5 superposition [repaired]

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Well Test Analysis, © UTP – MAY 2011

The Principle of Superposition




Principle of Superposition

At this point, the most useful solution to the flowequation, the Ei-function solution, appears to beapplicable only for describing the pressure distribution in aninfinite reservoir, caused by the production of a single well inthe reservoir, and, most restrictive of all, production of the well at constant rate beginning at time zero. In this section,

we demonstrate how application of the principle ofsuperposition can remove some of these restrictions,and we conclude with examination of an approximation thatgreatly simplifies modeling a variable rate well.For our purposes, we state the principle of superposition inthe following way:The total pressure drop at any point in a reservoir is the sumof the pressure drops at that point caused by flow in each ofthe wells in the reservoir .




The simplest illustration of this principle is the caseof more than one well in an infinite reservoir. As an example, consider three wells, Wells A, B, and C,that start to produce at the same time from an infinite

reservoir.




Application of the principle of superposition shows that:

( P i-p wf ) total at well A=(P i-P)due to well A + (P i-P)due to well B +(P i-P)due to well CThen




Note that this equation includes a skin factor for Well A, but does notinclude skin factors for Wells B and C.Because most wells have a nonzero skin factor and because we are

modeling pressure inside the zone of Altered permeability near Well A,we must include its skin factor.However, the presence of nonzero skin factors for Wells B and C affects

pressure only inside their zones of altered permeability and has noinfluence on pressure at Well A if Well A is not Within the altered zone ofeither Well B or Well C.Using this method, we can treat any number of wells flowing at

constant rate in an infinite-acting reservoir.Thus, we can model so-called interference tests, which basically aredesigned to determine reservoir properties from the observedResponse in one well (such as Well A) to production from one or moreother wells (such as Well B or Well C) in a reservoir.




A relatively modern method of conducting interference

tests, called pulse testing, is based on these ideas.Our next application of the principle of Superposition is tosimulate pressure behavior in Bounded reservoirs.Consider this well in a distance, L, from a singleno-flow boundary (such as a sealing fault).




Mathematically, this problem is identical to the problem ofa well a distance 2L from an image well. (i.e., a well thathas the same production history as the actual well).

The reason this two-well system simulates the behavior ofa well near a boundary is that a line equidistant between

the two wells can be shown to be a no-flow boundary.(along this line the pressure gradient is zero). which meansthat there can be no flow. Thus, this is a simple two-wellin-an infinite reservoir problem:




Our final and most important application of the

superposition principle will be to model variablerate producing wells. To illustrate this application, consider theFollowing case




in which a well produces at rate q1 from time 0 to t1.

At t1, the rate changed to q2, and at t2 the rate changed to q3.The problem that we wish to solve is this:

At some time t > t2, what is the pressure at the sand-faceof the well?To solve this problem, we will use superposition as before,but, in this case, each well that contributes to the totalpressure drawdown will be at the same position in thereservoir-the wells simply will be "turned on" at differenttimes. The first contribution to a drawdown in reservoirpressure is by a well producing at rate q1starting at t=0.This well, in general, will be inside a zone of alteredpermeability; thus, its contribution to drawdown of reservoir pressure

is:-




Starting at time t1 , the new total rate is q2. We introducea Well 2, producing at rate (q2 –q1) starting at time t1 , sothat the total rate after t1 is the required q2.Note that total elapsed time since this well startedproducing is (t-t1); note further that this well is still inside azone of altered permeability.Thus, the contribution of Well 2 to drawdown of reservoirpressure is




Similarly, the contribution of a third well is

Thus, the total drawdown for the well with two changes inrate is:




Proceeding in a similar way, we can model an actual well

with dozens of rate changes in its history; we also canmodel the rate history for a well with a continuouslychanging rate (with a sequence of constant-rate periodsat the average rate during the period)-but, in many suchcases, this use of superposition yields a lengthyequation, tedious to use in hand calculations. Note,however, that such an equation is valid only for the totaltime elapsed since the well began to flow at its initialrate-i.e., for time (t) , ri must be less or equal re




Example

A flowing well is completed in a reservoir that has the followingproperties:pi = 2,500psia, B = 1.32 RB/STB, µ = 0.44cp, k = 25 md,h=43ft, C t=18x10-6 ,Φ =0.16

What will the pressure drop be in a shut-in well 500 ftfrom the flowing well when the flowing well has beenshut in for 1 day following a flow period of 5 days at300 STB/D?Solution: We must superimpose the contributions of two wellsbecause of the rate change:




Horner's Approximation:This approximation can be used in many cases to avoid the use of

superposition in modeling the production history of a variable-rate well. With this approximation, we can replace the sequence of Ei functions, reflecting rate changes, with a single Ei function that

contains a single producing time and a single producing rate.The single rate is the most recent nonzero rate at which the well wasproduced; we call this rate q last for now.The single producing time is found by dividing cumulativeproduction from the well by the most recent rate; we callthis producing time t p or pseudo-producing time:tp= 24( Np / q last ), where N p is cumulative production(STB) andqlast is production rate (STB/day)




Then, to model pressure behavior at any point in a

reservoir, we can use the simple equation

Two questions arise logically at this point:(1) What is the basis for this approximation?(2) Under what conditions is it applicable?The basis for the approximation is not rigorous , butintuitive, and is founded on two criteria:(1) If we use a single rate in the approximation, the clearchoice is the most recent rate; such a rate, maintained forany significant period, determines the pressure distribution

nearest the wellbore and approximately out to the radius




2- Given the single rate to use, intuition suggests that wechoose an effective production time such that the product

of the rate and the production time results in the correctcumulative production. In this way, material balances willbe maintained accurately.But when is the approximation adequate? If we

maintain a most-recent rate for too brief a time interval,previous rates will play a more important role indetermining the pressure distribution in a tested reservoir.

We can offer two helpful guidelines.First, if the most recent rate is maintained sufficientlylong for the radius of investigation achieved at this rate toreach the drainage radius of the tested well, then Horner'sapproximation is always sufficiently accurate.




This rule is quite conservative, however.

Second, we find that, for a new well that undergoes aseries of rather rapid rate changes, it is usually sufficientto establish the last constant rate for at least twice as longas the previous rate.

When there is any doubt about whether these guidelines Are satisfied, the safe approach is to use superposition tomodel the production history of the well. Application of Horner 's Approximation:This approximation can be applied only when ∆t last greater or equal to 2(∆t next to last )




Pressure Build Up Test:Its basically, conducted by producing a well at constant

rate for some time, shutting the well in (usually at thesurface), allowing the pressure to build up in the wellbore,and recording the pressure(usually down hole) in the wellbore as a function of time.

From these data, it is frequently possible to estimate1- formation permeability2- current drainage-area pressure,3- To characterize damage or stimulation4- Reservoir heterogeneities or boundariesThe analysis method discussed in this course is basedlargely on a plotting procedure suggested by Horner. Analysis technique using type curves, is discussed as well.




To derive an equation describing a pressure buildup test.Test is in an infinite, homogeneous, isotropic reservoircontaining a slightly compressible, single-phase fluid with constant fluid properties.

Any wellbore damage or stimulation is considered to beconcentrated in a skin of zero thickness at the wellbore; atthe instant of shut-in, flow into the wellbore ceases totally.No actual buildup test is modeled exactly by this idealized




description, but the analysis methods developed for thiscase prove useful for more realistic situations if we

recognize the effect of deviation from some of theseassumptions on actual test behavior. Assume that:1- a well is producing from an infinite-acting reservoir

(one in which no boundary effects are felt during theentire flow and later shut-in period).2-The formation and fluids have uniform properties, sothat the Ei function (and, thus, its logarithmicapproximation) applies, and3-That Horner's pseudo-producing time approximation isapplicable.




If the well has produced for a time tp at rate q before

shut-in, and if we call time elapsed since shut- in ∆t, then, using superposition we find that following shut-in




The last equation suggest that plotting of shut in bottom

hole pressure ( P ws ) against {(tp+∆t)/∆t}, yields a straight Line with slope of:m= -162.6 qBµ/ kh




Thus the value of permeability can be obtained from

the slope of the straight line.In additional to that the straight line can be extrapolated toinfinite shut in time {(tp+∆t)/∆t}=1, yields the value of original formation pressure (P i ) .conventional practice in the industry is to plot Horner plot

on semi- logarithmic paper with values of {(tp+∆t)/∆t},decreasing from left to right.The slope (m) on such plot is found by simply subtract thepressures at any two points on the straight line that areone cycle apart on the semi-log paper.

The skin factor (s) can be determined from the dataavailable. At the instant a well shut in , the flowing BHP,Pws is




Combining these equations and solving for the skin factor(s), we have

It is conventional practice in the petroleum industry tochoose a fixed shut- in time, ∆t of 1 hour and the correspondingshut-in pressure, P 1hr to use in this equation (although any shut-intime and the corresponding pressure would work just as well).The pressure, , P 1hr must be on the straight line or its extrapolation.

We usually can assume further that:log (tp + ∆t)/ tp is negligible.With these simplifications




Note again that the slope m is considered to be apositive number in this equation.

In summary, from the ideal buildup test, we candetermine formation permeability (from the slope m of theplotted test results), original reservoir pressure, pi, andskin factor, s, which is a measure of damage orStimulation.Example: A new oil well produced 500 STB/D for 3 days;it then was shut in for a pressure buildup test during

which the data in following table were recorded.




For this well, net sand thickness, h, is 22 ft; formationvolume factor (B) is 1.3 RB/STB, porosity ɸ=0.2, totalCompressibility is 20x10 -6 oil viscosity µo is 1.0 cp and

wellbore radius r w is 0.3 ft.estimate formation per meability, k, initial reservoir pressure, pi, and skin factor( s).




To estimate permeability, original reservoir pressure, andskin factor, we must plot shut-in BHP, p ws, vs.log (t p + ∆t)/ ∆t; measure the slope m anduse equation of the slope to calculate formationpermeability, k; extrapolate the curve to [(t p +∆t) / ∆t] = 1and read original reservoir pressure, p i




and use the equation of skin to calculate the skin factors.Producing time, t , is given to be 3 days, or 72 hours.

(in this case, Horner's approximation is exact because the well was produced at constant rate since time zero).Thus, we develop the following table.

We plot these data, and they fall along a straight linesuggested by ideal theory. The slope m of the straightline is 1,950- 1,850= 100 psi (units are actually psi/cycle).The formation permeability k is:




From extrapolation of the buildup curve to [(tp + ∆t) / ∆t]=1, pi = 1,950 psig. The skin factor s is

The value for p ws , is P1hr hr on the ideal straight line at(t +∆t) /∆t = (72+ 1)/1= 73; this value is P 1hr =1764 psig.Thus




Actual Buildup Tests

Encouraged by the simplicity and ease of applicationof the ideal buildup test theory, we may test an actual well and obtain a most discouraging result: Insteadof a single straight line for all times, we obtain a curve with

a complicated shape. To explain what went wrong, theradius-of-investigation concept is useful. Based on thisconcept, (Figure in below)




we logically can divide a buildup curve into three regions1- an early-time region during which a pressure transient is

moving through the formation nearest the wellbore;2- a middle-time region during which the pressuretransient has moved away from the wellbore and intothe bulk formation; and3- a late-time region, in which the radius of investigation has

reached the well's drainage boundaries.Let us examine each region in more detailEarly-Time Region:

As we have noted, most wells have altered permeabilitynear the wellbore.Until the pressure transient caused by shutting in the wellfor the buildup test moves through this region of altered




permeability, there is no reason to expect a straight-lineslope that is related to formation permeability,

(We should note that the ideal buildup curve, i.e. one witha single straight line over virtually all time is possible fora damaged well only when the damage is concentrated ina very thin skin at the sand-face.)

There is another complication at earliest times in a pressurebuildup test.Continued movement of fluid into a wellbore (after-flow, a form of wellbore-storage) following the usual surface shut-in compresses thefluids (gas, oil, and water) in the wellbore. Why should this affect the character of a buildup curve at earliest

times? Perhaps theclearest answer lies in the observation that the idealizedtheory leading to the equation




P ws=Pi -m log [( tp +∆t) /∆t] explicitly assumed that, at ∆t=0 flow rate abruptly changed from q to zero. In practice, q declines towardzero but, at the instant of surface shut-in, the down-holerate is, in fact, still q. Thus, one of the assumptions wemade in deriving the buildup equation is violated in theactual test, and another question arises. Does after-flowever diminish to such an extent that data obtained in apressure buildup test can be analyzed as in the ideal test?The answer is yes, fortunately, but the important problemof finding the point at which after-flow ceases distortingbuildup data remains. This is the point at which the earlytime region usually ends, because after-flow frequently




lasts longer than the time required for a transient to movethrough the altered zone near a well. We will deal with thisproblem more completely when we discuss a systematicanalysis procedure for pressure buildup tests.Middle-Time Region: When the radius of investigation has moved beyondthe influence of the altered zone near the tested well,and when after flow has ceased distorting the pressurebuildup test data, we usually observe the ideal straight line whose slope is related to formation permeability.This straight line ordinarily will continue until the radius ofinvestigation reaches one or more reservoir boundaries,massive heterogeneities, or a fluid/fluid contact.




Systematic analysis of a pressure buildup test using theHorner method of plotting P ws, vs. log[(tp+∆t) /∆t] requires

that we recognize this middle-time line and that, inparticular, we do not confuse it with false straight lines inthe early- and late-time regions. As we have seen,determination of reservoir permeability and skin factor

depends on recognition of the middle-time line; estimationof average drainage-area pressure for a well in adeveloped field also requires that this line be defined.Late-Time Region:Given enough time, the radius of investigation eventually will reach the drainage boundaries of a well. In this latetime region pressure behavior is influenced by boundaryconfiguration, interference from nearby wells,




significant reservoir heterogeneities, and fluid/fluidContacts.Qualitative Behavior of Field Tests:

We now have developed the background required to understand thequalitative behavior of commonly occurring pressure buildup curves.There is an important reason for this examination of behavior.It provides a convenient means of introducing some factors thatinfluence these curves and that can obscure interpretation unlessthey are recognized.In the figures that follow, the early, middle, and late time regions aredesignated by ETR, MTR, and LTR, respectively.In these curves, the most important region is the MTR.




Interpretation of the test using the Horner plot[ P ws , vs. log (t +∆t)/∆t] is usually impossible unless the

MTR can be recognized.

This figure illustrates the ideal buildup test, in whichthe MTR spans almost the entire range of the plotteddata. Such a curve is possible for an undamaged well




(Curve 1, with the level of pwf, the flowing pressure

at shut-in, is shown for reference) and for a damaged well with an altered zone concentrated at the wellbore.This latter situation, shown in Curve 2, is indicated bya rapid rise in pressure from flowing pressure at shut-in to

the pressures along the MTR.Neither case is observed often in practice with asurface shut-in because after-flow usually distorts theearly data that would fall on the straight line.




The above figure illustrates the pressure buildup testobtained for a damaged well. Curve 1 would be obtained with a shut-in near the perforations (minimizing theduration of after-flow); Curve 2 would be obtained with themore conventional surface shut-in.




Note in this figure that the flowing BHP at shut-in, Pwf, is thesame for either case, but that the after-flow that appears with the surface shut-in

1- completely obscures information reflectingnear-well conditions in the ETR and

2- delays the beginning of the MTR. A furthercomplication introduced by after-flow is that severalapparent straight lines appear on the buildup curve. Thequestion arises, how do we find the straight line (the MTRline) whose slope is related to formation

permeability? We will deal with this question shortly.




The last figure shows characteristic behavior of a builduptest for a fractured well without after-flow. For such a

well, the pressure builds up slowly at first; the MTRdevelops only when the pressure transient has movedbeyond the region influenced by the fracture.In a buildup test for a fractured well, there is a possibilitythat boundary effects will appear before the ETR hasended (i.e., that there will be no MTR at all).The coming figure illustrates two different types ofBehavior in the LTR of a buildup test plot. Curve1illustrates middle- and late-time behavior for a wellReasonably centered in its drainage area; Curve 2Illustrates behavior for a well highly off center in itsdrainage area.




For simplicity, the ETR is not shown in either case.Many curve shapes other than those discussed aboveappear in practice, of course. Still, these few examplesillustrate the need for a systematic analysis procedure thatallows us to determine the end of the ETR (usually, thetime at which after-flow ceases distorting the test data)and the beginning of the LTR. Without this procedure,there is a high probability of choosing the incorrectstraight-line segment and using it to estimate permeabilityand skin factor.




Effects and Duration of After-flow: We have noted several problems that after-flow causes

the buildup test analyst. Summarizing, these problemsInclude:1-Delay in the beginning of the, MTR, making itsrecognition more difficult.2-Total lack of development of the MTR in some cases, With relatively long periods of after-flow and relativelyearly onset of boundary effects. 3- Development of several false straight lines, any one of which could be mistaken for the MTR line. We note further that recognition of the middle-time line isessential for successful buildup curve analysis based onthe Horner plotting method ( P ws , vs. log [ ( t p + ∆ t )/ ∆ t]




because the line must be identified to estimate reservoirpermeability, to calculate skin factor, and to estimate static

drainage-area pressure.The need for methods to determine when (if ever) After-flow ceased distorting a buildup test is clear; thisSection fills that need.The characteristic influence of after-flow on a pressurebuildup test plot is a lazy S-shape at early times, asshown in following figure. In some tests, parts of theS-shape may be missing in the time range during whichdata have been recorded - e.g., data before time A maybe missing, or data for times greater than time B may beabsent . Thus, the shape of the build up test is not sufficientto indicate the presence or absence of after-flow.

-




A log-log graph of pressure change, P ws-P wf , in a builduptest vs. shut-in time, ∆t, is an even more diagnosticindicator of the end of after-flow distortion.For this method type curve will be used.To use this type curve the followings must be prepared.




Where CsD is dimensionless well bore storage constant. Where

for a well with a rising liquid/gas interface in the wellbore, And

for a wellbore containing only single-phase fluid we define




wellbore storage distortion (after-flow in the case of a buildup test)has ceased when the graphed solutions for finite C sD become

identical to those for C sD =0. Also , a line with unit slope (45" line)appears at early times for most values of C sD and s.The meaning of this line in a buildup test is that the rate of afterflow is identical to the flow rate just before shut-in.If the unit-slope line is present, the end of after-flowdistortion occurs at approximately one and a half logcycles after the disappearance of the unit-slope line.Regardless of whether the unit-slope line is present, theend of after-flow distortion can be determined byoverlaying the log-log plot of the test data onto the Rameysolution (figure in slide no.49) -plotted on graph or tracingpaper with a scale identical in size to the Ramey graph -




finding any pre-plotted curve that matches the test data,

and noting when the pre-plotted curve for finite value ofC sD becomes identical to the curve for C sD =0.This point, on the actual data plot, is the end of after-flowor wellbore storage distortion.

If the unit-slope line is present, we can establish the value of C SD that characterizes the actual test.There, we noted that any point on the unit-slope line mustsatisfy the relationship:

which, in terms of variables with dimensions, leads to




where ∆t and ∆p are values read from a point on the unit-slope line. If we can establish Cs in this waya less acceptable alternative is to use the actualmechanical properties of the well, e.g.,Cs = 25.65/ ρ Wb for a well with a rising liquid/gasInterface. we then can establish C sD and thus determinethe proper curve on which to attempt a curve match.(It is difficult to Interpolate between values of C sD on thiscurve; accordingly , many test analysts prefer to find aMatch with the pre-plotted value of C sD closest in valueto the calculated value.) With C sD established, andpermeability, k, and skin factor, s, determined fromcomplete analysis of the test, we can use the empirical




relationships below to verify the time, t Wbs , marking the

end of wellbore storage distortion.

Example:

The data in following table obtained in a pressure builduptest on an oil well producing above the bubble point.The well was produced for an effective time of 13,630hours at the final rate.




other data include the following




From the data given, Answer the following:

1- At what shut-in time ( ∆t) does after-flow ceasedistorting the pressure buildup test data?2- At what shut-in time ( ∆t) do boundary effects appear?3- Estimate the

permeability

Solution:First we plot PWS Vs.(t+∆t)/ ∆t




From the plot, it seems plausible that after-flow distortiondisappears at (t+ ∆t) /∆t =2,270 or ∆t=6 hours because ofthe end of the characteristic lazy-S-shaped curve.However, other reservoir features can lead to this

same shape, so we confirm the result with the log-loggraph.This log –log graph is a plot of After plotting ∆p which isequal to (P ws- P wf ), vs. ∆t e which is equal to∆t/(1+∆t/t p)




From the log-log graph we find that:The actual data fit well* curves for s= 5 for several

values of C sD (e.g., C SD = 103, 10

4, and10

5).

In each case, the curve fitting the earliest data coincidesWith the C sD =0 curve for s=5 at ∆t e=∆t=4 to 6 hours. This, then, is the end of wellbore effects:twbs

=6 hours.The data begin to deviate from the semi-log straight line

at (t p +∆t) /∆t=274 or ∆t= 50 hours. On the log-log graph, data begin falling below the fittingcurve at ∆t=∆t e , =40 hours , consistent with the semi-log

graph. In summary, basing our quantitative judgmenton the more sensitive semi-log graph, we say that theMTR spans the time range of ∆t = 6 hours to ∆t = 50 hours




This judgment is verified qualitatively by the log-log graphcurve matching. Even though the semi-log graph is moresensitive (i.e., can be read with greater accuracy),it alone is not sufficient to determine the beginning andend of the MTR: matching Ramey's solution is a criticallyImportant part of the analysis.The log-log curve-matching analysis was performedwithout knowledge of C SD . Note that C SD can beestablished in this case, at least approximately:from the curve match, we note that the data are near theunit-slope line on the graph of Ramey's solution; the point∆p= 100, ∆t=0.1 is essentially on this line.Thus:




Then




Permeability estimation:First of all we need to detect MTR( the beginning and the

end) of the straight line.In our example MTR spans the time range of ∆t = 6 hours to ∆t = 50 hours [2,270≥ (tp + ∆t) /∆t ≥274] The slope m of this straight line ism=4437-4367= 70 psi/cycle.

Then

It is of interest to determine the portion of reservoirsampled during the MTR; that region is given roughly bythe radius of investigation achieved by the shut-intransient at the start and end of the MTR.




Thus, a significant fraction of the well's drainage area hasbeen sampled; its permeability is 7.65 md




For the same example: 1-Calculate the skin factor for the tested well.

2- Calculate the effective wellbore radius r wa3- Calculate the additional pressure drop near the

wellbore caused by the damage that is present.4- Calculate the flow efficiency.5- Verify the end of wellbore storage distortionSolution:1- In the skin factor equation, we need P 1hr fromextrapolation of the middle-time line to a shut-in time of1 hour (∆t = 1 hour), (t p + ∆t) /∆t = 13,631. From an

extrapolation the middle-time line to this timeP 1hr =4,295 psi. (Note the difference in this with the onegiven in the table ( actual pressure) which is = 4,103 psi




This value got a very good agreement with the value which beengot from the matching.

2-




The physical interpretation of this result is that thetested well is producing 250 STB/D oil with the samepressure drawdown as would a well with a wellboreradius of 0.00034 ft and permeability unaltered up tothe sand-face.

To understand the significance of this quantity, note that




This shows that the effect of S on the total pressure

drawdown is the same as that of a well with no altered zone butwith a wellbore radius of rwa .Calculation of effective wellbore radius is of special value foranalyzing wells with vertical fractures.

Model studies have shown that for highly conductive vertical

fractures with two equal-length wings of length Lf,Lf=2 rwaThus, calculation of skin factor from a pressure buildup or fallofftest can lead to an estimate of fracture length-useful in a postfracture analysis.However, this analysis technique for a fractured well isfrequently oversimplified; more complete methods are discussedlater.




Calculation of Additional Pressure Drop Near Wellbore:We defined additional pressure drop (∆p), across the altered zone as:

In terms of the slope m of the MDR

Calculation of this additional pressure drop across the altered zonecan be a meaningful way of translating the abstract skin factor intoa concrete characterization of the tested well.For example, a well may be producing 100 STB/D oil with a

drawdown of 1,000 psi. Analysis of a buildup test might show that (∆p s), is 900 psi and,

thus, that 900 psi of the total drawdown occurs across the alteredzone.




This implies that if the damage were removed, the wellcould produce much more fluid with the same drawdownor, alternatively, could produce the same 100 STB/D witha much smaller drawdown.Calculation of Flow Efficiency:

The final method that we will examine for translating(s) into a physically meaningful characterization of a wellis by calculation of the flow efficiency,(E). We define flow efficiency as the ratio of actual or observedPI of a tested well to its ideal PI.




For rapid analysis of a pressure buildup or fall off test, thelast equation can be written in approximate form as

Where p*, the extrapolation of the middle-time line to(tp +∆t) /∆t =1 , is found more readily than p ─ , whichrequire lengthy analysis. Flow efficiency is actually timedependent unless a well reaches pseudo-steady stateduring the producing period (only then is p ─ -p wf

is constant

Flow efficiency is unity for a well that is neither damagednor stimulated.For a damaged well, flow efficiency is less than one




for a stimulated well, flow efficiency is greater than one. A damaged well with a calculated flow efficiency of 0.1 isproducing about 10% as much fluid with a given pressuredrawdown as it would if the damage were removed; astimulated well with a calculated flow efficiency of two isproducing about twice as much fluid with a given pressuredrawdown as it would had the well not been stimulated.Back to the example:3- (∆p) s=0.869(m)(s)

=0.869x70x6.37=387psi4- p* = 4,437 + 2 (70) =4,577 psi




5-End of Wellbore Storage Distortion:




Static Drainage-Area Pressure:For a well in a reservoir in which there has been some pressure

depletion, we do not obtain an estimate of original reservoirpressure from extrapolation of a buildup curve.Our usual objective is to estimate the average pressure in the

drainage area of the well; we will call this pressure staticdrainage-area pressure.There two useful methods for making these estimates:1-The Matthews- Brons-Hazebroek(MBH) p * method.2-The modified Muskat method.The p * method was developed by Matthews et al.

by computing buildup curves for wells at various positionsin drainage areas of various shapes and then, from theplotted buildup curves,




comparing the pressure (p *) on an extrapolated middle timeline with the static drainage-area pressure (p ─ ), which is the

value at which the pressure will stabilize given sufficient shutin time.The buildup curves were computed using imaging techniques andthe principle of superposition.The results of the investigation are summarized in a seriesof plots of kh (p* - p ─ )/70.6qBµ vs. 0.000264 kt p /ɸµct A[Note that kh(p* -p)/70.6 qBµ can be written as2.303 (p* - p ─ )/m

Also, the group 0.000264 ktp/ ɸµct A is a dimensionless

time and is symbolized by t DA.The group kh (p* - p ─ ) / 70.6 qBµ is a dimensionless

pressure and is given the symbol p DMBH




Steps to use p * method:1- Read the value of p * from the extrapolation of

Middle time line.2-Estimate the drainage area shape.3- Choose the proper curve (just like the one in below) for

the drainage-area shape of the tested well.



4- Estimate 0.000264 kt p /ɸµct A , and find2.303 (p *- p ─ )/m =p

DMBH(it is essential that t

pused in this

calculation be the same used in Horner plot.)5- Calculate p ─ = p * - m p DMBH /2.303

5 superposition [repaired]

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