Page 78 - Petroleum Production Engineering, A Computer-Assisted Approach

Guo, Boyun / Computer Assited Petroleum Production Engg 0750682701_chap06 Final Proof page 70 3.1.2007 8:40pm Compositor Name: SJoearun

6/70 PETROLEUM PRODUCTION ENGINEERING FUNDAMENTALS
6.1 Introduction then the operating flow rate q sc and pressure p wf at the
bottom-hole node can be determined graphically by plot-
Well deliverability is determined by the combination of
well inflow performance (see Chapter 3) and wellbore ting Eqs. (6.1) and (6.2) and finding the intersection point.
flow performance (see Chapter 4). Whereas the former The operating point can also be solved analytically by
describes the deliverability of the reservoir, the latter pre- combining Eqs. (6.1) and (6.2). In fact, Eq. (6.1) can be
sents the resistance to flow of production string. This rearranged as

chapter focuses on prediction of achievable fluid produc- p 2 ¼ p q sc 1 n : (6:3)
2
p
tion rates from reservoirs with specified production string wf C
characteristics. The technique of analysis is called ‘‘Nodal Substituting Eq. (6.3) into Eq. (6.2) yields
analysis’’ (a Schlumburger patent). Calculation examples
are illustrated with computer spreadsheets that are p p q sc 1 n Exp(s)p 2
2
provided with this book. C hf
2
2
4
6:67 10 [Exp(s) 1] f M q z T 2
T
z
5 sc ¼ 0; (6:4)
6.2 Nodal Analysis D cos u
i
Fluid properties change with the location-dependent pres- which can be solved with a numerical technique such as the
sure and temperature in the oil and gas production system. Newton–Raphson iteration for gas flow rate q sc . This
To simulate the fluid flow in the system, it is necessary to computation can be performed automatically with the
‘‘break’’ the system into discrete nodes that separate sys- spreadsheet program BottomHoleNodalGas.xls.
tem elements (equipment sections). Fluid properties at the Example Problem 6.1 Suppose that a vertical well
elements are evaluated locally. The system analysis for produces 0.71 specific gravity gas through a 2 ⁄ 8 -in.
7
determination of fluid production rate and pressure at tubing set to the top of a gas reservoir at a depth of
a specified node is called ‘‘Nodal analysis’’ in petroleum 10,000 ft. At tubing head, the pressure is 800 psia and
engineering. Nodal analysis is performed on the principle the temperature is 150 8F, whereas the bottom-hole
of pressure continuity, that is, there is only one unique temperature is 200 8F. The relative roughness of tubing is
pressure value at a given node regardless of whether the about 0.0006. Calculate the expected gas production rate
pressure is evaluated from the performance of upstream of the well using the following data for IPR:
equipment or downstream equipment. The performance
curve (pressure–rate relation) of upstream equipment is Reservoir pressure: 2,000 psia
called ‘‘inflow performance curve’’; the performance IPR model parameter C: 0.1 Mscf/d-psi 2n
curve of downstream equipment is called ‘‘outflow per- IPR model parameter n: 0.8
formance curve.’’ The intersection of the two performance
curves defines the operating point, that is, operating flow Solution Example Problem 6.1 is solved with the
rate and pressure, at the specified node. For the conveni- spreadsheet program BottomHoleNodalGas.xls. Table 6.1
ence of using pressure data measured normally at either shows the appearance of the spreadsheet for the Input data
the bottom-hole or the wellhead, Nodal analysis is usually and Result sections. It indicates that the expected gas flow
conducted using the bottom-hole or wellhead as the solu- rate is 1478 Mscf/d at a bottom-hole pressure of 1059 psia.
tion node. This chapter illustrates the principle of Nodal The inflow and outflow performance curves plotted in Fig.
analysis with simplified tubing string geometries (i.e., 6.1 confirm this operating point.
single-diameter tubing strings).
6.2.1.2 Oil Well
6.2.1 Analysis with the Bottom-Hole Node Consider the bottom-hole node of an oil well. As discussed
When the bottom-hole is used as a solution node in Nodal in Chapter 3, depending on reservoir pressure range, dif-
analysis, the inflow performance is the well inflow per- ferent IPR models can be used. For instance, if the reser-
formance relationship (IPR) and the outflow performance voir pressure is above the bubble-point pressure, a straight-
is the tubing performance relationship (TPR), if the tubing line IPR can be used:

shoe is set to the top of the pay zone. Well IPR can be q ¼ J ( p p wf ) (6:5)
p
established with different methods presented in Chapter 3.
TPR can be modeled with various approaches as discussed The outflow performance relationship of the node (i.e., the
in Chapter 4. TPR) can be described by a different model. The simplest
model would be Poettmann–Carpenter model defined by
Traditionally, Nodal analysis at the bottom-hole is car-
Eq. (4.8), that is,
ried out by plotting the IPR and TPR curves and graph-
ically finding the solution at the intersection point of the k k L
r
two curves. With modern computer technologies, the p wf ¼ p wh þ r þ r r 144 (6:6)
solution can be computed quickly without plotting
the curves, although the curves are still plotted for visual where p wh and L are tubing head pressure and well depth,
verification. respectively, then the operating flow rate q and pressure
p wf at the bottom-hole node can be determined graphically
by plotting Eqs. (6.5) and (6.6) and finding the intersection
6.2.1.1 Gas Well point.
Consider the bottom-hole node of a gas well. If the IPR of The operating point can also be solved analytically by
the well is defined by combining Eqs. (6.5) and (6.6). In fact, substituting
n
2
2
q sc ¼ C( p p ) , (6:1) Eq. (6.6) into Eq. (6.5) yields
p
wf

and if the outflow performance relationship of the node q ¼ J p p p wh þ r þ k k L , (6:7)
r

(i.e., the TPR) is defined by r r 144
p 2 ¼ Exp(s)p 2 which can be solved with a numerical technique such as the
wf hf Newton–Raphson iteration for liquid flow rate q. This
4
2
2
T
6:67 10 [Exp(s) 1] f M q z T 2 computation can be performed automatically with the
z
þ sc , (6:2) spreadsheet program BottomHoleNodalOil-PC.xls.
5
d cos u
i