Where the work of others has been used, it has been duly acknowledged in the text. The geometry of partially penetrating wells differs from the simple radial geometry discussed above in that only part of the reservoir is intersected by the well (see Figure l.2a). Although the non-radial geometry of partially completed wells introduces complications in pressure analysis, pressures also contain additional information about reservoir anisotropy.
Fluid Flow in Porous Media
Understanding errors is essential in determining the limits of applicability of the techniques developed. This avoids unnecessary constants other than those that arise naturally through the physics and geometry of the problem at hand. Although Darcy's law was originally defined in a one-dimensional form (as given in equation 1), it can be extended to three dimensions, where the permeability is normally assumed to be a tensor of diagonal shape properties.
Problems with Analytical Solutions
To obtain these values, they numerically evaluated an analytical solution of the partial completion problem derived by Nisle6. Both the Brons and Marting approach and the Odeh approach suffer from the disadvantage of assuming constant flux along the length of the wellbore. Their method can be applied to partially completed wells by a simple rotation of the coordinate system.
Resolution of Analytical Problems using Simulation
Unit Convention
34;standard” indicates that the volumes are measured at the surface at standard conditions of 14.7psia and 60°F.
Aims
Parameter Symbol Dimensions SI Units Darcy Units Field Units
Derivation of Equations
- Conservation of Mass
- Darcy's law
- Diffusivity Equations
- Typical Boundary Conditions
- Analytical Solutions of the Diffusivity Equation
- Black-oil Fluid Model
- Compositional model
Most of the fluid behavior in the reservoir can be considered to occur under isothermal conditions. At reservoir conditions, some of the surface gas dissolves in the reservoir fluid and vaporized surface fluid is present in the gas phase. To have validity, it is necessary to match the behavior of the compositional model to experiments performed in the laboratory with the reservoir fluid.
Numerical Scheme
- Gridding
- Finite Difference Equations
- Calculation of Transrnissibilities for a Radial Grid
- Boundary Conditions and Well model
- Solution of the Implicit Finite Difference Equations
- Solution of the on-linear Equations
- Direct Solution of the Linear Equations for a One-Dimensional Grid
- Iterative Solution of the Linear Equations using Orthomin
- Nested Factorisation
- Chapter Summary
This equation assumes that the grid point is at the center of pressure of the grid block. Note that this is not the same as the center of mass of the lattice block. Where this face corresponds to the edge of the grid, this is nevertheless the default situation.
Note that there may be gaps in the bands that correspond to the edge of the grid. This recursive definition appears to be a natural extension of the factorization used in the Thomas algorithm.
Application to a Hypothetical Problem
- Definition of Hypothetical Problem
Only the first 20 feet of the intersection is perforated, the rest of the well is shut off from the formation. The lateral extent of the reservoir is assumed to be infinite, although in practice it only needs to be far enough that the lateral boundaries have a negligible effect on the pressure response.
34;- WELL
Modelling the Hypothetical Problem
In this grid, the radial width of the inner cell is 0.5 feet with each subsequent cell doubling in width. The grid consists of a total of 14 cells in the radial direction giving an outer radius of the grid of approximately 8200 feet. The perforated interval is modeled by defining well connections between the wellbore and the inner radial cells of the first 5 layers.
Relative permeability tables are available which provide that the relative permeability for oil is 1.0 for water saturations of less than 0.5 pore volume. Most diagnostic plots show some function of pressure versus a logarithmic function of time. The simulation results are shown in Figure 4.2, where they are compared with two analytical models.
The two analytical solutions used in Figure 4.2 are the method implemented in Intera's WELTEST program and an implementation of the Hantush solution described in Equations 10 through 13, Chapter 1. A list of the computer program, written with Borland Turbo C++ 3.0, is given in Appendix 2. Gringarten and RameylO state that the results arising from the constant flux approximation are equivalent to those from the infinite conductivity solution, provided that the pressure is evaluated at the correct position along the length of the well.
For the present problem, the correct position, which was used in the preparation of Figure 4.2, is 72.4% of the length along the perforated interval.
Improvements to the Initial Model
This problem will be greatest near the end of the perforated section of the well, where the potential gradient may be sharply inclined to the grid axis. It is also expected that the iso-potential lines converge towards the bottom of the perforated section. At best, the approximation of the pressure gradient in terms of the pressure difference at two points will be first order.
That is, the dominant term in the error function will be proportional to block size, probably multiplied by a second-order derivative of pressure with respect to distance. One tries to find a formula that relates o\fl/or at the block boundary to the potential difference between the grid points, ~\fI =\fill - \fIi' For simplicity, it is assumed that the permeability in both blocks is constant. Thus, the part of the potential that does not obey steady-state radial flow will be subject to errors dominated by the first term.
This is not necessarily the case since the choice of transmissions made in section 3.3c completely eliminates the remaining term for the part of the potential corresponding to steady-state radial flow. For the part of the network away from the well, the radial assumption is good. Thus, the fraction of the potential that does not obey these conditions is often small.
Another option would be to explore alternative formulations for portability, particularly in problem areas near the well and at the bottom of holes.
Use of a Refined Grid
After flowing for 4 hours, the shape of the contours in the inner part of the grid stabilized. An inner region where the pressure has been reduced, but the shape of the pressure contours has stabilized. Under these circumstances, the transmissibility calculation will be exact for the point in the middle of the grid block plane.
This prompted an investigation into the differences in the assumed preconditions for the two methods. As with partially penetrating wells, the flux distribution for the infinite conductivity case will change with time and the pressure disturbance from the well tip will increase. They claim that the pressure averaging should provide an exact solution in the case where the borehole radius tends to zero.
Initially, at the time of the velocity change, the flux distribution is uniform along the well. At an early stage, the pressure drop will be concentrated near the tip of the well and become more evenly distributed later. The consequence of the difference in tendon pressure would imply a small difference in the apparent skin.
To reduce the number of variables, the distance between the blocks in the z direction is assumed to be constant.
Chapter Summary
Application to a Real Life Problem
- Background
- Geology and Geophysics
- Reservoir Engineering
- Review of Well Test Using Analytical Techniques
- Simulation Modelling of the Well Test
Within each of the sandstone layers, vertical permeability is expected to be similar to horizontal permeability. A portion of the reservoir was cored and the rock sample brought to the surface for examination by geologists. The magnitude of the early time derivative is a function of horizontal and vertical permeability.
During construction the flow is negligible and the skin is therefore not taken into account. Towards the end of the build-up (at small r) the overall slope/derivative increases again. Most of the buildings have a slope that is too small considering the core permeability.
At the well site, the water contact falls near the base of the sandstone interval. Vertical permeability equal to half the horizontal permeability is used in the matching model. Reducing the vertical permeability beyond this point degrades the quality of the early time fit to the derivative.
The distance to the edge of the field is approximately 300 meters (considerably closer than mapped).
Conclusions This study has achieved its main aims of
The simulation thus provided a basis for testing hypotheses about the nature of pressure behavior. In the well examined it could be shown that the vertical permeability must be high in order to match the early time pressure behavior. M., "Numerical Simulations of the Combined Effects of Wellbore Damage and Partial Penetration", SPE paper 8204, presented at the.
King Hubbert, M., "Darcy's law and the field equations of the flow of subsurface fluids", Trans. Note that the change in the state vector at each nonlinear iteration in solving the finite difference equations is equivalent to the solution of the linear equations (sections 3.6 through 3.10 only). Compressibility - A measure of the extent to which the volume of a substance can be changed by the application of pressure, i.e.
Drainage - A dynamic process in which the saturation of the wetting phase (ie the phase that preferentially wets the surfaces of the grains) decreases with time. The slope of the line is inversely proportional to the permeability thickness of the interval contributing to flow (Sections 1.3 and 5.4). Imbibition - A dynamic process in which the saturation of the wetting phase (ie the phase that preferentially wets the surfaces of the grains) increases with time.
Partially completed well - A well with only part of the reservoir interval open to flow (Figure 1.2b).
Eclipse Command File for Hypothetical Problem This is the command file listing for the Eclipse reservoir simulation package that was
Computer Program Listings
Calculate the improper integer of the function f from x to infinity // by calculating the integer to larger and larger limits. Used to calculate pressure from the Yiidiz and Bassiouni analytical solution using the Stehfest algorithm for inverse Laplace transform. Use the Stehfest algorithm and the Laplace inverse of pd to calculate pd.
Eclipse Command File for Real-Life Problem
