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1. INTRODUCTION

The storage of CO2 in geological formations is currently

intensively discussed in the world of politics and scienceas an economical and environmentally sound way toreduce the emissions of greenhouse gases into the

atmosphere. Carbon capture and storage (CCS)technology aims at separating carbon dioxide from theflue gases of power plants (fossil-fuelled or other CO2emitting ones) with subsequent transport to a site where

it can be injected for storage into a deep geologicalformation, such as depleted oil and gas reservoirs,unminable coal seams, and deep saline aquifers [1,2].Suitable formations should be deeper than 800 m, have athick and extensive seal, have sufficient porosity for

large volumes, and be sufficiently permeable to permitinjection at high flow rates without requiring overly high

pressure [3]. Fig-1 shows a schematic of a suitable target

formation and also the primary migration of CO2.

Sequestration capacity estimates for saline aquifers andcoal beds are highly uncertain, although in the pastseveral years, there has been some progress in

developing standard methods for capacity estimation andimproving regional estimates [4].

ARMA 13- 631

Inclusion of Geomechanics in Streamline Simulation for

Hydromechanical Modeling of Underground CO2 Storage

Koohmareh Hosseini, B.1

Chalaturnyk, R.J.

1

, and Darcis, M.

2

University of Alberta, Department of Civil & Environmental Engineering, Edmonton, Alberta, Canada 1

University of Stuttgart-Department of Hydromechanics and Hydrosystem Modeling, Stuttgart, BW, Germany 2

Copyright 2013 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 47th

US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, USA, 23-26June 2013.

This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review ofthe paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, ormembers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMAis prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. Theabstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: Realistic hydromechanical simulation of carbon dioxide storage into deep saline aquifers is computationallyintensive and thereby time consuming. Large field heterogeneous models of stress-sensitive reservoirs with complex flow and

geomechanical behavior are sometimes required to be modeled, which is very demanding. Therefore, most recent studies on

simulation of CO2 sequestration in saline aquifers have been focused either on short-term migration or near field processes or large

scale models with highly idealized geometries. This paper presents a coupled geomechanics-streamline simulation technique for

rapid hydromechanical simulation of large heterogeneous reservoirs with elastic geomechanical constitutive relations. Streamline

trajectories represent a three-dimensional velocity field during injection of CO2 in porous medium, and therefore are helpful for

model order reduction and inclusion of geomechanics in sub domains where streamlines density is relatively high. To assess the

robustness and speed of the technique, a large reservoir-cap rock system with large number of grid-blocks was made. Accuracy and

speed of streamline-based method were compared to finite volume based flow geomechanical simulations for the same model with

the same geometry. Porosity and its relationship to absolute permeability were the primary geomechanically influenced variables

studied in the simulations. The effective stress principle was applied to characterize the stress state and governing geomechanical

differential equations were implemented based on mass and momentum conservation laws on a C++ platform with the Box Method

(subdomain collocation method) as the discretization technique. The streamline tool used was 3DSL which is developed on a

FORTRAN platform. Simulation results showed that streamlines can be helpful for fluid flow simulation or hydromechanical

coupling particularly during the injection process when the fluid flow mechanism is advection-dominated. The method is

demonstrated to increase model efficiency and reduce computational cost, particularly for heterogeneous reservoirs. The inclusion

of geomechanics in streamline simulation thus represents a key step in the direction of quantifying uncertainty in CO 2 storage

process for large scale, heterogeneous systems.


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Figure 1- Schematic of a reservoir target and CO2plume

migration in a host geological formation (Zhdanov et al.,2013 [4])

Reservoir processes during injection and post-injectionof CO2 involve several trapping mechanisms. Fig-2illustrates different mechanisms related to CO2 trappingmechanisms. Extensive discussion on physical and

chemical mechanisms related to CO2 storage insedimentary formations can be found in the works ofGunter et al. 2004. [5], Benson and Cole 2008. [3].

Figure 2- Conceptual sketch of sequential physical

mechanisms of CO2 storage (Zhdanov et al., 2013 [4])

The trapping mechanisms can be divided into two main

categories [6]: (1) physical trapping (2) chemical gastrapping. Physical trapping encompasses stratigraphicand residual trapping, which are the first two sequences

of trapping shown in Fig-2. Chemical trapping occurswhen CO2 dissolves in subsurface fluids (solubility) and

subsequent reactions with the mineral phases in the

reservoir (mineral trapping). The permanence andduration of these trapping mechanisms depends highly

on the long-term geological integrity of the targetunderground reservoir and the seal. Geomechanical

processes during CO2 injection (and potentially in thepost-injection period) can play an important role(particularly during stratigraphical CO2 trapping) on the

integrity of the storage formation and bounding seals.Geomechanics is likely of greatest importance during

physical trapping stages when CO2 is being injected into

the formation and the fluid flow mechanisms areadvection-dominated.

Hydromechanical (HM) processes generally play a

significant role during CO2 injection into deep salineaquifers. These saline aquifers are generally sandstoneformations that are initially saturated with saline fluid.Therefore, they can deform either due to variation ofexternal loads (dissolution or precipitation of mineral

phases) or changes in pore pressures or temperature inthe reservoir. The main topic of this work relates tostress-strain field redistribution and domain deformationdue to pore pressure changes both in the near wellboreregion and in the far field. Direct and indirect

hydromechanical coupling mechanisms can explainthese sorts of geomechanical changes, as suggested by

Rutqvist and Stephansson, 2003 [7]. Direct HM couplingrefers to both solid-to-fluid coupling and fluid-to-solidcoupling. The former refers to the phenomenon wherevariation in the applied load results in a change in

porosity and accordingly in fluid pressure and saturation;

and the latter takes place when a change in fluid pressurecauses a variation in the volume of the geological media(due to inducement of volumetric strain). Indirect HMcoupling however refers to changes in hydraulic ormechanical properties in response to strain changes.

According to this definition, the focus of this study is ondirect HM coupling.

Even though geomechanics play an important role inCO2 storage in saline aquifers, the computational burden

is much higher using hydromechanical coupling than forthe hydraulic problem alone [8], particularly when the

hydrodynamic and mechanics of the process are solvedsimultaneously (known as monolithic fully coupledHM). Although, there are some coupling strategies

available to avoid the full coupling, such as sequential oriterative coupling, these strategies are not still applicablein large domains with multi-million cells due to

computational burden. For more detailed discussions onthese techniques and discussions over computational

efficiencies refer to Mainguyand and Longuemare, 2002[9] , Settari and Walters, 1999 [10], and for frameworks

for coupling a flow and geomechanical code and inducedgeomechanical changes in reservoir properties, refer to

Chalaturnyk [11] and Li [12].

In recent years, a number of coupled fluid flow and

geomechanical numerical models have been developedfor analysis of various geomechanical issues associatedwith geological storage of CO2 (GCS). Examples of thesequentially coupled HM analysis of GCS includeFEMH by Bower and Zyvoloski 1997. [13], TOUGH-

FLAC by Rutqvist et al., [14] , OpenGeoSys by Wangand Kolditz [15] and also Goerke et al., ECLIPSE-VISAGE by Ouelletet al.[16] . However these schemes


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are not computationally efficient due to large number ofcoupling variables as well as computational loads, andcomplexity of multi-physics associated withconventional reservoir simulator. Simplified models

have been shown to be sufficient for long term or large-scale processes (i.e. single phase flow coupledgeomechanics for fault reactivation analysis in a largedomain).

This paper introduces a new HM coupling strategy forfaster coupled simulations and reduction ofcomputational loads for large underground reservoirs.This paper first studies the feasibility of coupling

between a newly developed geomechanical code and

streamline simulation, and later applies the developedscheme in the hydromechanical modeling of CO2 storage

in a large synthetic underground model.

2. STREAMLINE SIMULAITON

Streamlines are approximate methods for fluid flow in

porous medium. In case of an isotropic permeabilityfield, streamlines are orthogonal lines at each point to

the potential lines, and locally tangent to a definedvelocity field, and they represent the direction of fluidflow. The elegance of streamline simulation lies in their

power for fast forward flow simulations. They are themost accurate when the flow mechanism behavior is

closest to unit mobility ratio (e.g. tracer flow). Theefficiency of streamline simulation relies mainly on their

power in utilizing larger time-steps with fewer pressure

updates, which leads to faster flow simulations. Inscenarios where the geomechanical model changes (in

MEM or petrophysical properties) and streamlineschange and the process requires an iterative approach,streamline simulations scale well and are of significant

advantage over finite difference (FD) or finite volume(FV) simulation techniques. The other advantage ofstreamline simulations is their visualization power (i.e.flow directionality and streamline density from sourcesto a sink as a flow indicator). Also they mitigate the

numerical artifacts (numerical diffusion) due to theirdual-grid nature (streamline grid and original Cartesian3D grid). The beauty of streamline simulation lies in thetime-of-flight formulation that helps to decouple the

multidimensional transport equation to a series of 1Dsolution along streamlines. Particularly for highlyheterogeneous porous medium or large domains (i.e. one

million cells) streamlines are significantly fastersimulator tools compared to conventional flow

simulators in terms of CPU load and computationalefficiencies. Batycky (1997) and Batycky et al. (1997),was the first who developed a 3D two-phase flow

simulator for field scale scenarios such as heterogeneity,rearranging well conditions, and gravity [17]. A large

literature describes the development and application of

streamline simulation to prediction of flow in three-dimensional heterogeneous reservoirs. See the papers ofBatycky, Thiel, King and Datta-Gupta (1998) and Craneet al. (2000) for many references to the full range of

work on streamlines.

Streamlines also have been used as proxies for reservoir

model order reductions. For more details you can refer to

the work of Kovscek and Wang [18] and Jesmani,Koohmareh Hosseini, and Chalaturnyk [19]. Streamlinesimulation technology and its application in undergroundCO2 storage has also been extensively discussed by

Blunt, and Ran Qi [20].

3. MODEL GEOMETRY AND BOUNDARYCONDITIONS

For model geometry selection as well as petrophysicalmodeling, standard concepts that have been suggested

for suitable formations as target reservoirs for CO2storage have been used. Benson and Cole [3] suggestedan injection depth of at least 800 m, and also a thick and

extensive seal that has sufficient porosity (capacity) andpermeability (infectivity) to permit injection at high

rates.

Most geomechanical analyses of CO2 storage considerthe reservoir seals as impermeable at the top of theformation to be able to trap CO2 in saline underground

aquifers such as Sleipner (Korobol and Kaddour, 1995).

The sealed formation scenario has also been consideredfor scenarios related to oilfields such as Weyburn,

Canada (Chalaturnyk [21]; White [22]). In a similarfashion, the model shown in Fig-3 considers no-flow

boundaries at the top and bottom (i.e., Neumannboundary conditions). The lateral boundaries howeverare considered to be aquifers with varying heads based

on the depth from the ground surface. Both aquifers onthe right and left have the same conditions (i.e., Dirichlet

boundary condition). The geomechanical (deformation)boundaries were considered as roller boundary on thelateral sides and the bottom side of the reservoir was

considered as no-displacement boundary conditionwhere displacement in all three directions is set to zero.

And the overburden was replaced with a constantvertical stress at the top of the injection formation. The

injection well was in the middle of the model and theinjection depth was chosen to be larger than 1000 m.This injection depth is important and necessary in order

to have sufficiently high pressures at these depths. Atthese depths (>800 m) CO2 is supercritical and its

density is high enough to allow efficient pore filling and


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also no significant buoyancy difference is observedcompared with in situ fluids. [23]

Figure 3- Schematic description of the model geometry and

flow and geomechanical boundary conditions

4. GEOMECHANICS

Poroelasticity concepts proposed by Biot [24] are

usually adopted to resolve the (geo) mechanicalproblem. In this initial phase of the research exploringthe inclusion of geomechanics in streamline simulation,the theory of poroelasticity has been adopted. Therefore,the geomechanical code was developed based on the

assumption of existence of isotropic linear elasticmaterial. The code is based on a linearized approach formomentum balance equation with the final form of themomentum balance equation for calculation of linear

elastic deformations during CO2 injection into salineaquifers as:

0.)(.).( ' gSIPdiv wnneff (1)The assumption behind the formulation is that Sw is

initially equal to one, which is the case for all CO2injection scenarios in saline aquifers, constant solid

density and small changes in porosity. As shown in Eqn-1, when gravity is considered in the calculations, notonly saturation and pressure from flow simulation has to

be fed into geomechanical simulator, but also porosity(from previous cycle for sequential coupling) as well as

phase densities, need to be inputted to thegeomechanical simulator.

There are several issues to be addressed whengeomechanics is coupled (directly or indirectly) to flow

simulation, for geological storage mechanisms such asfault reactivation, limits on injection pressure, induced

seismicity, injection induced strains, ground surfacedisplacement, or caprock integrity. The geomechanicalfocus of this study is on induced strain and stress field

redistribution which result in changes in petro-physical

properties, and consequently on subsurface fluidmigration.

5. FLUID FLOW MECHANISM

The aquifer was considered to be initially fully saturated

with saline water. The injected phase was CO2. The fluidsystem was therefore a two-phase (brine-CO2) systemwith no component exchange. The process was assumed

to be isothermal; therefore no thermal strain inducementis assumed due to cooling of the formation, although it is

straightforward to include this in the analyses. Densitywas assumed to change with compressibility and pore

pressures in the domain. The fluid flow mechanism was

performed by two approaches: Box method (incorporatesboth finite element and finite volume) and streamline

method, both on a two-dimensional and three-dimensional scale. For calculation of capillary pressure

and saturation, Van Genuchten functions were selected,which were developed on the basis of a bundle ofcapillary tubes model [25].

Under conditions where the fluid phases are immiscible,

the pressure needed to inject CO2, the rate at which theleading edge of the CO2 plume moves and the fraction ofthe pore space filled with CO2 are all governed bymultiphase flow relationships as suggested by Bear [26].

6. DISCRETIZATION SCHEME

The fully coupled flow-geomechanics code is a moduleof the open source code simulator Dumux (Flemisch etal., 2011) which is based on the Distributed and Unified

Numeric Environment DUNE (Bastian et al.). Ageomechanical code was developed based on the fullycoupled scheme, but explicit from fluid flow module.The discretization scheme used was the Box-Method(collocated finite volume and finite element method) for

decoupled geomechanical simulation. In the decoupledscheme, the fluid flow numerical scheme was fullyimplicit, where both pressure and saturation equations(as primary variables for two phase isothermal flowsystem) are solved at the same time with the same

Jacobian matrix. The time discretization is Euler time-integration and the spatial discretization is based on

vertex-centered control-volume finite-element method(also called box method, (Huber and Helmig)). Boxmethod is briefly explained below.

6.1. Box MethodBox method is a control-volume finite-element method

(CVFEM). The advantage of the FE method is thatunstructured grids can be used for simulation, and theadvantage of FV method is that it is locally massconservative. As such the computational domain is


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discretized by a dual mesh, as illustrated in Fig-4. Onemesh handles the finite volume scheme (shown asdashed blue line in the middle of the larger square), andthe other mesh is used for FE. Each FE mesh contains

four sub-control volumes from four neighboring boxes(FV meshes).

6.2. Fully Coupled SchemeIn this type of coupling the variables of flow andgeomechanics are solved simultaneously through asystem of equations with displacements (geomechanical

primary variables) and pore pressure and saturation (asprimary flow variables). The technique is also calledimplicit coupling as a single discretization scheme isused [27]. In fully coupled schemes, the geomechanicalequations are solved by the finite element technique

while the fluid flow equations are solved by the Boxmethod. Therefore the primary variables of fluid flow

equations (pressure Pw, saturation Sn) are solved on thebox centers (finite volume mesh) and the flow gradientsproperties (i.e., fluxes) are solved on integration points(represented by IP in Fig-4). The geomechanicalequations and momentum balance equations on the other

hand, are solved on FEM points (vertices), usingstandard Galerkin finite element technique.The gradient

properties of primary variables (solid displacement in

three directions) are solved on Gaussians points, andtherefore effective porosity and permeability are also

calculated on these points. As such a mapping algorithmis needed to map the calculated displacements on theGauss points back onto FEM points to be able to becoupled fully with fluid flow. The mapping is based on a

sub-control volume face area average; therefore a loopover all sub-control volumes is needed to obtain anaveraged value on the FE points (vertices). It is notedthat use of the box method for both mass and momentum

balance equation has been tried but lead to instabilityand oscillation in pressure. More about the discretization

scheme can be found in the literature [28].

Figure 4-Schematic of spatial discretization (Fully coupled

Scheme)

6.3. Sequential Coupling SchemeIn the decoupled scheme, both fluid flow andgeomechanical equations are solved by the Box method.

Therefore flow gradients (fluxes) are solved onintegration points as well as gradient geomechanicsterms (i.e. strain, stress). It has been shown that effectivedynamic porosity and permeability are functions ofvolumetric strain (Tohidi-Baghini [29]); as such these

values are also obtained on integration points in thedecoupled scheme, and therefore need to be mapped

back onto grid vertices (box centers) by a mappingtechnique .The mapping technique adopted here wassimilar to the fully coupled scheme where the properties

were multiplied by the surface area of sub-controlvolumes edges, and averaged to the box centers.Decoupled fluid flow equations are solved by fully

implicit approach assuming a two-phase (saline waterand supercritical CO2) isothermal system.

Figure 5-Schematic of spatial discretization (decoupled Flow

and Geomechanics)

6.4. Streamline Simulation Discretization SchemeUnit mobility displacement with constant boundaryconditions over all time steps are the most suitable

scenarios for one-dimensional mapping of analyticalsolutions along streamlines, otherwise the numericalsolutions are better approaches for solving transportequations along streamlines as they accommodate easilynon-uniform initial condition and is totally general


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(unlike analytical solutions). The numerical approach forsolving saturation equation along streamlines was first

proposed by Bommer and Schecter (1979), [30].

Streamline numerical solutions are an IMPES typereservoir simulation where pressures are solvedimplicitly, but explicitly from the saturation equation(transport equation). Therefore, the main idea behind thediscretization technique in streamline simulation is the

decoupling of the pressure equation from the saturationequation. Although the pressure field is solved in thesame way as in conventional FD reservoir simulators,the discretization technique and the grid dimensionalityfor solving saturation/conservation equations is different.

For solving the transport equation, a three-dimensionalheterogeneous domain is decomposed into a series of 1D

grid (along streamlines). Therefore streamlinesimulations deal with a dual-grid system. Thediscretization technique (for time and space) for 1D griduses a standard explicit scheme and single-pointupstream weighting for solving one-dimensional

problem along streamline as the equation belowrepresents (Blunt & Rubin, 1992) :

)( 1,,,

1

,

1

,

n

ij

n

ij

isl

n

sln

ij

n

ij fft

SS

(2)

Prior to this step, streamlines need to be traced based oninstantaneous velocity field obtained from pressuredistribution in last step:

va =Kkram (gradP- r a g) (3)

The independent variables (such as pressure andsaturation) are mapped from the underlying 3D grid onto

the 1D streamline for initialization purposes at eachglobal time-step when streamlines are updated. In 3DSL,the solutions to the numerical problems are providedeither by the Algebric Multigrid Method known asAMG or Algebraic Multigrid Methods for Systems

known as SAMG, both developed in Fraunhofer SCAI(Scientific Computing Institute) a German researchcenter (Stben k. et al.) [31]. The solution obtained fromthe second grid (1D grid) is then mapped back onto the

initial 3D grid to represent phase distribution. Pressure isthen solved again and the process continues.

Since streamline models are based on IMPES (Implicitin Pressure and Explicit in Saturation) scheme, andtherefore, a time-varying velocity field can be modeled

by a series of successive stationary velocity fields

(Gupta and King [32]). For unsteady state flow,streamline configurations change in time, generatingtransverse flux to the origin of flow.

7. HYDROMECHANICAL COUPLING

Injection of CO2 can result in high-pressure buildup,leading to low effective stresses and ultimately to

hydraulic fracture conditions. This can lead topotentially large deformations and subsequent

redistribution of in situ stresses. Changes in the stressfield in the domain can lead to changes in petrophysical

properties of the aquifer (i.e. porosity and absolute

permeability), surface displacement and aquiferdeformation. These types of interactions between fluid

flow and rock mechanics are known as hydromechanicalcoupling. Classical reservoir simulators do not considerthis interaction, therefore to a proper strategy is neededto include the geomechanical impacts of fluid flow onreservoir simulation process. Ideally the flow problem

needs to be solved at the same time in one set ofequations known as fully hydromechanical coupling.

The basis of this system formulation is Biots poroelasticproblem [24]. However this approach is computationallyexpensive and therefore cannot be applied to large

domains with large number of grids. In many couplingstrategies, porosity and permeability are updated

(simultaneously or at each time step) due to a change involumetric strain, Eqn-4 as proposed by Touhidi-Baghini [29]:

f = f 0 + e v1+ e v (4)

or Eqn-5, proposed by Rutqvist and Tsang [33]:

k= koexp(22.2(

ff

0

- 1)) (5)

Since the reservoir medium was not fractured sandstone,a porosity-based permeability update approach isadequate and physically meaningful; for our study. The

permeability field was initially considered homogenous

and isotropic. In both equationsv

is the volumetric

strain at each time step,0

ando

k are the initial

porosity and permeability before inclusion of

geomechanics respectively.

7.1. Coupling strategyThe basic concept behind inclusion of geomechanics instreamline simulation lies in the streamline timestepping. Streamline time stepping is different thanconventional flow simulation time steps. There is aconvective time step that is simply the pressure updates

due to changes in flow regime. There is another timestep, which is technically a user provided time step andis mainly defined based on the existing productionhistory. The other time step is streamline sub-intervaltime step, which is basically the time span for a particle


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to travel along the streamline. The size of this time stepdepends on the Courant-Fredrich-Levy (CFL) condition,which is a formulation of a corrector step that leads to amechanism to ensure numerical stability, to solve one-

dimensional pressure transport equation alongstreamlines. To perform all coupling strategies on a solid

platform two sets of code-interfaces were developed.Fig-7 provides a schematic illustration of the couplingstrategy and how the parameters are exchanged between

each simulator at each time step.

With the same strategy as conventional couplingsequential strategies, streamline simulation can be

coupled to geomechanics, with the structure illustratedbelow

Figure 6-Schematic of coupling strategy for sequential

Streamline Based Flow and Geomechanical simulator

The red line in the figure illustrates the forward flow and

geomechanical simulations and the time interval of eachcycle. The diagonal blue lines show the step duringcoupling at each cycle where fluid flow simulatoroutputs (pressure, saturation, phase density difference)are fed into geomechanical simulator. And the yellow

lines show the step (and timing) where geomechanicaloutputs being inputted into flow simulator; each blue

block represents a cycle of fluid flow-geomechanicscoupling from ti to ti+1.

Time steps are kept constant in this stage of the research,however the time steps can be adaptive based on

convective time steps of streamline simulation, beforeeach pressure update. Even with constant fixed timesteps, streamlines (for large domains) are faster

techniques in hydromechanical coupling strategiescompared to conventional coupling techniques.

7.2. Interface DevelopmentThe main interface that was developed is a C++ based

interface which links the decoupled geomechanical code(in Dumux and on the Dune platform) to the fluid flow

code. This interface, unlike the interface betweencommercial software packages, does not work withrestart files as an exchangeable file for initializationstep of next sequential time steps. The interface

exchanges properties between flow and geomechanicscode by solution vectors within the frame of the code.Therefore a C++ routine has to be written inDumux_Geomechanics to output the vectors forDumux_Flow, and at the same time a routine was

developed to read the solution vectors outputted by

Dumux_Geomechanics (, K).

Fig-7 illustrates a snapshot of the interface developed for

hydro mechanical purposes.

Figure 7- Graphical User Interface Developed by QT

Programming Language for hydromechanical coupling of

streamline simulation with C++ Geomechanical code

Fig-8 shows the interactive structure of the sequentiallycoupled interface between streamline simulation and theC++ geomechanical code, as well as the exchange

parameters.

Figure 8-Schematic of interactive coupling between streamline

simulation and geomechanical code

8. GEOMECHANICAL INFLUENCE ONSELECTION OF TIME-STEPS


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Changing streamlines also requires the ability to solvetransport problems with generalized initial conditionsalong each streamline (suggested by Batycky et al. [17]).Changing porosity and permeability means a pseudo-

change of boundary conditions and general initialconditions (at each time-step), and therefore streamlinesrequire to be updated as the global pressure of thedomain is updated. Therefore treating the geomechanicalinfluences as an explicit factor, which changes porosity

and permeability (due to inducing volumetric strains inthe reservoir with high pressure gradients) in the courseof simulation, requires that streamlines be updated in thecourse of coupling. The frequency of pressure updates,and hence streamlines spatial configuration update,

dictates the time-step sizes of coupling. Because themain and most important factor common in fluid flow

and geomechanical simulation is thepore pressure andpressure gradient all over the domain. It should be notedthat if the streamline simulation was performed at thecoupled hydromechanical simulation, where significantheterogeneity is induced by geomechanics, the

streamline simulator performs faster.

Geomechanical effects are not only limited to thepetrophysical properties (such as porosity andpermeability, MEM properties, etc.) but can also cause

changes in the relative permeability curves and changethe end point mobility ratio, which is a source ofnonlinearity in flow problem. As such the flowsimulator is expected to function slower as it needs toemploy smaller time-steps and a larger number of

pressure updates, and therefore results in a slowerhydromechanical coupling simulation. This part however

is not the main focus of this study, yet can be aninfluential factor in all flow-geomechanical couplingschemes including streamline-based schemes,

particularly where rigorous capillary flow exists. Apreliminary study has already been done byOjagbohunmi and Chalaturnyk [34] for conventional

flow simulator coupled to FLAC geomechanicalsimulator. For these types of geomechanical influences

on streamline simulation, the forward transport equation(saturation) must be solved numerically. For this study

however time-step size is kept constant.

9. RESULTS AND DISCUSSIONS

To investigate the migration of CO2 in saline aquifers,

fluid flow simulations were performed with both a finitevolume based flow simulator and a streamline based

simulator. Two geometries were proposed: a two-dimensional (a cross-sectional view along the aquifer, asshown in Fig-3) and a three-dimensional model.

The injection process was considered to be immiscible,yet compressible (densities of brine and CO2 bothchanges with pressure, and were taken from a look-up

table based on existing correlations). Reservoir hydraulicproperties are provided in Table 1. For comparison ofcomputational efficiency sensitivity, on the number ofgrid cells, the simulations were performed on two

different grid refinement levels. Viscosity was keptconstant for the formation brine and injected CO2 forstreamline simulation.

Table 1.Reservoir hydraulic parameters

Input Parameter Unit Values

Porosity % 0.2

Permeability mD 100

Initial Gas saturation % 0.05

Initial Water Saturation % 0.3

Van Genuchten Pa 0.0037

Van Genuchten n - 4

Temperature K 315

CO2 viscosity c.p 0.06

Brine viscosity c.p 0.5

Brine density kg/m3 1024

Table 2.Reservoir mechanical parameters

Input Parameter Unit Values

Rock density kg/m 2500

Youngs modulus Pa 5.e9

Shear Modulus Pa 3.e9

Constitutive model - Linear elastic

9.1. Two-Dimensional ModelTo understand the mechanism of fluid flow using a

streamline simulation technique, a two-dimensional

model was initially considered for simulation, whichprovides a better visualization for monitoring of CO2

plume in the course of injection. Furthermore as ourflow simulation is streamline-based, for visualization

reasons (streamline paths, and flow directionality fromsink to source in the model) the geometry wasconsidered as a two-dimensional one. (Visualization of

streamlines in a 3D domain can sometime be misleading,as streamlines seem to be intersecting each other.)

9.1.1 High resolution model

The simulation results for a homogenous domain arepresented in Fig-9. The injection rate was considered as

800 kg/m3, to be able to capture the convective

migration of CO2 plume and ensuring the flow

mechanism scales well with streamline simulation,which is the fastest technique for convective flow

processes. The number of cells are NX=100, NZ=100,

and the domain is 2000 m inx direction and 200 m inzdirection.


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t1 t2

t3 t4

t5 t6

Figure 9 - Cross-sectional CO2 saturation map in different

time steps of simulation: t1=100 days, t2=200 days, t3=300

days, t4=400 days, t5=700 days t6 = 800 days

The streamline paths and configuration for the last two time

steps (t5, t6) are represented in the following Fig-10:

(a)

(b)

Figure 10- streamline configuration represented after 500 days

(a) and 900 days (b) of injection; time of flight as a streamline

property is also mapped along streamlines which is the

colourful region around the wellbore.

As it can be seen from the picture, the only sink andsource in the system are the injection well, and twomirror aquifers (boundaries with constant head).Therefore streamlines trajectory and drainage area are

separated into two parts between these pairs: 1) injectionwell-right aquifer and injection well-left aquifer.Streamlines not only show the flow directionality

between each two source-sink terms, but also show thedensity of flow (i.e., around the injection well more

streamlines exist which means the flow activity andpressure gradients are higher in those zones) andtherefore can provide even visually a quickunderstanding of the regions that undergo moregeotechnical changes and are affected more by effective

stress and thus more volumetric strain is expected toevolve in the regions with large density of streamlines.

Also the signature of plume can be roughly seen visuallyfrom the streamlines paths.

9.1.2 Low resolution modelAssuming the same injection scenario with the same

fluid and reservoir rock properties, the number of cells ineach was lowered 10 times. Fig-11 illustrates the resultsobtained from this analysis.

Figure 11-Evolution of CO2plume in the aquifer for the same

model as before with coarse grids. t1=100, t2=200, t3 =300,

t4=400, t5= 700, t6=800 days.

As shown Fig-11, the low resolution model providesresults which might not be reliable in long term.

However they are still informative as an approximateassessment of the migration process (i.e. when the CO2arrival time will be, and what will be the pressure value

t5 t6

t3 t4

t1 t2


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after 3 years of injection in the domain). In terms ofcomputational efficiencies, the computational loadrequired to carry out either of simulations is notsignificantly different, compared to the expected

difference between two levels of resolutions performedby conventional reservoir simulators. A rule of thumb(about the number of grid-blocks in the domain and itsimpact on CPU time) is that CPU time is expected toincrease to the power of two, if the number of grid

blocks is doubled. As it is presented in the below figure,it is not the case for streamline simulation as instreamline simulation the huge computational load ismitigated by decomposition of SL grids from underlyinggrids.

Figure 12 -Impact of grid refinement on computational load

for pure streamline simulation (decoupled from flow)

9.2. Three-Dimensional ModelA hydro-mechanical study of a three-dimensional model

was also considered. The geometry was kept the samebut the model was extended iny direction with the samelength as in x direction. The result of long-term

simulation of the three-dimensional model is shown inFig-13a and 13b, which represent the top view from a

three model (NY=20, NX=20, NZ=50) and the CO2saturation map and corresponding streamlines bundleafter 900 days of injection, respectively. The streamline

patterns are homogenous and symmetric around theinjection well from the top view, since porosity and

permeability are assigned as equal and constant in thecourse of simulation, however streamlines arecategorized into four groups based on their drainage areaand the boundaries they are travelling to. Eachstreamline belonging to a particular region (source-sink)

is shown by a different color. Fig-14a and 14b illustratethe results from a lateral perspective of the threedimensional model.

9.3. Coupled Streamline-Based Hydromechanicalsimulation

With the same coupling strategy explained before

streamline-based flow simulation for the same physicsand model was coupled to a decoupled purelygeomechanical code. A graphical user interface was

developed which was able to couple both tools atdifferent cycle time intervals as well as adaptive time-

(a)

(b)Figure 13- Representation of model top view: (a) shows

signature of CO2 after 900 days, and (b) shows the

corresponding streamlines configuration for the same time of

injection.

(a)

(b)

Figure 14-Lateral view of 3D-model representing (a) CO2

saturation map, and (b) shows corresponding streamline

configuration after 1500 days of injection. Injection well is

located in the middle of the domain.


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stepping, which is suggested by streamline simulation(based on CFL condition and auto-time-stepping

technique) at each cycle. Model geometries can also beadjusted in the interface, as the snapshot of the interface

shows in Fig-7). However the coupling for our study wasbased on a constant time step interval for each cycle. Thecycle forwarding time interval was considered as 100

days for all cycles. The reason for this selection comesback to the long term nature of CO2 injection in

underground formations, that can be on the order ofdecades, as well as the power of streamline simulation incapturing large time step and providing reasonably

accurate results specially for convective and linear (orclose to linear) flow mechanisms. However 100 days is

relatively large time interval for conventional sequentialhydro-mechanical coupling of such physics which is

computationally expensive and can cause huge materialbalance errors which might stop the flow simulator toeven initiate next cycle simulation.

The target model is the same as previous three-dimensional example except the number of gridlocks inzdirection was reduced to NZ=10. The coupled flowsimulation was performed much faster than a fully

coupled simulation of the same physics, and as expectedin the high gradient pressure zone due to changes in

effective stress, strain (volumetric) field was disturbedand two primary geomechanical parameters changed;

porosity and permeability. Keeping the injection

parameters and boundary conditions the same, changesin petrophysical parameters will cause some changes in

dynamical parameters such as pressure and phase

pressure and saturations. Fig-15 and Fig-16 illustratehow these variables change using the streamline-basedhydromechanical coupling scheme.

(a)

(b)Figure 15-Histogram of CO2 saturation after 500 days of

injection: a) streamline-based hydromechanically coupled

scheme and b) standalone streamline simulation.

Figure 16- Variogram of gas saturation after 500 days of

injection

Fig-17 illustrates the changes in permeability. Fig-18and Fig-19 provide a scatter plot of petrophysical

properties at two different injection times plotted against

each other.

Figure 17- Permeability difference map between cycle 9 (900

days after injection) and cycle 1 (100 days of injection).


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The maximum permeability difference after 900 days ofinjection is about 9-10 mD at some cells from differentlayers and depths.

Figure 18- Scatter gram between porosity (in all cells) in cycle

1 (100 days) vs. porosity values at cycle 9 (900 days) days

Figure 19- Scatter gram between permeability (in all cells) in

cycle 1 (100 days) vs. permeability values at cycle 9 (900

days)

Even though the plots show the changes have not be

very significant (as expected in a linear elastic model), itshows there is severe hysteresis in the update of porosityand permeability especially at small values of the two

properties (right left corner of the two diagrams).

Fig-20 and Fig-21 show how displacements in thex andz direction, respectively, change after 400 days of

injection. Both values are reported in meters. Due to theporoelastic formulation and the chosen formationproperties, the displacements are small. Changes in

displacements are associated with changes in volumetricstrain and accordingly to porosity and permeability as

reservoir geomechanically influenced parameters.

Figure 20-Histogram of vertical displacement (uzz in z)

direction after 4 cycles (400 days of CO2 injection)

Figure 21-Histogram ofuxxdisplacement inx direction after 4

cycles (400 days of CO2 injection)

9.4. DiscussionsComparison between fully coupled and sequentialcoupled simulations is reasonable provided the

simulators engine (grid or multi-grid solvers) as well asthe core code and programming language behind the

tools are the compatible. On this basis, a decoupledgeomechanical code (from fluid flow code) based on thesame fully coupled code which was developed on thesame platform (DUNE), with the same programminglanguage ( C++), and the same discretization scheme

(Box method), and the same simulation grid managerwas developed. Comparison of a computational loadand third party simulator with different simulationtechnique is also reasonable as long as the simulationsare both performed with the same CPU and the solvers

use the same technique for system solving. The solverthat streamline code is based on can be either AMG or

SAMG, and Dumux is also able to switch (to becompiled with) to Algebric MultiGrid (AMG). Howeverfor problem simplicity the library embedded in our

approach was SuperLU [35] as system solver in Dumux;still the computational load to carry on the simulations

are significantly different (in streamline-basedgeomechanical coupling than finite volume based flow-geomechanical coupling) that switching to even a fastersolver (i.e. SAMG) does not change the comparisonscenario significantly.


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Inclusion of the CFL (CourantFriedrichsLewy)condition and assignment of proper CFL values andcriteria, in both simulators (streamline based and FV-FEM based) is another important key parameter which

might change the simulation computational load as wellas resolutions of the results. Auto time stepping alsomight be taken into account if FV-FEM technique iscompared versus streamline based geomechanicalcoupling, as fixed time steps can sometimes be bottle-

neck in forward flow and geomechanical simulations. Inboth of our coupling strategies we considered CFLcondition (0.75) and adaptive time-stepping.

All approximate methods in reservoir simulation are

associated with some disadvantages in capturing fullphysics of simulation or in model accuracy. (e.g., proxy

methods, IMPES scheme, streamline simulations,convex optimization techniques, model order reductionapproaches, upscaling, etc.). All these class oftechniques have been offered to the world of reservoirsimulation to enhance simulation CPU inefficiencies, but

on the other hand they result in a reducedmodel accuracy, depending on the physics of the

problem (compositional, diffusion, capillary, multi-phase or single phase, gravity, etc.), model size andsimulation time The main disadvantages associated with

streamline-based hydromechanical coupling, are thedisadvantages in the streamline simulation, as they areapproximate methods. For highly nonlinear systems suchas compositional miscible flood simulations, gravitydominant systems, systems with rigorous capillary

pressure effects, etc., streamline-based hydromechanicaltechniques are not expected to provide accurate results.

It is expected that the CPU efficiencies achieved byinclusion of geomechanics in streamline simulation,would be reduced by increasing model non-linearity.

Furthermore, since streamline simulation deals with adual-grid system (Cartesian grids to solve pressure andstreamline grid to solve saturation equation), the

mapping in the streamline workflow between these twogrids is associated with some material balance error.

However the technique is not associated with numericalsolution divergence during iterative simulation, in

contrary to traditional coupling techniques. Whenporosity is significantly updated in traditional coupling

schemes, the solution diverges, (forward simulation willbe interrupted) and a divergence criterion must bedefined for the simulation.

For the dynamic hydro-geomechanical problems, whereboth fluid flow and geomechanics problems areconsidered dynamic, the solid material velocity term inDarcy equation becomes important, which leads to atime derivative compressibility term (bulk volume for

each node) and the fluid flow formulation needs to beextended to a two-phase flow system with deformable

porous medium, and a displacement velocity term is

required to be added to saturation transfer equation. Thistechnique is mainly important where some wave isinduced in the medium, or the problem is studied aroundthe wellbore vicinity with high-pressure gradient, (e.g.

sand production). Our problem, however, is considereddynamic in fluid flow and quasi-static in geomechanics,and therefore this term is neglected in our formulation,considering that the problem is for a large field (notnecessarily around the wellbore vicinity). Furthermore a

quick comparison between solid displacement velocityand fluid velocity (pressure gradient) for a time stepshowed that these two are different by orders ofmagnitude and therefore assumption of zero solidvelocity is physically meaningful and would not affect

our solution.

CONCLUSIONS

Simulation of underground CO2 injection in salineaquifers was studied. Geomechanics was introduced as

an important piece of physics that is influential in thestorage mechanism. Different existing strategies ofcoupling were introduced. Above that, a novel hydro-mechanical strategy was introduced. Initially weinvestigated the feasibility of the approach and if thecoupling is doable. For that purpose, a geomechanical

code was developed (based on an existing research opensource code) and was linked to a FORTRAN based toolfor streamline simulation. This approach wasimplemented on the CO2 storage scenario as it was wellsuited to the nature of streamline simulation and

dimensionality and large size of aquifers. Streamline

simulation was coupled to an elastic geomechanical codeand was shown to be computationally robust andefficient especially for larger reservoirs. The approachwas also compared to the fully coupled technique, which

is generally expected to provide more precise numericalsolutions but can come with severe computational

burdens, particularly for large domains with largenumber of grid blocks. Streamlines are most suited whenthe process of fluid flow is convective, which is the most

relevant to the primary trapping mechanism that isoccurring during injection process of CO2.

Geomechanics also is best fitted to this trapping

mechanism, therefore linking of these two pieces ofphysics at the time span of several years was found to berelevant.

Initial assessment of fully coupling technique forscenario of CO2 injection is completed; however the

current challenges in all the coupling strategiesdiscussed in this paper, are how the three couplingstrategies explained will result in approximately the

same fluid flow and stress redistribution results fordifferent model geometries and complexities of physics.


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