4.6 Dynamic reservoir study - Treccani, il portale del sapere I / EXPLORATION, PRODUCTION AND TRANSPORT 4.6.1 Oil recovery processes Natural drive mechanisms The main drive mechanisms

VOLUME I / EXPLORATION, PRODUCTION AND TRANSPORT

4.6.1 Oil recovery processes

Natural drive mechanismsThe main drive mechanisms in oil reservoirs are

(Dake, 1978): a) solution gas; b) gas cap; c) naturalwater influx; and d) compaction (see Chapter 4.3).

In a reservoir with solution-gas drive, oilproduction results from a change in volume as thereservoir pressure is reduced. As the pressuredecreases, the oil and associated dissolved gasexpand resulting in an increase of the volume whichis recovered by the production wells. If the reservoiris above the bubble-point pressure, pb (i.e.undersaturated oil), then there is no free gas in thereservoir, and hence there is no gas cap overlyingthe oil. Thus, in undersaturated oil reservoirs wherethere is no active aquifer support, the primary drivemechanism is solution gas. In undersaturated oilreservoirs, one must account for the decrease in theHydroCarbon Pore Volume (HCPV) as the pressuredecreases. This decrease in the HCPV is due to theexpansion of the connate, or interstitial, water andthe reduction of the pore volume due to rock (pore)compressibility.

As the reservoir pressure falls below the bubblepoint, gas is liberated from the oil phase, and a freegas phase develops. Gas compressibility is larger thanthat of liquid oil and significantly larger than thecompressibility of water and rock (pore volume). As aresult, the free gas dominates the effectivecompressibility of the system, and the contributionsfrom the compressibility of rock and water may beneglected as a first approximation. In solution-gasreservoirs, the reservoir pressure typically declinesrapidly until the bubble point is reached; subsequently,the pressure decline continues, but at a slower rate, dueto the high compressibility of the free gas. The gasdissolved in the oil is the only source of primary

energy for gas production at the surface until thebubble-point pressure is reached. Shortly thereafter,the presence of a highly mobile free-gas phase in thereservoir usually leads to a significant increase in theproducing gas/oil ratio. The sharp increase in theproducing gas/oil as the reservoir pressure falls belowthe bubble point further depletes the reservoir of themain energy source. Later in the life of the reservoir,the producing gas/oil ratio itself declines as theavailable reservoir energy diminishes.

If the reservoir is at the bubble point whendiscovered, then it is likely that a gas cap, which lieson top of the oil region, may be present. Expansionof the gas cap provides significant pressure supportand slows down the rate of pressure decline in thereservoir compared to the case in which only asolution-gas drive is present. Of course, solution gascontributes to oil production in reservoirs with a gascap. A natural water drive occurs when the oilreservoir overlays, or is flanked by, a water aquifer.As the pressure in the oil reservoir decreases due tofluid production, the aquifer water expands, leadingto water influx from the aquifer into the oil reservoir.For a given pressure drop at the boundary betweenthe oil reservoir and the water aquifer, however, theinflux of large volumes of water into the oil reservoiris necessary in order to provide adequate pressuresupport. This is due to the much smaller watercompressibility compared to that of the reservoirfluids. Consequently, active aquifer support requiresrelatively large aquifer sizes. When active aquifersupport is present, relatively high pressures can besustained in the oil reservoir, thus leading toimproved oil recovery.

In the case of gas reservoirs the main drivemechanism is the gas expansion which could lead, inbounded reservoirs, to very high gas recovery (80-90%of the Original Gas In Place, OGIP). By contrast, with

575

4.6

Dynamic reservoir study

an active water drive the gas recovery coulddecrease considerably, due to the waterbreakthrough and the residual gas saturation leftbehind the waterfront.

Improved oil recovery Natural drive mechanisms present in the

reservoir system lead to the flow and recovery of oilto production wells. However, only a small fractionof the Original Oil In Place (OOIP) is usuallyrecovered using natural drive mechanisms. In orderto improve oil recovery, pressure support byinjection processes can usually prolong theproductive life of a reservoir. Common approachesare peripheral water injection into the aquifer tomaintain the pressure level in the oil reservoir whilefluids are being produced. Similarly, gas injectioninto the original or secondary gas cap may be usedto support the pressure and increase the effectivecompressibility of the reservoir system.Unfortunately, pressure support is only a partialremedy to declining oil production.

Secondary recovery displacement processes areusually employed to increase the amount of oilrecovery. Waterflooding, a technique in which wateris injected into the reservoir to displace the residentoil, is the most common secondary recovery process.Waterflooding operations are usually conductedusing patterns, which are groups of injection andproduction wells arranged in a certain arealgeometric shape. Well spacing, injection allocationschedules, and operating conditions are case specific.The design and operation of a waterflood is usuallyobtained from simulation studies augmented by fieldexperience. Waterflooding can add significantly tothe cumulative oil recovery. However, it is often thecase that the sweep efficiency of a waterflood may bequite low. Moreover, a significant amount of residualoil is usually left behind, or trapped, in the floodedregions. Of course, the performance of a waterfloodstrongly depends on the specific case under study. Ingeneral, however, at the end of waterfloodingoperations significant oil reserves remain in placebecause of complete bypassing and trapping by theinvading water.

Enhanced Oil Recovery (EOR) displacementprocesses have been developed to recover more oil.EOR processes involve the injection of a fluid intothe reservoir in order to displace the resident oil, andthey fall in one of three general categories (Lake,1989): thermal, chemical, and solvent methods. Ineach category, a wide variety of displacementprocesses have been devised and applied. Themechanisms for improving the displacement of theresident oil depend on the particular EOR process

used. The objective is usually to improve the sweepefficiency and/or the local displacement efficiency.For that purpose, EOR schemes attempt to improvethe mobility ratio of the displacing fluid to that ofthe displaced oil. For example, polymer floodingaims to improve the mobility ratio by increasing theviscosity of the injected water, while steam injectionattempts to improve the mobility of the resident oilby reducing its viscosity. Solvent flooding, includingmiscible and near-miscible gas injection, is usedwith the goal of improving the local displacementefficiency by achieving miscibility with the residentoil, which allows the miscible mixture to be driventowards the production wells. Of course, theviability of an EOR project depends on theeconomics of the specific process for a particularsite. EOR techniques will be discussed in detail inVolume 3.

The physics of EOR processes must beaccurately described by the equations, which aresolved numerically by the reservoir flow simulator.The mathematical statement of many EORprocesses can be quite involved; as a result, precisemodelling of EOR processes usually requiressophisticated numerical solution methods andimplementations. A detailed understanding of thephysics at work in each particular case, which weare trying to model using a reservoir simulator, isvery important. Using a simulator that does notrepresent the important physical processes thatdictate the flow behaviour for the system inquestion is one of the major misuses of reservoirsimulation. Thus, in addition to field experience,practitioners of reservoir simulation must also havean in-depth understanding of the relevant physicalprocesses being modelled.

It should be remembered that before performingany 3D-reservoir-simulation modelling, it could beuseful to carry out material balance calculations inorder to preliminarily investigate the main drivemechanisms acting in the reservoir. This can be easilyand quickly done using a personal computer equippedwith available windows-based material balancesoftware. Material balance calculations can also helpin evaluating the efficiency of the displacementprocesses.

4.6.2 Reservoir flow simulation

Reservoir simulation is a broad discipline ofpetroleum engineering concerned with predictingreservoir performance using computer programs thatgenerate numerical solutions of the equationsgoverning the complex physical processes in oil

576 ENCYCLOPAEDIA OF HYDROCARBONS

OIL FIELD CHARACTERISTICS AND RELEVANT STUDIES

reservoirs. It is an essential tool in contemporaryreservoir management. Performance predictionsobtained from reservoir simulation can be used toresolve a wide range of issues related to bothoperational and design considerations during all stagesof field development.

Reservoir simulation is a blend of physics,applied mathematics, computational fluid dynamics,numerical analysis, and computer programming. Theobjective is to provide an accurate andcomputationally efficient quantitative description ofcomplex behaviours in geologic formations of largeextent lying at great depths. It is often the case,however, that only limited data about the reservoirunder study are available to the engineer.Consequently, engineering experience and judgementlie at the core of reservoir simulation practice.Significant human and capital resources arenecessary for the development and operation of oilreservoirs. Typically during the life of an asset,reservoir management decisions must be made eventhough very limited information is usually available.Consequently, uncertainty quantification and riskmanagement are at the core of sound reservoirmanagement. Reservoir simulation is used to makepredictions of flow performance and quantify theuncertainty associated with these predictions; it is aprimary tool for making quantitative decisions. Inbroad terms, the important uses of simulation are: a) forecasting reservoir performance under a varietyof scenarios; b) improving reservoir descriptionthrough history matching; c) analysing physicalexperiments; d) understanding the complexmechanisms that dictate the flow and transport;e) developing simple, yet effective, models andcorrelations. Within each category, a wide range ofactivities and applications may be listed. In thefollowing, the core components of reservoir flowsimulators are described. Namely: reservoir flowequations; discretization of the equations;formulations; numerical solution. Subsequently abrief discussion about gridding methods, which playa central role in reservoir simulation, is presented.

Reservoir flow equationsEquations used to model isothermal flow and

transport in reservoirs are obtained by combining theconservation of mass with Darcy’s law. An energybalance equation is needed when the assumption ofconstant temperature is not valid. The equations thatdescribe the physics under consideration are usuallyexpressed in continuous (differential or integral)form. Reservoir characterization models oftendisplay geometric complexity and are populated withspatially variable properties. Moreover, highly

non-linear behaviours are usually associated withreservoirs flows. As a result, numerical solutiontechniques are normally the only choice, and workingwith a discrete form of the governing equation isnecessary.

A discrete form of the governing equations alongwith appropriate constraints, constitutive relations, andwith boundary and initial conditions is then solved bynumerical techniques to yield predictions of reservoirperformance. This computational tool, a reservoirsimulator, can then be used to model differentscenarios and operating conditions.

Here, we derive the flow equations, both indifferential and discrete forms, and introduceappropriate nomenclature. Important assumptionsinvolved in the derivation will be emphasized(see Aziz and Settari, 1979).

Compositional flow equations Consider an isothermal reservoir system with nc

components (c is the component index) and np phases(p is the phase index). Normally np�3, but nc can bearbitrary in compositional models. The differentialform of the conservation equation for component c,which may exist in np phases, is:

�Cc[1] 12��Fc�0

�t

Eq. [1] is a differential equation that describes themass balance of component c over an arbitrary controlvolume. It states that for component c, net mass fluxinto the control volume is balanced by accumulation.In Eq. [1], Cc is the overall mass concentration ofcomponent c in the arbitrary control volume overwhich the mass balance is written, and it can beexpressed as:

[2] Cc�F�rpSp yc,pp

where F is the porosity of the porous medium, and rpand Sp are the density and saturation of phase p,respectively. The mass fraction of component c inphase p is denoted by yc,p. The overall mass flux (mass per unit area per unit time) of component cacross a control-volume boundary is denoted by Fc,which we write as:

[3] Fc��Fc,p��rp yc,pupp p

where up is the Darcy velocity of phase p.Note that in the treatment here, the flux expressionaccounts only for transport by convection. This is acommon simplification in general-purposeapplications of reservoir simulation. However, in somecases, accurate description of the flow process understudy may require explicit modelling of additionalmechanisms, such as diffusion and dispersion.

577VOLUME I / EXPLORATION, PRODUCTION AND TRANSPORT

DYNAMIC RESERVOIR STUDY

Substitution of the expressions from Eqs. [2] and [3]into Eq. [1] gives

�[4] 1 (F�Sp rp yc,p)��(rp yc,pup)�0

�t p p

The Darcy velocity of phase p is given by

krp[5] up��k13 �(�pp�gp�D)mp

where k is the absolute permeability of the porousmedium, which can be a full tensor, and krp, mp, pp andγ p denote, respectively, the relative permeability,viscosity, pressure and density of phase p. The verticaldepth (assumed positive downward) is denoted by D.Substitution of Eq. [5] into Eq. [4] yields

�[6] ��13 �FSprp yc,p��rp yc,pklrpp �t

(�pp�gp�D)��0

where λ rp = krp/mp is the relative mobility of phase p.We have one conservation equation like Eq. [1] foreach component in the system. Therefore, there are nc

conservation equations like Eq. [1], one for eachcomponent. However, there are many more unknownvariables than nc. They are:

Unknown Number ycp nc np

pp np

Sp np

Total nc np + 2 np

The remaining equations to complete thedescription of our system are obtained from: a) capillary pressure relations; b) phase equilibriumrelations; c) phase constraints; and d) saturationconstraints. Thus, we have:

Equation Number conservation nc

capillary pressure np�1 equilibrium nc(np�1) phase constraints np

saturation constraint 1 Total ncnp+2np

Only the flow, or component conservation,equations have flux terms that connect the gridblock(control volume) to its neighbouring gridblocks. Theremaining equations are referred to as constraintrelations, and they depend only on the variables of thegridblock itself. We can use np�1 capillary pressurerelations, one saturation constraint and np phaseconstraints to eliminate 2np variables from thecomponent conservation equations. This reduces thenumber of equations and unknowns to nc np. Ingeneral, a component can partition, or exist, in any of

the np phases; as a result, there are nc (np�1) phaseequilibrium relations. To describe componentpartitioning for the general case where np�3, a three-phase flash is needed. However, since general three-phase flash computations are difficult, it is oftenreasonable to ignore the mutual interaction of waterwith the oil and gas phases, i.e. we assume that thewater phase contains only the water component andthe hydrocarbon phases do not contain any watercomponent. With this simplification, we are left withnh equilibrium relations, where nh is the number ofhydrocarbon components. Together with the nc�nh�1component conservation equations, the total numberof variables becomes 2 nh�1.

Black-oil modelIn standard black-oil models, which are the most

common models used in the oil industry, there are onlytwo hydrocarbon components (nh�2): a non-volatileoil and a volatile gas, which is soluble in the oil.Moreover, the hydrocarbon components do not interactwith the water phase, which is made up of the watercomponent only. The three components in the standardblack-oil model, namely, water, oil and gas, aredefined as phases at standard conditions, and simpleempirical relations are used to describe the phasebehaviour.

Here, we describe a generalized black-oil model,where each of the three phases can contain up to twocomponents, and each component is associated with aparticular (master) phase. For example, the masterphase for the gas component is the gas phase. Black-oil models are simplified versions of thecompositional statement. When working with thesemodels, however, it is important to keep a cleardistinction between components (phases at standardconditions) and phases. The relationships betweencomponents and phases are illustrated in Fig. 1.

The following notation is introduced: Vp is thevolume of phase p at reservoir pressure andtemperature, and Vc,p is the volume of component c atstandard conditions that is liberated from phase p. Toreduce the general compositional statement to thecommon black-oil form, the concepts of formationvolume factor and solubility are introduced (seeChapter 4.2). The formation volume factor of phase pis defined as the ratio of the volume of phase p atsome specified conditions (usually reservoirconditions) to the volume of the component associatedwith that phase at standard conditions:

Vp[7] Bp�

133, p�o, w, gVp,p

where o, w, g refer to the oil, water, and gas phases,respectively. The solubility of component c in phase p,



measured with respect to a reference reservoircondition, is defined as

Vc,p[8] Rc,p�

133

Vp,p

which is a ratio of two volumes at standard conditions.Namely, the ratio of the volume of component c inphase p to the volume of the component associatedwith phase p. Obviously, the solubility of a componentin its master phase is unity (i.e. Rp,p�1).

Relationships for mass fractions and equilibriumratios can be derived as functions of formationvolume factors, solubilities, and densities. It can beshown that

rc�Rc,p[9] wc,p�yc,p rp�

13213

Bp

where ω c,p is the concentration of component c inphase p (mass, or moles, per unit volume), and rc isthe density of component c at standard conditions.Substituting Eq. [9] into the general compositionalequation, Eq. [6], and dividing by rc yields

� Rc,p Rc,p[10] 1�F�Sp

12 ��12 up��0�t p Bp p Bp

for c�w, o, or g. Substitution of Darcy’s law, Eq. [5],into the above equation yields the form commonlyused to describe the generalized black-oil model.

Discretization of the equations The conservation equations for reservoir flows are

non-linear. As a result, they are not amenable tomethods involving an analytical solution. Instead,numerical techniques are used to solve these systemsof equations. The first step is to express the governingequations in discrete form as opposed to thedifferential, or continuous form, given in Eq. [1].

Consider the control volume, or gridblock, shownin Fig. 2. The conservation of mass for component c,which may be transported via np flowing phases, ingridblock i can be written as:

1[11] 1 (Mc

n�1�Mcn)i��mci,l

�mci

wDt l

This equation is a discrete form of Eq. [1]. In Eq. [11], ∆ t is the timestep size over which the massbalance is written, and Mc�ViCc is the mass ofcomponent c in gridblock i whose bulk volume is Vi.Superscripts n and n�1 denote previous and currenttime levels, respectively. The mass flow rate ofcomponent c through a cross-sectional area A is

[12] mc�Fc A

where Fc is the overall flux of component c definedpreviously. In Eq. [11], ∑ l indicates summation overall l connections to gridblock i. Note the additionalsource term, mci

w, which accounts for the presence of a



OIL, Vo

reservoir orarbitrary conditions

GAS, Vg

WATER, Vw

OIL, Vo, o

componentassociated withreservoir phase

GAS, Vg, g

WATER, Vw, w

GAS, Vg, o

componentdissolved

OIL, Vo, g

GAS, Vg, w

Fig. 1. General three-component black-oilsystem.

well production

influx from l

il

Fig. 2. One-dimensional flow in a simple Cartesian system.

well in the gridblock. The left hand side of Eq. [11] isthe rate of mass accumulation, and the right hand sideis the rate of mass influx into the block (production ispositive). Overall component quantities are obtainedby summing over the individual phases. Specifically,Mc��p Mc,p and mc��pmc,p, mc

w ��p mwc,p, and we

have

1[13] 1��Mc,p

n�1�Mc,pn�i��mc,pi,l

��mc,pi

wDt p l p w p

The mass flow rate of component c in phase p at theinterface between blocks i and l can be written as:

[14] mc,p i,l=Tc,p i,l

[Fp,l�Fp,i]

where

[15] Tc,p=wc,p lp Ts

is the transmissibility,

[16] wc,p=rp yc,p

is the concentration of component c in phase p, lp isthe relative mobility, and

kA[17] Ts=

13

Dx

is the static component (geometry and permeability) ofthe transmissibility. In Eq. [14], F is the potentialdefined as

[18] Fp,l�Fp,i�( ppl�ppi

)�gpi,l(Dl�Di)

where

[19] gp� grp

with g being the acceleration due to gravity. The wellterm can be expressed as

[20] mc,pw �wc,pqp

w

So far, we have not specified the time level atwhich the flow terms in the discrete equations areevaluated. If an explicit treatment is employed, thenthe flow terms are evaluated at the previous time level,n. In contrast, fully implicit treatment requires that theflow terms be evaluated at the current (new) timestep,n�1. The spatial and temporal discretization appliedto the governing equations play a central role in theconvergence, stability, and accuracy of the computedsolutions.

We presented the differential and discrete forms ofthe basic equations that describe flow and transport inreservoirs. These equations form the foundation ofmost of the reservoir simulators used in the industry.Proper use of a reservoir simulator requires anunderstanding of the physics being modelled, and ofthe assumptions and limitations associated with theequations being solved.

FormulationsFor the case of isothermal compositional flow with

simple water treatment (see above), we used the linearconstraints to finally arrive at the 2nh�1 equationswith an equal number of variables. In order to solvethe system, we do not have to solve for all of thesevariables simultaneously. Instead, we must solve forthe primary set of variables. The primary variables arethose degrees of freedom that, when fixed, define aunique state of thermodynamic equilibrium. Theremaining non-primary variables make up thesecondary set. So, we must choose and solve a primaryset of equations and unknowns. The secondaryvariables can be obtained from the secondary equationset and the primary variables.

For isothermal compositional flow, the number ofprimary variables is equal to the number ofcomponents, nc, and several different primaryvariable sets have been proposed in the simulationcommunity (Cao, 2002). On the other hand, the ncconservation equations are the natural choice asprimary equations.

Formulations are usually referred to as explicit orimplicit depending on whether the primary variablesare treated implicitly or explicitly. A variable isimplicit if it is evaluated at the current time level,n�1, and explicit if it is evaluated at the previous timelevel. Two major formulation types are common(Coats, 2000): Fully Implicit Method (FIM) andIMplicit Pressure, Explicit Saturations (IMPES). InFIM, all the primary variables in a gridblock aretreated implicitly. In IMPES, pressure is the onlyimplicit primary variable.

The Adaptive Implicit Method (AIM), which is ahybrid, is a third type (Thomas and Thurnau, 1983;Cao, 2002). In AIM systems, a primary variable in agridblock can be labelled as implicit or explicit.Variables, other than pressure which is alwaysimplicit, are treated implicitly only when and wherenecessary. Criteria based on stability analysis can beused to label unknowns in a gridblock as implicit orexplicit (Coats, 2003). Compared to FIM, an AIMformulation offers two major advantages: timestepsizes comparable to those obtained with FIM can beused at a significantly lower computational cost; andthe quality of AIM numerical solutions is superior tothat obtained with FIM, assuming similar timestepsizes, due to lower levels of numerical dispersion.Realizing these advantages, however, depends onachieving a good balance between the timestep sizeand the fraction of gridblocks requiring implicittreatment.

For compositional flows, a general AIMformulation with three levels of implicitness, namely,IMPES, IMPSAT (IMplicit Pressure and SATurations)



and FIM, has been proposed by Cao (2002). The costadvantage for this general AIM formulation, withoptimal labelling of the unknowns, is expected toincrease dramatically as the number of componentsincreases, especially for highly detailed field modelswith large numbers of gridblocks. Most reservoirsimulators are wedded to a particular formulation andchoice of primary variables. This is because the detailsof computing the Jacobian, flash calculations, andissues related to the appearance and disappearance ofphases, are related to the choice of the primaryvariable set. Cao (2002) built a general-purposeresearch simulator that allows for flexibility inchoosing the primary variables. That simulator may beused to study the merits of the various choices that arein common use.

Numerical solutionThe conservation equations for a given timestep,

can be written in residual form as

[21]→R � →u n�1��0

→,

where →R represents the vector of the non-linear

residual equations written for each gridblock in themodel. The objective is to find the primary unknownsfor all the gridblocks of the current timestep (i.e. then�1 time level), →u n�1 that satisfy Eq. [21]. TheNewton-Raphson method is an iterative solutionscheme for non-linear algebraic equations, in whichthe following linear system of equations must besolved in each iteration:

[22] J (v) d→u (v�1)��→R (v)

where J(ν ) is the Jacobian matrix at iteration level n,and d→u (v�1)�→u (v�1)�→u (v) represents the change in theunknowns during the iteration. The

→R (v) is the residual

vector at the end of the previous, nth, iteration. Theiterations are performed until we converge (withinsome specified tolerance) to the solution of the non-linear system, Eq. [21]. The Jacobian matrix is definedas the partial derivative of the residual equations withrespect to the unknown variables. In other words, theelement in the i th row (equation) and j th column(unknown) of the Jacobian matrix is given by

�Ri[23] J ij�12

�ui

The elements of R and u are usually expressedusing gridblock-priority ordering, where the equationsand unknowns associated with a particular gridblockare kept together. For example, for the numberingscheme of the simple two-dimensional (2D) grid ofFig. 3, the resulting non-zero structure of the Jacobianmatrix is shown in Fig. 4. A given row in the Jacobian matrix corresponds to the equations of that

particular gridblock number. The non-zero element atrow i and column j, which is indicated by a bullet, is asmall block matrix representing the derivatives of theequations of gridblock i with respect to the vector ofunknowns in the jth gridblock.

Simulation gridThe Partial Differential Equations (PDEs) that

describe the flow process under consideration andthe appropriate initial and boundary conditions mustbe specified. The simulation grid is a discretedecomposition of the reservoir volume into a largenumber of small volumes, which are often referredto as cells, or gridblocks. For each gridblock, anappropriate discrete form of the PDEs that governthe flow process of interest is written, and anumerical solution is sought. The coefficients thatmake up the discrete form of the PDEs areassembled using properties defined on thesimulation grid. Examples include static data, suchas gridblock geometry, volume, and permeability, aswell as dynamic data, such as fluid and rock/fluidproperties.

The simulation grid describes the geometry of theporous medium and the spatial distribution of thereservoir properties. The number of cells, orgridblocks, used to discretize the reservoir depends onthe geometric complexity of the features (layers,faults, wells), the level of detail in propertydescription, and the accuracy requirements for thenumerical solutions of the specific governingequations.

Different types of grid may be used (Aziz, 1993).Grid types are often labelled based on their structure.A globally structured grid is one where theconnectivity of any gridblock is described using asimple generic stencil. Dividing a regular domain intonx, ny, and nz segments, in the x, y and z directions,respectively, yields n�nx ny nz structured gridblocks.In this case, away from the reservoir boundaries, the



11

j=1

i=1 2 3 4 5

2

312 13 14 15

6 7 8 9 10

1 2 3 4 5

Fig. 3. Natural ordering of a two-dimensional grid.

standard, or nearest-neighbour, discretization of theflow equations gives a seven-point stencil. Namely, thegridblock itself and its connection (transmissibilitycoefficient) to gridblocks in the positive and negativecoordinate directions.

In reservoir simulation practice, globally structuredCartesian grids are quite common. If a higherresolution is desired only in a few locations, such as inthe vicinity of wells, one can retain the globalCartesian grid, and refine that grid only locally whereneeded. This so-called Local Grid Refinement (LGR)approach can be quite effective. The type of grid in theLGR region does not have to be Cartesian. Forexample, a radial grid around the well may be used(Aziz, 1993).

A structured stratigraphic grid that uses corner-point geometry to conform more accurately to thecomplex shapes of geologic layers, may also beused. In that case, however, while the logicallyrectangular structure is retained, the grid is non-

orthogonal. Special discretization techniques of theflow equations are required to obtain accuratenumerical solutions when the grid is non-orthogonal and/or the permeability is a full-tensor(Lee et al., 2002).

The geometric complexity (sloping faults,geologic layers, deviated and multilateral wells),wide variation in the length-scales, and level ofdetail used in Reservoir Characterization Models(RCMs) continue to grow. It is difficult, if notimpossible, to represent such complexity accuratelyusing globally structured grids. Complete flexibilityin representing these complex and highly detailedRCMs can be achieved using (completely)unstructured grid.

Multi-block grids, which are locally structured,but globally unstructured, offer an intermediatesolution with significant flexibility (Lee et al.,2003). Examples of these grids are shown in Figs. 5and 6.

Accurate gridding for simulation posessignificant challenges. It is important, however, torecognize that the impact of the simulation-gridchoice permeates throughout the major componentsof the reservoir simulator. This is expected since wework with a discrete representation of the governingflow equations on the simulation grid. Thecharacteristics of the algebraic systems of equations(non-linear and linear), such as convergence,stability, and solution accuracy, are tightly linked tothe properties of the grid and the discretizationmethods employed.

The steps involved in a reservoir flow simulator: • Read input data (problem definition). • Initialize (set initial conditions). • Prescribe conditions at reservoir boundaries. • Start timestep calculations: a) set an initial

timestep size; b) specify production/injection ratesfor the current timestep; c) linearize the governingdiscrete equations; d) start the iteration loop(Newton iterations); e) solve the linear systems of equations; f ) test for solution convergence;g) repeat Newton iterations if necessary.

• Output simulation results as required. • End if the specified constraints are violated. • Increment time and go to the third step if the

ending conditions are not reached. • End when the time period of interest is covered.



1123456789

101112131415

2 3 4 5 6 7 8 9 10 11 12 13 14 15

Fig. 4. Jacobian matrix structure for the example of Fig. 3.

Fig. 5. Faulted model.

4.6.3 Simulation workflow

An overall description of the general workflow forperforming a reservoir simulation study is shown inFig. 7. Here, we discuss the input data requirements,followed by a quick description of the historymatching and performance forecasting phases of adynamic reservoir study.

Input data In order to perform a reservoir simulation, the

following input data are required:• The RCM: this includes a detailed description of

the reservoir geometry (see Chapter 4.5) and gridinformation (spacing and dimensions) andassociated (static) formation properties (porosityand permeability). The input data should allow forcomputation of the transmissibility field.Moreover, rock compressibility must be supplied(see Chapter 4.1).

• Fluid properties: the specific data requirementsdepend on the selected fluid model. In black-oilmodels, the variations of the formation volumefactors, solubility ratios, and viscosities as afunction of pressure are required (see Chapter 4.2).For compositional models, tables for theequilibrium ratios (K-values) may be adequate.When an Equation Of State (EOS) is used todescribe the phase behaviour, the parameters of thefluid system being modelled must be supplied asthese parameters are required by the EOS model.

• Rock/fluid properties: these include relativepermeability relations and capillary pressurecurves (see Chapter 4.1).

• Well data including the location, completion(perforation) information, operating conditions,and appropriate constraints must be specified.

• Complete specification of the boundary conditions,both on the reservoir boundaries and wells.

• Specification of the initial conditions: enoughinformation should be provided to allow for thecomputation of the initial distribution of thepressures (see Chapter 4.4), saturations, and ifnecessary, the compositions and temperatures.Often, the initial state of the system is provideddirectly to the simulator. This may be apreprocessing step, or a restart run, which refers toa continuation from the point where a previoussimulation run left off. Each of the input items listed above can range from

a simple minimal set of values to a highly detailedspecification with many levels of options and sub-options.

The quality of the predictions obtained from thereservoir flow simulator depends strongly on thequality of the input data. As a result, data samplingand collection schemes must be carefully planned andexecuted. Furthermore, experts should be deeplyinvolved with the carrying out of laboratory tests andexperiments conducted in the process of providing theproperties of interest (formation volume factors,solubility, compositions, relative permeability, etc.).The properties needed for flow simulation are oftenobtained indirectly, or interpreted, using informationfrom a wide range of data sources of varying qualityand quantity. As a result, in order to place the flowsimulation results in context, it is important to have aclear grasp of the assumptions and limitationsassociated with the interpretation methods that areused to construct the input data set for flowsimulation.

The RCM and the methods used to obtain therequired fluid properties are now discussed:

The RCM. The RCM describes the architecture ofthe reservoir, the geometry of the geologic features,



A

B

Fig. 6. Inclined well crossing two fault blocks: regular view (A) and inside view (B).

and detailed spatial distributions of the reservoirproperties, such as porosity and permeability. Theavailable information used to construct the RCM,often referred to as the geomodel, comes from manydifferent sources with significant variations in bothquality and quantity (see Chapter 4.5). Examplesinclude seismic data, electronic well logs, coremeasurements, well tests, and production histories.Moreover, the length scales associated with availablemeasurements, whether direct or inferred, varygreatly.

Fluid and rock/fluid properties. The fluidproperties necessary are obtained from Pressure-Volume-Temperature (PVT) laboratory analysis offluid samples from the reservoir under investigation.Ideally, a fluid sample should be collected when thereservoir is originally discovered. The reservoir fluidsamples may be collected either downhole or at thesurface. In either case, it is a challenge to make surethat the composition of the collected sample isrepresentative of the reservoir fluids. Here, we touchon the subject very briefly; details on samplecollection and analysis are given in Chapter 4.2.

When a sample is collected at the surface, thecorrect proportion of surface fluids (i.e. oil and gas)must be recombined such that the composite sample,when taken to reservoir conditions of pressure andtemperature, matches the in situ reservoir fluid. On theother hand, downhole sample collection may bedirectly representative of the reservoir fluid if thereservoir itself and the downhole conditions are bothabove the bubble-point pressure, which ensures single-phase liquid (oil plus dissolved gas) flow intothe wellbore where the sample is captured. This is notpossible when dealing with a saturated-oil reservoirsince the pressure drawdown at the well is likely tocause some gas to leave the oil as the fluids move to

the wellbore making it difficult to collect a properfluid sample.

For the purpose of PVT analysis, speciallaboratory equipment is used to place the reservoirfluid sample in a holding cell with environmentalcontrols for both pressure and temperature.Subjecting the fluid sample to a liberation processis a basic PVT test. A liberation, or expansion, teststarts with the reservoir fluid sample above thebubble-point pressure; then the pressure is reducedin small steps. As the pressure falls below thebubble point, gas is liberated from the liquid oil. Ifthe liberated gas is removed from the cell, theprocess is referred to as differential liberation,while if the liberated gas is not removed andremains in contact with the oil, then the process isreferred to as a flash liberation process.Differential and flash liberation processes can leadto different amounts of oil and gas at the finalconditions of the test, and this why fluid samplesare usually subjected to both tests.

The basic PVT analysis consists of three parts(Dake, 1978): bubble-point pressure determinationfrom a flash expansion of the fluid sample, calculationof the formation volume factors of oil and gas and thesolubility of gas in the oil as functions of pressureusing a differential liberation process, andperformance of flash liberation tests for a set ofseparator conditions of pressure and temperature sothat the PVT data can be recalibrated as needed tomatch operating field conditions. More complex PVTtests include detailed compositional analysis of thereservoir fluid sample. Additional importantmeasurements which are usually performed include,among others, measurement of the gas compressibilityfactor, gas specific gravity, and oil viscosity asfunctions of pressure.



cores

logs

seismic survey

other

multiple models

geological survey

production data

well tests

data gathering

data integrationgeomodels

grid generationupscaling

simulationhistory matching

information fordecision making

Fig. 7. Reservoir simulation workflow.

The functional relations of rock/fluid properties,such as relative permeability and capillary pressure,are usually obtained from laboratory displacementtests performed on reservoir core samples. The relativepermeability plays a very important role in reservoirdisplacement processes, and obtaining representativerelative permeability curves for a particular reservoirformation is a challenging problem. Details of themethods used to obtain these important properties willnot be presented here (see Chapter 4.1). We emphasizehere that while discussions of reservoir heterogeneityusually focus on the spatial variation of (absolute)permeability and porosity, significant heterogeneity inrelative permeability and capillary pressure is alsocommonly observed and is usually correlated with thecomplex lithology of reservoir rock.

History matchingHistorical dynamic data include pressure

measurements, oil production, Water/Oil Ratio(WOR), and Gas/Oil Ratio (GOR) as a function oftime, both on a well-by-well and field basis. Thefrequency and accuracy of the measurements varieswidely and depends on, among other things, the size ofthe reservoir asset, prevailing regulations, location,and the field operator.

Historical flow performance data are used tovalidate and update the RCM so that it is consistentwith all available information about a specific site.This process of integrating the dynamic data into theRCM is referred to as history matching. Confidence inpredictions of future performance using reservoir flowsimulation hinges in a large part on the quality of thehistory match. Quality in this case is measured interms of accurate representation of the relevantreservoir properties and physical processes that dictatethe flow and transport behaviours. This importantvalidation phase is often the most time consuming partof a simulation study because the history matchingprocess itself usually requires running the reservoirflow simulator repeatedly to guide the calibration ofthe RCM.

Significant progress in history matching methodshas taken place in the last few years. Given thecomplexity of the problem, however, engineeringknowledge and experience are essential in order toarrive at a history match that is both consistent withthe available information, and more importantly, ahistory match that is actually useful in making morereliable predictions of future performance. After all,reservoir simulation is intended to be a predictivereservoir management tool, both from the perspectiveof long term development plans as well as the abilityto rationalize and optimize the relatively short termoperating conditions of existing reservoir assets.

Obviously, history matching can only be carriedout when the model is run during the production life ofa field and not before the production begins.

Forecasting performance Reservoir simulation can be used: a) to update,

or calibrate, the RCM based on dynamic historicaldata; b) to make performance predictions based onthe existing development strategy; c) to explore awide range of recovery processes and futuredevelopment plans; d) to optimize the placementand operations of existing and future wells; ande) to quantify the uncertainty associated with thepredictions.

A reservoir simulator is a computational tool.Thus, the range of possible uses is difficult to cover.The utility and value of this computational tooldepends strongly on the engineer using it, and thecontext in which it is applied and integrated. Broadlyspeaking, a reservoir simulator allows the explorationof limitless ‘what-if ’ scenarios related to recoveryprocesses and development plans for a specificreservoir. Given the limited and incompleteinformation that is available in practice when makingdevelopment decisions, a reservoir simulator alsoplays a vital role in the quantification of theuncertainty associated with predictions of futurereservoir performance. In that sense, a reservoirsimulator is a vital tool for quantitative reservoirmanagement. We should also not overlook the utilityof numerical reservoir simulation in exploring andunderstanding the complex physics associated withreservoir flow processes.

4.6.4 Practical considerations

In this section, we discuss model selection, in terms ofthe fluid model and grid selection.

Fluid model choice The choice of the fluid model (e.g. black-oil,

compositional, thermal-compositional) depends on theproperties of the reservoir fluids and the recoveryprocess under study. As described earlier, the standardblack-oil model is based on a two-component (oil andgas) description of the hydrocarbon, which arepartitioned between the oil and gas phases accordingto simple PVT relationships. These PVT relationshipsare in the form of pressure dependent formationvolume factors and gas solubility. Generally speaking,a compositional model is only necessary when themass transfer between the components cannot bedescribed using the simple PVT relations employed inthe black-oil model.



Despite its simplicity, the black-oil model iswidely used in reservoir simulation practice. Unlessthe oil is volatile (e.g. if it is mainly composed oflight components along with high gas solubility),natural depletion processes of oil reservoirs underthe various drive mechanisms, including solutiongas, gas cap, and aquifer support, can usually bemodelled using a black-oil approach. The black-oilmodel is also popular for the simulation of variousinjection processes, including waterflooding andimmiscible gas injection. However, a black-oilmodel is inadequate for accurate simulation of EORprocesses where complex mass transfer between theinjected and resident fluids takes place. Examples ofsuch processes include the injection of gas that ismiscible or near-miscible with the resident oil. Forsuch recovery processes, a compositionalformulation is necessary to describe the physics thatdictates the behaviour of the flow and transportprocesses accurately.

In thermal processes, such as steam injection, theenergy equation must be added to the mathematicalstatement that describes the flow and transport in thereservoir. A thermal, black-oil approach may beadequate for modelling steam injection. However, it isoften the case that a thermal-compositional model isrequired to accurately describe the complex masstransfer and physical interactions that dictate thedisplacement behaviour.

The physics and equations that describe EORdisplacement processes, such as polymer, andsurfactant flooding, can be quite complex; as a result,special-purpose simulators capable of modellingchemical and surfactant injection processes have alsobeen developed. It is not uncommon to have a differentsimulator for each fluid model: black-oil,compositional, miscible, thermal, etc. Recently,however, there has been a concerted effort to buildmulti-purpose reservoir simulators with the ability tomodel a wide range of recovery processes.

Grid selection In reservoir simulation, it is common to

categorize the simulation grid in terms of itsgeometry (e.g. Cartesian, stratigraphic, radial,multi-block, unstructured), dimensionality (e.g. one, two, or three dimensions), and overallextent (single-well, pattern, sector, field). Selectionof the appropriate grid for a particular simulationstudy is a difficult matter, and it depends on severalinterrelated factors. These comprise: a) theobjectives of the study; b) the complexity of therecovery process; c) the specific character of thetarget region of interest (geometry, fluids,heterogeneity, number and configuration of wells,

etc.); d) the quality and quantity of availableinformation; e) the available human and computingresources. Thus, it is difficult to give specificguidelines for grid selection. As a result,engineering judgement is crucial for building thesimulation model, including grid selection.

Generally speaking, however, whether theobjective is to simulate near-well or field-scalebehaviour, there is a growing trend in gridcomplexity and resolution. This is the result ofgrowing demand for accurate descriptions ofcomplex geometrical features in three spatialdimensions, of both geologic structures andcomplex wells (e.g. deviated and multilateral), andto meet the resolution requirements imposed by theRCM (property description). Of course, we shouldkeep in mind that we need to accurately resolve therelevant length and time scales that govern thephysics of the flow process being modelled.

4.6.5 Prediction uncertainty

Building a coherent RCM that integrates all availableinformation is closely tied to the predictive reliabilityof the computed predictions of future flowperformance. However, the integration of data fromdifferent sources with varying quantity, quality, andscale is a challenging problem.

Populating the RCM with permeability is of centralimportance because it has a significant impact on thepredicted behaviour of flow and transport. Detaileddescription of the spatial distribution of permeabilityis complicated because permeability of natural porousformations, such as oil reservoirs, often displayscomplex, multiscale spatial correlation structures andhigh levels of variability, and we typically havemeasurements at a few locations only (Dagan, 1989).To deal with the sparse availability of data on thesehighly heterogeneous natural geologic formations, astochastic, or probabilistic, framework is usuallyemployed (see Chapter 4.5). In this frameworkreservoir properties are treated as random spacefunctions. In addition to available measurements, theproperty, e.g. permeability, is usually described by itsmean, variance, and correlation structure. Such adescription is possible when available data allowaccurate delineation of large-scale features, but whenonly a weaker form of information (statisticalmoments) is available about spatial property variationwithin these large-scale features.

The uncertainty in the RCM due to incompleteknowledge of the formation properties leads touncertainty in the performance predictions obtainedfrom reservoir flow simulation. Above, we focused



on permeability uncertainty, however, we emphasizethat other sources of uncertainty must also beconsidered.

The Monte Carlo Simulation (MCS) approachis widely used in the oil industry to quantify theprediction uncertainty (Haldorsen and Damsleth,1990). In the MCS approach, high-resolution,equiprobable RCM realizations are generated. EachRCM realization is constructed such that availabledeterministic and statistical information arehonoured (Deutsch, 2002). A flow simulation isthen performed for each RCM realization in theensemble. The number of realizations in theensemble should be large enough to span theuncertainty space of interest. Statistical post-processing of the simulation results can thenbe used to generate a report on the predictions andthe associated uncertainty for quantities of interest(e.g. mean, standard deviation, and confidenceintervals for well pressures, production rates, andcumulative recovery). It should be noted that inaddition to the RCM, the Flow Simulation Model(FSM) also includes the fluid properties(e.g. density, viscosity, and phase behaviour), andthe rock/fluid properties (e.g. capillary pressureand relative permeability relations). Moreover,initial and boundary conditions and applicablecontrols must be specified.

When performing MCS in practice, time and costconsiderations, in terms of both human andcomputational resources, usually limit both theresolution of the individual realizations as well as thenumber of realizations in the ensemble. Theresolution gap between the RCM and the FSM iscommonly bridged using scale-up techniques. Scale-up methods attempt to coarsen the RCM to a size thatis more amenable to reservoir flow computations.The resulting coarse model should capture theessential effects of the fine-scale details of the flowbehaviours. The process of scale-up often includesregridding for the purpose of flow simulation(Durlofsky et al., 1997).

A different approach to quantifying predictionuncertainty is based on solving the equations thatdescribe the statistical moments of the quantities ofinterest (pressure, saturation, production rate). TheseStatistical Moment Equations (SMEs) are derivedfrom a stochastic statement of flow and transport inthe heterogeneous porous medium. It is nowdocumented that SME methods are applicable for awide range of variability levels and correlationlength scales (Li et al., 2003). However, significantprogress is necessary before SME methods can beused for practical general-purpose reservoirsimulation.

4.6.6 Development plans

Development plans are usually based on detailedsimulation studies that are linked with economicanalysis of the development stages. For cases in whichthe recovery process is well defined, the developmentplan may include, for instance, the specification ofoptimal completion intervals for existing wells inaddition to the drilling of new in-fill wells, their typesand locations, and specific completion intervals.Reservoir simulators coupled to surface facilities canbe used to specify the capacity and infrastructurerequirements during the various development stages.Examples include water and gas handling facilities.

As mentioned earlier, with the help of reservoirsimulation, development plans, whether modest andshort term or ambitious and long term, focus onmaking predictions and providing a quantification ofprediction uncertainty along with assessments ofeconomic risk.

4.6.7 Development schemes

The location and the spacing of production wells andthe interval to be open to production (perforations)depend upon a number of different factors: a) thestructural configuration of the reservoir; b) thepetrophysical characteristics of the reservoir rock andthe type of fluid it contains; c) the presence or lack ofa gas cap; d) the nature of the hydrocarbon-watercontact (bottom contact or edge contact); e) theprevailing drive mechanism.

For example, in an undersaturated oil reservoir withno water drive, in which the principal drive mechanismis represented by solution gas, the wells may be spacedaccording to a square grid, where the distance betweenproduction lines (normally variable between 400-500 mto over 1 km) depends exclusively upon thepetrophysical characteristics of the reservoir rock (Fig. 8). The interval to be open to production may berelatively wide and also close to the Oil-Water Contact(OWC) but need to avoid perforations in the upperstructural layer, given that these types of reservoirs mayform a secondary gas cap during the course of itsproductive life. This occurs when the pressure of thereservoir has dropped below the bubble point.

A similar strategy may also be adopted whendealing with a saturated oil reservoir with a gas cap,under weak or no water drive at all, and with oil-waterbottom contact (the contact is referred to as bottomcontact when the thickness of the level is greater thanthe hydrocarbon column, i.e. the difference in heightbetween the upper structural area and the depth of oil-water contact).



In the case of a natural gas reservoir withcharacteristics similar to those described above,given equal properties of the rock, the spacingbetween the wells is greater as a result of the highermobility of natural gas with respect to oil. Thelocation of the wells and the perforations in this casemust, however, be limited largely to the upperstructural area (in the presence of either weak orstrong water drive), in order to avoid an early waterbreakthrough in the wells and in order to recover allof the gas contained in the trap.

In the event of a saturated oil reservoir with a gascap and edge oil-water contact (edge contact occurswhen the hydrocarbon column is greater than thethickness of the level), the production wells can besituated mainly along an external ring located in thestructural flanks (Fig. 9). In this case, the perforationscan be placed at an adequate distance from the oil-water contact and closer to the Gas-Oil Contact(GOC), if the water drive is strong and the gas cap isof limited extent, in order to avoid the formation ofwater coning. The perforations will instead be locatedcloser to the oil-water contact, and far away from thegas-oil contact, if the water drive is weak or zero andthe gas cap is large, so as to avoid of the risk of gasconing. The perforations will be located in anintermediate zone, between the two contacts, in theevent of the concomitant presence of an wide gas capand a strong water drive.

In the case of an undersaturated oil reservoir and astrong water drive, the production wells will be largelyconcentrated in the upper structural area (as far as thepetrophysical characteristics of the reservoir rock

allow it) in order to avoid, as much as possible, theearly water breakthrough in the wells following thenormal raising of the oil-water contact and/orphenomena of water coning and water fingeringduring the productive life of the field.

The above examples can be considered to be thebasic development schemes; of course, in reality aseries of intermediate cases with respect to thosedescribed above is possible, each of which must beindividually considered, bearing in mind, in any case,the criteria explained here.

The schemes described above are based on theexclusive consideration of vertical wells. In the case ofhorizontal or multilateral wells (with a number ofhorizontal sections that branch out from a singleborehole) these schemes may be slightly modified,especially with respect to the number of wells.

4.6.8 Injection schemes

In the processes of secondary recovery an attempt ismade to increase the recovery factor by injecting fluidsthat are immiscible with oil into the reservoir in orderto almost totally, or at least partially, restore the energydissipated during production and to create or increasethe process of displacement.

The methods of injecting immiscible fluids includethe use of natural gas (gas injection) or water(water injection). The choice of which of these types of injection will be used may depend on thecharacteristics of the reservoir and the type of oil present,though in many cases it is determined by the availabilityof the fluid to be injected and its cost (natural gas is without doubt more costly than water).It would in fact be illogical to choose to inject natural gas into a reservoir if this fluid is not availablein the area. According to statistics water injectioncurrently represents the most widely used techniqueof secondary recovery.

The gas injection technique calls for the use ofspecific wells (existing or to be drilled), located in theupper portion of the structure, through which naturalgas can be injected into an existing gas cap (original orsecondary) or through the creation of an artificial gascap. The displacement of oil towards the productionwells by the injected gas is the same as that whichoccurs in the natural drive mechanism generated bythe expansion of a naturally occurring gas cap (seeChapter 4.3).

The gas to be injected may be that which has beenrecovered in the separator (gas dissolved in oil), it maycome from a gas layer that is present in the samegeological structure, located above or below the levelof oil production, though not in hydraulic



wells

oil

water

crosssection

map

OWC

Fig. 8. Example of a development scheme in an anticlinal oil field whose main drive mechanism is solution gas.

communication with it, or it may come from nearbyfields, whether it be a solution gas or a gas fromgasiferous horizons.

Prior to being injected, the gas must undergo aseries of treatments (normally it is dehydrated and insome cases the condensates may be removed); theinjection is to be made using compressors or a seriesof compressors (see Chapter 5.3).

The injection of water into a reservoir, as part ofthe technique of secondary recovery, may take placewithin the aquifer below the oil zone, in a peripheralarea ( peripheral injection) or directly within the oilzone (dispersed injection), according to a variety ofdifferent injection patterns.

The choice to use peripheral injection ordispersed injection depends primarily upon the petrophysical characteristics of the reservoir rock (above all its permeability) and the sizeof the reservoir. In fact, peripheral injection is used in most cases for relatively permeablereservoirs, while dispersed injection is used for reservoirs with low levels of permeability. In both cases the injection of water has the objective� other than that of providing energy to the reservoirin order to maintain the pressure above a certainvalue (above the bubble point), thus preventing the gas liberation into the reservoir � of creating a displacement front in order to push the oil towards the production wells and, therefore, maintainan acceptable level of production whilesimultaneously increasing the recovery factor. In most cases the injection takes place at balance, i.e.by injecting a quantity of water that is comparable tothe amount of oil being produced.

The peripheral water injection (Fig. 10) occurs,as has been stated, in the aquifer in an area that isperipheral to the reservoir, and is undertaken, inmost cases, during the productive life of the field.The number, location and the injection rate of thewells is established by means of numericalsimulations and injectivity tests (see Chapter 4.4).The action and the effectiveness of a peripheralwater injection are, in all respects, very similar tothose of the expansion of a strong natural aquiferand allow very high levels of recovery (greaterthan 50%) to be obtained.

In very large reservoirs where the reservoir rockpresents low levels of permeability, the action of aperipheral injection (and also that of a naturalaquifer) would be very slow and, therefore,ineffective in terms of keeping the production of oilat an economically sustainable level. In this case it isthus preferable to inject water directly into the oilzone (dispersed injection). In this technique ofinjection the production wells and the injection wellsare uniformly distributed across the area of thereservoir, using a regular grid that may assumevarious configurations (injection patterns). The mostcommonly used patterns are those of parallelalignment (direct line drive), staggered alignment(staggered line drive), or geometries with fourpoints ( four spot), five points ( five spot), sevenpoints (seven spot) or nine points (nine spot), asillustrated in Fig. 11. In addition to maintaining thepressure of the reservoir, injection into the oil zonecreates a number of displacement fronts movingtowards the production wells and it is possible, insome cases, to obtain recoveries that are greater thanthose obtained with a peripheral injection (for details on the phenomenon of displacement seeChapter 4.3).

The technique of dispersed injection will bemore effective when dealing with fairlyhomogeneous and, above all, non-fractured reservoirrocks, given that the injected water may bechannelled into preferential paths and arrive tooquickly at the production wells, leaving behindnotable quantities of oil. Naturally, the injectionpattern to be adopted must be tested usingmathematical simulation models, with respect to thelocation and spacing of the wells, as well as theirrates. In most cases before undertaking any patternof dispersed injection it is standard practice to testthe efficiency of the injection by means of pilotplants, for a fixed period of time in a particular area,in order to later extend the project to the entire areaof the reservoir.

The dispersed injection pattern � given that,on the basis of the reservoir characteristics, it



crosssection

wells wells

oil

oil and water

gas - oil - water

gas and oil

gasGOC

OWC

OWC (top)

GOC (top)

OWC (bottom)

water

map

Fig. 9. Example of a development scheme in an anticlinal oil field with weak water drive.

must be assumed that the injection of water willtake place during a relatively early period of itsproductive life � requires that also the injectionwells be drilled during the initial developmentphase of the reservoir and put in production for acertain period of time, together with theproduction wells. Later, they will be convertedinto injection wells when the pressure has droppedto values that are close to the bubble point; in thisway the process of injection will be assisted by thefact that it will not be necessary to pump the waterat high pressures.

The water to be used for injection, which may beconsiderable in quantity, can be taken fromsuperficial aquifers, from the sea if the reservoir islocated offshore or on land but near the coast, orfrom surface waterways (in addition to the waterproduced by the reservoir, if the quantities areconsiderable).

Prior to proceeding with water injection it isnecessary to chemically analyse the compatibilitybetween the water to be injected, the reservoir rockand the fluids contained, in order to avoid theformation of precipitates (usually sulphates) whichcan plug the rock pores. Furthermore, the watermust be filtered using appropriate filters in orderto remove any suspended solids, as well as beingde-oxygenated and treated with bactericides inorder to prevent the formation of bacterialcolonies, which in turn may lead to the creation ofgelatinous masses which plug the rock pores; andfinally it must also be treated with chemical

additives in order to prevent scales and corrosionin the equipment being used. With regard to thesystems for water pumping and water treatment,reference should be made to Chapter 5.3.

References

Aziz K. (1993) Reservoir simulation grids. Opportunities andproblems, «Journal of Petroleum Technology», 45, 658-663.

Aziz K., Settari A. (1979) Petroleum reservoir simulation,London, Applied Science Publishers.

Cao H. (2002) Development of techniques for general purposesimulators, PhD Dissertation, Stanford University, Stanford(CA).

Coats K.H. (2000) A note on IMPES and some IMPES-basedmodels, «Society of Petroleum Engineers Journal», 5,245-251.

Coats K.H. (2003) IMPES stability. Selection of stable timesteps,«Society of Petroleum Engineers Journal», 8, 181-187.

Dagan G. (1989) Flow and transport in porous formations,Berlin-New York, Springer.

Dake L.P. (1978) Fundamentals of reservoir engineering,Amsterdam, Elsevier.

Deutsch C.V. (2002) Geostatistical reservoir modelling,Oxford-New York, Oxford University Press.

Durlofsky L.J. et al. (1997) A non-uniform coarseningapproach for the scale up of displacement processesin heterogeneous porous media, «Advances in WaterResources», 20, 335-347.

Haldorsen H.H., Damsleth E. (1990) Stochastic modelling,«Journal of Petroleum Technology», 42, 404-412.



map cross section

A

A

A

A

gas zone

oil zone

water zone

area swept by the water front

injection wells

production wells

GOC

OWC after water injection

OWC prior to water injection

Fig. 10. Example of peripheral waterinjection.



a

d

d a

a

d

h a

A. Direct line drive pattern

B. Staggered line drive pattern

C. Four spot pattern

D. Five spot pattern

E. Seven spot pattern

D. Nine spot pattern

coefficient of areal sweep efficiencyproduction wells

injection wells

d/a = 1 h a = 57%d/a = 4 h a = 90%

d/a = 0.5 h a = 72%d/a = 1.5 h a = 80%d/a = 4.0 h a = 90%

h a = 74% d/a = 0.5 h a = 72%

h a = 74% h a = 52%

Fig. 11. Main oil injection patterns.

Lake L.W. (1989) Enhanced oil recovery, Englewood Cliffs(NJ), Prentice Hall.

Lee S.H. et al. (2002) A finite-volume method with hexahedralmultiblock grids for modelling flow in porous media,«Computational Geosciences», 6, 353-379.

Lee S.H. et al. (2003) New developments in multiblock reservoirsimulation. Black-oil modelling, non-matching sub-domains,and near-well upscaling, in: Proceedings of the Society ofPetroleum Engineers reservoir simulation symposium,Houston (TX), 3-5 February, SPE 79682.

Li L. et al. (2003) Perturbation-based moment equationapproach for flow in heterogeneous porous media.

Applicability range and analysis of high-order terms,«Journal of Computational Physics», 188, 296-317.

Thomas G.W., Thurnau D.H. (1983) Reservoir simulationusing an adaptive implicit method, «Society of PetroleumEngineers Journal», 23, 759-768.

Khalid Aziz

Hamdi A. TchelepiDepartment of Petroleum Engineering

Stanford UniversityStanford, California, USA



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4.6 Dynamic reservoir study - Treccani, il portale del sapere I / EXPLORATION, PRODUCTION AND TRANSPORT 4.6.1 Oil recovery processes Natural drive mechanisms The main drive mechanisms