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Summary Determining the rate distribution over multilayer injection zones (i.e., injection profiling) is critical to the optimization of injection operations. With the recent advancements in deployment of fiber-optic distributed-temperature-sensing (DTS) technology, temperature data can be achieved at high resolution and at relatively low cost along the wellbore length. During injection of cooler fluids into a higher-temperature injection zone, the temperature at the wellbore and near-wellbore region decreases. During a shut-in period when the injection operation is paused, the temperature at the wellbore sandface and near-wellbore region experiences “warm back” that is caused by the heat flux from the warmer inswept region of the injection zone. A slower warm back is observed for a layer that admits larger amount of cooler fluid during injection. As a result, the sandface warm-back temperature can be analyzed to determine the injection rate per layer, and hence the thermal-front extent per layer. In this work, we develop an analytical model to determine the temporal and spatial temperature variation for a single-phase reservoir during a warm-back period following a constant-rate-injection period. The analytical solution is developed for a single-layer reservoir and extended to multilayer reservoirs. The solution considers heat transfer by conduction and convection during the injection period and conduction during the shut-in warm-back period. The solution is verified by comparison with synthetic numerical-simulation results obtained using a thermally coupled numerical simulator for single-layer and multilayer cases. Graphical interpretation techniques are introduced by recasting the analytical solution into desirable forms. The graphical techniques are applied to synthetic warm-back data to illustrate their application and accuracy in obtaining the injection rate, thermal-front extent, and initial geothermal temperature per each layer.

Horizontal wells have been used for water flooding operations for the purpose of pressure maintenance and improved oil recovery, e.g., in tight oil reservoirs. Injection profiling along the well lateral is required to determine the effectiveness of water flooding operations and to maintain high sweep efficiency. Recent improvements in downhole temperature monitoring tools allow monitoring the transient temperature along horizontal waterflooding wells at a relatively low cost. Temperature data can be analyzed to infer the injection profile along wellbore and locate high permeability regions that cause non-uniform rate distribution along the wellbore. In this work, an analytical model is developed to describe the transient temperature in the reservoir during cold fluid (water) injection via a horizontal well during injection and warmback periods. The analytical solution assumes linear flow in the reservoir and accounts for heat transfer by different mechanisms including convection, conduction, and heat gain from the surrounding impermeable strata. The inversion procedure is introduced to evaluate injection profiling using analytical solution-based type curves. Different cases are considered using a thermally coupled numerical reservoir simulator to verify the analytical approach and illustrate the potential application of the inversion technique. We show that the analytical approach introduced in this work is an efficient and robust technique to infer the injection profile along horizontal wells. Keywords Injection profiling . Horizontal wells . Temperature warmback analysis . Analytical modeling Abbreviations α t thermal diffusivity of reservoir, L 2 /t,ft 2 /s[m 2 /s] β dimensionless convection parameter BVP boundary value problem C r , C w , C s heat capacity of reservoir system, reservoir fluid, and solids, L 2 /(t 2 T), Btu/lbm F [J/kg K] DTS distributed temperature sensing D f hydraulic front position, L, ft [m] Erfc (−) complementary error function h reservoir thickness, L, ft [m]

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