THE PRESSURE DERIVATIVE

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Nickolas Fields
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1 Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable. The field curves may be mached agains wo ye curves raher han one. The logarihmic ressure derivaive has he advanage of being dimensionwise consisen wih ressure. Hence hey may be dislayed in he same coordinae sysem. Drawdown During radial flow he following equaion alies: B k kh ln 4π ϕµ cr w The above equaion will show u as a sraigh line on a versus ln lo. This imlies ha he logarihmic ressure derivaive will be consan. S d d ln B The above equaion will show u as a horizonal line in any coordinae sysem. By use of he chain rule he logarihmic derivaive may be exressed in erms of he ime derivaive: d d ln d d During he wellbore sorage eriod sorage eriod he following equaion alies: qb s The logarihmic derivaive becomes: d d ln qb s which is he same righ hand side as for he ressure difference,. Flow eriod diagnosics is bes done in a log-log coordinae sysem. Taking he logarihm on boh sides of he above equaion yields:

The logarihmic ressure derivaive has he advanage of being dimensionwise consisen wih ressure. Hence hey may be dislayed in he same coordinae sysem.

2 log ' [ ln] qb log + log The above equaion will show u as a sraigh line wih sloe one. Wellbore sorage followed by radial flow has a characerisic shae on diagnosic (log log) lo which is shown below. s Figure : Pressure and ressure derivaive diagnosic lo for drawdown This examle illusraes he diagnosic roeries of he ressure derivaive. The wellbore sorage may be recognized as a sraigh line wih uni sloe and he radial flow eriod will show u as a horizonal line. Buildu The dimensionless ressure change for build u is: ( ) ( + ) + ( ) DBU D D D D D D D Suose here are wo simulaneous radial flow eriods. Then he buildu may be described by he modified Horner equaion B k (ln S) ϕµ c r ( + ) BU + w where BU ws. The ressure difference is illusraed below. Noe ha is he las recorded flowing ressure. Tom Aage Jelmer THE PRESSURE DERIVATIVE

The wellbore sorage may be recognized as a sraigh line wih uni sloe and he radial flow eriod will show u as a horizonal line.

3 Figure : The develomen of P BU wih ime. One may define an equivalen ime: e + Subsiuion of he equivalen ime ino he buildu equaion yields: B k e (ln ϕµ c r BU + S w Observe ha he above equaion has he same aearance as he corresonding drawdown equaion. The ressure derivaive becomes: ) which will lo like a horizonal line. d w d ln e B Immediaely afer he well has been closed, fluid will coninue o flow ino he well. This henomenon has been called aferflow. From he definiion of equivalen ime, we conclude ha for small values of shuin ime, here is essenially no difference beween he shuin ime,, and equivalen ime, e,. Hence: e Tom Aage Jelmer THE PRESSURE DERIVATIVE 3

d w d ln e B Immediaely afer he well has been closed, fluid will coninue o flow ino he well. This henomenon has been called aferflow.

4 Suose he well was closed during a radial flow eriod. Then he dimensionless buildu equaion during aferflow will simlify o: DBU D sd ln D + D D Noe ha he laer erm is essenially be zero for small values of. Hence he buildu equaion during aferflow is he same as for wellbore sorage during a drawdown es. DBU D SD The equivalen dimensional equaion becomes: qb BU S The above equaion will show u as a sraigh line wih uni sloe on a log-log lo. The diagnosic lo for a ransiion from aferflow o radial flow is shown below. Figure 3: The buildu diagnosic lo has he same aearance as for drawdown. For an old well, he condiion ha shuin ime is small in comarison wih he roducion ime is valid during he enire es, hen he buildu equaion will simlify o: B k BU ln s ϕµ crw Tom Aage Jelmer THE PRESSURE DERIVATIVE 4

Hence he buildu equaion during aferflow is he same as for wellbore sorage during a drawdown es.

5 The ransiion beween wellbore sorage and radial flow will show u as he revious curves when loed agains he shuin ime. Figure 4: Pressure and ressure derivaive for buildu in an old well, >> Derivaion algorihm A derivaive is usually aroximaed by a finie difference quoien. As such i is sensiive o noise. Some smoohing may be necessary o filer ou high frequency noise. Mos numerical filers deend on an even samling rae. This is rarely he case in well esing. For a non-even sacing beween samled he oins, he derivaive may be aroximaed by he sloe of a leas square line hrough neigbouring oins. All oins in he inerval conribue o he comued derivaive. Tom Aage Jelmer THE PRESSURE DERIVATIVE 5

As such i is sensiive o noise. Some smoohing may be necessary o filer ou high frequency noise. Mos numerical filers deend on an even samling rae.

6 Figure 5: Schemaic of a 7 oin leas square filer An alernaive algorihm uses hree oins only, he oin of ineres, one o he lef and one o he righ. Figure 6: Schemaic of a derivaion algorihm The derivaive is aroximaed as he weighed average beween he lef and righ hand sloes. d dx i + + Tom Aage Jelmer THE PRESSURE DERIVATIVE 6

Figure 6: Schemaic of a derivaion algorihm The derivaive is aroximaed as he

7 A smaller difference,, on he horizonal axis leads o a beer aroximaion of he derivaive. Hence he highes weigh facor is alied o he finie difference quoien wih he smalles difference. I is recommended o sar wih consecuive oins. If he degree of noise is unacceable, hen he disances and have o be increased. The oins and are seleced as he firs oins such ha If x ln e, hen and, L ln L. ln i i L L has been called he smoohing arameer. The degree of smoohing will increase wih increasing smoohing arameer. On he oher hand, he accuracy of he derivaive aroximaion will decrease. Hence L should be ke as small as ossible. Exerience has shown ha L 0. is ofen a naural choice. Any derivaion algorihm has a roblem near he ends of he inerval. The calculaion scheme is hen running ou of oins. This henomenon has been called he end effec of he ressure derivaive. These regions have o be reaed searaely. One soluion is o reduce he smoohing arameer close o he ends of he inervals. Tom Aage Jelmer THE PRESSURE DERIVATIVE 7

ln i i L L has been called he smoohing arameer. The degree of smoohing will increase wih increasing smoohing arameer. On he oher hand, he accuracy of he derivaive aroximaion will decrease.

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