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Keywords:

  • comment;
  • inverse problem;
  • SWIW;
  • simulation of temperature transients (signals);
  • thermal injection-backflow tests

[1] The authors Jung and Pruess [2012] are highly commended for addressing two important issues in the context of the inverse problem of heat transport in discrete fractures, i.e., thermal injection-shut-in-backflow tests and thermal injection shut in tests. Specifically, the modeling of thermal injection-shut-in-backflow tests is an important extension of the simpler thermal injection-backflow tests as they are important simulations of sensible energy storage in aquifers [Palmer et al., 1992; Sauty et al., 1978, 1982]. The injection shut-in tests solutions, on the other hand, are valuable since they serve as modeling tools for those geothermal wells exhibiting good injectivity but very poor productivity. For poorly productive wells, backflow rates are low, and hence the injection shut-in solutions serve as the most convenient tools of modeling. In their thorough literature review, the authors cited several publications focusing on analytical [e.g. Kocabas, 2005, 2010] and numerical solutions [Pruess and Doughty, 2010] for single-well injection-withdrawal tests (SWIW) using temperature as tracer.

[2] There are four issues pointed out in Jung and Pruess [2012], which we would like to address. First point is related to the efficiency of the solution method as “… the analytical solution (15) for SWIW tests involving injection, quiescent, and withdrawal phases includes a six-dimensional improper integral…” and “… the solution (16), which is for SWIW tests with no quiescent period,… requires numerical integration only in two and three dimensions.” Second, in their discussion of the literature review, they pointed out the typographical errors in derivation process and final equations of Kocabas [2005, 2010]. Third issue they addressed is the independence of return profiles from injection flow rate: “… for the withdrawal phase is that the variation of fluid return temperatures with time will be independent of the flow rate.” Finally, they discussed the computational efficiency of the analytical and numerical solutions stating “For the case of no quiescent time, the analytical solution evaluation is much faster than TOUGH2 (<10 min versus several hours), but with a rest period, TOUGH2 becomes much faster than the analytical solution evaluation. The solution (15) converges extremely slowly, and it takes up to 1 week for computing one point.”

[3] First, we would like to discuss concerns regarding the efficiency of solution methods. The Jung and Pruess [2012] real-space closed-form solution, which includes a shut-in period, is a considerable improvement to that of Kocabas [2005, 2010] developed for injection-backflow tests only. However, to develop the analytical solutions, the solution method employed by Jung and Pruess [2012] appears to be significantly less efficient than the iterated Laplace transform [De Smedt and Wierenga, 1981; De Smedt et al., 2005; Kocabas., 2011] employed by Kocabas [2010] and Kocabas and Kucuk [2011]. To verify this statement, one should consider equation (11a) of Jung and Pruess [2012] and equation (1) that we developed using the double and iterated Laplace transform as the injection shut-in solution:

  • display math(1)

[4] Apparently, computational speed of equation (1) which has a semi-infinite integral will be much higher than the equation (11a) in Jung and Pruess [2012], which contains a two-dimensional integral.

[5] In their review, Jung and Pruess [2012] stated “However, the earlier analytical solutions unfortunately include several typographical errors in their derivation process and final equations….” We would like to clarify that the final solutions in Kocabas [2005, 2010] have indeed small typing errors and to avoid any future confusion, the correct equation in Kocabas [2010] reads

  • display math(2)

[6] Equation (2) assumes a simple injection and backflow test with injection to backflow rate ratio equal to λ. For equal injection to backflow rates, i.e., λ = 1, equation (2) reduces to the correct solution for Kocabas [2005].

[7] Shortly after publication of Kocabas [2010], we worked out that the same solution can be put into a much simpler expression, which is also valid for radial flow geometry [Kocabas and Kucuk, 2011]. The new and simpler form is reproduced below for the discussion that follows below and future referencing:

  • display math(3)

where

  • display math(4)
  • display math(5)

[8] Note that equation (3) is the solution for a simple injection-backflow test where the ratio of injection rate to backflow rate is λ. Also note that, while equation (2) was developed by using direct inversion of double Laplace transform, we additionally employed the iterated Laplace transform to reach equation (3). Mathematically equations (2) and (3) were identical (Figures 1 and 2). While computing time of equation (3) was several times faster than that of equation (2) in MATLAB programming, it was at least 10 times faster in Fortran programming.

image

Figure 1. Comparison of the results from Kocabas (equation (2), λ = 1, subscript K, Fortran) and the results from Maier (equation (3), λ = 1, subscript M, MATLAB) for α variation.

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image

Figure 2. Comparison of equation (2) (α = 1) with equation (3) (α = 1, various λ). For λ = 1, the results are identical.

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[9] Considering the influence of flow speed allows extension of the illuminative discussion given by Jung and Pruess [2012] for SWIW with a fixed injection/withdrawal rate. Pumping rates are directly connected to the transport equation as they determine the flow velocity in a given fracture system. Equation (3) allows considering different flow velocities for injection and withdrawal expressed by λ. The conclusion that the breakthrough curve (BTC) is independent of flow velocity is given for the special case λ = 1. That implies that the pumping rate equals the injection rate and that the fracture geometry remains stationary during the SWIW. The more generalized case λ ≠ 1 is able to account for two basic processes, namely, a change in the pumping rates or a change in the fracture geometry. Both result in a change of flow velocity during the runtime of SWIW, where only the latter case could overcome the discussed insensitivity using SWIW as a universal tool to detect enhancing fracture-matrix interface area. While for static reservoirs, during a SWIW, the only changing geometry parameter, which could be assessed at the injection/withdrawal well with consecutive SWIWs, is the fracture width.

[10] By expanding the solution to different injection and production rates, more information can be extracted from the BTC. A closer look at Figure 3 shows that the BTCs could be split into two parts. The early times, for which λ * t_pump < t_inject, and the late times where λ * t_pump > t_inject. The BTC varies most during early times, while for late times, all curves are more or less equal and not distinguishable should noise in experimental data also be considered. By applying a lower pumping rate, we are able to gain more early times data points (by the factor of λ), and therefore the resulting BTC can be interpreted better and more uniquely.

image

Figure 3. Influence of λ on the BTC, subscripts indicate λ = 2 and λ = 0.5. For a given experimental time and sampling rate, we obtain more early times data for higher λ values.

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[11] Furthermore, for the rare case when a nearby observation well is located directly along the flow path, it is possible to expand the iterated Laplace solution, which now allows computing both spatial and temporal distributions of temperature:

  • display math(6)

where inline image and inline image are given by equation (4). Note that aforementioned equations (6) and (16) in Jung and Pruess paper represent the same simple injection backflow tests with the former being more efficient computationally. The introduction of a third dimensionless number xDn (dimensionless distance) in the solution will give us the opportunity to investigate the influence of the monitoring location (distance from the injection well) on the return profiles at well and also the direct influence of flow speed, which is included in xDn. In addition, the problem description via two dimensionless numbers α (measure for fracture aperture) and λ (measure for flow velocity ratio) provides the opportunity to examine physically more versatile configurations.

[12] The next issue concerns the computational efficiency. Equations (3) or (6) contain only single and double integrals, and hence computing equation (3) is several orders of magnitude faster than computing equation (16) in Jung and Pruess [2012], which involves double and triple integrals. We achieved a significant reduction in computational time from <10 minutes stated by Jung and Pruess [2012] to few seconds on a state-of-the-art desktop computer. Hence, the solutions given by equations (3) and (6) are quite suitable for the nonlinear regression procedures employed for parameter estimation (Figure 4). The discussed analytical solutions were cross checked with finite element simulations using Comsol Multiphysics. The simulated BTC was compared with the analytical solution and used to test an optimization tool (Figure 4). The performance of the whole parameter estimation was in the order of only 10 s.

image

Figure 4. Plot section of the optimization tool written in MATLAB. Data originate from FE-Modeling using Comsol Multiphysics. The synthetic data are for α = 0.36 and the 95% confidence interval of regression is 0.35 < α <0.39.

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[13] In conclusion, equations (3) and (6) are the proposed improvements over resolving three issues raised by Jung and Pruess [2012]. They are simple in mathematical terms and computationally more efficient compared to equation (16) in Jung and Pruess [2012]. They also avoid the return profile being independent of flow rate by considering different injectivity and productivity for the test well, which is a common problem in almost all reservoirs. Finally, equation (6) allows to explore the influence of distance between observation and injection well during backflow period.

References

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  2. References
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  • Jung, Y., and K. Pruess (2012), A closed-form analytical solution for thermal single-well injection-withdrawal tests, Water Resour. Res., 48, W03504, doi:10.1029/2011WR010979.
  • Kocabas, I. (2005), Geothermal reservoir characterization via thermal injection backflow and interwell tracer testing, Geothermics, 34, 2746.
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