34;- WELL
4.2 Modelling the Hypothetical Problem
The hypothetical problem was modelled using the Eclipse program from Intera which implements the numerical scheme discussed in Chapter 3. A radial grid was defined with grid-block dimensions typical of a detailed coning study. In this grid the radial width of the innermost cell is 0.5 feet with each subsequent cell doubling in width.
Choosing a geometric progression for the cell widths ensures an equal pressure drop across each cell for purely radial steady state flow. This in turn minimises the truncation errors that result in the discretisation process. The grid is made up of a total of 14 cells in the radial direction giving an outer radius of the grid of approximately 8200 feet. This is considerably further than the radius of investigation and therefore the model will behave as though it has infinite lateral extent. Vertically
the grid is split into 20 layers of 4 feet in thickness giving a total of 280 cells in the grid. The perforated interval is modelled by defining well connections between the wellbore and the innermost radial cells of the first 5 layers.
The rock and fluid properties are defined according to Table 4.1. The oil compressibility is not entered explicitly but is instead defined in terms of the rate of change of the formation volume factor for oil according to the formula:
oB);;P = -c"B" . Relative permeability tables are provided which ensure that the relative permeability to oil is 1.0 for water saturations less than 0.5 of the pore volume. Similarly the relative permeability to water is zero over the same saturation range. Capillary pressure curves have no effect on the flowing behaviour of the model because they are only used in determining the water phase pressure and hence flow rate which is zero at all times. The curves are therefore defined in such a way as to ensure that the initial saturation is 0.1 of the pore volume throughout the grid, with little regard for how the capillary pressure changes as a function of saturation.
The time steps are defined in such a manner that the first time-step is very small (0.00004 Days~ 0.001 Hours = 3.6 Seconds) and each subsequent time step increases by a constant factor thereafter. The rationale behind this choice lies in the desire to minimise the time discretisation (or truncation) error that is introduced in approximating the derivatives with respect to time. Examination of Equations 52 and 53 shows that the error is given by:
(157) where the first term will dominate for small time-steps, !1t. Initially the changes are rapid and a small time-step is needed. As the flow proceeds the system becomes closer to steady state behaviour with changes that are almost constant from time-step to time-step. Thus the magnitude of
if- M/ a
2 becomes smaller as time goes on and so the same degree of accuracy can be achieved with a much longer time-step.A second reason for choosing increasing time-steps is a purely practical one. Most of the diagnostic plots display some function of pressure versus a logarithmic function of time. Allowing the time-step to increase gives a more even distribution of points on such a plot than a constant time-step would.
The control file containing the Eclipse commands for the simulation model has been included in Appendix 1.
The results of the simulation are displayed in Figure 4.2 where they are compared with two analytical models. It can be seen that the analytical models agree closely with each other, but that the Eclipse simulation has pressures that are too high soon after the flow begins and too low towards the end of the flow period. It is also clear that the rate of change of pressure with respect to time is too high at small time values for the Eclipse simulation. At large time values, when the slope is related to the permeability thickness product, kh, of the entire interval, the slopes agree.
FIGURE 4.2: COMPARISON OF SIMULATED PRESSURES WITH TWO ANALYTICAL SOLUTIONS
3340 3320 3300 3280
Pressure (psia) 3260
3240 3220 3200 3180
•••... ECLIPSE I
Hantush
0.001 0.01 0.1
Time (hours)
10
The two analytical solutions used in Figure 4.2 are the method implemented in the WELTEST program from Intera and an implementation of the Hantush solution described by Equations 10 to 13, Chapter 1.
As mentioned in Chapter 1, the WELTEST program uses the horizontal well solution of Odeh and Babu18to calculate the pressure response for partially completed wells.
A computer program was written to evaluate the Hantush solution. In this program, the exponential integral is evaluated using the well known series equivalent:
00 ( ) "
Ei(x)=-Ye -lnx-I~,
11=1 n.n! (158)
where
re
=0.5772156. The series is terminated when the terms become too small to have a significant influence on the result.The integral,
W(
u,13) ,
in the Hantush method eEquation 12) is calculated by splitting the improper integral into a series of definite integrals and using a Simpson's rule procedure to evaluate each of the definite integrals.An improper integral such as W can be defined in terms of limits as:
00
flex) dx =
L
u
lim
flex) dx.
U~O L
(159)
This can in turn be used to rewrite the improper integral as a series,
00 ~ 00
flex) dx = flex) dx
+I t" lex)
dx,L L n=! n-I
(160)
where Un has been chosen to equal IOUn-l' The numerical sum is terminated when the term to be added becomes sufficiently small. Each definite integral in the series is evaluated using an adaptive form of Simpson's rule where the number of intervals used for the integration is doubled until the required accuracy is reached.
A listing of the computer program, which was written using Borland Turbo C++ 3.0, is given in Appendix 2
The Hantush solution is based on the assumption that the flux along the borehole is constant. As a result the pressure along the wellbore varies. The Eclipse simulation models the infinite conductivity solution, where pressure is constant along the wellbore. Gringarten and RameylO state that the results arising from the constant flux approximation are equivalent to those from the infinite conductivity solution provided the pressure is evaluated at the correct position along the length of the well. They provide a chart that relates the correct position to the dimensionless quantity
hwD
=~/f"
rw k= .For the current problem the correct position, which was used in preparing Figure 4.2, is 72.4% of the length along the perforated interval.
The good agreement between the two analytical solutions is encouraging and they will be regarded as representing the true solution that the Eclipse simulation must try to duplicate.