Calculation of Wellbore Pressures and Rate Distribution in Multilayered Reservoirs
- Michael Prats (Michael Prats and Assocs. Inc.) | J.P. Vogiatzis (Consultant)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 1999
- Document Type
- Journal Paper
- 307 - 314
- 1999. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 2.4.3 Sand/Solids Control, 5.6.4 Drillstem/Well Testing, 4.1.5 Processing Equipment, 5.1.1 Exploration, Development, Structural Geology
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An analytic procedure in Laplace space is developed to calculate the wellbore pressure and rate distribution, and their time rate of change, in anisotropic multilayer formations with axial symmetry and interlayer crossflow, in the presence of skins and wellbore storage, for any number of layers, any number of intervals open to the formation, for wells of nonzero radius, for finite external radii, and for no flow at the outer boundaries. Numerically, the procedure appears to be robust and trouble free. A number of applications to which the procedure can meaningfully be used are identified, and some elements of the procedure are illustrated through two brief examples.
This paper aims at providing a general analytic procedure in Laplace space to calculate the wellbore pressures and rate distributions arising in finite cylindrical reservoirs composed of any number of layers of different properties, with crossflow, for a variety of completion intervals at a well of nonzero radius, and with skins and wellbore effects. Several authors1-5 have determined pressure distributions in reservoirs composed of any number of layers from analytic solutions of differential equations such as Eqs. 1, but not for the conditions described above. Ehlig-Economides and Joseph1 provide an excellent literature review, but do not consider vertical pressure distributions within a layer. References 2-5 consider transients in infinite acting reservoirs. Additionally, Abbaszadeh et al.4 consider a well of zero radius, and do not consider different layer compressibilities, wellbore storage, or skin. In all cases,1-5 the solution procedures differ from the one developed here.
Although the emphasis of this tract is on the development of the analytic procedure, its use is illustrated through brief discussions of two example applications. One considers the effect of a non-uniform skin distribution on the injection pressure in a homogeneous anisotropic reservoir. The other example discusses the application of the Prats6 vertical permeability test to a three-layer reservoir. Other applications are identified. The intent is to provide a general-purpose procedure to investigate similar problems without the need to rederive the solution.
The reservoir under consideration consists of n horizontal layers, each of uniform and constant thickness ?hi porosity ?i horizontal permeability kHi vertical permeability Vi small total compressibility ci skin si with i=1,2, . . ., n, and containing a common fluid of viscosity ?. These properties can have any reasonable value.
The reservoir extends from the wellbore of radius rw>0 to a finite external radius re across which there is no flow. The borehole is in communication with the formation through a number of open intervals.
Injection is into one or more open intervals, which may be within a layer or may straddle several boundaries between layers. Injection at a total rate q (t) RB/D starts at t=0. This causes a pressure increase. (For production rather than injection, the sign of the pressure response is reversed.) The distribution of the total rate q(t) among the layers open to the injection interval(s) is an integral part of the solution.
The pressure response is calculated at the measuring intervals, which may coincide with the injection intervals. Usually, the intervals coincide except in tests aimed at determining vertical communication near the wellbore, in which case they are separated (e.g., by a packer). Measuring intervals may be within a layer or may straddle several boundaries between layers.
Any layer may be subdivided into a number of sublayers having the same reservoir properties (except for thickness). No distinction is made in the mathematical development between physical layers and their subdivisions.
A number of assumptions are generally made to arrive at a solvable set of linear differential equations within the reservoir. In addition to the ones discussed in the preceding section, the following assumptions are noted:
- The reservoir is initially at a uniform and constant pressure.
- The only disturbance to the pressure in the reservoir is that due to injection.
- Open intervals are treated as circular cylindrical sources of finite height.
- The well has only one injection string and only one measuring string, which may coincide.
- Wellbore storage coefficients are constant, and may differ in the injection and measuring strings.
- Flow is two dimensional in r and z.
- The pressure response of short well tests in infinite reservoirs may be obtained by choosing an adequately large external radius.
- Skin values (rather than physically altered zones near wells) are used.
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