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# Modeling Transient Wellbore Temperature During Diagnostic Fracture Injection Tests

## This paper offers an analytical model for estimating the transient temperature at a given depth and timestep, for computing the BHP.

Diagnostic fracture injection tests (DFITs) have gained widespread usage in the evaluation of unconventional reservoirs. In typical field operations, pressure is measured at the wellhead, not at the bottom of the hole. The bottomhole pressure (BHP) is obtained by adding a constant hydrostatic head of the water column to the wellhead pressure (WHP) at each timestep. One can question the soundness of this practice because of significant changes in temperature that occur in the wellbore, leading to changes in density and compressibility throughout the fluid column. This paper offers an analytical model for estimating the transient temperature at a given depth and timestep, for computing the BHP.

## Temperature Model During Pressure-Falloff Test

After a well is shut in at the surface, afterflow at the sandface is negligible because of low formation permeability. Upon cessation of injection, the cold injection water begins to gain heat from the surroundings. Heat transfer during flowing and shut-in conditions has been modeled with an energy-balance equation, accounting for various resistances to heat transfer. The resulting differential equations are solved numerically. However, with robust assumptions, an analytical expression can be developed for transient fluid temperature during falloff. The heat flow from the formation into the wellbore raises the internal energy of the fluid and of the composite tubing/casing/cement material. Please see the complete paper for the quantitative expression.

## Field Example

Modeling Temperature Transient During Falloff Test. The pressure-computation algorithm entails two steps. First, the temperature is evaluated as a function of time at various depths throughout the wellbore. As expected, the temperature at well bottom will exhibit the largest excursion. Second, the fluid properties of density and compressibility are evaluated at each depth step corresponding to the temperature profile, leading to the BHP estimation at a given timestep.

Fig. 1 above presents the falloff data measured during a test. The rapid rise in temperature over the first 10 hours suggests potentially large changes in compressibility and density of water. Therefore, if the fracture closure occurs within this time period, uncertainty in BHP estimation may affect analysis. Fig. 2 shows the quality of match obtained for temperature data with the square-root-of-time model presented by the equations found in the complete paper.

Establishing time-dependent temperature profiles with the equations is a first step toward computing BHP from WHP. Thereafter, the fluid’s expansivity and compressibility are estimated. Fig. 3 displays the temperature and density profiles. Note that the error between the corrected and the uncorrected pressures begins to diminish with time because the change in density or expansivity is counteracted by the fluid’s compressibility.

Analysis of Pressure and Temperature Transients During Falloff Test. Fig. 4 exhibits the pre- and post-closure responses, where the fracture-closure time of 13 hours is indicated. This plot comparing the BHP and WHP responses suggests that the pressure-conversion issue becomes moot for the problem at hand because the two curves converge well before the closure time.

The analysis of temperature transients also suggests the dominance of linear flow because of the slow thermal-diffusion process. A fracture-closure time of 12 hours is estimated from the diagnostic plot, which is in good agreement with its pressure counterpart. The authors surmised that the upward shift in the temperature derivative near the 0.8‑hour mark is a manifestation of nonidealized fracture geometry. However, the smooth transition from the higher elevation appears to be a reflection of the fracture closure.

Analysis of Injection Pressure With Modified Hall Method. Analysis of injection data has faced uncertainty because of complex mechanisms of fracture initiation and propagation, variable fluid loss, and fluid efficiency that are all in play over a short period of time. The modified Hall approach can be used to establish the formation-breakdown pressure with injection data. The numerical derivative is a good way to arrive at the breakdown pressure. The break-over point agrees closely with that of the numerical derivative. As expected, neither the radial-flow model nor the log-time derivative show any point of inflection. Despite the new semianalytical formulation with linear flow, the numerical derivative provides a clearer picture of the breakdown pressure and is, therefore, recommended.

Another interesting observation emerges when the same modified-Hall-method data are graphed on the log-log plot. The expected half-slope response emerges after the fracture breakdown occurs but over a very short time span. The earlier unit-slope line suggests that the wellbore was being loaded with the injection water, which may be construed as wellbore storage, used in the context of transient-pressure testing.

The authors note that the degree of separation between the derivative of the Hall-integral curve and the Hall-integral curve is a measure of fracture conductivity in conventional formations. However, such separation cannot provide clues about the pressure behavior during the shut-in period. In fact, analysis of injection data of several wells from reservoirs of diverse geomechanical and fluid-efficiency considerations suggested that the short-duration linear flow followed by the unit-slope response is the norm. The authors speculate that the dominance of the late-time unit-slope response, preceded by the half-slope signature, suggests that the fluid has not had time to diffuse into the formation given the high-rate injection over a short time span in a very tight formation.

## Discussion

The transient-temperature model and the computational approach presented in the complete paper are primarily intended for the falloff test run in any DFIT in an unconventional setting to account for large changes in fluid temperature at early times. As expected, the early-time injection data for the modified Hall formulation also require the linear-flow treatment in micro- and nanodarcy formations.

Questions arise when downhole pressure measurements become a necessity to avoid any uncertainty arising from the potential transient-temperature issue. Using Fig. 4 as a guide, one can surmise that, if the fracture closure occurs within the first log cycle (1–10 hours) or earlier, one needs to consider running a downhole gauge. Otherwise, wellhead measurements suffice. To that end, this study provides practical guidelines on pressure measurements.

Although the conventional modified-Hall-method plot diagnoses the fracture-breakdown pressure quite well, graphing the same data on a log-log plot is even more illuminating because it clearly delineates both the early-time unit-slope and the late-time half-slope responses. The half-slope response is in accord with linear flow. However, the derivative of the modified Hall formulation on a log-log plot suggests that the half-slope period is short-lived. The subsequent development of the unit-slope response is speculative in that it may be associated with fluid storage within the fracture.

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 166120, “Modeling Wellbore Transient Fluid Temperature and Pressure During Diagnostic Fracture Injection Testing in Unconventional Reservoirs,” by B. Nojabaei, SPE, The Pennsylvania State University, A.R. Hasan, SPE, Texas A&M University, and C.S. Kabir, SPE, Hess, prepared for the 2013 SPE Annual Technical Conference and Exhibition, New Orleans, 30 September–2 October. The paper has not been peer reviewed.