5. FracDim: Fractional Dimension Type Curve Analysis

5.1 Algorithm
5.2 User Interface
5.3 FracDim Verification

5. FracDim: FRACTIONAL DIMENSION TYPE CURVE ANALYSIS

Hydraulic pathways through fractured rock are frequently formed by a combination of matrix permeability, flow in planar features such as fractures and fracture zones, and flow through one-dimensional channels such as those formed by selective mineralization, dissolution, and fracture intersection processes (Figure 5-1). This combination of flowing features of different dimensionality is referred to as "fractional dimension response" (Barker, 1988; Doe and Chakrabarty, 1996), as illustrated in Figure 5-2.

An approach was developed for analysis of DFN models to obtain simulated distributions of flow dimension to ensure that the simulated DFN has the same connectivity and heterogeneity structure as the in situ reservoir rock. This method provides an integrated approach to analysis of hydraulic tests in fractured rocks exhibiting this type of "fractional dimension" (Barker, 1988) and heterogeneously connected behavior.

The approach developed combines fractional dimension type curve analysis (Doe and Chakrabarty, 1996) with discrete fracture network simulation (Dershowitz et al, 1996).

5.1 Algorithm

The main geometric feature which distinguishes different flow dimensions is the power law change in flow area with radial distance. There is a second-power relationship for spherical flow, first power relationship for cylindrical flow, and zero power (or constant) for linear flow. The dimension is simply the power of the radial variation plus one. As pointed out by Doe and Geier (1990), power law variability of hydraulic properties can also produce dimensional behavior. The combined effects of area and property variation define a conductance, which is the product of area and hydraulic conductivity.

The dimension of the well test contains very fundamental and useful information about the hydraulic geometry of fracture networks. One-dimensional flow may indicate a single channel within a fracture, or a chain of channels forming a linear network. Two-dimensional flow may indicate a single fracture normal to the borehole, or a network of fractures that is confined to a planar zone, such as a fracture zone or a highly fractured sedimentary bed. Three-dimensional flow may indicate a well-connected, space-filling network of discrete fractures or channels. Finally, non-integer dimensions will appear between these cases where the fracture pattern does not fill a particular space, as in a fractal or power-law network geometry.

Differently dimensioned flow systems have significantly different behavior. In addition, since the systems are fractured, they can be both scale dependent and heterogeneously connected. Research was carried out toward development of procedures for analysis of fractional dimension type curve responses, using Laplace transform solutions for the equation of fractional dimensional flow.

The main assumptions made in the course of developing the models for transient rate and pressure behavior in a two-zone composite system are as follows:

1. Transient Darcian flow takes place in the system, and the near-well flow direction is radial
2. The i^th zone is characterized by flow dimension n_i (i = 1 for the inner zone and i = 2 for the outer zone), where n_i is not necessarily an integer; the source well is an n_1-dimensional "sphere" projected through three-dimensional space
3. The i^th zone is characterized by hydraulic conductivity and specific storage K_i and S_si, respectively
4. The system is infinite, and either a constant-rate or a constant-pressure condition is imposed at the source well
5. Wellbore/source storage capacity is non-negligible

The radial flow behavior of water in a two-zone composite system is governed by the following equations (Barker, 1988):

(5-1a)
and
(5-1b)

respectively, where

. (5-1c)

In terms of the dimensionless variables, the initial and boundary conditions become

, (5-2a)

, (5-2b)

, (5-2c)

, (5-2d)

and

, (5-2e)

where

, (5-2f)

, (5-2g)

and

. (5-2h)

Laplace transforms can be used to solve the system of partial differential equations. The subsidiary equations are

(5-3a)
and
. (5-3b)

After transforming the boundary conditions, Equations (2-3a) and (2-3b) are solved simultaneously. The solutions in Laplace space are

(5-4a)

and

(5-4b)

where

, (5-5a)

, (5-5b)

, (5-5c)

, (5-5d)

and

. (5-5e)

Using equations 5-1 through 5-5 and the related type curves of Figures 5-3, 5-4 and 5-5, it is possible to derive both transmissivity, storativity, and flow dimension as a function of distance from the well bore from well tests. Of these, the flow dimension as a function of distance may prove to be the most important for reservoir design, since lower flow dimensions indicate that only a small portion of the reservoir is being accessed.

5.2 User Interface

FracDim uses the generalize radial flow solution for fractional dimension flow, first developed in the groundwater flow literature by Barker (1988). Barker (1988) defined the radial flow diffusion equation for variable dimensioned flow as

, and (5-6)

,

where

r* = dimensionless radius (r/r_w),

h* = dimensionless head,

t* = dimensionless time, tK/4Sr_w²,

D = flow dimension,

Q = flow rate (m²/s),

h = head (m),

K = hydraulic conductivity (m/s),

b = thickness, r_w = well radius, and

_ = gamma function.

Barker (1988) noted that the solution of Equation 5-6 for transient head or head versus radius during constant-rate well tests is the incomplete gamma function which reduces to the exponential integral (Theis) curve for two-dimensional flow.

Results expressed as time t (seconds) and drawdown head s (meters) are related to dimensionless time t_D and dimensionless head h_D using the expressions:

(5-7)

(5-8)

where: r = radial distance, m

T = transmissivity, m²/s

S = storativity, -

Q = flow rate, m³/s

D = flow dimension

The transmissivity describes how much fluid flows through a fracture when a certain pressure gradient is applied. It is similar to the term permeability thickness kh commonly used by reservoir engineers, with the difference that it already includes fluid properties. It can be related to the permeability thickness using:

(5-9)

where: k = effective permeability, m²

_ _ fluid density, kg/m³

g = acceleration of gravity, m/s²

h = thickness, m

_ = fluid viscosity, Pas

For a porous media h represents the net thickness of the aquifer or reservoir layer. In a fractured rock h can be interpreted as the sum over the hydraulic apertures of all fractures intersecting the test interval. Obviously this is a simplification that does not take into account the connectivity of the fracture network away from the wellbore.

Like the transmissivity the term storativity also already contains fluid properties. Otherwise, for a porous media, it can be interpreted as the product of porosity, total compressibility and layer thickness:

(5-10)

where: _ = fluid density, kg/m³

g = acceleration of gravity, m/s²

_ = porosity, -

c_t = total compressibility, 1/Pa

h = thickness, m

Obviously the concepts of 'porosity' and 'layer thickness' are not directly applicable to fractured rocks. In a fractured reservoir the storativity should be interpreted as a function of the fracture intensity and the fracture aperture distribution.

The quotient of the transmissivity and the storativity is called the hydraulic diffusivity d:

(5-11)

The user interface for the FracDim spreadsheet is illustrated in Figure 5-6. Since the program is implemented as an Excel spreadsheet, its use is consistent with Excel user interface conventions. User instructions for FracDim are as follows.

1. Data File Preparation: Prepare a data file for the well test carried out. Use the format specified in Table 5-1.
2. Parameter Setting: Set the parameters for the well test in the worksheet "User Interface." Enter the values defined in Table 5-2:
3. Data File Load: Click button to load datafile into spreadsheet
4. Manual Fit: Use arrow buttons to manually adjust dimension, transmissivity, and diffusivity to obtain a type curve match.
5. Automatic Fit: Click autofit-enable checkbox, and Click button for automatic fit of type curve using Excel's solver based on RMS error.

Table 5-1 Input file for FracDim Spreadsheet

Table 5-2 File Format for FracDim

5.3 FracDim Verification

FracDim was verified by comparison of type curves produced by FracDim against type curves produced by the FORTRAN code, INCGAM (Doe and Geier, 1990). The verification cases are summarized in Table 5-3. The type curves for both codes are identical, as shown in Figure 5-7.

Table 5-3 FracDim Verification Case