A Generalized Method for Analyzing Well Tests in Complex Flow Systems
Authors: Roberts R.M. , Ruskauff G.J. , Avis J.D. 
Published: December 2007

Abstract
Log-log and semilog plots of pressure-transient test data often reveal features characteristic of a particular type of formation response. These features are identified in the analysis to determine the most appropriate interpretive model. These models are usually linked to some specific system geometry (i.e. channel sands), and unless geologic information exists to support the interpreted feature its meaning is questionable. This is particularly true in fractured rocks, where a change in fracture geometry can suggest erroneous interpretive models. A completely general method of handling complex pressure-transient responses, the Generalized Radial Flow model, is discussed. Application to two synthetic cases illustrates the flexibility of the method.

The Generalized Radial Flow Model
Barker presented the basic development of the generalized radial flow model. Familiar systems with integer flow dimensions include linear flow such as a channel at long times (dimension of 1), radial flow (dimension of 2), and spherical flow (dimension of 3). A quarter-slope straight line may be produced by either bilinear flow or by a flow dimension of 1.5 The spatial dimension of a pressure-transient test is determined by the power by which either the surface area of the pressure transient or reservoir properties change with distance from the pressure disturbance. Varying flow dimension is a general way of describing how changes in formation properties influence the pressure-transient propagation.

Approach
Two synthetic test cases were constructed. The first is a three- strip model (a generalized channel sand model) where different properties can be assigned to each strip and the pumping well assigned to any location in the domain. Fig. 1 shows the configuration of the pumping and observation wells, and the formation properties of each strip. Fig. 2 shows the drawdown and derivative at the pumping well, along with the fitted interpretation and flow dimension. The first stabilization (region 1 on the derivative, Fig. 2) is associated with radial flow at early times near the well (region 1 Fig. 2). A transition occurs (region 2 Fig. 2) when the pressure, transient encounters the lower permeability region of domain 3 (Fig. 1). since the lower permeability restricts flow, the flow dimension decreases. This results in subradial flow (region 2 Fig. 2). The final stabilization (region 3 Fig. 2) occurs as the pressure transient reaches region 1. Hyper-radial flow results since the permeability of region 1 is higher.
The second test case is a three-dimensional simulation of a test in hypothetical fracture system (Fig. 3). The fractures were at least 2 orders of magnitude more conductive than the matrix; local variability was introduced through geostatistical simulation. Fig. 4 shows the response at an observation well cluster to the north of the pumping well. The flow dimension required to fit the data are 0.2 at 10 m, 2.7 at 100 m, and 1.8 at 180 m. The hydraulic conductivity values determined from GTFM analysis are all consistent with the rock-mass properties in the general area of each well.

References
Barker, J.A.: "A Generalized Radial Flow Model for Hydraulic Tests in Fractured Rocks", Water Resources Research, 24, No. 10, 1796–1804. Liu, W.Z. and Butler, J.J., Jr.: Software for the Evaluation of Analytical and Semianalytical Solutions for Pumping Induced Drawdown in Complex Geologic Settings, Kansas Geological Survey, Computer Program Series 90-4.
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