A solution to interference pumping tests in fractal fractured media of Euclidean dimension two has been developed. It is proposed in dimensioned variables with a pre-conditioning of its most sensitive parameter and a Gauss-Newton inversion. The method allows for a rapid identification of hydrodynamic parameters by fitting experimental data. The fractal dimension and the scale exponent of the hydraulic diffusion are also determined without any other calculation or reference to a theoretical medium. Thus, the results provide a reliable appraisal of how the hydrodynamic parameters evolve with the size of the system. This feature has important applications in hydrology and petroleum engineering especially when up-scaling approaches are needed.
 Hydraulic interference tests have been widely used in hydrogeology and petroleum engineering to assess the hydraulic diffusivity of underground reservoirs. The basic principle is to pump a constant flow rate in a well and to monitor the pressure transients in another well. Assuming a 2D radial flow in a homogeneous medium, the pressure transients at the observed well obey the Jacob analytical solution [Cooper and Jacob, 1946], which states that the pressure draw-down is linear with the logarithm of time (ln t). This solution appears valuable and reliable for numerous concrete cases in porous reservoirs, even heterogeneous ones, provided that the pumping times are long enough for the pressure depletion to propagate over a large portion of the reservoir. In fractured rocks however, the Jacob solution may appear unsuited because the pressure draw-down often increases more rapidly than linearly with ln t [see, e.g., Bogdanov et al., 2003]. Models assuming a fractal behavior of the fractured reservoir have been developed to overcome this problem [e.g., Barker, 1988; Chang and Yortsos, 1990; Acuna and Yortsos, 1995; Leveinen, 2000]. The model by Chang and Yortsos is probably the most complete and rigorous one since it assumes a generalized radial flow into a shell of Euclidean dimension d (d = 1, 2, 3) and scaling power laws for the permeability kf and the fracture porosity ϕf of the shell. Dimensionless variables are used to derive analytical solutions, which entails certain difficulties in evaluating kf and ϕf unambiguously. We propose therefore an alternative approximation combining a 2D Jacob analytical solution with scaling laws for kf and ϕf. It is written in dimensioned variables with a limited number of parameters that are fitted easily on experimental data by inversion using a Gauss-Newton procedure. An application to the calcareous aquifer of the hydrogeological site in Poitiers (France) has been carried out. The method has been found to work for simulating observed pressure transients and to identify scaling laws for kf and ϕf that are consistent with 2D percolation networks. This feature may appear logical for this aquifer of wide horizontal extension, vertically fractured along two principal regional directions.
2. Analytical Model and its Inversion Procedure
 Assuming that flow only occurs in the fractures and not in the matrix, the basic equation of radial flow in a fractured shell of Euclidean dimension d may be derived from the following mass-balance principle:
with r the radial coordinate in the shell of bulk volume Brd−1dr (B = symmetry constant), ϕf (r) the fracture porosity of the shell (volume of open fractures/bulk volume of the shell), ρ the fluid bulk density, Q(r) [M.L−3] the flow rate through the shell of surface area Brd−1. Using Darcy's law, the flow rate writes:
with kf(r) [L2] the permeability of the shell, μ [M.L−1.T−1] the fluid dynamic viscosity, p [M.L−1.T−2] the pressure head. Assuming the first-order approximation (f(r + dr) − f(r))/dr ≈ ∂f/∂r for any scalar function f and substituting (2) into (1), we can write:
which yields for weakly compressible media (classical assumptions: ρ(r) almost constant and ∂ρ/∂t ≈ ρcf ∂p/∂t):
with cf [M−1.L.T2] the fluid compressibility or a composite one (fluid + fractures) if the fracture walls are assumed to warp under pressure variations. For d = 2, note that expression (4) is the classical 2D radial flow equation that obeys the Jacob analytical solution for confined reservoirs and constant values of kf and ϕf:
with pi the initial constant pressure head, Q the pumping rate at the well, e the thickness of the reservoir. Now let parameters kf(r) and ϕf (r) be those of a fractal fractured network; they will scale as power laws of r. Since ϕf is the ratio of the volume of the fractures to the bulk volume and assuming that the mass fractal dimension of the network is D, then ϕf (r) scales as rD−d. Another relevant scale exponent for fractal objects is the so-called transport exponent θ [Stauffer and Aharony, 1994]. It depicts how the main parameter of transport scales, i.e., for our concern, the hydraulic diffusion coefficient DH = kf/(μϕfcf) ∝ . This implies that kf(r) scales as . Consequently, the following relationships ϕf (r) = ϕ0rD−d, kf(r) = k0 can be used and put in the mass balance equation (4). This yields:
As stated above, Chang and Yortsos  derived analytical solutions for the pressure draw-down Δp(r, t) = pi − p(r, t) from an expression equivalent to equation (6). Despite the correctness of these solutions, their form in dimensionless variables makes them difficult to use in concrete cases for the identification of both key parameters kf(r) and ϕf (r). For fractal networks of Euclidean dimension d = 2, we propose an alternative that can be viewed as a mix between the Jacob solution to expression (4) and the scaling laws for kf(r) and ϕf (r). This results into a non-exact solution to equation (6) since the Jacob form is strictly valid for constant values of kf and ϕf. However, one may expect for long-time behavior and/or large radii experienced by the diffusing pressure wave that the decrease of kf and ϕf will be small enough to get an acceptable approximation. Scaling laws for kf and ϕf need first to be expressed in time. Since the hydraulic diffusion DH has the dimension [L2.T−1] but scales as , time and space coordinates follow the relation T ∝ . 2+θ is also called the fractal dimension of the random walker and is of course 2 for non-fractal media. Using this relationship between space and time coordinates, the behavior of kf and ϕf in time can be modeled as:
which is an expression in dimensioned variables with four parameters k0, ϕ0, α, γ to be identified by fitting experimental data. Reliability of equation (8) can be assessed in different ways. The first one is to note that a rough approximation of the derivative of Δp with respect to ln t may be written as follows (assuming t′α ≈ tα):
This expression, which scales as tα, is equivalent for d = 2 (i.e., α = 1 − D/(2 + θ)) to the one found by approximation of the Chang and Yortsos solution for long times [Aprilian et al., 1993]. Another possibility is to check the solution (8) against numerical results. The latter can be calculated on well known fractal media such as Euclidean 2D regular percolation networks. Theoretical results [e.g., Adler and Thovert, 1999] show that α ≈ 0.34, γ ≈ 0.036 at the percolation threshold and these values decrease rapidly when the fraction of open bonds increases. We made several calculations over percolation networks by averaging how both XA, the fraction of bonds in the spanning cluster (bonds connected to the edges of the network, including dandling ends but excluding isolated clusters), and k, the macroscopic permeability, scale: XA ∝ L−β/υ, k ∝ L−μ/υ with L the size of the network. From the know values β/υ and μ/υ are derived the mass fractal dimension D = 2 − β/υ and the transport scaling factor θ = (μ − β)/υ, which in turn enable the parameters α, γ to be calculated. A numerical pumping test is then performed over the network and the pressure draw-downs at different locations are fitted with expression (8). The fitted values of α and γ are in good agreement within 15% with those calculated over the percolation networks. Errors on α and γ can be translated into errors on D and θ using the relations: α − γ = θ/(2 + θ) and D = d − 2γ/(1 − α + γ). For the values α ≈ 0.34, γ ≈ 0.036, we get D = 1.87 − 1.91 and θ = 0.68 − 1.12. Even though these ranges may appear somewhat wide, they are relatively narrow as compared to those obtained from statistics on sets of equiprobable percolation networks with the same topology (D = 1.5 − 2.0 and θ = 0 − 2.1, F. Delay, unpublished computations, 2004). Reliability of k0 and ϕ0 cannot be tested since numerical results on networks have no sense for very short pumping times and also because there is no straightforward relationship between k0, ϕ0 and the permeability of a bond in the network (set to one) or the fraction XA.
 These comparisons as well as direct tests have shown expression (9) to be very sensitive to the α exponent. Thus, for concrete applications to experimental data, a precise evaluation of α is required. This can be done with the calculation of the pressure draw-down increment between times t and t′. Using the approximations ln(z + ɛ) ≈ ln z + 2ɛ/(2z + ɛ) and (z + ɛ)α ≈ zα exp (2αɛ/(2z + ɛ)) with ɛ ≪ z, one obtains from (8) and for z = t, ɛ = t′ − t:
If the above expression is left unchanged, C encloses the unknown parameters k0 and ϕ0, which prohibits the pre-evaluation of α. However, for classical values in natural homogeneous media, i.e., k0 = 10−10 − 10−15 m2.s−1, ϕ0 = 10−2 − 10−4 and times = 104 − 106 s, the value of ln C can be dropped. Thus, we can write:
A log-log plot of the left-hand-side with respect to time is therefore a straight line of slope α. Basically, this method is non-linear since α is present in ξ, but the problem converges rapidly (3–5 iterations) by calculating αk+1 with ξ(αk), k being the iteration index. Direct calculations performed with equation (8) and evaluations of α with equation (11) show that for common values of C and times, the error on α is less than 5%. Once the most sensitive parameter α has been estimated independently, the remaining part, i.e., γ, k0, ϕ0, is inverted by a classical procedure. An objective function built as the sum of the square errors between observed and simulated draw-downs is minimized using a Gauss-Newton algorithm. To avoid possible discrepancies because of “concurrent” effects between α and γ, the latter is allowed to vary within a range 0–15% of the pre-evaluated α value. This range is consistent with values observed in 2D regular percolation networks (e.g., α = 0.34 as compared to γ = 0.036 predicted from theory). In the cases where the fitting is just fair, a second run is performed by allowing α to vary within 10% of its pre-evaluated value.
3. Application to a Calcareous Aquifer
 The analytical method and its inversion as depicted above have been applied to a series of interference pumping tests carried out over the hydrogeological experimental site in Poitiers (France). It encloses about 40 wells spatially set up as nested five-spot systems (a well at the center and four wells at the corners of a square) with lag distances between wells ranging from 50 to 300 m. The wells are bored in a vertically fractured Jurassic limestone of about 100 m in thickness that typically behaves as a regional confined aquifer. All wells are full-penetrating and wide enough in diameter to dive a pump of max flow rate of 70 m3.h−1. In the test series reported here, four wells have been pumped sequentially at a constant flow rate set up between 30 and 60 m3.h−1 to avoid a possible drying of the pump. For each well pumped, about twenty others have been monitored. An example of simulation + inversion on experimental data is given in Figure 1. The experimental data show a pressure draw-down that evolves more rapidly than linearly with ln t (Figure 1a). This behavior is typical of pressure transients with hydrodynamic parameters kf and ϕf varying in time (or equivalently in space) during the propagation of the pressure depletion. Note that some experimental curves have a particular behavior for early pumping times, say 104 s. The pressure draw-down may show a rapid increase followed by a slightly increasing plateau. This has already been reported by Acuna et al.  for numerical fracture networks and probably corresponds to a transition regime when the fractal dimension of flow is above 2. At early times, the pumping into the well does not stress evenly all the aquifer thickness and the problem is basically 3D. Afterwards, when the stress has propagated enough, its fluctuations along the vertical direction are negligible as compared to horizontal ones and the problem becomes 2D. As stated above, the pre-evaluation of α requires a log-log plot of the modified derivative of the pressure draw-down with respect to ln t (expression (11)). After average oscillations at early times (Figure 1b), the curve stabilizes and its slope gives the value of α (e.g., 0.288), which scales the permeability in time. Then, γ, k0, ϕ0, or more exactly ϕ0 cf, since no predetermined value is assigned to cf, are directly identified on Δp versus ln t by the Gauss-Newton algorithm. Here, α is left at its pre-evaluated value and one gets γ = 0.020, k0 = 1.2 × 10−10 m2.sα, ϕ0cf = 5.4 × 10−11 k−1.m.s2+γ. This solution is stable but not unique; let α be slightly varied around its initial value, one may obtain a good fitting for instance with α = 0.260, γ = 0.028, k0 = 8.0 × 10−11, ϕ0cf = 1.05 × 10−10. This solution is obviously different but does not modify the orders of magnitude of the sought parameters. Another way to evaluate stability and accuracy is to look at the covariance matrix of the error derivatives with respect to parameters. This matrix is calculated in the Gauss-Newton procedure [see, e.g., Delay and Porel, 2003]. In the present case, when the optimal solution is reached, all the matrix coefficients are of the same order of magnitude. There is no preferential direction able to improve the solution; the latter is stable. The decomposition of this matrix gives also an estimate of errors on parameters around the optimal solution. We have calculated that relative errors on parameters are almost the same whatever the parameter and of about 5%.
Figure 2 shows the interpretation from four tests in terms of long-time permeability: k(t) = k0t−α with t = 72 hours. The values are reported as a function of the distance r between pumped and observed wells. Even if the values of k0 are in the range [7.0 × 10−12, 2.2 × 10−10] and α in the range [0.08, 0.35], the long-time permeability is almost constant. This could mean that even if the medium is fractal, a pseudo-homogenization scale has been reached. By recalculating the transport parameter θ from the relation α − γ = θ/(2 + θ) and using the equivalence T ∝ , this pseudo-homogenization scale is about 50 to 250 meters, which is consistent with the range of distances experienced by the interference tests performed. This fact is specific to permeability and, for instance, is not observed for the storage capacity ϕf cf, which continues to lower with r even for long times (not reported here). Note that with the values obtained for α and γ, the calculated fractal dimension D = d − 2γ/(1 − α + γ) is in the range 1.85–1.98, which is consistent with expected values for 2D percolation networks just above the percolation threshold. It is indeed impossible to prove indisputably that the Poitiers aquifer is a percolation network. Nevertheless, its hydraulic behavior is compatible with the general idea that this reservoir is of weak thickness as compared to its horizontal extension (i.e., 2D) and is regularly fractured by vertical planes along the two major regional directions N000 and N130.
 The mixing of a classical 2D radial flow solution and scaling laws for the hydrodynamic parameters appears as a good approximation of pressure transients from interference pumping tests in fractal fractured rocks. The associate inversion procedure is handy and rapid for concrete case applications. The parameters are identified in dimensioned variables that allow for direct comparison with other values drawn from classical methods. The scaling law exponents of hydraulic properties, the fractal dimension of the medium and the exponent of anomalous hydraulic diffusion are also identified by inversion, which lowers subjectivity and gives them reliability. These features are directly drawn from data and there is no need for additional modeling exercises (e.g., in synthetic fracture networks). The method is therefore relatively simple and of valuable interest for practical applications in hydrology and petroleum engineering.
 We are grateful to the “French National Program for Research in Hydrology” and the “Poitou-Charentes Water Research Program” for the financial support of this work.