Constant-Rate Drawdown Solutions Derived for Multiple Arbitrarily Oriented Uniform-Flux, Infinite-Conductivity, or Finite-Conductivity Fractures in an Infinite-Slab Reservoir

A new analytical pressure transient solution (constant rate) for a well containing multiple arbitrarily-oriented uniform-flux, infinite-conductivity, or finite-conductivity fractures in an infinite-slab reservoir is presented. The multiple-fracture solution is derived using a new uniform-flux solution for a single arbitrarily-oriented fracture in an anisotropic reservoir. The variables in this solution are: fracture half-length, fracture conductivity, and fracture angle of rotation for each fracture relative to the primary fracture. Example constant-rate type curves are provided for two intersecting fractures – cruciform or oblique – and three intersecting fractures – trifracture. Introduction Fracture imaging has changed the concept of a well producing from an infinite-slab through a single planar fracture. Microseismic fracture imaging strongly suggests complex fracture patterns can develop during primary fracture treatments, and fracture imaging during subsequent refracturing treatments demonstrates that secondary fractures are oriented in a plane(s) other than the primary fracture. An analytical solution for a well producing from an infinite-slab reservoir through multiple arbitrarily-oriented finiteor infinite-conductivity fractures is presented that was developed as part of a new pressure-transient test for refracture-candidate assessment. Obviously, this solution is also applicable when interpreting conventional pressure transient tests in reservoirs with multiple arbitrarily-oriented fractures. The purpose of this paper is to derive the new analytical solution; evaluate a pressure-averaging infinite-conductivity solution versus a semianalytical solution; illustrate the solutions by generating type curves for typical configurations of two intersecting – cruciform or oblique – fractures and three intersecting – star-shaped – tri-fractures; and present a model and type curves for a pressure-transient test in a formation exhibiting complex fracturing patterns. Uniform-Flux Solution for a Fracture Rotated at Arbitrary Angle Developing a multiple-fracture solution requires writing a uniform-flux solution for a single fracture rotated at any arbitrary angle from a reference axis. The plane-source solution for a fracture aligned with the reference axis can be written in the Laplace domain as 2 2 ( ) ( ) 0 2 D L fD q p K u x y d fD D D sL L fD fD α α ⌠ ⎮ ⎮ ⌡ ⎡ ⎤ = − + ⎢ ⎥ ⎣ ⎦ − , ................ (1) where dimensionless pressure for a single fracture is defined in the Laplace domain as 2 kh p p fD q π μ Δ = , ..................................................................... (2) and dimensionless production rate is defined as t q qD q = . .............................................................................. (3) Dimensionless fracture half-length is defined as f c L L fD L = , ........................................................................... (4) with the dimensionless coordinates written as c x xD L = , ............................................................................. (5) c y yD L = , ............................................................................. (6) with Lc defined as a characteristic length. The Laplace transform variable is denoted s, and u = sf(s) where for a single-porosity reservoir f(s) = 1. For a dual-porosity case with pseudosteady-state interporosity flow, ( ) f s is written as (1 ) ( ) (1 ) s f s s λ ω ω λ ω + − = + − , ......................................................... (7) for transient interporosity flow with slab matrix blocks, (1 ) 3(1 ) ( ) tanh 3 s f s s λ ω ω ω λ − − = + , ............................... (8) and for transient interporosity flow with spherical matrix blocks, SPE 100586 Constant-Rate Drawdown Solutions Derived for Multiple Arbitrarily Oriented UniformFlux, Infinite-Conductivity, or Finite-Conductivity Fractures in an Infinite-Slab Reservoir D.P. Craig, Halliburton, and T.A. Blasingame, Texas A&M U.