A New Pseudosteady-State Model for Dual-Porosity/Dual-Permeability Aquifers and Two Interconnected Single-Porosity Aquifers
- Woon F. Leung (Gulf R and D Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- September 1986
- Document Type
- Journal Paper
- 511 - 520
- 1986. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.8.6 Naturally Fractured Reservoir, 4.3.4 Scale, 4.1.2 Separation and Treating, 5.5 Reservoir Simulation
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Summary. A new pseudosteady-state (PSS) aquifer model for simulation of the interaction of reservoirs with finite-extent aquifers in dual-porosity/ dual-permeability formations has been developed. The fast convolution method, which does not require lengthy data manipulation as with the superposition method, is used to implement the analytical solution of the model for time-dependent reservoir/aquifer boundary pressure. The model does not require a table lookup of the rate or pressure function; thus it is computationally pressure function; thus it is computationally efficient and lends itself readily to hand calculations or numerical reservoir simulations. A numerical example on water influx from a naturally fractured aquifer is used to show the utility of the model. The example shows the correct behavior of fractured aquifers quantitatively. The model is also applicable to a system of two interconnected single-porosity aquifers. An example demonstrates that the complex behavior associated with an interconnected aquifer system can be studied with the current model.
Water influx from aquifers with more than one set of porosity and permeability values can be very complicated. An example is a naturally fractured system with both matrix and fractures. Fig. 1 is a schematic of water influx from a naturally fractured aquifer that shows the three components of fluid flow: (1) matrix macroscopic flow to the reservoir, (2) fracture macroscopic flow to the reservoir, and (3) matrix/fracture interporosity flow. The macroscopic matrix flow is governed by the intrinsic matrix permeability and the continuity of the matrix pores. The continuity of the matrix media depends on the distribution of the fractures. Thus it is possible in the case where the matrix is completely isolated by the fracture system for the macroscopic matrix flow to be zero while the intrinsic matrix permeability remains finite. Similarly, the macroscopic fracture flow is governed by the intrinsic fracture permeability as well as the continuity and density of the fracture network. On the other hand, the matrix/fracture interporosity flow, which is on a smaller scale, is determined solely by the intrinsic matrix permeability and the matrix-block geometry. permeability and the matrix-block geometry. In view of this system, the flow behavior of a fractured aquifer may be complicated by the interactions of the three flow components with the matrix and fracture fluid-storage elements. The PSS aquifer model for single porosity is generalized to aquifers with two types of porosities and permeabilities (i. e., dual-porosity porosities and permeabilities (i. e., dual-porosity /dual-permeability aquifers). A fractured-aquifer example is used to illustrate the use of the model. It will be shown that water influx from two interconnected single-porosity aquifers are mathematically analogous to a dual-porosity/ dual-permeability system. As such, the solution developed for the latter is also applicable for the former. An example illustrates the application of the current model for interconnected aquifers.
Fractures typically have very small storage capacity but are the main avenues for fluid flow, whereas the matrix porosity accounts for almost all of the fluid storage of the aquifer but the permeability is usually very small. The aquifer matrix flow to the reservoir may be neglected, however, depending on how well the fracture system is connected. In cases where the matrix permeability is substantially large or the fractures are sparse and isolated the matrix permeability plays an important role in the fractured aquifer. As such, the matrix flow from the aquifer to the reservoir is included in the formulation of the current model. An estimate based on realistic parameters shows that the flow from the matrix to the fractures and the flow from the fractures in the aquifer to the reservoir can be approximated by the PSS relation. Because the permeability of the matrix is usually small, the water influx to the reservoir from the matrix is in a transient state in most instances; therefore, the matrix flow, strictly speaking, cannot be studied by a PSS model. Because the matrix influx is normally negligible, however, the PSS assumption of the matrix flow would not introduce significant error to the total (matrix and fracture) water influx. The spacing of the fractures, which provides a measurement of the size of the matrix block Lmb (see Fig. 1), depends on the rock geology and ranges between a few inches for highly fractured formations to several hundred feet for sparsely fractured formations. In the current model, the density of the matrix blocks, mb, is measured in percentage of matrix that are identifiable as blocks (with fractures as boundaries) in the formation.
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