CHAPTER 5

1

Chapter 5 - Well Rates and PressuresCHAPTER 5

Well Rates And Pressures

l Well Equationsl Comparing Simulator Pressure to Pressure Build-Up Datal Summary of Well Rate and Pressure Equations

Well Rates And Pressures

Well Equations

The simulation equations use pressures at the center of the gridblocks. These pressures represent material balance average pressures in the gridblock. However, if a well is located in the center of a gridblock, the gridblock pressure, pi,j, is not the wellbore flowing pressure, pwf. These equations compute the gas flow from gridblock to gridblock but do not model the very high pressure gradients near the wellbore. So if a well is located in a cell, we need additional equations to relate the well performance to the cell variables. We assume that steady state flow occurs within a cell and use the following three equations:

(1)

(2)

(3)

The fluid and rock properties are the same as for the cell.

We now have 3 equations with 4 unknowns: qo, qw, qg and pwf. This means that the user must specify one of these unknowns. This is the condition under which the well will produce. For example, if the user specifies qo, then qw, qg and pwf are calculated from Eqs. 1-3. Similarly if we specify pwf , we can calculate qo, qw and qg from the above equations.

Peacemans Equations. Jmodel is a "productivity index" or "well index" that has a special meaning which will be discussed later. It is not the same as the well's actual productivity index, J. Jmodel is calculated as follows:1(4)

where "ro" is calculated using following equations :

(1) when Dx = Dy, kx = ky

ro = 0.2 Dx (5)(2) otherwise

(6)(3) for Radial Systems

2 (7)

Sometimes the user may want to specify total reservoir rate "qt". Eqs. 1-3 can be substituted into the "qt" equation(8)3to give:(9)4

A number of other variations are possible to simulate actual field operating conditions. Notice that a Bn+1/Bn factor has been omitted from each term in Eq. 9. These can be added, of course, if a new estimate of Bn+1 is available.

Implicit Draw-Down. Notice that the draw-down in Eqs. 1-3, (pi,jn+1 - pwf), uses the n+1 value of cell pressure. This is important if pwf is specified as a producing condition. Otherwise, the rates will oscillate and may blow up for sequential timesteps.

Rate Specified. If qo, qw, qg, or qt are specified, then qo, qw, and qg are calculated before the timestep and put into the right-hand-side of the IMPES pressure equation. At the end of the timestep, pwf can then be calculated. This is simple and stable (as long as krn does not cause stability problems).

pwf Specified. If the user specified pwf, then none of the rates are known until after the timestep. It then becomes necessary to solve for the three-phase rate in the IMPES pressure equation. The IMPES pressure equation is in units of reservoir flow rate. The rate term is qt. Thus, the relationship in Eq. 9 can be used, rewritten as:

5 (10)

where rt is the total relative mobility, (11)

The term, Jmodel lrt , is added to the main diagonal term, ac. The other term Jmodel lrt pwf stays on the right hand side.

At the end of the timestep, qo, qw and qg are calculated.

Comparing Simulator Pressure to Pressure build-up data

It is possible to match bottom-hole flowing pressure, pwf. However, that data is usually not available and is also not very reliable because of possible inaccuracies in rate data. It is more common and much more reliable to match pressure build-up data when it is available. The problem is - how to match the pressure build-up data. The time scale of the pressure build-up test is usually too short to model accurately with a field scale grid because the gridblocks are too large.

Peaceman has provided a method for comparing simulator gridblock pressures to pressure build-up pressure. Fig.1 shows a profile of pressure in a gridblock containing a producing well. The pressure profile is assumed to be at pseudo-steady state. It is seen that the gridblock pressure (the material balance average pressure inside the gridblock) is somewhere between pwf and the average reservoir pressure. Fig. 2 shows the corresponding pressure build-up curve from field data. The gridblock pressure corresponding to the proper field build-up pressure lies on the semi-log straight line at time of Dto which is calculated by the following equation:

(12)6

The "match pressure", po corresponds to the steady state pressure at 0.2Dx . If po is the same as the gridblock pressure, pi,j, then the simulator is properly matching field behavior.

Fig.1 - Pressure profile in a grid-block containing a producing well.

Example Problem. Find the "match pressure" from the following field pressure build-up test.

q= 23,000 scf/Dk= 0.15 mdf= 0.18ct= 5 x 10-6 psi-1 = 0.216 cp Shut-in time Build-up pressure Dt ps (hrs) (psia) ___________ __________ 0.10 2854.5 0.23 2861.5 0.39 2865.5 0.84 2871.5 1.56 2875.6 3.50 2881.0 7.38 2886.2 15.11 2891.0 30.53 2895.5 61.31 2900.0 122.72 2904.1 245.24 2907.1 489.71 2909.3 840.00 2910.4

Model data:Dx= 100 ft

Solution.The solution follows these simple steps:

(1) plot the build-up data on a log Dt plot(2) draw a "semi-log straight line". (3) calculate Dto,(4) find po at Dto on the "semi-log straight line", extrapolated if necessary. The value of po "match pressure" which will be compared to the simulator gridblock pressure at the time of the pressure build-up test.

Fig. 2 shows the field build-up data plotted on a semi-log plot. The match pressure is found by calculating Dto as follows:

7

Then we find the corresponding pressure on the semi-log straight line: po = 2872 psig.This pressure is then compared to the simulator gridblock pressure when evaluating a history match run.

Fig. 2 - Pressure build-up curve for example problem.

Summary of Well Rate and Pressure Equations

A.Bottom-hole pressure, pwf, and rate:

8

B. Productivity Index or Well Index :

9

where, s = skin factor

C.Value of ro:

(1) when Dx = Dy, kx = ky

ro = 0.2 Dx

(2) otherwise

10(3) for Radial Systems

11

D.Comparison of cell pressure, pcell, with Field Build-up Test

12 EXERCISES

PROBLEM NO.1

Calculate the following for a producing well. Use the data of Problem no. 4.3 with DRs/Dp = 0.2 scf / STB.

(a) pwf(b) producing GOR (scf/scf)(c) shut-in time at which the cell pressure is on the semi-log straight line of a field build-up test, po.

PROBLEM NO.2

A build-up test gives the following pressure behavior;

(a) calculate Jmodel(b) calculate Dto(c) determine pcell (d) calculate pwf.

Dtps(hrs)(psia)

0.31600 0.51690 1.01782 2.51852 5.0188410.0191820.0194830.01968

Other data

qo= 617.65 scf/Dct = 15.0E-6 psi-1Bo= 1.3f = 0.15mo= 1.9 cp kro= 0.7rw = 0.25 ft s = 2.0h = 25 ft Dx= 200 ft