00077569 Analysis and Interpretation of Water-Oil-Ratio Performance

Copyright 2002, Society of Petroleum Engineers Inc.

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AbstractThis work presents the development and validation of a multi-variate relation for the behavior of the water-oil -ratio (WOR)and/or water cut (fw) functions. This new model incorporatesthe reservoir and fluid properties for both phases (oil and wa-ter) and is based on the assumption of pseudosteady-state flowconditions.

This work is an extension of traditional (i.e., steady-state)methods for the case of pseudosteady-state flow — for boththe oil and water phases. In this work, our pseudosteady-statemodel reproduces observed field performance substantiallybetter than any of the steady-state models. We propose thatthis approach can be applied to any reservoir system undergo-ing waterflood.

The specific tasks achieved in this work include:

1. Development of a rigorous model for the simultaneousflow of oil and water during pseudosteady-state flowconditions. This model has been validated using sever-al field cases and gives an excellent representation ofthe WOR (or fw) data trend(s).

2. Development of a "reciprocal rate plot" for the estima-tion of both the original oil -in-place (N), as well as theEstimated Ultimate Recovery (EUR) at current produc-ing conditions.

3. Development of a diagnostic technique for assessing(qualitatively) the efficiency/effectiveness of a water-flood. This technique involves the use of the followinglog-log format plots: WOR-derivative, the WOR-inte-gral, and the WOR-integral-derivative functions.

4. Application/interpretation of the following extrapola-tion methods for the water-oil-ratio (WOR) and the wa-ter cut (fw) functions:

—qo versus Np,— log(fw) versus Np,—1/fw versus Np,— fo versus Np, and— log(WOR) versus Np.

While our new formulation of the two-phase (oil -water),pseudosteady-state flow relation does not provide for a simpleextrapolation formula for the estimation of recoverable oil , wedo prove the utili ty of this relation as an interpretation me-chanism. We also provide insight into the existing and pro-posed techniques for estimating the ultimate oil recovery.

OrientationNatural water drive and/or water injection are two of the mostcommon drive mechanisms in oil production. A detailedanalysis of past performance should be conducted in order topredict the future performance of the well/ reservoir system —as well as to estimate the volume of in-place and recoverablefluids.

The logarithm of the water-oil -ratio (WOR) and/or water cut(fw) functions plotted versus the cumulative oil production arecommonly used tools for the evaluation and prediction ofwaterflood performance. This presumed (actually empirical)log-linear relationship of WOR (or fw) and oil recovery allowsfor the extrapolation of the observed straight-line to anydesired water cut as a mechanism for determining the cor-responding oil recovery.

Such straight-line extrapolation methods are essentiall y em-pirical, and when theory is used to validate such techniques,we must make prohibitive simplistic assumptions. Two suchassumptions that have been documented in the literatureinclude — the assumption that the mobility ratio is equal tounity and that a plot of log (krw/kro) versus So is linear.

Our goal in this work is to develop and validate a multivariaterelation to represent the behavior of the water-oil-ratio (WOR)and/or water cut (fw) functions — the only significant assump-tion that we make in this work is that pseudosteady-state flowconditions must exist in the entire reservoir system.

SPE 77569

Analysis and Interpretation of Water-Oil-Ratio PerformanceV.V. Bondar, ChevronTexaco Corp., T.A. Blasingame, Texas A&M U.

2 V.V. Bondar and T.A. Blasingame SPE 77569

The objectives pursued in this work include:

1. Derivation and validation of a pseudosteady-state mo-del for the simultaneous flow of oil and water. Thismodel is validated against 28 different field cases —and in all cases, the new model gives an excellentrepresentation of the data.

2. Development and validation of the "reciprocal rateplot" for the estimation of original oil -in-place (N) andestimated ultimate recovery (EUR). This method isvalidated against field data and yields an appropriatetrend in virtually all cases

3. Development and application of a series of log-logdiagnostic plots for waterflood evaluation — these in-clude:

—WOR functions versus production time,—WOR functions versus Np/qo, and—WOR functions versus (Np+Wp)/(qo+qw).

The WOR functions include: WOR-derivative, WOR-integral, and WOR-integral-derivative functions.

4. Application/interpretation of several different extrapo-lation approaches for the water-oil-ratio (WOR) and thewater cut (fw) functions. The new pseudosteady-stateWOR model is given in terms of both Np and Wp, and,as such, is not amenable to extrapolation.

As this issue could not be resolved (i.e., the use of thepseudosteady-state model as an extrapolation method),we chose to focus on several extrapolation techniques— all of which use the cumulative oil production (Np)as the x-axis plotting function. These extrapolationplots include:

—qo versus Np (constant pressure (liquid) case)— log(fw) versus Np (steady-state approach)—1/fw versus Np (new approach)— fo versus Np (field approach — not documented)— log(WOR) versus Np (steady-state approach)

As noted above, the formulation of the two-phase (oil-water),pseudosteady-state flow relation does not provide for a simpleextrapolation formula for the estimation of recoverable fluids— we believe that this is an area for further investigation.

Introdu ctionHistorically, it has been difficult to analyze and predict oilproduction behavior in water-drive or waterflood reservoirsystems (here we distinguish "waterdrive" as a natural condi-tion of water influx — and waterflood as a manufactured con-dition of water injection). Difficulty arises in how to charac-terize two-phase (oil-water) flow performance using analyticalsolutions based on single-phase flow theory, or by usingsimplified, steady-state solutions to represent two-phase (oil -water) flow. Neither approach is correct — and yet both areused regularly for the analysis of data acquired from water-drive/waterflood reservoir systems.

As a matter of practice, we must be able to evaluate and pre-dict waterflood performance in petroleum reservoirs — in-

place and recoverable oil volumes are required for evaluationand reservoir management purposes. A number of essentiallyempirical methods have been proposed over time for theevaluation of waterflood performance, and these methods aretypically assumed to give acceptable results.

A plot of the logarithm of the water-oil ratio (WOR) (or watercut function (fw)) versus cumulative production (Np) is themost widely used technique for the evaluation and predictionof waterflood performance.1 This simple (and, we will note,empirical) method is applicable for the analysis of "late time"production behavior and the technique allows us to estimatethe recoverable oil volumes by extrapolating the straight-linetrend of the fw function to an arbitrary value of water cut (oftenfw=1, or some other "high" value, such as fw=0.95).

This approach is only used when a straight line canapproximate the function of interest (WOR or fw). Unfortu-nately, in most cases, this method is not applicable for theearly stages of a waterflood (e.g., a rule of thumb is that thelog(WOR or fw) versus Np plots can not be used for values ofwater cut function (fw) less then 0.5). Misuse of these em-pirical techniques can yield substantial errors in the extrapola-tion of recoverable reserves.

We recognize that the use of a model based on the assumptionof steady-state flow behavior is an approximation at best —our motivation for this work is the development of a modelthat can be used to represent the pseudosteady-state behaviorof a water-oil reservoir flow system. We also recognize thatthe pseudosteady-state model is an approximation as well —and that mobility components can (and do) change substan-tially with time (which is a condition that we do not explicitlyconsider). However, our primary goal remains the develop-ment of a WOR (or fw) relation for pseudosteady-state flowconditions.

Aside from the estimation of reservoir volumetric properties(N, Np,max, etc), WOR data can be plotted versus time (or the"material balance time" functions) on a log-log plot and usedas a diagnostic tool to identify the dominant reservoir perfor-mance mechanism (uniform displacement, water coning, orwater channeling).2

Our intention is to develop a methodology that combines theclassic techniques for well test and production data analysis(pressure derivative, pressure integral, and pressure integral-derivative functions) with our proposed model for the analysisof oil and water production data (in this case the WOR, WOR-derivative, WOR-integral, and WOR-integral-derivative func-tions). We believe that the qualitative analysis of water-oilratio (WOR) performance data will significantly improve ourevaluation and assessment of well problems, assist in injec-tion/production balancing, and aid in the identification of thedominant reservoir drive mechanism.

Development of the Pseudo steady-State WOR ModelIn this section we provide the derivation of the pseudosteady-state WOR equation. We begin with the rigorous, single-phasepseudosteady-state flow equation used to describe the beha-vior of oil and water production. This derivation makes

SPE 77569 Analysis and Interpretation of Water-Oil-Ratio Performance 3

several (potentially) limiting assumptions — but the pseudo-steady-state relation does not require the same assumptions asthe steady-state case (which is much more restrictive). Speci-fically, he primary assumptions used in the conventional WOR(steady-state) analysis are:

� The pressure and the flowrate throughout the systemremain constant.

� The mobili ty ratio is assumed to be equal to unity.� The logarithm of relative permeability ratio versus water

saturation relationship is linear.

The rule of thumb for the existing WOR models is that thesemodels can be applied only after the WOR (or fw) functiondevelops a straight-line trend — in most of the cases this oc-curs when the value of the fw function approaches 0.5-0.7 (orhigher). Our goal is to extend the conventional WOR analysesto include the case of pseudosteady-state flow behavior — andto develop a relation that can represent the entire spectrum ofoil and water production performance. We do not expect sucha relation to represent the entire production history for aparticular well — but we do anticipate significantly improvedbehavior when we incorporate pseudosteady-state flow char-acteristics into the WOR model.

We begin by using the relation presented by Blasingame andLee3 for single-phase variable-rate, pseudosteady-state flow ina bounded reservoir. This result is given as:

mbtwA

tAhc

B

rCe

A

kh

B

q

p

φµ

γ 2339.0

4ln 6.70

Ä2

+= ............(1)

Eq. 1 is subject to the following assumptions:

� Pseudosteady-state (i.e., boundary-dominated) flow con-ditions must exist in the entire reservoir system.

� Homogeneous and isotropic reservoir.�Constant porosity and permeabili ty.� Small and constant fluid compressibili ty.� Constant fluid viscosity.� Small pressure gradients.� Negligible gravity forces.

For simplicity, Eq. 1 can be written in the following, morecompact form:

pssmb bmt

pq

+∆= (general form) .....................(2)

where:

∫=t

dtqq

tmb0

1

(material balance time)........(3)

Ahc

Bm

tφ 0.2339 = (pss slope term) ...................(4)

24

ln70.6

wApss

rCe

A

kh

Bb

γµ= (pss intercept term)..............(5)

Using the general form of the pseudosteady-state flow equa-tion (Eq. 2), we can write the following relations for single-phase oil and water flow:

pssoooo btm

pq

+∆= (oil form).............................(6)

psswwww btm

pq

+∆= (water form) ........................(7)

where the "oil " and "water" variables are defined as:

opo qNt = (oil material balance time) .. (8)

Ahc

Bm

t

oo φ

2339.0= (oil pss slope term)..............(9)

24

ln70.6 wAo

oopsso

rCe

A

hk

Bb

γµ

= (oil pss intercept term) (10)

wpw qWt = (water material balance time)... (11)

Ahc

Bm

t

ww φ

2339.0= (water pss slope term) .............(12)

2

4ln70.6

wAw

wwpssw

rCe

A

hk

Bb

γµ

=

(water pss intercept term).........(13)

Recall ing the definition of the water-oil ratio function, WOR,we have:

o

wq

qWOR= (water-oil ratio).................(14)

Substituting Eqs.6 and 7 into Eq. 13, we obtain an expressionfor the WOR function:

psswww

pssooo

btm

btmWOR

+

+= ..................................................(15)

Eq. 15 is specifically valid for pseudosteady-state flow beha-vior — and we implicitly assume that the entire reservoir is atpseudosteady-state flow conditions (i.e., both the oil and waterphases). To extend this concept, we can re-write Eq. 15 interms of the fractional flow of oil and water functions (fo, fw(respectively). Re-writing Eq. 15, we obtain:

pssooo

psswwww

btm

btmf

+

++

=1

1...............................................(16)

psswww

pssoooo

btm

btmf

+

++

=1

1................................................(17)

It is important to note that we have made no assumptionsregarding a relationship between the relative permeabil ityfunctions and water saturation — although we do note that in


Eqs. 15-17 we presume that the mobilit y ratio is constant (butis not necessarily unity).dominant reservoir drive mechanism.

Application of the Pseudo steady-State WOR ModelIn this section we use the new pseudosteady-state WOR modelto analyze and interpret two field case examples. The firstexample is the case of a vertical well i n a thick, low perme-abili ty dolomite sequence in West Texas, and the secondexample is the case of a vertical well i n a high permeabil ity,moderately consolidated reservoir in South Louisiana. In bothcases the fields are undergoing waterfloods, but in the case ofthe wells from South Louisiana, water influx is also suspected.

In these examples we use a standard suite of plots to evaluateour regression-based analysis of the WOR function using theproposed pseudosteady-state WOR model. The followingsuite of plots is used:

Late-Time Extrapolation for Recoverable Oil

� log(fw) versus Np

� 1/fw versus Np

� fo versus Np

� log(WOR) versus Np

Regression Analysis — Global Summary Plot

� log(WORcal) versus log(WORmeas)

Example 1: NRU Well 3106Well 3106 from the North Robertson Unit (NRU Well 3106)in Gaines County Texas is the first case we consider, and theanalysis/interpretation plots for this case are provided in Figs.1-5. NRU Well 3106 was completed on July 9 1989. Thetotal depth of the well is 7,350 ft with two perforated intervals— 6,667-7,185 ft and 5,964-6,538 ft. The well was acidizedand hydraulically fractured in two stages on 15 July 1989 andagain on 17 July 1989.

In Figs. 1-4 we find that the pseudosteady-state WOR modelrepresents the production data functions extremely well —especially in the late-time region, but surprisingly, the WORmodel also matches all but the very earliest production data.For comparison we have plotted the "conventional" (i.e.,steady-state) straight-line extrapolation models on each analy-sis plot — and we note very good correlation of these straight-line models with the late-time production data.

Based on our observations in Figs. 1-4, we can conclude thatthe proposed pseudosteady-state WOR model best representsthe performance of the WOR functions for NRU Well 3106.We can also conclude that the conventional straight-line extra-polation models also work well for this case.

As a consistency check, we also provide a direct comparisonof the calculated and measured WOR functions for NRU Well3106 in Fig. 5. We note that with the exception of the veryearly time data (i.e., low WOR), the correlation of the calcu-lated and measured WOR functions is excellent.

Example 2: WWL Well B41In this case we consider the production data from Well B41 inthe West White Lake Field (WWL Well B41) in South Louis-iana. It is relevant to note that West White Lake Field is aMiocene sandstone reservoir sequence with good porosity andpermeabil ity characteristics. This is relevant because it mayhelp us to understand the water and oil production perfor-mance more clearly — and this case is in significant contrastto the case from the North Robertson Unit (NRU Well 3106)where the reservoir in that case is a low permeabili ty dolstonewith very heterogeneous reservoir properties.

WWL Well B41 was completed on 2 July 1987. The totaldepth of the well is 7,300 ft and is perforated from 7,038 to7,064 ft.

The analysis plots for this case are presented in Figs. 6-9, andwe immediately note good agreement between the pseudo-steady-state WOR model and the measured production data.We also note good agreement for the conventional straight-line (i.e., steady-state) extrapolation models. In this case, wedo observe that the proposed pseudosteady-state WOR modelbest represents the "late time" performance.

Our emphasis on the late-time data is warranted — this regionis where both the steady-state and pseudosteady-state modelswould be best expected to work. While we do not expect thepseudosteady state WOR relation to work at early times, it isworth noting that the early time data are dominated by oil f low(see Fig. 8). The pseudosteady-state relation presumes a con-stant mobili ty ratio, and this may not be the case where such adramatic change in WOR performance occurs. Regardless, weare satisfied that the proposed pseudosteady-state relation doesaccurately represent the data for this case.

As in the previous case (NRU Well 3106), we provide a directcomparison of the calculated and measured WOR functions forWWL Well B41 in Fig. 10, and we note also note in this casethat the correlation of the calculated and measured WOR func-tions is excellent. For reference, the early time WOR data arenot shown on Fig. 10, we have only plotted the WOR data thatwere actually used in the regression process.

SummaryOur goal in this section was to present and validate the pro-posed pseudosteady-state WOR relation. Recall ing this result,we have:

psswww

pssooo

btm

btmWOR

+

+= ..................................................(15)

Substituting the definitions for material balance time for eachphase, Eq. 15 becomes:

pssww

pw

pssoo

po

o

w

bq

Wm

bq

Nm

q

qWOR

+

+== .....................................(18)


In reviewing Eq.18, we immediately recognize that the rateand cumulative production functions can not be uncoupled, wecan rearrange the variables to perhaps yield a more usefulform, but the rate and cumulative functions remain.

Multiplying through Eq. 18 by the (qo/qw) ratio gives:

wpsswpw

opssopo

qbWm

qbNm

1

+

+=

or,

opssopowpsswpw qbNmqbWm +=+ ........................(19)

We note that although Eq. 19 could be considered a "moresimple" form of Eq. 15, we can not reduce Eq. 19 further intoa direct "analysis" relation.

The point of this discussion is that the proposed pseudosteady-state WOR model (Eq. 15) is valid (at least for pseudosteady-state flow, presuming a constant mobili ty ratio). However, wecan not reduce this approach into a direct analysis methodo-logy — we must use Eq. 15 as an analysis model, and fit thatmodel to the field performance data.

We are quite pleased with the performance of the pseudo-steady-state WOR model (and its auxili ary models), and webelieve that this approach is both robust and appropriate.While we did not obtain a direct solution or even an extrapo-lation formula from Eq. 15, we strongly recommend itsapplication. At this point our recommendation is to use re-gression analysis — it is our hope that future efforts yield adirect analysis method, in the form of a single-variablerelation, or (perhaps) in the form of a "type curve" or someother type of comparative solution technique.

ANALYSIS OF OIL AND WATER PRODUCTION DATAIn this chapter we focus our efforts on a discussion of the"conventional" analysis methods for water and oil productiondata. In particular, we present methods that are commonlyused in petroleum industry to estimate the recoverable oilvolume (Np,max). The primary advantage of the methods wediscuss is that these methods are straightforward — we onlyrequire production data in order to estimate the recoverable oilvolume and to make a production forecast. No tedious calcu-lations are required, and most of the methods discussed arewell accepted in the industry.

The disadvantage is that most of these methods are not rigor-ous, and can fail (sometimes in a spectacular fashion) — how-ever, we believe that the practical value of these simple ap-proaches makes an important contribution to the area of pro-duction data analysis.

Analysis MethodsThis subsection addresses the inventory of analysis techniquesthat are used to quantitatively and qualitatively assess oil andwater production data.

Exponential Decline Curve MethodsWe begin our discussion with a focus on the estimation of therecoverable oil volume, Np,max, (or the estimated ultimate

recovery (EUR)) using the simple exponential rate declinerelation. In particular, we utilize the following plottingtechniques that are derived from the exponential rate declinerelation:

� log(qo) versus t� qo versus Np

The log(qo) versus t plot is the most common productionanalysis mechanism — unfortunately this plot is only rigor-ously valid for the case of liquid flow at a constant bottomholeflowing pressure. The qo versus Np plot is derived from theexponential rate decline relation, and also requires the assump-tion of a constant bottomhole pressure.

The governing relation for the exponential production declinecase is given by: (flowrate identity)

tDoio

ieqq −= ..............................................................(20)

Integrating Eq. 20 to yield the cumulative production, thenrearranging, we obtain the following relation:

pioio NDqq −= .........................................................(21)

Conventional WOR Extrapolation MethodsWe will again discuss the various straight-line models pro-posed for the water-oil ratio (WOR) functions: WOR, thefractional flow of oil (fo), and the fractional flow of water (fw).These functions are plotted in various formats versus thecumulative oil production (Np) as indicated in the list ofplotting functions given below.

� log(WOR) versus Np

� log(fw) versus Np

� fo versus Np

The application of these functions was discussed previously— however, our present goal is to apply and compare thesefunctions with other analysis techniques.

In order to utili ze the fractional flow relations (fo and fw) werequire these definitions. As such, the definitions of fo and fware:

wo

oo qq

qf

+= ...............................................................(22)

wo

ww qq

qf

+= ..............................................................(23)

For the case of extrapolation using WOR versus Np and fwversus Np, we have the following implicit models for thesepresumed behaviors:

pbNaeWOR= .............................................................(24)

pbNw aef = ................................................................(25)

Another extrapolation technique that has been used exten-sively (but not investigated) is the case of a plot of fo versusNp, which should extrapolate to the recoverable reserves at fo=0. The presumed relationship for this case is


po bNaf −= ...............................................................(26)

New Methods for the Analysis of Production DataIn addition to the methods previously identified for the analy-sis of water and oil production data, we also present two newtechniques that can be used to estimate the recoverable oil ,Np,max, or the oil -in-place, N. These new techniques use thefollowing plotting functions:

� 1/fw versus Np (yields an estimate of Np,max)� 1/qo versus to (yields an estimate of N)

The 1/fw versus Np plotting function yields an apparent lineartrend that can be extrapolated to provide an estimate of therecoverable oil , Np,max. In contrast, the 1/qo versus to plottingfunction yields a linear trend (predicted by pseudosteady-statetheory), where the slope of the trend is proportional to 1/N.

For a plot of 1/fw versus Np we implicitly assume the followingrelationship:

pw bNaf −= 1/ ...........................................................(27)

For a plot of 1/qo versus to = Np/qo we begin with Eq. 6, andupon rearranging, we have:

)( 1 opo qNbaq += ...................................................(28)

At the condition qo = 0, we can rearrange Eq. 28 to yield therecoverable reserves, Np,max. This result is given by:

bN max,p 1/ = ...............................................................(29)

Put simply, Eq. 29 specifies that we construct a plot of 1/qo

versus Np/qo and obtain the slope (b) in order to estimate therecoverable reserves (Np,max).

Qualitative Methods for the Analysis of Production DataWe also provide a qualitative analysis of the behavior of theWOR, WOR derivative, integral, and integral-derivativefunctions plotted versus production time (t), oil materialbalance time (to), and total material balance time (tt). Weprovide a systematic presentation of examples for these plotslater in this section. In particular, we review typical trendsand characteristics of these plotting functions.

The specific plotting functions considered in this discussionare as foll ows:

Log-Log Diagnostic Plots:

� log(WOR) and log(WORd) versus log(t)� log(WORi) and log(WORid) versus log(t)� log(WOR) and log(WORd) versus log(to)� log(WORi) and log(WORid) versus log(to)� log(WOR) and log(WORd) versus log(tt)� log(WORi) and log(WORid) versus log(tt)

Cumulative WOR and Cumulative fw Plots:

� log(WORc) versus Np

� log(fwc) versus Np

Total Production Function Plots:

� log(WOR) versus total production (Np+Wp)� log(fo) versus log(tt)

Ershaghi X-Plot TechniqueOur final effort is to demonstrate the Ershaghi "X-plot"technique (refs. 4-5), and to compare the results of this methodwith the "conventional" (steady-state) straight-line extrapola-tion techniques.6 In all of the cases we considered, the X-plottechnique gave the least consistent results compared to theother methods used. Based on our results, we believe that"conventional" straight-line extrapolation methods will pro-duce more accurate (and more consistent) estimates of therecoverable oil than the X-plot technique.

Analysis ExamplesIn this subsection we demonstrate the analysis of productiondata using the techniques described above. The data for theseexamples are taken from corporate archives and deemed to berepresentative of the waterflood/water influx process.

Given the large numbers of figures in this section, discussionwill be limited to summary results for each "family" of analy-sis methodologies — with exceptions as warranted.

NRU 3106 WWL B41

Exponential Decline Curve Methods

� log(qo) versus t Fig. 11 Fig. 12� qo versus Np Fig. 13 Fig. 14

Conventional WOR Extrapolation Methods

� log(WOR) versus Np Fig. 15 Fig. 16� log(fw) versus Np Fig. 17 Fig. 18� fo versus Np Fig. 19 Fig. 20

New Methods for the Analysis of Production Data

� 1/fw versus Np Fig. 21 Fig. 22� 1/qo versus to Fig. 23 Fig. 24

Qualitative Methods for the Analysis of Production Data

Log-Log Diagnostic Plots:

� WOR, WORd vs. t Fig. 25 Fig. 31� WORI, WORid vs t Fig. 26 Fig. 32� WOR, WORd vs to Fig. 27 Fig. 33� WORI, WORid vs to Fig. 28 Fig. 34� WOR, WORd vs tt Fig. 29 Fig. 35� WORI, WORid vs tt Fig. 30 Fig. 36

Cumulative WOR and Cumulative fw Plots:

� log(WORc) versus Np Fig. 37 Fig. 39� log(fwc) versus Np Fig. 38 Fig. 40

Total Production Function Plots:

� log(WOR) vs (Np+Wp) Fig. 41 Fig. 43� log(fo) versus log(tt) Fig. 42 Fig. 44

Ershaghi X-Plot Technique

� Np versus X (refs. 4-5) Fig. 45 Fig. 46


Exponential Decline Curve MethodsIn Figs. 11 and 12 we confirm the log(qo) versus time tech-nique for these data cases — note that both cases have well-established semilog trends. Similarly, the rate-cumulativeplots (Figs. 13 and 14) confirm the validity of the exponentialde-cline model for these cases.

Conventional WOR Extrapolation MethodsThe WOR versus Np plots for these cases are shown in Figs. 15and 16 — we note that both cases (NRU Well 3106 and WWLWell B41) ill ustrate a strong semilog trend. Similarly, thelog(fw) versus Np plots (Figs. 17 and 18) also show very strong"late time" semilog straight line trends. Finally we considerthe fo versus Np (Cartesian) plots (Figs. 19 and 20). Here weagain find strong linear trends that yield consistent estimatesof recoverable reserves (Np,max).

New Methods for the Analysis of Production DataIn this effort we "test" our proposed extrapolation/interpreta-tion methods that can be used to estimate the recoverable re-serves (Np,max). The first technique involves a plot of 1/fw ver-sus Np (see Figs. 21 and 22) — in both cases we again findstrong linear trends in the data.

In the second approach we use 1/qo versus Np/qo as the plottingfunction (Figs. 23 and 24). We find that this technique pro-vides a remarkable "straightening" of the data for all casesconsidered in this work. The recoverable reserves are esti-mated using the slope of this trend, which can be problematicwith erratic data (though this has generally not been the casein our experience).

Qualitative Methods for the Analysis of Production DataIn this effort we plotted the data functions for each case, thencompared the variety of plots for each case (this is a "quali -tative" comparison so our only effort is "interpretation," notanalysis). In the first suite we consider Well NRU 3106 (Figs.25-30) in terms of the "qualitative" log-log plots. This case isgenerally well behaved — the "time" format plots are slightlyerratic, though consistent (and changing) trends do evolve forthe WORi and WORid functions. The WOR functions behavesomewhat better for the "oil " and "total" material balance timefunctions, but the "shift" from one unit-slope trend to anotheris prevalent in each case.

In Figs. 31-36 we consider the qualitative performance ofWWL Well B41 — where we immediately note that these dataare not as well-behaved as the data for NRU Well 3106. Per-haps the most relevant observation is that a single unit-slopetrend evolved for all of the WOR functions in the case of the"oil" material balance time function. This is the strongesttrend observed for this case.

Interestingly, the "cumulative" WOR and fw functions (WORc

and fwc) provide apparent straight-line trends for both cases —but other than this linearity, there is no clear interpretation ofthis behavior. This behavior is shown in Figs. 37-40.

Our effort now shifts to comparisons of WOR and total pro-duction (Np+Wp) (see Figs. 41 and 43), as well as fo versus(Np+Wp)/(qo+qw) (see Figs. 42 and 44). The most relevant

comment for the log(WOR) versus (Np+Wp) trend is that anapparent semilog straight-line evolves — however, the inter-pretation (and use) of this trend is unclear. It could be arguedthat the log-log plot of fo versus (Np+Wp)/(qo+qw) is simply ananalog for a "rate-material balance time" plot since the trendsare very similar to what we find when attempting to matchproduction data to decline type curves. There may be an even-tual application in that regard.

Ershaghi X-Plot TechniqueAs we note in both Figs. 45 and 46 the X-plot trends are verywell established. The issue of analysis is relevant, at whatvalue of fw do we extrapolate the plot? To demonstrate a fail-ure of the X-plot method we provide the production history forWWL Well A17 in Fig. 47 — note that not only is there nolinear trend, the data appear to trending in the opposite direc-tion of the proposed methodology suggests. This is hardly anindictment of the X-plot method, but we do believe that thiscase is "typical."

Summary and ConclusionsIn this work we present empirical and semi-analytical modelsfor the analysis and interpretation of oil and water productiondata. In particular, we provide the development, verification,and application of a new water-oil performance relation forpseudosteady-state flow conditions. This model was found tobe superior to all other models considered for the represen-tation of field production data (WOR, fw, and the water and oilflowrate functions).

We provide a broad spectrum of demonstrative analyses for oiland water production data — from "conventional" extrapola-tion plots to a several new analysis and diagnosis plots, in-cluding the new pseudosteady-state flow relation for thesimultaneous flow of oil and water. The oil and water produc-tion data used in this study were obtained from the NorthRobertson Unit (NRU) located in Gaines County (West Texas)and the West White Lake Field located in Southwest Louisi-ana.

Pseudosteady-State WOR ModelWe presented the development and validation of a pseudo-steady-state water-oil ratio model. This model does not re-quire the limiting assumptions of the conventional (steadystate) WOR model (the mobilit y ratio is not required to beunity, nor are any assumptions made regarding the relativepermeabil ity functions).

We utili zed the new pseudosteady-state oil-water flow modelin the following analysis/interpretation plots:

� Fractional flow of oil (fo) versus cumulative oil produc-tion (Np)

� Logarithm of fractional flow of water (fw) versus cumula-tive oil production (Np)

� Reciprocal of fractional flow of water (1/fw) versus cumu-lative oil production, (Np)

� Water-oil ratio (WOR) versus cumulative oil production(Np)


In each of these plots the pseudosteady-state flow model isplotted as a comparison to the given data function. We note insummary that the new pseudosteady-state model is anexcellent interpolation model — but we (regrettably) mustalso note that our new pseudosteady-state model does notprovide a practical approach for the estimation of reserves byextrapolation. This is an area for future research considera-tion.

Estimation of Recoverable OilThough often considered an empirical (and sometimes suspectapproach), the estimation of recoverable oil reserves (Np,max)by extrapolation of a function versus cumulative production(or some other production-related function) is a very popular(and effective) method for estimating such reserves. It is ourbelief that, because of the uncertainty in the accuracy of the"conventional" extrapolation methods, as well as the lack of acompletely rigorous mathematical foundation, the best ap-proach is to apply as many of these extrapolation techniquesas possible. Such an approach will provide duplication (hencevalidation), and though there is no single "perfect" extrapo-lation technique, the comparison of results obtained fromdifferent approaches does provide consistency as well as anelement of validation.

Our process of estimating the ultimate recovery using straight-line extrapolations of production data function incorporates asequence of simultaneous analysis for all data functions (e.g.,qo, fo, fw, WOR, 1/fw, 1/qo, and other functions versuscumulative production). The goal for such an analysis is todevelop estimates of recoverable oil that best represent most,if not all , of the extrapolation methods.

Specifically, we use the following extrapolation techniques forthe simultaneous estimation of recoverable oil (or EUR) atcurrent producing conditions:

� Oil rate (qo) versus production time (t)� Oil rate (qo) versus cumulative oil production (Np)� Fractional flow of oil (fo) versus cumulative oil produc-

tion (Np)� Logarithm of fractional flow of water (fw) versus cumu-

lative oil production (Np)� Reciprocal of fractional flow of water (1/fw) versus cumu-

lative oil production (Np)� Reciprocal of oil production rate (1/qo) versus oil material

balance time (to)

Qualitative Analysis of Oil and Water Production DataIn addition to the estimation of recoverable oil , we also canalso perform a "qualitative" analysis of the water and oil pro-duction data. These techniques provide a visual analysis ofthe data and may provide a qualitative assessment of the per-formance. While our focus is on the qualitative analysis ofdata, some of these techniques may also provide a quantitativeestimate of recoverable oil .

In particular, we considered the foll owing analysis techniques:

� Logarithm of cumulative fractional flow of water (fwc)versus the cumulative oil production (Np)

� Logarithm of cumulative water-oil ratio (WORc) versusthe cumulative oil produc-tion (Np)

� Reciprocal of the water rate (1/qw) versus water materialbalance time (tw)

� Logarithm of the water-oil ratio (WOR) versus totalproduction (Np+Wp)

� Logarithm of the fractional flow of oil (fo) versuslogarithm of total material ba-lance time (t)

� Logarithm of the oil production rate (qo) versus logarithmof oil material balance time (to)

� Cumulative oil production versus X-function (refs. 5-6)

ConclusionsThe following conclusions are derived from this study:

Pseudosteady-State WOR Model

1. The proposed pseudosteady-state WOR model is acombination of the analytical solutions for single-phaseoil and water flow at pseudosteady-state conditions, andthe only significant assumption that arises in thisrelation is that of a constant (non-unity) mobili ty ratio.While this may seem a limiting assumption, wesuccessfully matched all of the production data casesconsidered in this work using this new model.

2. The proposed pseudosteady-state WOR model clearlyprovides the best representation for the oil and waterproduction data cases that we investigated.

3. The major limitation to our new model is that it doesnot provide a mechanism for the prediction of futureproduction — this new model is expressed in terms ofoil and water production rates and cumulative pro-duction, and it is not possible to write a predictiveformula (or even an extrapolation formula) using thisrelation.

Estimation of Recoverable Oil

1. In this work we provide a compilation of the "conven-tional" straight-line extrapolation methods used in theindustry for the estimation of recoverable oil . Thetechniques considered are:

—log(qo) versus t—qo versus Np

— log(fw) versus Np

—log(fo) versus Np

These techniques should be applied simultaneously inorder to obtain consistent estimates of the recoverableoil . Application of only one or two techniques willli kely lead to substantial misinterpretations of the cor-rect model behavior, which, in turn, will l ead to over-or underestimation of recoverable reserves.

2. We proposed two new methods for estimating recover-able oil reserves. These techniques are:

—1/fw versus Np

—1/qo versus Np/qo

A plot of 1/fw versus Np should yield a straight-linetrend that can be extrapolated to 1/fw=1, which yields


the recoverable reserves, Np,max. Similarly, a plot of1/qo versus Np/qo should yield a straight-line trendwhere the slope of this trend is equal to 1/Np,max. Theresults obtained by these new methods correspond quitewell to the results obtained by the "conventional" WORtechniques (i.e., the straight-line extrapolation methodsdiscussed in the previous point).

Analysis of Oil and Water Production Data

1. We have found that plotting the cumulative water-oilratio (WORc) function and/or the cumulative fractionalflow of water (fwc) functions versus the cumu-lative oilproduction (Np) does typically yield a straight-line trend(in most cases we considered). However, theextrapolation of the observed straight-line trend doesnot yield a consistent estimate of recoverable re-serves(when compared to other techniques). This behaviorpresents an opportunity for future work.

2. We have extended the diagnostic plots proposed byChan2 (i.e., the WOR and WOR derivative, integral, andintegral-derivative functions plotted versus productiontime, oil material balance time, and total material ba-lance time). The following observations are noted:

—The WOR and WOR integral, and integral-deriva-tive functions typically exhibit a clearly definedunit slope trend when plotted versus time or mater-ial balance time.

—The WOR derivative function is typically veryerratic, and can not be used for routine analysis dueto poor overall behavior.

Log-log plots of the WOR functions versus t (or to)tend to be reasonably well behaved — with the notedexception of the WOR-derivative function, which istypically very erratic. It is our strong recommendationthat these plots only be used for diagnostic purposes— although we would encourage a future study focus-ing on a "type curve" analysis approach for WOR data.

3. We believe that the X-plot method (refs. 4-5) providesno substantive advantage over the "conventional" ex-trapolation techniques discussed previously. In fact,the X-function plot typically does not develop a clearstraight-line trend.

Of most concern is the observation that the estimates ofrecoverable oil obtained using the X-plot do not typi-cally correspond to the estimates obtained by the con-ventional and proposed extrapolation techniques.Finally, the extrapolation of X-function tends to signifi-cantly overestimate the value of recoverable oil .

Nomenclaturea = intercept of log(kro/krw) versus Sw plot, fractionA = reservoir drainage area, ft2 [m2]ac = intercept of conventional log(WOR) versus Np

plot, fractionaer = intercept of Ershaghi4,5 Np versus X-function plot,

STB [std m3]B = formation volume factor, RB/STB [res m3/std m3]b = slope of of log(kro/krw) versus Sw plot, 1/STB

[1/std m3]bc = slope of conventional log(WOR) versus Np plot,

1/STB [1/std m3]ber = slope of Ershaghi4,5 Np versus X-function plot,

STB [std m3]

bpss =2

4ln6.70

wArCe

A

kh

Bγ

µ, intercept of ∆p/q versus t

plot for general variable-rate case, psi/STB/D[kPa/std m3/d]

CA = reservoir shape factor, dimensionlessct = total system compressibil ity, psia-1 [kPa-1]Di = decline oil rate, STB/D [std m3/d]Er = Np/N, overall reservoir recovery, fractionfo = fractional flow of oil , fractionfoc = cumulative fractional flow of oil , fractionfw = fractional flow of water, fractionfwc = cumulative fractional flow of water, fractionh = total formation thickness, ft [m]k = effective formation permeabili ty, mdkr = relative formation permeability, md

mpss =Ahc

B

tφ2339.0 , slope of ∆p/q versus t plot for

general variable-rate case, psi/STB/D/hr [kPa/stdm3/d/hr]

Np = cumulative oil production, STB [std m3]Np,max = recoverable oil , STB [std m3]OOIP = original oil-in-place, STB [std m3]∆p = pi- pwf, pressure drop, psi [kPa]p = pressure, psia [kPa]pi = initial reservoir pressure, psia [kPa]pss = pseudosteady-statepwf = flowing bottomhole pressure, psia [kPa]q = production flowrate, STB/D [std m3/d]qi = initial production flowrate, STB/D [std m3/d]re = reservoir drainage radius, ft [m]rw = wellbore radius, ft [m]Sw = water saturation, fractionSwc = connate water saturation, fractionSwirr = irreducible water saturation, fractiont = time, daystm = material balance time, daysto = Np/qo, oil material balance time, daystt = (Np+Wp)/(qo+qw), total material balance time, daystw = Wp/qw, water material balance time, daysWi = cumulative water injection, STB [std m3]WOR = water-to-oil ratio, fractionWORc = Wp/Np, cumulative WOR function, fraction


WORd = etc.) , , , ( )( to tttxWORdx

dx = , fraction

WORi = etc.) , , , ( 1

0to

xtttxdxWOR

x=∫ , fraction

WORid= etc.) , , , ( )( toi tttxWORdx

dx = , fraction

Wp = cumulative water production, STB [std m3]

X = ww ff

11

1ln −

− , Ershaghi4,5 X-function

Greek:γ = 0.577216, Euler’s constantφ = reservoir porosity, fractionµ = fluid viscosity, cp [Pa*s]

Subscripts:i = initialo = oilw = watert = total

References

1.Robert S. Thompson, John D. Wright: Oil Property Eval-uation, Thomson-Wright Associates, Golden, CO, 1985.

2.Chan, K.S.: "Water Control Diagnostic Plots," paper SPE30775 presented at the 1995 Annual Technical Con-ference and Exhibition, Dallas, TX, 22-25 October..

3.Blasingame, T.A. and Lee, W.J.: "Variable-Rate Reser-voir Limits Testing," paper SPE 15028 presented at theSPE Permian Basin Oil and Gas Recovery Conference,Midland, TX, 13-14 March 1986.

4. Ershaghi, I. and Omoregie, O.: "A Method for Extrapola-tion of Cut vs. Recovery curves," JPT (February 1978)203-204.

5. Ershaghi, I. and Abdassah, D.: "A Prediction Techniquefor Immiscible Processes Using Field Performance Data,"JPT (April 1984) 664-670.

6. Startzman, R.A and Wu, C.H: "Discussion of EmpiricalPrediction Technique for Immiscible Processes," JPT(December 1984) 2192-2194.

AcknowledgementsThe authors would like to acknowledge the in-kind contibu-tions of Fina Oil and Chemical Co. (now TotalFinaElf) andUnocal (now Spirit Energy) for the field data used in thiswork.

Figure 1 – Fractional Flow of Water Versus Cumulative OilProdu ction, NRU Well 3106.

Figure 2 – Reciprocal of Fractional Flow of Water VersusCumulative Oil Produ ction, NRU Well 3106.

Figure 3 – Fractional Flow of Oil Versus Cumulative OilProdu ction, NRU Well 3106.


Figure 4 – Water-Oil Ratio Versus Cumulative Oil Produ ction,NRU Well 3106.

Figure 5 – Comparison of the Calculated WOR Performance(Pseudo steady-State Model) with the MeasuredWOR Performance, NRU Well 3106.

Figure 6 – Fractional Flow of Water Versus Cumulative OilProdu ction, WWL Well B41.

Figure 7 – Reciprocal of Fractional Flow of Water VersusCumulative Oil Produ ction, WWL Well B41.

Figure 8 – Fractional Flow of Oil Versus Cumulative OilProdu ction, WWL Well B41.


Figure 9 – Water-Oil Ratio Versus Cumulative Oil Produ ction,WWL Well B41.

Figure 10– Comparison of the Calculated WOR Performance(Pseudo steady-State Model) with the MeasuredWOR Performance, WWL Well B41.

Figure 11– Oil and Water Produ ction Rate History, NRU Well3106.

Figure 12– Oil and Water Produ ction Rate History, WWL WellB41.

Figure 13– Oil Production Rate Versus Cumulative Oil Produ c-tion, NRU Well 3106.


Figure 14– Oil Production Rate Versus Cumulative Oil Produ c-tion, WWL Well B41.

Figure 15– Water-Oil Ratio Versus Cumulative Oil Production,NRU Well 3106.

Figure 16– Water-Oil Ratio Versus Cumulative Oil Produ ction,WWL Well B41.

Figure 17– Fractional Flow of Water Versus Cumulative OilProdu ction, NRU Well 3106.

Figure 18– Fractional Flow of Water Versus Cumulative OilProdu ction, WWL Well B41.

Figure 19– Fractional Flow of Oil Versus Cumulative Oil Pro-duction, NRU Well 3106.


Figure 20– Fractional Flow of Oil Versus Cumulative Oil Pro-duction, WWL Well B41.

Figure 21– Reciprocal of Fractional Flow of Water VersusCumulative Oil Produ ction, NRU Well 3106.

Figure 22– Reciprocal of Fractional Flow of Water VersusCumulative Oil Produ ction, WWL Well B41.

Figure 23– Reciprocal of Oil Rate Versus Oil Material BalanceTime, NRU Well 3106.

Figure 24– Reciprocal of Oil Rate Versus Oil Material BalanceTime, WWL Well B41.

Figure 25– Water-Oil Ratio and Water-Oil Ratio Derivative Ver-sus Production Time, NRU Well 3106.


Figure 26– Water-Oil Ratio Integral and Integral-Derivative Ver-sus Production Time, NRU Well 3106.

Figure 27– Water-Oil Ratio and Water-Oil Ratio Derivative Ver-sus Oil Material Balance Time, NRU Well 3106.

Figure 28– Water-Oil Ratio Integral and Integral-Derivative Ver-sus Oil Material Balance Time, NRU Well 3106.

Figure 29– Water-Oil Ratio and Water-Oil Ratio Derivative Ver-sus Total Material Balance Time, NRU Well 3106.


Figure 30– Water-Oil Ratio Integral and Integral-Derivative Ver-sus Total Material Balance Time, NRU Well 3106.

Figure 31– Water-Oil Ratio and Water-Oil Ratio Derivative Ver-sus Production Time, WWL Well B41.

Figure 32– Water-Oil Ratio Integral and Integral-Derivative Ver-sus Production Time, WWL Well B41.

Figure 33– Water-Oil Ratio and Water-Oil Ratio Derivative Ver-sus Oil Material Balance Time, WWL Well B41.


Figure 34– Water-Oil Ratio Integral and Integral-Derivative Ver-sus Oil Material Balance Time, WWL Well B41.

Figure 35– Water-Oil Ratio and Water-Oil Ratio Derivative Ver-sus Total Material Balance Time, WWL Well B41.

Figure 36– Water-Oil Ratio Integral and Integral-Derivative Ver-sus Total Material Balance Time, WWL Well B41.

Figure 37– Cumulative Water-Oil Ratio Versus Cumulative OilProdu ction, NRU Well 3106.

Figure 38– Cumulative Fractional Flow of Water Versus Cumu-lative Oil Production, NRU Well 3106.


Figure 39– Cumulative Water-Oil Ratio Versus Cumulative OilProdu ction, WWL Well B41.

Figure 40– Cumulative Fractional Flow of Water Versus Cumu-lative Oil Production, WWL Well B41.

Figure 41– Water-Oil Ratio Versus Total Production, NRU Well3106.

Figure 42– Fractional Flow of Oil Versus Total Material BalanceTime, NRU Well 3106.

Figure 43– Water-Oil Ratio Versus Total Production, WWL WellB41.


Figure 44– Fractional Flow of Oil Versus Total Material BalanceTime, WWL Well B41.

Figure 45– X-Plot, NRU Well 3106.

Figure 46– X-Plot, WWL Well B41.

Figure 47– X - Plot, WWL Well A17 (example failure of the X-Plot Method).

Documents

00077569 Analysis and Interpretation of Water-Oil-Ratio Performance