# SPE Reservoir Evaluation & Engineering Volume 11, Number 5, October 2008, pp. 940-947

SPE-110984-PA

### Stabilized Well Productivity in Double-Porosity Reservoirs

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DOI  10.2118/110984-PA http://dx.doi.org/10.2118/110984-PA

### Citation

• Hagoort, J. 2008. Stabilized Well Productivity in Double-Porosity Reservoirs. SPE Res Eval & Eng11 (5): 940-947. SPE-110984-PA.

### Discipline Categories

• 6 Reservoir Description and Dynamics
• 5 Production and Operations

### Summary

In this work, we present a simple analytical formula for the stabilized productivity index (PI) of an arbitrary well in an arbitrary closed, naturally fractured reservoir that can be modeled as a double-porosity reservoir. The formula relates the PI to the PI of a well in a single-porosity reservoir with permeability equal to the effective fracture permeability of the double-porosity reservoir. We have elaborated this formula for a vertical and a horizontal well in a rectangular reservoir made up of orthogonal matrix blocks. It appears that the double-porosity characteristics, along with reservoir flow capacity and reservoir volume, combine into a single factor, which plays the same role as the familiar skin factor. For small values of this double-porosity skin, well productivity is dominated by fracture permeability; for large values, the controlling parameter is matrix permeability. Inflow-performance relationship (IPR) curves of wells in double-porosity gas reservoirs can be created readily with the help of the new PI formula. An example is included showing the production profiles of a vertical and a horizontal well in a double-porosity gas reservoir with cubes and square pillars as matrix blocks. The results of this work are particularly relevant for the reservoir engineering of naturally fractured gas reservoirs.

### Introduction

The productivity of a well is commonly indicated by a PI, which is defined as the ratio of the production rate to the difference between average reservoir pressure and well pressure (i.e., pressure drawdown). In the case of compressible liquid flow in a bounded reservoir, the PI starts out with a relatively high value but declines rather rapidly down to a constant value reflecting flow that is dominated by the outer reservoir boundary. This is true regardless of whether production takes place at a constant rate or at a constant well pressure. The constant PI value is known as the stabilized PI. Most, if not all, of the analytical work on stabilized PIs concerned wells in nonfractured, single-porosity reservoirs.

In this work, we address the stabilized PI of wells in naturally fractured reservoirs that can be modeled as double-porosity reservoirs. A double-porosity reservoir comprises two distinct and coupled flow systems: (1) a highly conductive, continuous fracture network with little storage capacity that surrounds (2) a system of poorly conductive matrix blocks with a high storage capacity. Fluid transport to wells occurs exclusively through the fractures, while the matrix blocks act as sources of fluid that feed into the fractures.

In the early 1960s, Warren and Root (1963) set down the theoretical basis for the description of compressible-fluid flow in double-porosity reservoirs. Their work and later work by others focused on the interpretation of transient-pressure well tests and as a consequence were restricted mainly to the early-time transient-flow regime. To the best of our knowledge, no formulas exist in the open petroleum-engineering literature for the PI of wells in double-porosity reservoirs in the stabilized-flow regime.

The outline of this paper is as follows. We begin with the general flow equations for stabilized flow of a slightly compressible fluid with constant viscosity in double-porosity reservoirs. This leads to a general formula for the stabilized PI of an arbitrary well in an arbitrary double-porosity reservoir. We then elaborate this formula for a vertical and a horizontal well in a rectangular-box-shaped reservoir that comprises orthogonal matrix blocks with arbitrary aspect ratios. We further show how the PI formula is to be modified to include wells in fractured gas reservoirs. Finally, we present an example calculation of the production profiles of a vertical and a horizontal well in a double-porosity gas reservoir with cubes and square vertical pillars as matrix blocks.

The results of this work are particularly relevant for the reservoir engineering of naturally fractured gas reservoirs where stabilized-flow conditions often prevail during most of their producing lives. The formulas presented enable quick-and-easy estimations of well productivities and related sensitivities to the double-porosity characteristics. In addition, the results may serve as an analytical benchmark for numerical simulations of complex naturally fractured reservoirs.

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### History

• Original manuscript received: 8 March 2007
• Revised manuscript received: 18 February 2008
• Manuscript approved: 23 February 2008
• Version of record: 25 October 2008