Reservoir Justification of Stimulation Treatments
Michael J. Economides
Dowel1 Schlumberger
l-l INTRQDUCTION Well and reservoir evaluationsare of particular importancein thejustification of stimulation treatments.Pressure transient and well performance analyses are the obvious available tools. This chapter will include two parts: (1) an outline of the fundamentals pressuretranof sient analysis, and (2) skin characterization,idiosyncrasiesof tight reservoirs,and recommended methodologies applicableto wells and reservoirs that are candidatesfor stimulation. l-2 FUNDAMENTALS OF PRESSURE TRANSIENT ANALYSIS Pressuretransient analysis, long establishedin groundwater hydrology, has becomea major tool in petroleum reservoir engineeringduring the last 40 years. van Everdingen and Hurst (1949) introduced what is believed to be one of the first major works in modem pressuretransient analysis.In their paper, they usedLaplacetransformations in formulating analytical mathematicalsolutions to fluid flow problems in porous media. Their work extended and transferred to the petroleum literature much of the analytical work done in the heat conduction area. Homer (1951) presenteda practical methodologywhich has becomethe mainstay in pressurebuildup analysis. Homer, using the principle of superposition, developed a simpleinterpretation graphingtechnique allowed and that the calculation of the permeability, skin effect, and average reservoir pressure. It was not until 1970 that Agarwal, Al-Hussainy, and Rameypresenteda major work that usheredin a new era in the field. Type-curve matching and well responsediagnostics became widely usedapproach pressuretrana to sient analysis. Since 1970, numerous publications on different reservoir and well geometries, effects of dual porosity, fractures,two-phaseflow, and multilayer reservoirs have been written.
In this chapter, a fundamentaltreatmentof the subject is offered, along with a step-by-stepmethodology and mathematical basisof the most widely usedequationsand interpretation techniques. 1-2.1 Solution Of The Diffusivity Equation For A Well Producing At Constant Rate In An Infinite-Acting Reservoir Much of the work donein modempressuretransientanalysis beganwith someform of the diffusivity equationa direct analogto the convection/diffusion equationwidely used in other engineering disciplines. The diffusivity equation, the result of the continuity equation, a rate equation (Darcy‘s law), and an equation of state for low compressibility and constant viscosity fluids (liquids), is normally given by
l-l
RESERVOIR STIMlJLATloN
Using “infinite-acting” boundaryandinitial conditions, and with the aid of a Boltzman transform, the solution to Eq. l-l is
Pr,t = Pi * Ei(x) Pwf
Pwf=Pi - * ln ,QErw2 . (l-11)
t
Similarly to othercases pseudosteady andsteady for state state, the skin effect can be introduced.
+ 2 s) (1-12)
or, more conveniently, if pw, = p1 hr which is found on the extension of the straight line at log t (1 hr), then s = 1.151
a9 1 F+yJy= ap
+w, - ap
k
at -
(l-1)
Introduction of the following dimensionlessvariables (cast in traditional “oil-field” units) khAp pD = 141.2 qBp ’ tD= 0.000264 kt h-$ rw2
l-2
Eqs. 1-14 and l-15a are significant in well analysis for drawdown testing. In a semiloggraphof field data,the slopeof the straightline portion of the datawill yield the value of the permeability, k (if all other variables are known); the value of the skin effectcanbe calculated from Eq. l-15a. The value of p1 hr can be obtainedfrom the graphical construction. Of course, the major problem is to identify the correct “semilog straightline,” because datamay cause field wide confusionasto which is the correct line. For the purposes of this review, the following notes will suffice. In a drawdowntest of an infinite acting reservoir, three regimes of flow may be evident: l wellbore storage effects,
Pi - PI hr m 1%
k
(&ctrw2 + *
(l-15a)Байду номын сангаасThe skin effect, first introducedby van Everdingenand Hurst (1949) defines a steady-state pressuredifference aroundthe wellbore. A positive value indicatesa restriction to flow while a negativevalueindicatesflow enhancement, usually a result of stimulation. At times, a negative value may be the result of a natural fracture. In oil-field units and changing natural log to log base 10, Eq. 1-12 is transformed into