Asymptotics, Saturation Fronts, and High Resolution Reservoir Characterization

Transport in Porous Media42: 315–350, 2001.c© 2001Kluwer Academic Publishers. Printed in the Netherlands. 315

Asymptotics, Saturation Fronts, and HighResolution Reservoir Characterization

D. W. VASCO1 and AKHIL DATTA-GUPTA2

1Earth Sciences Division, Berkeley Laboratory, Berkeley, CA 94720, U.S.A.2Department of Petroleum Engineering, Texas A & M University, College Station, TX, U.S.A.

(Received: 1 March 1999; in final form: 1 May 2000)

Abstract. Using an asymptotic methodology we formulate a fast, accurate algorithm for the inver-sion of multi-phase flow data. The approach is appropriate for many common reservoir productionstrategies such as CO2 and water flooding. The technique compares well to a purely numericalmethod with a significant reduction in computation time. In an application to fractional flow datafrom the North Robertson field in West Texas, 100,000 permeability and porosity parameters aredetermined on a workstation. Generally, higher permeability, approaching 1 milli-Darcy, is found inthe eastern portion of the reservoir. The permeability estimates agree with type curve analysis formaterial and volumetric balances and a previous numerical pilot-point inversion.

Key words: reservoir characterization, multi-phase flow, fractional flow, streamlines.

1. Introduction

Understanding the nature of flow within a reservoir is key to reservoir managementand the clean-up of containments at hazardous waste sites. Controlling factors forflow are the permeability and porosity distributions within the reservoir. However,only a small portion of the data obtained at either oil fields or waste sites is directlyrelated to reservoir flow properties. One useful class of data, measured system-atically at almost all sites, are the components of the fluids extracted at a well.Examples of such data are: water-cut, the ratio water/(oil+water) gathered at a pet-roleum reservoir under waterflood, and extraction concentrations during cosolventflushing (Raoet al., 1997). Unfortunately, the relationship between fractional flowdata, such as water-cut, and the porosity and permeability structure of a reservoir ishighly nonlinear, governed by the equations of multi-phase flow. Such informationis only now being used to infer permeability variations between shallow boreholesor deep wells (Finsterle and Pruess, 1995; Vasco and Datta-Gupta, 1997; Xue andDatta-Gupta, 1997).

The inversion of multi-phase flow data is computationally intensive (Finsterleand Pruess, 1995; Huang and Kelkar, 1996), generally requiring an order of mag-nitude or more computation than does forward modeling. By forward modelingwe mean simulation: one is given a model of reservoir structure with the task of

316 D. W. VASCO AND AKHIL DUTTA-GUPTA

predicting what would be observed during production. Calculating the fractionalflow at a well can be done using finite difference, finite element, or streamlineapproaches (Lake, 1989). Depending on the sophistication and assumptions, thecalculations can be substantial. The inverse problem of using two-phase data toestimate the spatial distribution of reservoir permeability requires the repeatedsolution of the forward problem of reservoir simulation. For example, stochasticapproaches such as simulated annealing require numerous reservoir simulations(Datta-Guptaet al., 1995). Alternatively, most gradient-based formulations of theinverse problem require computation of the sensitivity of multi-phase flow data tochanges in model parameters. That is, we must compute the change in the frac-tional flow at an observation point induced by a deviation in subsurface hydrologicproperties, such as porosity, or permeability. Currently, there are three main ap-proaches for estimating sensitivities: perturbation methods, direct algorithms, andadjoint state methods (Carteret al., 1982; Yeh, 1986; McLaughlin and Townley,1996). Each approach has distinct advantages and drawbacks, but all require eitherextensive computation and/or significant code development. At present all methodsare limited by computational considerations to 1,000–10,000 parameters (Xue andDatta-Gupta, 1997).

The asymptotic approach we shall describe is closely related to streamline meth-ods, front tracking techniques, and the method of characteristics. All of these meth-ods have been utilized in order to provide efficient numerical approaches to sim-ulate multi-phase flow. Streamline or streamtube reservoir simulation has a longhistory in petroleum engineering (Muskat, 1937; Fay and Pratts, 1941; Higginsand Leighton, 1962; Martin and Wegner, 1979; Hewett and Behrens, 1991; Datta-Gupta and King, 1995; Bratvedtet al., 1996; Batyckyet al., 1997; Emanuel andMilliken, 1997). An overview of streamline simulation has been provided by Kingand Datta-Gupta (1998). Front tracking techniques and related level set methodshave been applied to reservoir engineering (Glimmet al., 1983; Bratvedtet al.,1992; Tijink et al., 1994) as well as a wide range of other applications (Sethianet al., 1996) including the viscous Navier–Stokes equations (Zhu and Sethian,1992) and groundwater flow (Schafer-Perini and Wilson, 1991). The method ofcharacteristics has been mixed with both finite difference and finite element tech-niques in order to model reservoir processes (Douglas and Russell, 1982; Ewinget al., 1984; Russell, 1985). Characteristic techniques have also been combinedwith operator splitting to model gravitational effects (Dahleet al., 1990). Gen-erally, these efficient techniques have not been fully utilized in the inversion ofmulti-phase flow data. In particular, quantities inherent in these approaches such asfront positions and streamline geometries have not been used to great advantage inimaging reservoir flow properties.

As shown in this paper, an asymptotic approach leads to an extremely effi-cient formalism for subsurface imaging, applicable to three-dimensional problems.Using this technique we are able to obtain higher resolution models of reservoirstructure, of the order of 100,000 or more parameters. Asymptotic methods are

HIGH RESOLUTION RESERVOIR CHARACTERIZATION 317

effective in such fields as electromagnetics and optics (Kline and Kay, 1965), seis-mology (Nolet, 1987), fluid mechanics (Gersten, 1995), and hydrology (Grundyet al., 1994; Jaekelet al., 1996; Vasco and Datta-Gupta, 1999). For example,asymptotic formulations are now used in seismology to solve very large problemsinvolving millions of travel time data and hundreds of thousands of unknowns(Vasco et al., 1999). The key point is that the sensitivities needed for solvingthe inverse or imaging problem result from very few forward runs. Thus, suchalgorithms are orders of magnitude faster than current techniques which typicallyrequire a very large number of forward runs. Using the formalism, large scale three-dimensional imaging problems may be solved in hours on a workstation, ratherthan days.

A final reason for pursuing the asymptotic formulation is that it offers valu-able insight into subsurface processes. This is particularly true for the applica-tion to multi-phase transport. The Eikonal equation, derived below, governs thepropagation of a multi-phase front. Its analytical form clearly indicates the parti-cular combination of flow properties (saturation, porosity, permeability, relativepermeabilities) that influence the front travel times. Such understanding is criticalin determining the trade-offs and uncertainties associated with using multi-phaseflow data to image a reservoir.

2. Methodology

The process of interest in reservoir modeling, multi-phase flow, is described bya governing set of partial differential equations (Bear, 1972; Peaceman, 1977;Helmig, 1997). A basis for the formulation of the reservoir imaging problem isan asymptotic form for the solution to these governing equations. In essence, anasymptotic representation emphasizes the rapidly varying spatial and temporalattributes of the solution. Distinct scalar equations governing travel times andamplitudes are produced. In this section we outline how the asymptotic approachis applied to the inversion of multi-phase reservoir data to map variations in per-meability and porosity.

2.1. GOVERNING EQUATIONS FOR TWO-PHASE FLOW

Our starting point is the set of simultaneous partial differential equations describingthe flow of an aqueous (wetting) phase and a non-aqueous (non-wetting) phase(Peaceman, 1977),

∇ ·[ρwK(x)krw

µw∇(Pw(x, t)− ρwgz)

]= φ(x)∂(ρwSw)

∂t,

∇ ·[ρnK(x)krn

µn∇(Pn(x, t)− ρngz)

]= φ(x)∂(ρnSn)

∂t,


whereSw andSn denote the saturations of the aqueous and non-aqueous phasesrespectively. The relative permeabilities of the aqueous and non-aqueous phases arerepresented bykrw andkrn while the absolute permeability is given byK(x). Therespective densities areρw andρn, the gravitational constant isg and the porosity isφ(x). The pressure associated with the aqueous phase isPw(x, t) while the pressurefor the non-aqueous phase isPn(x, t), the viscosities areµw andµn. The aboveequations are coupled because the saturations are assumed to sum to unity

Sw + Sn = 1.

Making use of the fact thatSw = 1−Sn, we may derive a single equation describingthe evolution of the saturation of the aqueous phase which we shall denote byS(x, t) in all that follows,

∇ · [ρwH(S)∇S] −∇ · [ρwF(S)u+ ρwG(S)z] = φ(x)∂(ρwS)

∂t, (1)

wherez is the unit vector in the downward direction,u denotes total velocity, thesum of the velocity of the aqueous(uw) and non-aqueous(un) phases:u = un+uw,andF(S),H(S), andG(S) are specific functions of saturation (Peaceman, 1977).In particular,F(S) is the fractional flow of the aqueous phase, given by

F(S) = λw

λn+ λw

for non-aqueous and aqueous phase mobilitiesλn andλw,

λn = K(x)krn(S)

µn

and

λw = K(x)krw(S)

µw.

Note thatF(S) itself does not depend explicitly on the absolute permeability dis-tribution,K(x). Similarly, the functionH(S) is given by

H(S) = − λnλw

λn+ λw

dPc

dS

for capillary pressurePc = Pn− Pw. Finally we have the termG(S)

G(S) = F(S)λn(ρw − ρn)g

which contains gravitational effects.While it is entirely feasible to apply the asymptotic analysis, described in the

next subsection, to the very general equation (1) we choose not to. In our treatmentwe shall consider incompressible fluids and assume that capillary pressure effects


are dominated by convection. There are several reasons why we prefer to considera more limited equation, based upon these simplifying assumptions. First, becauseour methodology is new we wish to avoid the complications of the fully generalformulation which could distract from our discussion of the asymptotic approach.Relaxation of these assumptions will be the topic of future work. Furthermore,there are many important applications, such as cosolvent flushing of hazardouswaste sites and waterflooding in petroleum reservoirs, in which our current treat-ment will be appropriate. In fact, our approach is suitable for the North Robertsonfield application presented below. Finally, by including only the essential elementsin our formulation and considering field situations for which the formulation isaccurate we can validate the approach and develop a foundation upon which toconsider more general conditions.

Let us now consider the consequences of our simplifying assumptions. For in-compressible fluids we may factor out the density termsρw. Negligible capillarityimplies (Bear, 1972; Peaceman, 1977) thatH(S) vanishes and that the pressures inthe aqueous and non-aqueous phases are equal, hence

∇Pn(x, t) = ∇Pw(x, t) = ∇P(x, t),P (x, t) shall denote the reservoir pressure, andu is the total velocity

u = −κ(x)K(x)∇P(x, t), (2)

whereκ(x) is the total mobility

κ(x) = krn

µn+ krw

µw.

As a result of the preceding assumptions, Equation (1) reduces to

φ(x)∂S

∂t+ dF

dSu · ∇S +∇ · [G(S)z] = 0 (3)

(Peaceman, 1977; Lake, 1989; Bedrikovetsky,1993; Helmig, 1997), a quasi-linearhyperbolic partial differential equation. Defining the composite vectorU,

U = dF

dSu+ dG

dSz (4)

we may write Equation (3) as

φ(x)∂S

∂t+ U · ∇S + E(S) = 0, (5)

where

E(S) = F(S)(ρw − ρn)gkrn(S)

µn

∂K(x)∂z

.

The termE(S) explicitly accounts for spatial variations in permeability. The spatialvariations in the relative permeability parameters,krn andkrw, are assumed to be


primarily through saturation variations. We may normalize by the porosity to arriveat the equation

∂S

∂t+ V · ∇S + C(S) = 0, (6)

whereV = U/φ(x) andC(S) = E(S)/φ(x).

2.2. ASYMPTOTIC SOLUTIONS FOR TWO-PHASE FLOW

The motivation for our asymptotic solution is based upon a variation in scale.That is, we assume that the initial saturation distribution is a relatively slowlyvarying function of space and time when compared to the changes across the two-phase front. In effect, there is a scale, denoted byL, describing the variation inbackground saturation in time and space and a scale3 describing the spatial andtemporal variation across the propagating two-phase front. We are assuming thatL � 3 holds in the domain of interest. If we denote3/L by ε the condition is0 < ε � 1. An asymptotic expansion is the representation of the solution as aformal series in powers of the parameterε (Anile et al., 1993):

S(x, t) = S0(x, t) +∞∑n=1

εnSn(x, t, ω) (7)

whereS0(x, t) represents the background variation in saturation andω is the fre-quency of the wave. The frequencyω is assumed to have the form

ω = σ (x, t)ε

(8)

(Anile et al., 1993) whereσ (x, t) is the phase variation of the wave, a function de-scribing the geometric configuration of the propagating multi-phase front in spaceand time. In the next section we shall provide more details concerning the functionσ (x, t). We should note that the asymptotic representation is often given in themore explicit form

S(x, t) = eiω∞∑n=0

εnSn(x, t), (9)

whereω is defined in (8). The representation provided by Equation (9) may bethought of as a local plane wave expansion (Kline and Kay, 1965). There are otherapproaches to front and wave propagation which produce concepts similar to thosewe shall encounter. For example, explicit consideration of a propagating discon-tinuity in saturation or in the derivatives of saturation also leads to ideas such asphase, characteristics, and rays (Kline and Kay, 1965; Luneburg, 1966; Whitham,1974; Anileet al., 1993; Sethian, 1996). We must emphasize that even though wewill examine the asymptotic solutions of Equation (3), the techniques also apply


in the case of equations with dissipative or dispersive terms (Anileet al., 1993;Chapmanet al., 1999), such as Equation (1). Asymptotic methods have even beendeveloped for purely parabolic problems such as the diffusion equation (Cohen andLewis, 1967; Vascoet al., 2000).

In constructing our asymptotic representation we substitute the expansion (7)of S(x, t) into the various terms of Equation (6). For example, consider the com-ponents of the vectorV(S) which may be represented as a power series inS. Theexpansion is given by

V(S, x, t) = V(S0, x, t)+ ε ∂V∂SS1+O(ε2), (10)

where O(ε2) denotes terms of orderε2 and higher. Similarly, the power seriesrepresentation ofC(S) is given by

C(S) = C(S0)+ ε ∂C∂SS1+O(ε2).

The partial derivatives ofS(x, t) are given by

∂S

∂xi= ∂S0

∂xi+ ε

(∂S1

∂xi+ 1

ε

∂S1

∂ω

∂σ

∂xi

)+ ε ∂S2

∂ω

∂σ

∂xi+O(ε2) (11)

and

∂S

∂t= ∂S0

∂t+ ε

(∂S1

∂t+ 1

ε

∂S1

∂ω

∂σ

∂t

)+ ε ∂S2

∂ω

∂σ

∂t+O(ε2) (12)

where we have made use of (8). Substituting expansions (10)–(12) into Equa-tion (6) produces an equation containing an infinite sequence of terms

∂S0

∂t+ ε

(∂S1

∂t+ 1

ε

∂S1

∂ω

∂σ

∂t

)+ ε ∂S2

∂ω

∂σ

∂t+ · · ·

+[V(S0, x, t)+ ε ∂V

∂SS1+ · · ·

]×

×[∇S0+ ε

(∇S1+ 1

ε

∂S1

∂ω∇σ

)+ ε ∂S2

∂ω∇σ + · · ·

]+

+C(S0)+ ε ∂C∂SS1+ · · · = 0. (13)

Each term containsε to some power as a factor. Note that the powers range fromzero onward. Multiplying out the quantities in parenthesis we find

∂S0

∂t+ ∂S1

∂ω

∂σ

∂t+ V · ∇S0+ V · ∇σ ∂S1

∂ω+ C(S0)+

+ ε[∂S1

∂t+ V · ∇S1+ S1

∂V∂S· ∇S0+ S1

∂V∂S· ∇σ ∂S1

∂ω

]+

+ ε[∂C

∂SS1+ ∂S2

∂ω

∂σ

∂t+ ∂S2

∂ωV · ∇σ

]= O(ε2). (14)


2.2.1. Eikonal Equation and Arrival Time of the Two-Phase Front

Becauseε is a small number, the most significant terms are those which containε

to the lowest degree. Therefore, in this subsection we consider those elements inEquation (14) of orderε0 ∼ 1, ε to the power zero. Neglecting terms containingεof order one or greater in Equation (14) produces the expression

∂S0

∂t+ ∂S1

∂ω

∂σ

∂t+ V · ∇S0+ V · ∇σ ∂S1

∂ω+ C(S0) = 0. (15)

Because the background saturation distributionS0 satisfies Equation (6)

∂S0

∂t+ V(S0, x, t) · ∇S0+ C(S0) = 0, (16)

expression (15) reduces to[∂σ

∂t+ V(S0, x, t) · ∇σ

]∂S1

∂ω= 0. (17)

Assuming that the saturation changeS1 does not vanish nor is it uniform in bothtime and space, we have the constraint onσ (x, t)

∂σ

∂t+ V(S0, x, t) · ∇σ = 0. (18)

The quantityσ (x, t), known as the phase, governs the propagation of the multi-phase front. There are several possible interpretations ofσ (x, t). For example, itcan be thought of as a surface, in space and time, across which the saturation or itsderivatives may be discontinuous, the characteristic surface (Kline and Kay, 1965;Jeffrey, 1976). As noted by Kline and Kay (1965), we may write the equation forthis surface in a separated form

σ (x, t) = 9(x)− t = 0. (19)

Then Equation (18) becomes

V(S0, x, t) · ∇9(x) = 1. (20)

Equation (20) is a partial differential equation for the reduced phase function9(x).The coefficients of the differential equation generally vary as a function of spaceand time, but they only depend on the background saturation distributionS0. If wedefine a unit vectorn normal to the multi-phase front, we may write (20) as

V(S0, x, t) · n|∇9(x)| = 1. (21)

Letting Vn denote the projection ofV(S0, x, t) onto the normal vectorn, we maywrite (21) as

Vn|∇9(x)| = 1. (22)


Equation (22), known as the Eikonal equation, plays an important role in the studyof several types of propagating waves (Whitham, 1974; Anileet al., 1993).

It is possible to solve Equation (20) or (22) directly, using purely numericaltechniques such as fast marching methods (Sethian, 1996). However, we shall uti-lize the method of characteristics (Jeffrey, 1976) to rewrite (20) as an equivalentsystem of ordinary differential equations. First, let us define the vectorp withcomponents

p = ∇9(x). (23)

It is evident from Equation (22) that the magnitude ofp is V −1n . The solution of

Equation (20) is accomplished through the calculation of trajectories traced out bypoints on the characteristic surface, the multi-phase front, as a function of time.That is, we consider a pointr(t) = (x(t), y(t), z(t)) on the multi-phase front asit evolves over time. In what follows it will be assumed that the primary changein saturation over time is due to the front itself. That is, we shall assume that thebackground state is not changing significantly with time. It is straight-forward toinclude a time varying backgroundS0 (Anile et al., 1993). The tangent vector tothe curver(t) is normal to the propagating front. Hence, it is proportional top,

drdt= p, (24)

where we have set the proportionality constant to one. Equation (24) is the first ofour ray equations, used to solve (20). As shown in the Appendix, by consideringsecond derivatives ofr with respect tot we may derive the second set of rayequations:

dpdt= 1

2∇(

1

V 2n

). (25)

Integration of Equations (24) and (25) produces the trajectories normal to thecharacteristics, the bi-characteristic curves.

Thus, in order to solve the Eikonal equation we must numerically integratethe two sets of first order ordinary differential equations, Equations (24) and (25),subject to initial and boundary conditions. In applications we are often interestedin trajectories which extend from an injection well to a producer. The two-pointboundary conditions would be specified by providing the initialr i and final rppoints on the trajectory. There are well established computational techniques forsolving equations of this type (Keller, 1968). In many cases the trajectories may beassociated with streamlines in the reservoir simulation. In particular, the expressionfor V contains the pressure gradient as contained inu, see Equation (2). However,there are situations in which trajectories, computed using Equations (24) and (25),will differ from streamlines. In particular, due to the gravity term in Equation (4),our trajectories may deviate from streamlines in the vertical direction.


One can consider the characteristics surfaces and the ray trajectories, often re-ferred to as bi-characteristics, as forming a coordinate system in which we mayframe the two-phase flow problem. Simplifications follow from recasting the prob-lem in characteristic coordinates, as we shall see. For the moment, let us considerthe problem of calculating the arrival time of a two-phase front at a producingwell. In our new coordinate system we define two axes, tangent to the iso-surface,9(x) = C, for some constantC. The other coordinate is given byr the distancealong the trajectoryr(t). In these curvilinear coordinates9(x) only varies withr.Therefore, we have∇9(x) = d9/dr and from (20) we find that

d9

dr= 1

Vn. (26)

From Equation (19), integrating from the injector to the producer along the traject-ory, labeled6

T =∫6

1

Vn(r)dr. (27)

The variableT is the total travel time of two-phase front from the injector tothe producer. Thus, we have an expression relating the propagation time of thetwo-phase front to properties of the reservoir, as contained inVn(x). For isotropicpropagation in which the front moves in the directionV we have

Vn =√V 2

1 + V 22 + V 2

3 , (28)

where

V1 = 1

φ(x)dF

dSu1, V2 = 1

φ(x)dF

dSu2, V3 = 1

φ(x)

(dF

dSu3 + dG

dS

), (29)

and the components ofu are given by Equation (2). When we can neglect gravitythe front velocity reduces to

Vn = 1

φ(x)dF

dSκ(x)K(x)|∇P(x, t)|, (30)

where the quantities are evaluated with respect to the background distributionS0

(x, t). We re-iterate this point, as it is important in our development of an in-version methodology. Equations (28)–(30) depend in an analytic fashion on theflow properties of the reservoir. For example, both (29) and (30) depend linearlyon the absolute permeabilityK(x). Thus, we may use these expressions to relateperturbations in flow properties to deviations in the arrival time of the two-phasefront.

At this point we could consider terms of higher order inε as is done by Anileet al., (1993). In addition, it is possible to consider the influence of capillaryeffects. For now we wish to illustrate the approach with the fewest possible com-plications. Therefore, in the next section where we describe the perturbation


approach, we examine the situation in which we may neglect capillary effects andgravity. This simplifies the comparison of our analytic sensitivity estimates withpurely numerical sensitivities For example, we do not have to consider time varyingvelocities.

2.3. SENSITIVITIES TO VARIATIONS IN RESERVOIR STRUCTURE

In this section we shall derive the sensitivity coefficients associated with modelparameter perturbations. That is, we shall determine how perturbations in reservoirflow properties map into perturbations in observable quantities such as the multi-phase front arrival time and saturation history at a producing well. We shall makeuse of our velocity estimates, in Equation (29), when considering arrival time sens-itivities. In addition, we will need Equation (3) in our examination of saturationhistories. Throughout this section we shall consider the situation in which bothcapillary and gravitational effects may be neglected. As shown in Anileet al.,(1993) such effects may be included in general. For example, ray methods havebeen developed for convection–diffusion problems (Chapmanet al., 1999). How-ever, in order to simplify this initial presentation of our approach, we excludesuch influences. Furthermore, for our application to water-cut observations fromthe North Robertson Unit, the effects of capillary pressure and gravity are thoughtto be small.

We shall consider the equation for two-phase flow, Equation (3), for the situationin which the gravitational effect may be neglected:

φ(x)∂S

∂t+ dF

dSu · ∇S = 0, (31)

the Buckley–Leverett equation (Buckley and Leverett, 1942). It is more convenientto consider the partial differential equation in terms of saturation and the fractionalflow of the aqueous phase

φ(x)∂S

∂t+ u · ∇F(S) = 0. (32)

We may rewrite this equation in terms of the vector

W = uφ(x)

(33)

producing our basic relationship:

∂S

∂t+W · ∇F(S) = 0. (34)

2.3.1. The Two-Phase Flow Equation in Characteristic Coordinates

As alluded to earlier, there is some advantage in considering the two-phase flowEquation (34) in terms of the characteristic variables (Datta-Gupta and King, 1995;


King and Datta-Gupta, 1998). The underlying idea is to define a local coordinatesystem oriented with respect to the vectorW (King and Datta-Gupta, 1998). Wespecify the coordinate system by the functions(r, ψ, χ) such that bothψ andφ areorthogonal toW andr is oriented alongW. Then the gradient operator in physical(x, y, z) space is re-expressed as

∇ = ∇r ∂∂r+ ∇ψ ∂

∂ψ+∇χ ∂

∂χ. (35)

Because of the orthogonality of theψ andχ axes toW we have

W · ∇ = Wn

∂

∂r. (36)

Thus, we can write Equation (34) solely in terms of the position along thetrajectoryr

Wn

∂S

∂r+ ∂F (S)

∂t= 0, (37)

whereWn denotes the component ofW along the trajectory, normal to the satura-tion front.

Equation (37) takes a simpler form if we rewrite it in terms of the new variable

τ =∫6

p(x)dr (38)

with p(x) defined as

p(x) = 1

Wn(x)(39)

and whereWn(x) is given in Equation (33). Implicit in this procedure is the assump-tion that the total mobilityκ(x) is essentially constant. Thus, the transformation(38) does not depend on the saturation. The variableτ represents the time offlight, the time required for a neutral tracer to travel from the injector to the pro-ducer (Datta-Gupta and King, 1995). In terms of the new independent variable,Equation (37) simplifies to

∂F (S)

∂τ+ ∂S∂t= 0 (40)

a first-order quasi-linear hyperbolic equation forS(t, τ ) (Bedrikovetsky,1993).This equation is invariant with respect to coordinate scalings of the type:

t ′ = εt, τ ′ = ετ, ε > 0

which requires the solution to take the general form (setε = 1/t)

S(t, τ ) = S(τt

)(41)


if it is to be unique (Chorin and Marsden, 1990; Bedrikovetsky,1993). Importantpoints to note are that the saturation amplitude has the functional dependence onτ/t and that the temporal variation in saturation and fractional flow at a given wellis an integral or summation over all trajectories,6, reaching it.

2.3.2. Arrival Time Sensitivities

The arrival time sensitivities, in the absence of gravitational effects, follow fromEquations (27) and (30). As mentioned above, we shall assume that the total mo-bility is constant. Let us denote the reservoir properties associated with the back-ground model by a subscript zero. Hence, the travel time in the background modelis given by

T0 =∫60

1

κ0(x)

(dF

dS

)−1φ0(x)

K0(x)|∇P0| dr, (42)

where60 denotes the trajectory in the background medium. Consider a perturb-ation in the background porosity denoted byδφ(x). That is, we wish to calculatethe change in travel time to the producer in adopting the new porosity distributionφ(x) = φ0(x)+ δφ(x). The new travel time is given by

T =∫6

1

κ0(x)

(dF

dS

)−1φ0(x)+ δφ(x)K0(x)|∇P0| dr (43)

for a flow field close to steady state. The change in travel time is the differenceδT = T − T0 given by

δT =∫6

1

κ0(x)

(dF

dS

)−1φ0(x)+ δφ(x)K0(x)|∇P0| dr −

−∫60

1

κ0(x)

(dF

dS

)−1φ0(x)

K0(x)|∇P0| dr. (44)

It has been shown (Kline and Kay, 1965; Nolet, 1987) that the ray trajectories,6, computed using an Eikonal equation are second order in perturbations to thevelocity field. In our case, perturbations in velocity correspond to perturbations inVn(x) given by Equation (30). Thus, to second order inδφ(x), we may approximatethe new trajectory6 by the original path60. Equation (44) then becomes

δT =∫60

1

κ0(x)

(dF

dS

)−1 1

K0(x)|∇P0|δφ(x)dr. (45)

Sensitivities are functions relating perturbations in model parameters, such asδφ(x), to changes in observables. For Equation (45) the sensitivity is given by

∂T

∂φ= 1

κ0(x)

(dF

dS

)−1 1

K0(x)|∇P0| (46)


which may be written in terms ofp(x) in Equation (39)

∂T

∂φ=(

dF

dS

)−1p(x)φ(x)

(47)

where the quantities are evaluated with respect to the background model. We mayfollow a similar procedure with respect to the permeabilityK(x) and the pressuregradient magnitude|∇P | to find

∂T

∂K= −

(dF

dS

)−1p(x)Kx)

(48)

and

∂T

|∇P | = −(

dF

dS

)−1p(x)|∇Px)| (49)

respectively. Note that Equations (47)–(49) only describe the explicit dependenceof T onφ,K, and|∇P |. There is also an implicit dependence of|∇P | field on bothφ andK, through the pressure equation. We could account for this dependenceexplicitly by calculating the partial of the pressure gradient magnitude with respecttoφ andK and substituting into (49) to obtain additional terms for the sensitivities(47) and (48). However, this approach leads to complicated expressions and ana-lytic difficulties. Our approach is to take advantage of the fact that we recomputethe pressure field after each perturbation ofφ andK. Thus, for each perturbationof flow properties we obtain a numerical estimate of|∇P | which accounts for theimplicit dependence. The procedure is described in more detail in the section oninversion. Furthermore, in our discussion on the numerical calculation of sensitivityestimates we compare our estimates using Equations (47) and (48) to a purelynumerical perturbation approach. We find that sensitivities computed using (47)and (48) correspond quite closely to purely numerical sensitivities.

2.3.3. Saturation Amplitude Sensitivities

In order to infer reservoir properties from multi-phase data we need to know howthe saturation histories are changed if the reservoir properties are perturbed slightly.The equations describing the evolution of the saturation front along the trajectoriesmay be used to derive such sensitivities. That is, we can compute the changes infractional flow due to perturbations in reservoir properties directly from a singleforward simulation. The procedure is essentially identical to that followed forarrival time sensitivities.

Consider the saturation at a well which is located atrp along trajectories60.Consider a perturbation in reservoir properties along trajectories60 and the result-ing perturbation in time of flight(τ ), which is given by Equation (38),

τ = τ 0+ δτ, (50)


whereτ 0 is the value computed using the unperturbed reservoir structure. Therewill be a corresponding perturbation in the saturation at the observation well

S(τt

)= S0

(τt

)+ δS

(τt

), (51)

whereS0(τ/t) is the saturation history in the unperturbed reservoir. We have madeuse of the general form of the solution along the trajectories, as given in Equation(41). ExpandingS(τ/t) in a Taylor series about the pointτ 0/t we find that

S(τt

)= S

(τ 0+ δτ

t

)= S0

(τ 0

t

)+ 1

tS ′0

(τ 0

t

)δτ (52)

to first order inδτ , where the prime denotes the derivative with respect toτ . Hence,the perturbation in saturation,δS = S − S0, is given by

δS(τt

)= 1

tS ′0(τt

)δτ. (53)

Now the variableτ is given by Equation (38) andp(x) is the composite quantity

p(x) = φ

u= − φ(x)

κ(x)K(x)|∇P(x)| , (54)

where we have invoked Darcy’s law (Bear, 1972) in the form of Equation (2). Theperturbationδτ is the integral

δτ =∫6

δp(x)dr, (55)

where6 denotes the perturbed trajectory. From Equation (54) we may writeδp(x)

δp(x) = ∂p

∂φδφ(x)+ ∂p

∂KδK(x)+ ∂p

∂|∇P |δ|∇P(x)|. (56)

The partial derivatives may be calculated directly from Equation (54)

∂p

∂φ= p(x)φ(x)

,∂p

∂K= − p(x)

K(x),

∂p

∂|∇P | = −p(x)|∇P(x)| . (57)

If the trajectories are not significantly perturbed by the passage of the saturationfront then the perturbed trajectory6 may be replaced by the unperturbed traject-ory 60. This approximation is thought to be a good one for waterflood fronts(Datta-Gupta and King, 1995), and appropriate for the oil field application con-sidered below. Though we have derived the sensitivities in terms of saturationwe can convert them to fractional flow sensitivities by multiplying each term in(57) by dF/dS. The derivative is computed numerically, based upon the relat-ive permeability curves. We must emphasize that, althoughδ|∇P | is treated asan unknown parameter in our derivation it functions more as a correction term.That is, given estimates of flow properties, boundary conditions, and flow rates


we can compute the pressure field and the pressure gradient. And we do calcu-late the pressure field numerically for each iteration of our inversion. However,in solving for reservoir properties we do not know the internal pressure field ex-actly. Our pressure gradient perturbationδ|∇P | is a correction term which shouldminimize the mapping of deviations in the pressure gradient into reservoir proper-ties. At each iteration of the inversion we recalculate the pressure field given ourcurrent estimates of reservoir structure and overwriteδ|∇P | with the calculatedvalue.

3. Numerical Calculations

In this section, we explore both the accuracy and the utility of our approach. First,we compare sensitivities, the ratio of change in saturation to changes in permeabil-ity and porosity, computed numerically with those computed using our asymptoticapproach. Next, we illustrate how the sensitivity estimates form the basis for anextremely efficient inversion scheme in which fractional flow histories are used tomap lateral variations in reservoir properties, for example,K andφ.

3.1. SENSITIVITY ESTIMATES

The model parameter sensitivities are essential elements in our iterative inversionscheme described below. The sensitivities are the partial derivatives of the observeddata with respect to changes in the parameters representing reservoir flow proper-ties. For example, if a block- or cell-based parameterization is used, the modelparameters would be permeability, porosity, and pressure gradient in each cell,see Equation (56). If we wish to use Equation (56) to solve for perturbationsto these quantities, we require estimates of the partial derivative coefficients, thesensitivities.

In our previous work (Vasco and Datta-Gupta, 1997; Xue and Datta-Gupta,1997) the sensitivities were calculated in a purely numerical fashion. That is, fora given cell the permeability was perturbed by a small amount and a completesimulation was conducted. The changes in the predicted observations, such as thewater saturation at each well, were then recorded. The sensitivity associated witha particular observation due to changes in a particular cell is then the ratio of thesaturation change to the permeability change. While this approach proved generaland accurate it required extensive computation, a simulation for each parameter,limiting the number of parameters we could use to describe the reservoir. Contrastthis with the calculation of sensitivities as expressed in Equations (57). Just onesimulation is required to obtain all sensitivity coefficients. Consider sensitivitiesassociated with the observationj . First, a forward simulation is conducted using aninitial reservoir model. This initial model would typically be constructed based ondata such as well-logs or cores. The perturbation in fractional flow is given by com-


bining Equations (53) and (55)–(57). For illustration we consider the permeabilityparameters only,

δF j(τt

)= −1

t

dF

dS

N6∑n=1

S ′0(τt

)∫6n

p(x)K(x)

δK(x)dr, (58)

where the sum is over allN6 trajectories contributing to multi-phase flow obser-vationj and the integral is along each trajectory in the sum. The derivative dF/dSappears because we are now working in terms of fractional flow rather than interms of saturation. Equation (58) is an expression for the change in fractional flowdue to all permeability perturbations in the initial model. If we wish to isolate thechange due to a variation in celli say, we simply consider the increments of thetrajectory which pass through theith block. Denoting the volume of blocki by νiwe isolate that part of Equation (58) sensitive to changes inνi,

δF j

δKi= −1

t

dF

dS

N6∑n=1

S ′0(τt

) ∫νi

p(x)K(x)

dr (59)

using the fact that the perturbation in celli, δKi, is constant. As stated above, all thequantities appearing in Equation (59) are contained in the initial reservoir model orproduced by a single simulation.

The computation of sensitivities based upon Equation (59) is very differentfrom our previous purely numerical approach. In order to verify the correctnessof our asymptotic approach we compare it to the numerical perturbation sensitivitycalculations. For simplicity, the well configuration is a quarter five-spot with asingle injector-producer pair (Figure 1). Also shown in Figure 1 are 30 repres-entative trajectories connecting the two wells. The injector is pumping water intoan initially oil-saturated reservoir. Our reservoir model is a uniform layer with aporosity of 3% and a permeability of 5.5 milli-Darcy. A 21 by 21 grid of cellswas used in these simulations. The fraction of water appearing at the producingwell as a function of time is shown in Figure 2. The numerical perturbation andasymptotic porosity sensitivities are shown in Figure 3. Sensitivities correspondingto three fractional flow observations, at 270, 300, and 420 days, are displayed. Thesensitivity coefficients, plotted graphically in Figure 3 with each value located inits associated cell, are essentially identical. For early times (270 days) the sens-itivity is concentrated in an elongated elliptical region between the injector andthe producing well. Note that since a decrease in porosity produces an increasein fractional flow, the sensitivity coefficients are negative. With time (420 days)the maximum sensitivity migrates to the outer-most trajectories, approaching theboundaries of the quarter five-spot. The asymptotic porosity sensitivities are exactsince the steady state velocity field is insensitive to porosity variations and thus thetrajectories do not shift as a result of perturbing porosity.

Following a similar procedure we computed sensitivities associated withchanges in reservoir permeability (Figure 4). The numerical and asymptotic sens-itivity coefficients are again very similar. However, there are detectable differences


Figure 1. Well configuration used in the calculation of quarter five-spot sensitivities. Thepumping well is indicated by a filled-diamond and the injector by a filled-circle. For reference30 representative trajectories are also shown.

between the sensitivities. These differences are mainly due to the fact that unlikeporosity, the velocity field is influenced by the permeability variations and thusthe assumption of static trajectories is an approximation. Overall, the patterns arealmost identical, indicating that deviations in fractional flow data will be mappedinto permeability changes in the same regions by the numerical and asymptoticapproaches. The permeability sensitivity coefficients are almost inverse imagesof the porosity sensitivities (Figure 3). The negative correlation derives from theinherent trade-off between porosity and permeability indicated by Equations (30)and (54). Both the saturation arrival time and amplitude are influenced by the ratioof porosity to permeability. Changes in one parameter could be offset by changesin the other. For a non-uniform medium the sensitivity coefficients for porosity andpermeability are less dependent.

We should note that similar results are obtained in calculations with respect toa non-uniform reservoir model. In addition, a comment about computation timesis in order. The complete calculation of porosity and permeability sensitivitiesusing the asymptotic approach took approximately 7 CPU minutes. By contrastthe numerical perturbation approach required over 150 CPU minutes for both setsof parameters (permeability and porosity). Thus, the asymptotic approach is over20 times faster even for this modestly sized problem.


Figure 2. Synthetic water-cut fraction observed at the production well. Sensitivities arecalculated for the three points: (A) 270 days, (B) 300 days, and (C) 420 days.

3.2. INVERSION OF NINE-SPOT SYNTHETIC FRACTIONAL FLOW DATA

Given the sensitivity coefficients relating changes in reservoir parameters tochanges in multi-phase flow observations, we may devise an efficient inversionstrategy. In essence, our approach will be an iterative linearized inverse schemein which sensitivities are re-calculated at each step. For the moment, considerthe inverse problem of estimating permeability variations given a set of water-cutobservations. For example, in Figure 5 variations in the logarithm of permeabilityare shown for a stochastic realization of a single layer reservoir. The well configur-ation is a 9-spot pattern with a central injector and eight producers. The syntheticwater-cut for the eight producers is given in Figure 6(a) (solid line) along with thewater-cut, calculated with respect to a uniform initial permeability distribution of5.5 milly-Darcy. Thej th water-cut residual,δF j , is defined as the observed water-cut minus the predicted water-cut. It is related to the perturbations in reservoirpermeabilityδKi of theNK cells by

δF j =NK∑i=1

Sij δKi, (60)

where the sensitivity matrixSij consists of the elements

Sij =(δF j

δKi

)


Figure 3. (Numerical) Porosity sensitivities calculated using a purely numerical perturbationapproach. (Streamline) Porosity sensitivities calculated using the asymptotic technique.

given by Equation (59). In general, we will have many observations from a numberof wells and may write Equation (59) as a matrix equation for the vector of cellpermeability deviationsδK

δFw = SδK , (61)


Figure 4. (Numerical) Permeability sensitivities calculated using a purely numerical per-turba- tion approach. (Streamline) Permeability sensitivities calculated using the asymptotictechnique.

whereS is the sensitivity matrix andδFw is a vector of water-cut residuals. Estim-ates of changes in reservoir permeabilities are obtained by solving Equation (61)for δK . Because we typically have limited data and a large number of parameters,the problem is ill-posed and the equations may be singular or nearly singular.To circumvent the problem of numerical instability it may be necessary to aug-


Figure 5. (A) Permeability variation used to generate a set of synthetic water-cut values. Thenatural logarithm of permeability is plotted. The locations of the producing wells are indicatedby the unfilled circles. The central injection well is indicated by the star. (B) Permeabilityestimates produced by the iterative linearized inversion scheme.

Figure 6. (A) Initial match – the solid line denotes synthetic water-cut data corresponding tothe model shown in Figure 5. The dashed line indicates water-cut predicted using a homo-geneous reservoir model. (B) Final match – (solid line) ‘Observed’ synthetic water-cut data.(dashed line) Water-cut predicted by the permeability estimates produced by the inversionalgorithm.

ment Equation (61) with additional regularization equations (Parker, 1994) such asroughness penalty terms or norm penalty terms (Vascoet al., 1997). Regularizationmay be approached from either a least-squares or a Lagrange multiplier point ofview (Parker, 1994). In either case we obtain additional constraints of the formαPδK = 0, whereα is the regularization weighting andP are coefficients whichdetermine the type of regularization. In the case of model norm regularization,constraints on the amplitude of the perturbations,P = I . That is, the coefficientmatrix corresponds to the identity matrix. Physically, this implies that we do notwant to deviate too far from our initial model which already incorporates geologic,well log, and seismic information. An additional constraint we shall impose is a


Figure 7. Squared misfit to the synthetic water-cut values as a function of the number of itera-tions of the inversion algorithm. The first nine points correspond to the inversion of saturationfront arrival times.

roughness penalty,P= ∇δK . We desire the spatially smoothest model compatiblewith our observations. The roughness constraint takes into account the fact thatmulti-phase flow data contain information about the spatial averages of reservoirproperties. That is, these data are best suited to reproduce the large-scale featuresof the reservoir rather than small-scale variations in reservoir properties.

The iterative inversion scheme entails solving Equation (61) for permeabilitychanges in each cell. These changes are added to the current reservoir model,resulting in a new updated structure. A forward simulation through the updatedstructure produces a new sensitivity matrix and another system of equations asin (61). These equations are again solved for new perturbations with respect tothe updated structure. The entire procedure is repeated until suitable reduction inthe misfit to the data. For example, Figure 7 displays the squared residuals as afunction of iteration for the synthetic data set in Figure 6. After about 16 iterationsthe misfit is significantly reduced and the algorithm has converged. The resultingfit to the data (Figure 6(b)) is quite good and the final model (Figure 5(b)) containsthe large scale features of the original structure (Figure 5(a)).

A few words are in order regarding our approach to solving the linear systemof Equations (61) and any accompanying regularization equations. As might beguessed, for a high resolution reservoir model there will be many model para-meters. For example, in the North Robertson application discussed in the nextsection we require 100,000 porosity and permeability unknowns. The data andregularization constraints produce more equations than unknowns, resulting in anequally large number of equations. One property of these equations is their sparsity,


that is, each equation contains only a small proportion of nonzero coefficients.Because our algorithm for solving the nonlinear inverse problem is iterative weshall require repeated solutions of this large sparse system. There are several ap-proaches for solving such systems (Golub and Van Loan, 1989) and each approachhas benefits and drawbacks. One algorithm which is ideally suited for quicklyfinding approximate solutions to large sparse systems, is the LSQR algorithm ofPaige and Saunders (1982). This technique is now standard in seismic tomography(Nolet, 1987) and has been applied to problems containing of the order of a millionequations and a hundred thousand unknowns (Vascoet al., 1999).

There is a subtle but important point about convergence which we have notyet addressed. The above linearized iterative inversion scheme assumes that ourinitial estimate of reservoir structure is close to the actual reservoir structure. Inreality, we often have very little knowledge concerning reservoir properties, exceptperhaps at a limited number of wells. Our previous efforts to fit the multi-phaseflow observations directly suffered from poor convergence. The algorithm wouldconverge to a model which still produced a relatively large misfit. In synthetic teststhe model would be significantly different from the structure used to generate thedata. Such premature convergence has also been observed in attempts to fit tracerdata (Vasco and Datta-Gupta, 1999). Our solution, which appears to work wellin practice, is borrowed from seismic waveform fitting (Zhouet al., 1995). Theidea is to first match the arrival times of the saturation fronts to derive an initialbackground structure. Only then, after the breakthroughs have been ‘lined-up’, arethe histories themselves matched. An application to matching tracer observations,which relates directly to the work here, is presented in Vasco and Datta-Gupta(1999). The basic idea is to start with Equation (45), relating saturation arrival timeto the reservoir properties, and the sensitivities, given by Equations (47)–(49). Inplace of Equation (59), we end up with the arrival time sensitivities,

Sij = δT j

δKi= −

N6∑n=1

∫Vi

(dF

dS

)−1p(x)K(x)

dr (62)

and a system of equations equivalent to Equation (61), relating front arrival timesto permeability variations

δT = SδK (63)

with sensitivity matrix elements given by (62). The initial arrival time match hasproven critical in both reducing the misfit and in producing models which resemblethe original synthetic structures. In fact the first nine iterations in Figure 7, dur-ing which the greatest misfit reduction occurs, are associated with arrival timematching.

To close this section we wish to summarize the results: given fractional flowhistories from eight wells (Figure 6), the asymptotic algorithm is able to recoverthe large-scale permeability variations (Figure 5). In less than 10 CPU-minutes on


a standard workstation, we determined suitable permeability estimates for the 441cells (21× 21 grid) in our model. This is a dramatic improvement over our previousefforts, which required hours of computation for the same sized model, and oftendid not converge.

4. Application: The North Robertson Unit

4.1. OVERVIEW

The North Robertson Unit is a heterogeneous, carbonate reservoir located withinthe Permian Basin of West Texas. As with the majority of such reservoirs thereare production problems such as lack of reservoir continuity, low waterflood sweepefficiency, early water breakthrough, and water channeling. Furthermore, there islimited data available pertaining to reservoir structure and performance, partic-ularly with respect to older wells. For example, cores were only available for alimited number of wells within the North Robertson Unit. The North RobertsonUnit has been selected by the Department of Energy, as part of its Class II program,for the application of techniques for integrated management and reservoir charac-terization in order to optimize infill drilling (Doubletet al., 1995). In particular,an aggressive and systematic 20-acre water-flood and infill drilling program wasinitiated in 1987.

4.2. GEOLOGIC STRUCTURE AND RESERVOIR PROPERTIES

The depositional environment of the reservoir rocks is that of a carbonate platform.There are primarily two types of strata: the Lower Clearfork, and the Middle-UpperClearfork/Glorieta. Reservoir flow properties are dominated by variations in poregeometry and rocks containing similar total porosity may have significantly differ-ent permeability (Doubletet al., 1995). The non-reservoir rock types are relativelyimpermeable and form significant flow barriers. Fractures are also present in thereservoir, further complicating production efforts. Estimates of relative permea-bility parameters are based on 12 core samples from a single well in the NorthRobertson Unit (Doubletet al., 1995). The laboratory data themselves did notindicate large variations in relative permeabilities within the different lithologiescomprising the reservoir rocks. For our work we constructed a single representativecurve for these laboratory data.

4.3. WATER-CUT INVERSION

Our data set consists of water-cut data from two sections, 326 and 327, of theNorth Robertson Unit (Figure 8). There are several reasons why we limit ouranalysis to these sections. First, the boundaries between the lower–upper–middleClearfork, the Tubb, and the Glorieta sections are fairly flat in this region (Doubletet al., 1995). Thus, the depth variations of the interfaces are low amplitude and


Figure 8. Location map for sections 326 and 327 of the North Robertson Unit. Injection wellsare denoted by stars. Production wells are indicated by unfilled circles.

there will be fewer complications due to gravitational effects. In addition, thesesections are entirely surrounded by other producers (Figure 8), simplifying theboundary conditions. During the time period under consideration there were 15active injectors and 27 active producers in sections 326 and 327. To ensure anadequate representation of reservoir structure we require a sufficient number ofgrid blocks between each well pair. From experience, and test inversions, we foundthat 5–10 cells between wells is generally sufficient. Given the well pattern, wemust have a grid of 100 (east–west) by 50 (north–south) cells in each layer ofthe model. In the vertical direction, roughly 10 layers are necessary to realisticallyrepresent the variation observed in the wells. And so, 50,000 grid blocks are neededto represent permeability heterogeneity. Another 50,000 porosity parameters werealso included, for a total of 100,000 variables.

While there are some drawbacks associated with working in the NorthRobertson Unit, it is highly heterogeneous and fractured, there is one distinctadvantage: the development of this field has been systematic. Sets of injectorsand producers were drilled in more or less regular patterns (Figure 8) and sets ofinjectors began operation at approximately the same time (Figure 9(a)). Figure 9(a)denotes the injection rate as a function of time for the injectors in sections 326and 327. The synchronization of injectors reduces the number of times we mustrecalculate the pressure field in the reservoir. In fact, for this work, we assumethat all injection began at the same time (300 days) and was constant for eachinjector (equal to the average injection over the time interval). This assumption isadequate for our first attempt at delineating the large-scale structure of the reser-voir. From Figure 9(a) one observes deviations of 30–60 days from our start timeof 300 days. However, in Figure 9(b), the watercut breakthroughs, much largervariations of 100s of days are seen. In fact, for wells 21–27 water never arrivedduring the 615 days under consideration. On the other hand, water arrives within5 days of injection in wells 6, 7, and 9. We should note that the time scale on this


Figure 9. (a) Injection rate for the 15 active wells in the North Robertson Unit. (b) Water-cutfraction as a function of the number of days since injection began.

Figure 10. Observed water-cut (solid-line) and water-cut predicted using a uniform startingmodel (dashed-line).

figure refers to the number of days elapsed since the start of injection, the scaleorigin is different from that of Figure 6(a). This origin shift is present in all figuresdisplaying production matches.

Due to the paucity of core data from the North Robertson Unit we had dif-ficulty constructing a detailed initial model of the reservoir. One advantage ofour two stage inversion (arrival time/water-cut amplitude) is that convergence forthe arrival time matching is generally much less sensitive to the initial model. Inthis application, we initiated our inversion from a homogeneous structure with ahorizontal permeability of 0.1 milly-Darcy, a vertical permeability of 0.01 milli-Darcy, and an initial porosity of 0.1%. The initial fit to the observed water-cut datais shown in Figure 10. For plotting purposes, we only show the 20 wells in whichwater arrived within the 615 day interval. However, all 27 production histories are


Figure 11. Squared misfit as a function of the number of iterations of the linearized inversionscheme. The first 30 iterations correspond to arrival time matching.

Figure 12. Observed water-cut (solid-line) and water-cut predicted using the final inversionestimates of permeability and porosity (dashed-line).

used in the inversion. Note the early breakthroughs and the low values of water-cut predicted by the initial reservoir model. Clearly our starting permeability andporosity estimates are very different from the actual reservoir properties.

We began the inversion with 30 arrival time iterations, reducing the squaredmisfit by 62% (Figure 11). Following this, amplitude matching alternated with ar-rival time matching for the remaining iterations. The total inversion, 180 iterationsin all, took approximately 42 CPU-hours on a conventional workstation. By far, the


Figure 13. Lateral variations in reservoir permeability for three layers of the final model. Thenatural logarithm (lnK) of the permeability is shown here.

amplitude inversions required the most computation, taking many tens of minutesto complete a single iteration. By contrast, each travel time iteration needs just afew CPU-minutes. The final misfits, for the wells showing water breakthrough, aredisplayed in Figure 12. Comparing Figure 10 and 12 we see that the match is signi-ficantly improved when we use the reservoir properties estimated by our inversionscheme. The fit is a good one, given our simplifying assumptions (simultaneous


Figure 14. Lateral variations in reservoir porosity for three depth intervals in the final model.The natural logarithm (lnφ) of the porosity is plotted.

injector initiation, homogeneous boundary conditions), our simple starting model,and our assumptions concerning relative permeability in the reservoir.

The permeability and porosity distributions produced by the water-cut inversionare displayed in Figures 13 and 14. The total variation in permeability ranges froma high of about 1 milly-Darcy to values below 0.05 milly-Darcy. Permeabilities aregenerally higher in the east and lower to the west. The porosity is low throughoutthe reservoir (1% or less). Porosity appears to be somewhat higher to the north and


west. The extremely low porosity values in the south-eastern edge of the model maybe an artifact induced by boundary effects. The pattern of porosity and permeabilityheterogeneity is fairly similar in each layer of the model. This reflects the fact thatwe have little vertical control on heterogeneity. Each producer is open for most ofthe depth range of our model. Thus, we may not resolve detailed depth variationin reservoir properties without additional constraints. This brings up the topic ofresolution and spatial averaging which we have yet to discuss. It has been shown(Parker, 1994) that there is a trade-off between model parameter uncertainty andspatial resolution. Spatial averaging is particularly significant in the inversion offlow and transport data (Vascoet al., 1997) and assessments of inversion estimatesare important. While it is not possible to give a detailed treatment here, there areextensive discussions of these issues (Parker, 1994). We should point out that es-timation of spatial averaging and parameter uncertainty are feasible for large-scaleinversions, such as this one, and have been applied to inversions with of the orderof 100,000 parameters (Vascoet al., 1999).

It is useful to compare the variations in reservoir properties we infer with theresults of two previous studies. Doubletet al. (1995) produced maps of originaloil-in-place, permeability× reservoir thickness(Kh), based upon a type curve ap-proach. They estimated ultimate recovery for the reservoir, averaged over the entirereservoir depth interval. They found that section 326 of the Unit (the eastern-mosthalf of our model) had the ‘best’ production characteristics overall. The productionproperties of section 327 (the western portion of our model) were not as high assection 326. A map of 20-acre flow capacity (permeability× reservoir thickness)givesKh a factor of 3 higher in section 326 than in section 327 (Doubletet al.,1995). An earlier pilot-point inversion of North Robertson water-cut data (Xue andDatta-Gupta, 1997), based upon a purely numerical conjugate gradient algorithm,produced a pattern similar to ours: higher permeability in the eastern half of themodel region.

5. Conclusions

A reservoir model derived from geologic, well-log, and core data, will generallyresult in fluid flow predictions that do not necessarily match the observed fractionalflow history. Conditioning reservoir models with multi-phase flow data typicallyrequires the solution of an inverse problem. This can be computationally prohibi-tive, particularly for fine scale reservoir descriptions consisting of several thousandto a million or more grid cells. Our approach relies on a new technique for theinversion of fractional flow data for reservoir permeability and porosity variationswhich requires the same order of computation as does the forward problem. Ourforward model is a very general and in this work we make use of a fully three-dimensional streamline simulator which has been shown to be orders of magnitudefaster than conventional numerical simulators. We compute parameter sensitivities(the change in the observed production data given a change in reservoir structure)


using an asymptotic approach that exploits an analogy between the streamlinemodeling and ray tracing.

The asymptotic approach is applicable under several very common conditions,such as NAPL migration, CO2 and water flooding. The method of characteristics,which is strongly tied to the asymptotic approach (Kline and Kay, 1965), has beensuccessfully applied to multi-phase and shock front propagation problems (Chorinand Marsden, 1990; Bedrikovetsky,1993; Mitlin, 1993). Furthermore, streamlinesand trajectories may be defined for a propagating two-phase front and are foundto be relatively unperturbed by the passage of the fluid fronts (Datta-Gupta andKing, 1995). Finally, one-dimensional solutions along the trajectories are knownfor situations in which capillary effect are absent, the Buckey-Leverett equation(Peaceman, 1977), as well as when capillarity is active (McWhorter and Sunada,1990). These one-dimensional solutions may be efficiently solved along stream-lines in three-dimensions (Datta-Gupta and King, 1995). The asymptotic approachcan unite all these methods and tie them to related approaches in electromagneticand seismic wave propagation. The result is an efficient approach for imagingreservoir structure using fractional flow data. We have demonstrated the power ofthe method by inverting water-cut data from the North Robertson Unit for 100,000parameters describing the reservoir structure. The technique has also been utilizedto map watercut data from the Goldsmith San Andreas Unit in west Texas intothree-dimension permeability variations (Yoonet al., 1999). In this case, some41,000 permeability parameters were estimated. The entire inversion required ap-proximately 2 h of computation on a desktop workstation. To our knowledge, theseare the first such large-scale inversions of multi-phase production data.

In the future we would like to relax the assumption of incompressibility andincorporate gravity and capillary effects. Gravity has already been incorporatedinto our arrival time calculations given in Equations (29). Asymptotic methods areindeed applicable to problems in which dispersion and diffusion are present (Cohenand Lewis, 1967; Anileet al., 1993; Chapmanet al., 1999; Vascoet al., 2000)It would also be interesting to incorporate other information into the inversionscheme. For example, tracer data and transient pressure information are sensitive toreservoir structure. Moreover, tracer and pressure sensitivities differ from fractionalflow sensitivities and the various data sets are complementary. Given the efficiencyof the asymptotic approach for inverting multi-phase flow and tracer (Vasco andDatta-Gupta, 1999), a joint inversion of these data is certainly feasible.

Acknowledgements

This work was supported by a Laboratory Directed Research and Developmentgrant, Office of Energy Research, by the Division of Basic Energy Sciences, Engin-eering, and Geosciences, and by the Assistant Secretary for Fossil Energy, Office ofOil Gas and Shale Technologies, of the U.S. Department of Energy under contractDE-AC03-76SF00098. All computations were carried out at the Center for Com-


putational Seismology and the National Energy Research Scientific Computing(NERSC) Center of the Berkeley Laboratory.

Appendix

In this appendix we derive the additional equations needed in order to computetrajectories from an injector to a producer. The procedures are akin to those used totrace rays in optics. These techniques are widely used in many fields for imagingand inversion. More details my be found in texts such as Kline and Kay (1965).

The second and final set of ray equations is derived by considering secondderivatives ofr . Because thex component ofr is given by (24) andp is defined interms of9(x) we have

dx

dt= p1 = ∂9

∂x(A.1)

and similarly for the other componentsy(t) andz(t). Differentiating Equation (A.1)with respect tot and noting the dependence ofx, y, andz on t we find

d

dt

(dx

dt

)= ∂29

∂x2

dx

dt+ ∂29

∂y∂x

dy

dt+ ∂29

∂z∂x

dz

dt. (A.2)

Using Equations (23) and (24) we may rewrite this expression as

d

dt

(dx

dt

)= ∂29

∂x2

∂9

∂x+ ∂29

∂y∂x

∂9

∂y+ ∂29

∂z∂x

∂9

∂z. (A.3)

or, equivalently

d

dt

(dx

dt

)= 1

2

∂

∂x

[(∂9

∂x

)2

+(∂9

∂y

)2

+(∂9

∂z

)2]. (A.4)

Making use of the Eikonal equation, we may write this as

d

dt

(dx

dt

)= 1

2

∂

∂x

(1

Vn2

). (A.5)

Following a similar procedure for they(t) andz(t) components, we arrive at oursecond set of ray equations

dpdt= 1

2∇(

1

Vn2

)(A.6)

where Equation (24) has been invoked.

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Documents

Asymptotics, Saturation Fronts, and High Resolution Reservoir Characterization