly independent equations have an additional equation that makes the upscaled permeability as symmetric tensor. The added equation satisfies

. Eq. 21 can be solved using the

linear least square method as follow. First we define matrix , solution vector , and right hand side vector as

[ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩

]

................................................................................................ (22)

[

]

............................................................................................................................................ (23)

[ ⟨ ⟩

⟨ ⟩

⟨ ⟩

⟨ ⟩

]

........................................................................................................................................... (24)

Now we can approximate the solution vector as

................................................................................................................................................... (25)

where is the solution of a quadratic minimization problem that can be obtained as

................................................................................................................................... (26)

................................................................................................................................ (27)

The permeability tensor, which we can obtain from the computation, always satisfies symmetry. However it would not guarantee the positive definiteness. The resulted permeability tensors are mostly positive definite. However, if the upscaled permeability tensor is not positive definite then we a solve boundary value problem of the coarse grid that generate non-positive definite permeability tensor with different boundary conditions. Periodic boundary condition is a good choice since it always guarantees the positive definiteness.

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The mechanics solver provides the displacement solution that is calculated from the equilibrium equation written as

............................................................................................................................................. (28)

where is the total Cauchy stress tensor. Under the assumption of the isotropic material, the total Cauchy stress tensor can be expressed as

....................................................................................................................... (29)

where , and are the first Lamé’s constant, shear modulus (the second Lamé’s constant), the Biot coefficient, and the second order identity tensor . Note that there is no pressure term in the Cauchy total stress tensor since we only compute the deformation of solid material due to the imposed mechanical boundary conditions. Now, we define a space of solution and test function and as

.................................................................................................................................... (30)

We applied continuous Galerkin finite element discretization and this can be expressed as

( ) ................................................................ (31)

The displacement solution is approximated as

∑ ............................................................................................................................ (32)

where and are a shape function (or test function) and scalar coefficient at each degree of freedom. In addition is the total number of degrees of freedom of displacement solution, which is the number of nodes times the dimension ). The resulting linear system is

[ ][ ] [ ] .................................................................................................................................. (33)

In order to upscale heterogeneous elastic media we used the Hill condition (Hill 1963) that is the necessary and sufficient condition of the equivalence between the mechanically defined elastic material properties and the energetically defined effective properties written as

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ........................................................................................................................ (34)

where ⟨ ⟩

∫

and .

Eq. 34 indicates that the volume-weighted average of the double dot product of stress and strain in a coarse domain is equivalent to the double dot product of the volume-weighted stress and strain in the coarse domain. Strain energy is the elastic energy stored in the material under deformation and defined as

.................................................................................................................................... (35)

In order to obtain the upscaled elastic stiffness tensor from Eq. 35, we need to have 21 independent equations to solve. For the 2D domain we need to have 6 independent equations to solve. For the local upscaling problem, we can impose the prescribed displacement boundary conditions as

[⟨ ⟩ ] [

] [⟨ ⟩

] ................................................................. (36)

[ ⟨ ⟩ ] [

] [

⟨ ⟩

] ................................................................. (37)

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[ ⟨ ⟩ ] [

] [

⟨ ⟩

] ............................................................. (38)

where ,

, and are the equivalent strain energy in each coarse grid block with different boundary

conditions. Fig. 2 shows the displacement solution of an isotropic medium under the three different types of boundary conditions.

(a) (b) (c) Fig. 2—Displacement solution of a 2D isotropic medium under (a) x-direction tension, (b) y-direction tension, and (c) pure shear strain.

From Eq. 36, Eq. 37, and Eq. 38 we can obtain ,

, and . The other three components can be obtained

using ,

, and as

[⟨ ⟩ ⟨ ⟩ ] [

] [⟨ ⟩ ⟨ ⟩

] ........................................................... (39)

where

. Therefore, is calculated from

(⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ )

⟨ ⟩ ⟨ ⟩ .................................................................................. (40)

The equivalent strain energy under a combination of a pure shear strain and x-direction tension is calculated from

[⟨ ⟩ ⟨ ⟩ ] [

] [⟨ ⟩

⟨ ⟩

] ....................................................... (41)

where

. Therefore, is obtained as

(⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ )

⟨ ⟩ ⟨ ⟩ ................................................................................. (42)

Likewise, the equivalent strain energy under a combination of a pure shear strain and y-direction tension is calculated from

[ ⟨ ⟩ ⟨ ⟩ ] [

] [

⟨ ⟩ ⟨ ⟩

] ....................................................... (43)

where

. Therefore, is obtained as

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(⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ )

⟨ ⟩ ⟨ ⟩ ................................................................................. (44)

where the displacement field for Eq. 39 is obtained by adding the displacement field of Eq. 36 and Eq. 37. Using the displacement field for Eq. 39,

can be obtained from volume weighted averaging of the fine scale strain energy computed. Likewise,

and can be obtained by adding the displacement fields of Eq. 36 and Eq. 38, and Eq.

37 and Eq. 38, respectively. Note that the volume weighted average of the strains equals to the prescribed strains when we impose the displacement boundary condition (⟨ ⟩ | ).

Fully Implicit and Coupled Flow and Geomechanics Simulator. We present a finite element formulation to solve a coupled single-phase flow and geomechanics problem. We used a mixed finite element discretization to satisfy the local mass conservation. A standard Galerkin finite element discretization was used to solve the equilibrium equation. The fully coupled three equations (pressure, velocity, and displacement) were solved with the Newton-Raphson method.

Mass balance equation for a fluid flow in deformable porous media is

(

)

( ) ............................................................................... (45)

where , , , , , , and are the fluid density, porosity, the fluid compressibility, Biot’s coefficient, volumetric strain, and the solid grain stiffness, and source and sink term respectively. The Darcy’s equation is defined as

[ ] .......................................................................................................................... (46)

where , , , and are the Darcy velocity, permeability tensor, fluid viscosity and gravity vector. The governing equation that describes the poroelastic geomechanical deformation is

.................................................................... (47)

where and are the initial total stress tensor, the initial total pore pressure, the current total pore pressure, and the second order identity tensor, respectively. is the bulk density defined as

...................................................................................................................... (48)

where is the solid density. Governing equations are nonlinear since porosity is a function of pressure and displacement. In addition, we used the porosity dependent permeability defined as

( (

)) .............................................................................................................. (49)

where and are the reference permeability and porosity, respectively, and and are the current porosity and an experimentally determined constant (Moridis et al. 2008). Eq. 49 states that porosity is a function of pressure and volumetric strain. Therefore, permeability became a function of pressure and volumetric strain as well. Mass balance equation of gas flow in deformable porous media is

(

)

( ) ....................................................................... (50)

where is the gas density, (g/mol or kg/kmol) is the molar mass of the gas, is the z factor of the gas, (J/mol-Kelvin or J/kmol-Kelvin) is the gas constant and is the absolute temperature (Kelvin), is a nonlinear function of the z-factor and pressure.

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In order to make the finite element formulation, we can define the spaces of solutions ( ) and test functions ( ) as

{ ( ) } .................................................................................... (51)

{ } ............................................................................................. (52)

.......................................................................................................................................... (53)

.................................................................................................................................... (54)

Now, we can multiply the test functions to the governing equations and integrate the resulting functions over the domain which yields the following weak forms

( (

)

) (

) ..................................... (55)

( (

)

)

(

) ............................. (56)

( ) ................................................ (57)

( ) ( )

................................................................................................................... (58)

Then, the ail of the finite element formulation (weak formulation) is to find , , and , such that

( (

)

) (

)

.. (59)

( )

.................... (60)

( )

(

)

............................................................................... (61)

where the subscript i indicates the degree of freedom. To compute the finite element formulation we loop over all the degrees of freedom and compute the given governing equations. Therefore, the residual formulation has only the index i since the residual equations become the right hand side (vector) of the linearized equations. The solutions are approximated as

∑

............................................................................................................... (62)

∑

............................................................................................................... (63)

∑

............................................................................................................... (64)

where

are the unknown expansion coefficients that we need to determine (the degrees of freedom

of this problem), and are the finite element shape functions that we will use. In addition, e, a,

and i indicate an edge, a node and the center of an element, respectively.

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In order to deal with the nonlinear governing equations we used the Newton-Raphson method. We made derivatives of each residual equation with respect to the solution variables, which resulted in the Jacobian matrix. Then, the fully coupled linear system can be written as

[

]

[

]

[

]

................................................................................ (65)

The primary variables in each time are improved by

.............................................................................................................. (66)

.............................................................................................................. (67)

............................................................................................................. (68)

Numerical Experiments We conducted three numerical experiments to compare the numerical solutions of the fine scale and coarse scale models. The fine scale 2D model has 4096 cells and each cell has nodal vector solution for the displacement, pressure solution at the cell center, and the velocity vector on the center of each face. The fine scale model is upscaled with coarse cells so the resulting coarse model has 256 cells.

The first numerical experiment has a sink at the corner of the model that depressurizes the reservoir. The second experiment is a consolidation problem that has a drainage boundary condition (constant pressure boundary condition) at the left and the right side boundaries. The flow properties (permeability and porosity) were adapted from the SPE10 problem.

Fig. 3 shows the porosity (a) and permeability (b) fields adapted from the SPE10 problem. The figure shows that the permeability field has a channelized barrier in the middle of the domain which makes it difficult for fluid to move from the upper area to the lower area or vice versa. Since the SPE10 problem does not have data for the values of the elastic stiffness tensors we made a relationship between initial porosity and elastic stiffness tensor. The relation is expressed as

(

)

.................................................................................................................................. (69)

where is elastic stiffness tensor for the computation that is a function of porosity, is reference elastic stiffness tensor and n is a constant. We assumed that each fine scale cell has isotropic elastic stiffness tensor. Fig. 4 shows the Lamé’s first constant and shear modulus fields when n=1.5. Note that the values of the Lame’s first constant and shear modulus vary up to 1000 times.

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(a) (b)

Fig. 3—(a) Porosity field and (b) x-direction permeability field. We assumed that y-direction permeability field is same as x-direction permeability (isotropic). The permeability values are up to 10,000 times different. Note that the permeability field has logarithmic distribution and the unit for the permeability is millidarcy (md).

(a) (b)

Fig. 4—(a) The Lame’s first constant field and (b) the shear modulus field when the constant n=1.5. The values in each field vary up to 1000 times. The unit of the Lame’s first constant and shear modulus is pascal (Pa).

Production from a Sink. In this problem, we imposed a constant rate constraint at the lower-left corner of the domain (Fig. 5). For the fine scale model we used the permeability and porosity fields shown in Fig. 3. For the fine scale mechanical properties, we used the Lamé’s first constant and the shear modulus fields shown in Fig. 4. We obtained upscaled permeability tensors from the pressure solver and elastic stiffness tensors from the equilibrium equation (mechanics) solver. The fluid viscosity is 1 cp and compressibility is . An initialization was done before running the production simulation. The sink at the lower left corner of the domain has a rate of . The CPU time to run 60 days of simulation is 345.59 seconds for the fine scale model and 19.137 seconds for the coarse scale model. So the computation of the coarse scale model is about 18 times faster than the fine scale model.

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Fig. 5—A 2D reservoir domain for the coupled and uncoupled simulations. A no-flow boundary condition for the flow equation and a mixed boundary condition (traction and prescribed displacement) for the geomechanics equation are applied on the boundaries of the domain. The sink is located at bottom left corner and the observation point is located at the center of the domain.

Fig. 6 shows the pressure solution of the fine and coarse scale models at the observation point. To compare the pressure solutions of the fine and coarse scale models the fine scale pressure values at the observation point were upscaled. So, we can compare the pressure solutions of the two representative elements (fine scale and coarse scale) that have identical dimensions. Fig. 6 (a) indicates that the fine scale pressure solutions are in very good agreement with the coarse scale solutions. At the early stage of simulation (Fig. 6 (b)), both simulations arrived at a pressure that was higher than the initial pressure.

(a) (b)

Fig. 6—Reservoir pore pressure comparison between the fine scale and the coarse scale model at the observation point. (a) The 60 days of pressure profile indicates that the pressure solution of the coarse model matches well with the fine scale model. (b) At the early stage of the production, pore pressure of the coarse model became slightly higher than the fine scale model.

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Fig. 7 compares the local pressure solutions of the fine scale and the coarse scale models after 4.75 hours of production. It clearly shows that there is a good agreement between the fine scale and the upscaled coarse scale pressure solutions. Both models capture a little amount of pressure rise (the maximum pressure is larger than the initial pressure) and the pressure support in the reservoir due to the mechanical loading.

(a) (b)

Fig. 7—Pressure (Pa) distribution of (a) the fine scale model and (b) the coarse scale model after 4.6 hours of production. It shows that there is a good agreement between the fine scale solution and the coarse scale solution. Two models also capture a slight pore pressure increase due to mechanical loading. The pore pressure is supported by given traction.

Fig. 8 and Fig. 9 show x- and y-direction displacements respectively after 4.75 hours of production. The displacement solutions show that the magnitude of the displacement solutions is very high where the sink is located. This indicates that the largest displacement occurs where the pressure gradient is largest, and is caused by poroelasticity. Negative x- and y-displacements occur near the well, which indicates compression due to the pressure drop near the well.

(a) (b)

Fig. 8—X-direction displacement solution of (a) the fine scale model and (b) the coarse scale model after 4.6 hours of production. It shows that there is a good agreement between the fine scale and the coarse scale solutions. The magnitude of the x-direction displacement solution is the highest near the sink. This indicates that there is compression taken place near the sink. The unit of the displacement solution is meter (m).

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(a) (b)

Fig. 9—Y-direction displacement solution of (a) the fine scale model and (b) the coarse scale model after 4.6 hours of production. It shows that there is a good agreement between the fine scale and the coarse scale solutions. Like the x-direction displacement the magnitude of the y-direction displacement solution is the highest near the sink. This indicates that there is compression taken place by lowering the pore pressure near the sink. The unit of the displacement solution is meter (m).

Fig. 10 shows the pressure solutions after 59 days of production. The pressure solution indicates that the fluid in the lower half region was mainly depleted. This was because there was a very low permeability channel in the middle of the reservoir that made it difficult for the upper fluid to reach the sink. Therefore, the pressure at the upper half of the reservoir pressure remained relatively higher compared to the lower part. The fine scale pressure solution (a) and the coarse scale pressure solution (b) matched well.

(a) (b)

Fig. 10—Pressure (Pa) distribution of (a) the fine scale model and (b) the coarse scale model indicates the channelized low permeability zone in the middle of the domain acts as a barrier that makes the fluid in the upper half region difficult to reach to the sink. The fine scale pressure solution and the coarse scale pressure solution match well.

Fig. 11 and Fig. 12 show x- and y-direction displacements, respectively, after 59 days of production. The displacement solutions clearly show the compaction due to the mechanical loading. The compaction was the highest where the mechanical loading took place. Even though the pressure at the lower region is substantially less then the upper region, there is no significant difference in the displacements at two regions. It is because the elastic stiffness tensors of the lower region are relatively higher than the upper region. The higher elastic tensors indicate that the

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rock is less deformable. The x-displacement map in Fig. 11 shows the regions that have higher elastic stiffness tensors are less deformable (shown as green color) than the other regions. The upscaled x- and y-direction displacement solutions match well with the fine scale solutions.

(a) (b)

Fig. 11—(a) The fine scale x-direction displacement and (b) the coarse scale x-direction displacement. There is a good agreement between the fine scale and the coarse scale solutions. The unit of the displacement solution is meter (m).

(a) (b)

Fig. 12—(a) The fine scale y-direction displacement and (b) the coarse scale y-direction displacement. There is a good agreement between the fine scale and the coarse scale solutions. The unit of the displacement solution is meter (m).

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Consolidation Problem. In this work, we investigated the numerical solution of a consolidation problem. Boundary conditions of the consolidation problem are described in Fig. 13. The size of the reservoir domain is 100 m by 100 m. The domain has a constant pressure boundary condition on each side (the left end and the right end) and the top and bottom boundaries have no flow boundary conditions. For the geomechanics problem, it has a fixed roller boundary condition on the left, right, and bottom boundaries. The top boundary has a traction imposing the overburden stress. Due to the mechanical loading on the top, the system will subside and the fluid in the reservoir will be drained through each constant pressure boundary (left and right). We ignored the gravity term to investigate the strong impact of heterogeneity. The fluid viscosity is 1 cp and compressibility is . The initial reservoir pressure is 0.1 MPa. The mechanical loading imposed by the traction boundary condition instantaneously increased the reservoir pressure. Then the pressure continuously decreased due to the drainage boundary condition on the left and right boundaries of the domain. Observation point is located near the center of the domain. The CPU time for 7.6 days of simulation is 198.81 seconds for the fine scale model and 6.87 seconds for the coarse model. The computation with the coarse scale model is about 29 times faster than the fine scale model.

Fig. 13—Boundary condition for the flow and geomechanics of the consolidation problem. The flow problem has no flow boundary at the top and bottom. The left and the right boundaries have constant pressure boundary condition. The mechanics problem has fixed roller boundary conditions at the left, right, and bottom. It has specified traction boundary condition at the top.

Fig. 14 compares the pressure solutions of the fine and the coarse scale models at the observation point. The comparison for 6.5 days of simulations indicates (Fig. 14(a)) that the fluid drained through the drainage boundaries at which we imposed the constant pressure boundary condition. Early in the simulation (Fig. 14(b)), the pressure rose instantly from 1.0 MPa to 4.5 Mpa due to the overburden traction on the top. Then the pore pressure began to decrease, but increased again after several hours. This was because the effective stresses near the drainage boundaries increased which resulted in higher compression. In addition, the low permeability in the middle of the domain made it difficult for the fluid to flow to the side boundaries (it was difficult for the fluid to drain) causing the pore pressure began to increase (the mechanical response is faster the pressure propagation). The coarse scale pressure at the observation point matched very well with the pressure of the coarse scale model. Fig. 15 compares the local pressure solution of the fine scale and the coarse scale models about 6.2 hours after the loading. It clearly shows that the pressure at the middle of the domain remained the highest since the permeability at this region was the lowest.

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(a) (b)

Fig. 14—(a) The reservoir pressure of the fine scale and coarse scale models at the observation point with time. (b) At the early time of the simulation the pressure rose instantly from 1.0 MPa to 4.5 Mpa due to the traction on the top.

(a) (b)

Fig. 15—Pressure (Pa) solution of (a) the fine scale and (b) the coarse scale models about 6.2 hours after the mechanical loading. The highest pore pressure occurs where the permeability is the lowest. It is because the low permeability makes the fluid difficult to reach the drainage boundary so the pore pressure increases due to the mechanical loading. The coarse scale solution matches well with the fine scale solution.

Fig. 16 and Fig. 17 show the x- and y-direction displacement solutions. The x-direction displacement solution indicates that the magnitude of the displacement is large where the pressure gradient is large (Fig. 16). Since this was a consolation problem, the largest pressure gradient occurred along the x-direction of the top and bottom regions (Fig. 15). This was because the top and bottom regions have relatively higher permeability and lower values of the mechanical properties compare to the middle region. In the case of the y-direction displacement (Fig. 17) it clearly shows the consolidation (compression caused by the imposed traction). The coarse scale displacement solution matched well with the fine scale displacement solution.

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(a) (b)

Fig. 16—X-direction displacement solution of (a) the fine scale and (b) the coarse scale models about 6.2 hours after the mechanical loading. The magnitude of the displacement is large where the pressure gradient is large. The coarse scale solution has a good agreement with the fine scale solution. The unit of the displacement solution is meter (m).

(a) (b)

Fig. 17— Y-direction displacement solution of (a) the fine scale and (b) the coarse scale models about 6.2 hours after the mechanical loading. The solution clearly shows the consolidation (compression caused by the imposed traction) of the porous medium. The unit of the displacement solution is meter (m).

Production from a Tight Gas Reservoir. We applied numerical upscaling of the permeability and elastic stress tensors to a tight gas reservoir modeling. In order to produce gas from a tight gas reservoir, we need to induce fractures in the formation, which results in strong heterogeneity in the reservoir. In this study, we used the same heterogeneity pattern used in previous experiments. In order to make permeability and porosity fields of the tight gas reservoir, we simply decreased the order of porosity and permeability. For the porosity field in the tight gas reservoir we used half of the porosity in the previous experiment (production from a sink). For the permeability field, we multiplied to lower the order of permeability. Fig 18 shows the porosity and permeability fields. The values of porosity ranged from 0.05 to 0.2 and permeability values ranged from 130 nanodarcy to 1017 microdarcy. Like previous experiments, we used Eq. 69 to generate the fine scale mechanical property field. Therefore, higher mechanical properties were obtained from the lower porosity values, which is indicative of the high strength of rock.

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We performed updcaling to construct a coarse scale model and used methane properties and the Peng-Robinson equation of state to model gas flow. We imposed a constant rate constraint ( ) at the lower-left corner of the domain (Fig. 5). Boundary conditions are same as in the previous simulation (Fig. 5). The initial reservoir pressure is the same as the traction imposed on the boundaries (5.0MPa). The viscosity of methane is . We assumed the simulation model is under isothermal conditions with a reservoir temperature of 30 oC. The CPU time for 3.2 years of computation with a maximum time step size of 12 days is 1324.4 seconds for the fine scale model and 91.7 seconds for the coarse model. The computation with the coarse scale model is about 14.4 times faster than the fine scale model.

(a) (b)

Fig. 18—(a) Porosity field and (b) x-direction permeability field. We assumed that y-direction permeability field is same as x-direction permeability (isotropic). The permeability values are up to 10,000 times different. Note that the permeability field has logarithmic distribution and the unit for the permeability is microdarcy

(d).

Fig. 19 shows the reservoir pressure of the fine scale and coarse scale models at the observation point with time. During the 3.2 years of simulation the fine and coarse scale pressures at observation point matched very well (Fig.

19(a)). In the beginning of the simulation, both the fine and coarse scale models captured the pressure increase due to mechanical loading (Fig. 19(b)).

Fig. 20 shows the pressure distributions of the fine scale and coarse scale models after 146 days of production. Pressure drops occurred near the production well and the high permeability region. The low permeability zone in the middle of the domain makes the gas difficult to move toward the production well. The coarse scale solution matches well with the fine scale solution. The figure clearly shows that there is a region where its pressure is even higher than the initial reservoir pressure.

Fig. 21 shows the x-direction displacement solution after 146 days of production. The region near the production well clearly shows the compaction and both the fine and coarse scale models capture the behavior near the well. The x-direction displacement solution at the lower left corner of the domain shows relatively high compaction. This is because the region has relatively high flow properties (permeability and porosity) and low mechanical properties, which made the rock deform easily under given traction.

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(a) (b)

Fig. 19—The reservoir pressure of (a) the fine scale and (b) the coarse scale models at the observation point with time. Coarse scale pressure solution matches well with the fine scale solution. At the early time of the simulation the pressure solutions of both models became higher than the initial pressure.

(a) (b)

Fig. 20—Pressure (Pa) distribution of (a) the fine scale model and (b) the coarse scale model after 146 days of production. It shows that there is a good agreement between the fine scale solution and the coarse scale solution. Two models also capture a slight pore pressure increase due to mechanical loading. The pore pressure is supported by given traction.

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(a) (b)

Fig. 21—X-direction displacement solution of (a) the fine scale model and (b) the coarse scale model after 146 days of production. It shows that there is a good agreement between the fine scale and the coarse scale solutions. The magnitude of the x-direction displacement solution is the highest near the sink. This indicates that there is compression taken place near the sink. The unit of the displacement solution is meter (m).

Fig. 22 shows the y-direction displacement solution after 146 days of simulation. Like the x-direction displacement solution, there is a good agreement between the coarse and fine scale solutions. The magnitude of compaction along y-direction is higher than along the x-direction.

(a) (b)

Fig. 22—Y-direction displacement solution of (a) the fine scale model and (b) the coarse scale model after 146 days of production. It shows that there is a good agreement between the fine scale and the coarse scale solutions. Like the x-direction displacement the magnitude of the y-direction displacement solution is the highest near the sink. This indicates that there is compression taken place by lowering the pore pressure near the sink. The unit of the displacement solution is meter (m).

After 3.2 years of production, we observed that the pressure at the lower half region of the domain significantly reduced (Fig. 23). However, the pressure at the upper half region remains almost same. This is because of the gas is highly compressible which maintains the pore pressure in that region. Porosity reduction due to the pressure drop is approximately from 1.4% to 1.5%.

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(a) (b)

Fig. 23—Pressure (Pa) distribution of (a) the fine scale model and (b) the coarse scale model after 3.2 years of production.

Fig. 24 and Fig. 25 show x- and y-direction displacement solutions after 3.2 years of production. The x-direction displacement solution shows that compaction was affected by the sideburden and the magnitude of the compaction was the highest at the lower right corner of the domain. The y-direction displacement solution indicates that the domain was consolidated along the y-direction since the pressure of the lower half of the domain decreased considerably and the pressure differential along the x-direction became very small. Coarse scale displacement solutions match well with the fine scale solutions.

(a) (b)

Fig. 24—(a) The fine scale x-direction displacement and (b) the coarse scale x-direction displacement after 3.2 years of production. There is a good agreement between the fine scale and the coarse scale solutions. The unit of the displacement solution is meter (m).

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(a) (b)

Fig. 25—(a) The fine scale y-direction displacement solution and (b) the coarse scale y-direction displacement solution after 3.2 years of production. There is a good agreement between the fine scale and the coarse scale solutions. The unit of the displacement solution is meter (m).

Conclusions We conducted a numerical upscaling of coupled flow and geomechanics problem for a highly heterogeneous system. We used a flow solver, which has a mixed finite element discretization, to solve for both pressure and velocity, to conduct the flow based upscaling. We used a mechanics solver that has a continuous Galerkin discretization to solve for displacement. We conducted upscaling that reduces 4096 cells to 256 cells. The upscaled permeability tensor and elastic stiffness tensors were used for fully coupled flow and geomechanics simulations. In addition, fine scale directional permeability tensor and isotropic elastic stiffness tensor were used for the fully coupled flow and geomechanics simulations that provide the fine scale reference solutions.

We conducted three numerical experiments to compare the fine scale solution and the coarse scale solution. Three numerical experiments clearly showed that the numerical simulation with the coarse scale model captured the important multiphysics of the fine scale model. It also indicates that the heterogeneity is a critical factor that affects both the numerical solutions of flow and the geomechanics problems.

Comparison of the upscaled solution with the fine scale solution indicates that the upscaled solution matches well with the fine scale solution with very favorable computational efficiency. This result implies that numerical upscaling can be applied to a very heterogeneous reservoir system that we want to simulate a coupled flow and geomechanics problem. By performing upscaling we can obtain a more efficient computation for expensive multiphysics simulations.

Nomenclature = velocity = second order absolute permeability tensor = viscosity of fluid = gravity vector = second order stress tensor = The fourth order elastic moduli tensor = second order strain tensor = double dot product = dot product = source and sink = outward normal vector [ ] = right hand side of the linear system = transpose of the matrix A

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= The coarse scale 4th order elastic moduli tensor using index notation.

= stress tensor using index notation = strain tensor using index notation = fluid compressibility = Biot’s coefficient = volumetric strain = solid grain stiffness = density of the fluid = density of the fluid and solid mixture = Lame’s second constant = shear modulus = second order identity tensor = traction vector = ∫

⟨ ⟩ = volume weighted averaging on the domain = residual of the pressure equation = residual of the velocity equation = residual of the displacement equation = time step = number of time step number = number of Newtop –Raphson iteration = increment (i.e. : pressure increment) = reference initial elastic moduli = divergence operator = gradient operator = for all or for any

Subscripts

= domain = boundary

Superscripts

= dimension of the domain e = edge of the element i = node of the element = coarse scale

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