EP2772610A1 - Method for determining the inflow profile of fluids of multilayer deposits - Google Patents

Abstract

A method for determining the profile of fluids inflowing into multi-zone reservoirs provides for temperature measurement in a wellbore during the return of the wellbore to thermal equilibrium after drilling and determining temperature of the fluids inflowing into the wellbore from each pay zone after perforation at an initial stage of production. Specific flow rate for each pay zone is determined by rate of change of the measured temperatures.

Description

Field of the Disclosure
• The disclosure relates to the field of geophysical studies of oil and gas wells, in particular to determining the inflow profile of fluids inflowing into the wellbore from multi-zone reservoirs.
Background of the Disclosure
• Usually when estimating flow rate of individual pay zones by temperature data, temperature measurement along the entire wellbore is conducted, while temperature of a reservoir near the wellbore is assumed close to the temperature of the undisturbed reservoir.
• In particular, a method for determining relative flow rates of pay zones by quasi-stationary flow temperatures measured along a wellbore is known. This method is, for example, described in Cheremensky G.A. Applied geothermy, Nedra Publishers, 1977, p. 181. The main assumption of the traditional approach is that an undisturbed temperature of a reservoir near a wellbore must be known prior to the tests. This assumption is not performed if temperature is measured at a first stage of production shortly after perforation of the well. The influence of the perforation itself is not very significant, but as a rule the temperature of the near-wellbore part of formation is considerably lower than the temperature of the undisturbed reservoir due to the cooling resulting from previous technological operations: drilling, circulation and cementing.
Summary of the Disclosure
• The method for determining profile of fluid inflow from a multi-zone reservoir provides the possibility to determine the inflow profile at an initial stage of production, just after perforating a well, and in enhancing the accuracy of inflow profile determination due to the possibility of determining inflow profile by transient temperature data.
• The method comprises measuring temperature in a wellbore during a wellbore-return-to-thermal-equilibrium time after drilling and then perforating the wellbore. Temperature of fluids inflowing into the wellbore from pay zones is determined at an initial stage of production and a specific flow rate for each pay zone is determined by rate of change of the measured temperatures.
• In case of direct measurement of temperature of the fluids inflowing into the wellbore from each pay zone, specific flow rate of each pay zone is determined by the formula $$Q_i=\frac{4\pi }{\chi }\cdot a\cdot h_i\cdot \left(\frac{\stackrel{•}{T_{\mathit{in},i}}}{\stackrel{•}{T_s}}-1\right),$$

  

  
  where Q_i is a flow rate of an ith pay zone,
  Ṫ_s is a rate of temperature recovery in the wellbore before perforation,
  Ṫ_{in ,i} is a rate of change of temperature of the fluid inflowing into the wellbore from the ith pay zone at an initial stage of production,
  h _i is a thickness of the ith pay zone,
  a is a thermal diffusivity of a reservoir, $$\chi =\frac{c_f\cdot {\rho }_f}{{\rho }_r\cdot c_r},$$

  

  
  ρ _fc_f is a volumetric heat capacity of the fluid,
  ρ _rc_r = φ · ρ _fc_f + (1 - φ)·ρ _mC_m is a volumetric heat capacity of the rock saturated with the fluid,
  ρ _mc_m is a volumetric heat capacity of a rock matrix;
  φ is a porosity of the reservoir.
• In case of impossibility to directly measure temperature of the fluids inflowing into the wellbore from each pay zone, temperature of the fluids is determined with the use of sensors installed on a tubing string, above each perforated interval. A specific flow rate of a lower zone is determined by the formula $$Q_{}$$ $${}_1=\frac{4\pi }{\chi }\cdot a\cdot {\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">h</figure-callout>}}_1\cdot \left(\frac{\stackrel{•}{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_1}}{\stackrel{•}{T_s}}-1\right),$$

  

  
  where Q ₁ is a flow rate of a lower pay zone,
  Ṫ_s is a rate of temperature recovery in the wellbore before perforation,
  Ṫ ₁ is a rate of change of temperature of the fluid inflowing into the wellbore from the pay zone at an initial stage of production as measured above the lower perforated interval,
  h ₁ is a thickness of this pay zone,
  a is a thermal diffusivity of a reservoir, $$\chi =\frac{c_f\cdot {\rho }_f}{{\rho }_r\cdot c_r},$$

  

  
  ρ _fc_f is a volumetric heat capacity of the fluid,
  ρ _rc_r = φ·ρ _fC_f + (1-φ)·ρ _mC_m is a volumetric heat capacity of the rock saturated with the fluid,
  ρ _mc_m is a volumetric heat capacity of a rock matrix;
  φ is a porosity of the reservoir.
• Then by temperatures measured by the sensors installed on the tubing string, specific flow rates of overlying zones are determined, using values of flow rates determined for the underlying zones.
• The wellbore return-to-thermal-equilibrium time usually lasts for 5-10 days.
• Temperature of the fluids inflowing into the wellbore from pay zones at the initial state of production is preferably measured within 3-5 hours from start of production.
Brief description of the Figures
• The disclosure is illustrated by drawings where Figure 1 shows a scheme with three perforated intervals and three temperature sensors; Figures 2a and 2b show results of calculation of inflow profiles for two versions of formation permeabilities; Figure 3 shows temperatures of fluids inflowing into the wellbore and temperatures of the corresponding sensors for the case illustrated in Figure 2a, Figure 4 shows temperatures of the fluids inflowing into the wellbore and temperatures of the corresponding sensors for the case illustrated in Figure 2b; Figure 5 shows time derivatives of fluid temperature and temperature of sensor 1 for the case illustrated in Figure 2a, Figure 6 shows time derivatives of fluid temperature and temperature of sensor 1 for the case illustrated in Figure 2b; Figure 7 shows ratios of temperature growth rates $f_{21}=\frac{{\stackrel{•}{T}}_2}{{\stackrel{•}{T}}_1}$

  

  and $f_{32}=\frac{{\stackrel{•}{T}}_3}{{\stackrel{•}{T}}_{21}}$

  

  for Figure 5, Figure 8 shows the same ratios for Figure 6; Figure 9 shows correlation between the time derivative T_in and specific flow rate q.
Detailed description
• The method may be used with a tubing-conveyed perforation. It is based on the fact that a near-wellbore space, as a result of drilling, usually has a lower temperature than temperature of surrounding rocks.
• After drilling of a wellbore, circulation and cementing, temperature of a reservoir in a near-borehole zone is significantly (by 10-20 K and more) lower than an original temperature of the surrounding reservoir at a depth under consideration. After these stages, a relatively long period of wellbore-returning-to-thermal-equilibrium follows during which other working operations in the well are carried out, including installation of a testing string with perforator guns. In the process of wellbore-returning-to-thermal-equilibrium after drilling resulting cooling of near-wellbore formations, temperature measurements in the wellbore are conducted.
• After perforation, an initial stage of production follows - cleanup of the near-borehole zone of the reservoir. At the initial stage of production, when a significant change takes place in the temperature of fluids inflowing into the wellbore (usually during 3-5 hours), temperature of the fluids inflowing into the wellbore form each pay zone is measured.
• In case of a homogeneous reservoir, radial profile of temperature in the reservoir prior to start of the cleanup is determined with the use of some general relationship that follows from the equation of conductive heat transfer (1). $$\frac{\partial T}{\partial t}=a\cdot \left(\frac{{\partial }^{2}T}{\partial {\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">r</figure-callout>}}^2}+\frac{1}{r}\cdot \frac{\partial T}{\partial r}\right)$$

  

  where "a" is a heat diffusivity of the reservoir.
• From the physical viewpoint, it will be justifiable to suppose that with a long wellbore-returning-to-thermal-equilibrium time, some near-wellbore zone (r<r_c ) exists within which the rate of increase of temperature in the formation is constant, i.e. it does not depend on distance from the wellbore: $$\frac{\partial T}{\partial t}\approx \varphi \left(t\right)\equiv \stackrel{•}{T_s}$$

  
• Equations (1) and (2) have the following boundary conditions at the wellbore axis: $$T\left(r=0\right)=T_a;\frac{\mathit{dT}}{\mathit{dr}}{|}_{r=0}=0$$

  

  where T_a is temperature at the axis (r=0).
• The solution of the problem (1), (2), (3) is $$T\left(r\right)\approx T_a+b\cdot {\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">r</figure-callout>}}^2$$

  

  where $$b=\frac{1}{4\cdot a}\cdot {\stackrel{•}{T}}_s$$

  
• Formulas (4), (5) give an approximate radial temperature profile near the wellbore prior to start of production. A numerical simulation demonstrates that after 50 hours of borehole-return-to-thermal-equilibrium time, these formulas are adequate for r < 0.5 ÷ 0.7 m (with accuracy of 1÷5 %) for an arbitrary possible initial (before closure) temperature profile.
• Formulas (4), (5) do not take into consideration the influence of heat emission in course of perforation and radial non-uniformity of thermal properties of the wellbore and the reservoir, that is why after comparison with results of numerical simulation, introduction of some correction coefficient might be necessary.
• After start of production, the radial profile of temperature in the reservoir and transient temperatures of the produced fluid is determined, mainly, by convective heat transfer that is determined by the formula $${\rho }_r{c}_r\cdot \frac{\partial T}{\partial t}-{\rho }_f{c}_f\cdot v\cdot \frac{\partial T}{\partial r}=0$$

  

  where $$v=\frac{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">q</figure-callout>}}{2\pi \cdot r}$$

  
• is a velocity of radial filtration of the fluid, q [m³/m/s] is a specific flow rate, ρ _fc_f is a volumetric heat capacity of the fluid, ρ _rC_r = φ·ρ _fc_f + (1-φ)·ρ _mC_m is a volumetric heat capacity of the rock saturated with the fluid, ρ _mc_m is a volumetric heat capacity of the rock matrix, φ is a porosity of the reservoir.
• Equation (6) does not account for conductive heat transfer, the Joule-Thomson effect and the adiabatic effect. The influence of the conductive heat transfer will be accounted for below, while the Joule-Thomson effect (ΔT = ε₀ΔP) and the adiabatic effect are small due to a small pressure differential ΔP and a relatively big typical cooling of the near-wellbore zone (5-10 K) before start of production.
• Equation (6) has the following solution $$T\left(r,t\right)={\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_0\left(\sqrt{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">r</figure-callout>}}^2+\frac{\chi }{\pi }q\cdot t}\right),$$

  

  where T₀ (r) is an initial temperature profile in the reservoir (4), $\chi =\frac{c_f\cdot {\rho }_f}{{\rho }_r\cdot c_r}\mathrm{.}$

  
• Temperature of the fluid inflowing into the wellbore is (4), (8): $$T_{\mathit{in}}\left(t\right)={\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_0\left(\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="r\; w"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">r</figure-callout>}}_w}^2+\frac{\chi }{\pi }q\cdot t}\right)$$

  

  or $$T_{\mathit{in}}\left(t\right)\approx T_a+b\cdot \left({{\mathrm{<figure-callout\; id="2"\; label="r\; w"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">r</figure-callout>}}_w}^2+\frac{\chi }{\pi }\cdot q\cdot t\right)=\alpha +\beta \cdot q\cdot t$$

  

  where $$\alpha =T_a+\stackrel{•}{T_s}\mathrm{.}\frac{r_{\mathrm{<figure-callout\; id="2"\; label="w"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">w</figure-callout>}}^2}{4\pi \cdot a},$$

  

  $$\beta =\stackrel{•}{T_s}\mathrm{.}\frac{\mathrm{<figure-callout\; id="4"\; label="\chi "\; filenames="imgf0009.png"\; state="\{\{state\}\}">\chi </figure-callout>}}{4\pi \cdot a}\mathrm{.}$$

  
• In accordance with (9), rate of fluid temperature increase at the inlet is $$\frac{{\mathit{dT}}_{\mathit{in}}}{\mathit{dt}}=\beta \cdot q\mathrm{.}$$

  
• This formula for rate of temperature increase of the produced fluid is not fully correct because Equation (6) does not take into consideration the conductive heat transfer. Even in case of very small production rate (q→0), temperature of the inflow must increase due to the conductive heat transfer and the approximate formula accounting for this effect can be written in the following way $$\frac{{\mathit{dT}}_{\mathit{in}}}{\mathit{dt}}=\beta \cdot q+\stackrel{•}{T_s}$$

  
• Thus, with direct measurement of temperature of the fluid inflowing into the well, specific flow rate of each pay zone Q_i can be determined by the formula $$Q_i=\frac{4\pi }{\chi }\cdot a<figure-callout\; id="1"\; label="\cdot \; h"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">\cdot </figure-callout>h_{}{}_1\cdot \left(\frac{\stackrel{•}{T_{\mathit{in},i}}}{\stackrel{•}{T_s}}-1\right),$$

  
• For such cases where no possibility exists to directly measure temperature of the fluids inflowing into the wellbore from the pay zones, it is suggested to use results of temperature measurements above each perforated interval, for example, with the use of sensors installed on a tubing string utilized for perforating. In accordance with the numerical simulation, in 20÷30 minutes after start of production, the difference between temperature of the fluid inflowing into the wellbore T _in,1 and temperature T ₁ measured in the wellbore above a first perforated interval is practically constant: T _in,1 - T₁ = ΔT ₁ ≈ const, and $\frac{d{T}_1}{dt}=\frac{d{T}_{\mathit{in},1}}{dt}\mathrm{.}$

  

  In accordance with Formula (12), this means that a flow rate of the lower pay zone Q₁ can be determined (Q ₁ = h ₁ · q ₁) (h ₁ is a thickness of this pay zone) by temperature measured above the first perforated interval: $$\stackrel{•}{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_1}=\beta \cdot \frac{Q_1}{{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">h</figure-callout>}}_1}+\stackrel{•}{T_s}$$

  

  or, taking into consideration Formula (11), we find $${\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1=\frac{4\pi }{\chi }\cdot a<figure-callout\; id="1"\; label="\cdot \; h"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">\cdot </figure-callout>h_{}{}_1\cdot \left(\frac{\stackrel{•}{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_1}}{\stackrel{•}{T_s}}-1\right)$$

  
• All parameters in this formula can be approximately estimated ("a" and χ) or measured. The value of Ṫ_s is measured with the use of temperature sensors after installing the tubing string before the perforation. The value of Ṫ ₁ is measured above the first perforation interval at the initial stage of production.
• In case of three or more perforated zones, numerical simulation can be used for determining the inflow profile. For any set of values of flow rate {Q_i } (i=1,2..n, where n is number of perforated zones), transient temperatures of produced fluids can be calculated in the following way (9): $$T_{\mathit{in},i}={\alpha }_i+\left(\beta \cdot \frac{Q_i}{h_i}+\stackrel{•}{T_s}\right)\cdot t$$

  

  $${\alpha }_i=T_{a,i}+{\stackrel{•}{T}}_s\cdot \frac{r_{\mathrm{<figure-callout\; id="2"\; label="w"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">w</figure-callout>}}^2}{4\pi \cdot a},$$

  
• The parameter β (11) is one and the same for all zones; the parameters α _i are different because they depend on the temperature of the reservoir T_a,i recorded in the wellbore before start of production.
• For this set of flow rate values, the numeric model of the producing wellbore should calculate transient temperatures of the flow at each depth of placement of the sensor with consideration of heat losses into the surrounding reservoir, the calorimetric law for the fluids being mixed in the wellbore, and the thermal influence of the wellbore which is understood here as the influence of the fluid initially filling the wellbore. The flow rate is determined with the use of the procedure of model fitting that minimizes differences between the recorded and calculated temperatures of the sensors.
• An approximate solution of the problem can be obtained with the use of the above-described analytical model, which utilizes rates of increase of sensor temperatures.
• The calorimetric law for the second perforated zone is described by the equation $$\frac{T_{}^{}}{}$$ $$\frac{{}_1^{*}\cdot {\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+T_{\mathit{in},2}\cdot Q_2}{{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2}={\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_2^{*}$$

  

  where T ₁* are T ₂* are temperatures of the fluid below and above the perforated zone. In accordance with the numeric simulation, the difference between T ₁ and T ₁*, T ₂ and T ₂* remains practically constant and instead of Equation (18) we can use the following equation for time derivatives of the measured temperatures: $$\frac{\stackrel{•}{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_1}<figure-callout\; id="1"\; label="\cdot \; Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">\cdot </figure-callout>Q_{}{}_1+\stackrel{•}{T_{\mathit{in},2}}\cdot Q_2}{{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2}=\stackrel{•}{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_2}$$

  
• Taking into consideration the above-presented relationships (11) and (16), this formula can be written as an equation for the dimensionless flow rate y₂ of the second perforated zone y₂ = Q ₂/Q ₁ : $$\frac{1}{1+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2}\left\{1+\left[\frac{h_{12}\cdot {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2+y_a}{1+y_a}\right]\cdot {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2\right\}=\frac{{\stackrel{•}{T}}_2}{{\stackrel{•}{T}}_1}=f_{21}$$

  

  where $h_{12}=\frac{h_1}{{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">h</figure-callout>}}_2},y_a=\frac{4\pi \cdot a}{\chi }\cdot \frac{h_1}{{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1}\mathrm{.}$

  
• If Ṫ ₂ > Ṫ ₁ (f ₂₁ > 1), a unique solution exists. In the opposite version (f ₂₁ < 1), this equation has two solutions. The physical sense of this peculiarity is quite obvious for f ₂₁ =1, that corresponding to equal increase rates of temperatures T ₂ and T ₁. Indeed, this may take place in two cases: (1) Q ₂=0 (y ₂=0) and above the upper zone the behavior of the temperature is the same as below it; (2) Q ₂=Q ₁ (y ₂=1) - both zones are equal and they have the same rate of temperature increase.
• The possible solution of the problem of non-uniqueness of solution consists in the combination of two approaches. After evaluating Q ₁ with the use of Equation (12) and determining y₂ by Equation (20), the true value y₂ of can be chosen using the known total flow rate Q (for two perforated zones): $$Q={\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2={\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1\left(1+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2\right)$$

  
• Relative flow rates for perforated zones 3 and 4 can be calculated using the dimensionless values y₂, y₃ and so on, which were determined previously for the perforated zones located down the wellbore. $$\frac{1}{1+{\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_3}\left\{1+\left[\frac{{\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_3\left(1+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2\right)\cdot h_{13}+y_a}{\left(1+y_a\right)\cdot f_{21}}\right]\cdot {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_3\right\}=f_{32}$$

  

  $$\frac{1}{1+{\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_4}\left\{1+\left[\frac{{\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_4\cdot \left[1+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_3\left(1+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">y</figure-callout>}}_2\right)\right]\cdot h_{14}+y_a}{\left(1+y_a\right)\cdot f_{21}\cdot f_{32}}\right]\cdot {\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_4\right\}=f_{43}$$

  

  where $${\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_3=\frac{Q_3}{{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2},{\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="imgf0009.png"\; state="\{\{state\}\}">y</figure-callout>}}_4=\frac{Q_4}{{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="Q"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">Q</figure-callout>}}_3},f_{32}=\frac{\stackrel{•}{{\mathrm{<figure-callout\; id="3"\; label="T"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">T</figure-callout>}}_3}}{\stackrel{•}{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_2}},f_{43}=\frac{\stackrel{•}{{\mathrm{<figure-callout\; id="4"\; label="T"\; filenames="imgf0009.png"\; state="\{\{state\}\}">T</figure-callout>}}_4}}{\stackrel{•}{{\mathrm{<figure-callout\; id="3"\; label="T"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">T</figure-callout>}}_3}}\mathrm{.}$$

  
• The possibility of determining the inflow profile with the use of the suggested method for a case where direct measurement of temperatures of fluids inflowing into the wellbore from pay zones is impossible was checked up on synthetic examples prepared with the use of numerical simulation software package for the producing wellbore, which performs modeling of the unsteady-state pressure field in the "wellbore-formation" system, flow of non-isothermal fluids in a porous medium, mixing of the flows in the wellbore, and heat transfer in the "wellbore-formation" system, etc.
• Modeling of the process operations carried out under the following time schedule was performed:
• Circulation of the well during 110 hours. The temperature of fluids at the formation occurrence depth is assumed to be 40°C.
• Borehole-return-to-thermal-equilibrium time is 90 hrs.
• Production for 6 hrs with flow rate Q = 60 m³/day.
• Geothermal gradient equals 0.02 K/m. The temperature of the undisturbed reservoir at the depth of sensor 1 (274 m) is 65.5°C that at the depth of sensor 3 (230 m) is 64.6°C. Thermal diffusivity of the reservoir is a = 10^-6 m²/s and χ=0.86.
• Figure 1 shows the scheme of a well with three perforated intervals (#1: 280-290 m, #2: 260-270 m, #3: 240-250 m) and three temperature sensors: T₁ at the depth of 274 m, T₂ at the depth of 254 m and T₃ at the depth of 230 m.
• Two options were considered with different combinations of formation permeabilities and the following flow rate parameters:
• Option 1 (Figure 2a): Q₁=10 m³/day, Q₂=23.4 m³/day, Q₃=26.6 m³/day; and
• Option 2 (Figure 2b): Q₁=46 m³/day, Q₂=13 m³/day, Q₃=1 m³/day.
• During circulation and the return-to-thermal-equilibrium time, the reservoir/wellbore temperature is the same in both cases under consideration. At the end of the return-to-thermal-equilibrium time, the rate of temperature growth was Ṫ_s(200h) = 0.034 K/hr.
• Figures 3 and 4 show temperatures of the produced fluids (thin curves) and temperatures of the corresponding sensors (bold curves). The difference between T_in,1 and T ₁ remains practically constant after ~1 hr of production. Time derivatives of fluid temperature and temperature of sensor #1 are presented in Figures 5 and 6. One can see that approximately 3 hours after start of production, the difference between dT _in,1/dt and Ṫ ₁ amounts to about 6-8%, that confirming our assumption made in the analysis presented above.
• The correlation between time derivative T_in and specific flow rate q (data for all of the perforated intervals are utilized) is presented in Figure 9. For flow rate q tending to zero, the linear regression equation gives: Ṫi_n(q → 0) = 0.0374 K/hr. This value is close to the rate of temperature recovery Ṫ _s(200h) = 0.034 K/hr due to the conductive heat transfer. This result confirms Formula (14) suggested above for correlation between flow rate and rate of temperature growth of the produced fluid.
• Let us estimate the values of the flow rate from the lowermost perforated zone. With duration of production equaling 4 hours, Figures 5 and 8 give: Option #1 - Ṫ ₁ = 0.067 K/hr, Option #2 - Ṫ ₁ = 0.17 K/hr. Substituting these values in Formula (1), we find:
• Option #1: Q ₁=11 m³/day (the true value is Q ₁=10 m³/day);
• Option #2: Q ₁= 46.5 m³/day (the true value is Q ₁=46 m³/day).
• Flow rate values for other perforated zones are determined by Formulas (20), (23).
Option # 1:
• For the estimated value Q ₁=11 m³/day presented above, we find y_a =1.1. For production duration of 4 hours, Figure 7 gives f ₂₁ ≈ 1.45, while Equation (2) gives one positive solution y ₂ = 2.346 and flow rate Q₂ = Q ₁ · y ₂ = 25.8 m³/day.
• For the third perforated zone Figure 7 gives f ₃₂ ≈ 1.08 and from Equation (22) we find one positive solution y ₃ = 0.75 and Q ₃ = (Q₁ + Q ₂) · y ₃ = 27.6 m³/day.
• The total flow rate calculated by temperature data amounts to Q_e = Q ₁ + Q ₂ + Q ₃ = 64.4 m³/day (the true value is 60 m³/day).
• Using this value for determining relative flow rates, we find: ${\mathrm{<figure-callout\; id="1"\; label="Y"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Y</figure-callout>}}_1=\frac{Q_1}{Q_e}=0.17;$

  

  Y ₂ = 0.4; Y ₃ = 0.43
• The corresponding flow rate values for different zones are:
• Q ₁ = Q·Y ₁ =10.2 m³/day (the true value is 10 m³/day)
• Q ₂ = Q·Y ₂ = 24 m³/day (the true value is 23.4 m³/day)
• Q₁ = Q·Y ₁ = 25.8 m³/day (the true value is 26.6 m³/day)
• Relative errors (related to the total flow rate) are 0.3%, 1%, and 1.3%.
Option #2:
• For the above-estimated flow rate value Q ₁=46.5 m³/day, we find y_a = 0.25. Figure 8 gives for production duration of 4 hours f ₂₁ ≈ 0.85. In this case, Equation (20) has no solution and as the approximate solution we have to take the value of y ₂ that corresponds to the minimum value of f ₂₁ (f _21min ≈ 0.863), which provides for the real solution: y ₂ = 0.413.
• The corresponding flow rate is Q ₂ =19.85 m³/day.
• For the third perforated zone, Equation 8 gives f ₃₂ ≈ 0.96, while from Equation (22) we find two roots:
• y ₃ = 0.5, Q ₃ = (Q ₁ + Q ₂ )·y₃ = 34 m³/day and total flow rate Q_e = 102 m³/day, and
• y ₃ = 0.062, Q₃ = (Q ₁ + Q ₂) · y ₃ = 4.18 m³/day and total flow rate Q_e = 72 m³/day.
• As the approximate solution of the problem, we will take the value of y ₃ = 0.062, which gives the total flow rate value Q_e = 72 m³/day that is closer to the true value.
• In the second case the estimate of Q ₁ is more reliable than the estimate of Q₂ and Q ₃, hence, we fix the value of Q ₁ and use the previously determined values of Q ₂ and Q ₃ for distributing the remaining flow rate Q-Q ₁ between these zones: $${\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2ʹ=\frac{Q_2}{{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="Q"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">Q</figure-callout>}}_3}\cdot \left(Q-{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1\right)=11.2{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">m</figure-callout>}}^3/\mathrm{day}$$

  

  and $${\mathrm{<figure-callout\; id="3"\; label="Q"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">Q</figure-callout>}}_3ʹ=\frac{Q_3}{{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2+{\mathrm{<figure-callout\; id="3"\; label="Q"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">Q</figure-callout>}}_3}\mathrm{.}\left(Q-{\mathrm{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0007.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1\right)=2.3{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="imgf0001.png,imgf0009.png"\; state="\{\{state\}\}">m</figure-callout>}}^3/\mathrm{day}$$

  
• At last, the determined flow rate values are as follows:
• Q ₁ = 46.5 m³/day (the true value is 46 m³/day)
• Q ₂ = 11.2 m³/day (the true value is 13 m³/day)
• Q ₃ = 2.3 m³/day (the true value is 1 m³/day)
• Relative errors (related to the total flow rate) are 0.8%, 3% and 2.2%.
• For solving the inverse problem, this inflow profile (a low inflow rate of the uppermost zone) is the most complex. Nonetheless, results of solving the inverse problem are well consistent with the data utilized in direct simulation.
• In general case, the most reliable inversion of temperature measured among perforated intervals immediately after perforating can be made with the use of a specialized numerical model and fitting the transient temperature data with consideration of absolute values of temperature as well as time derivatives of temperature.

Claims (7)

1. A method for determining profile of fluid inflow from multi-zone reservoirs into a wellbore comprising:

   - measuring temperature in the wellbore during a wellbore-retum-to-thermal-equilibrium time after drilling,

   - perforating the wellbore,

   - determining temperature of the fluids inflowing into the wellbore from each pay zone at an initial stage of production, and
   determining a specific flow rate for each pay zone by rate of change of the measured temperatures.
2. The method of Claim 1 wherein the temperature of the fluids inflowing into the wellbore from the pay zones is determined by direct measurement of temperature of the fluids inflowing into the wellbore from each pay zone, and a specific flow rate of each pay zone is determined by the formula $$Q_i=\frac{4\pi }{\chi }\cdot a\cdot h_i\cdot \left(\frac{\stackrel{•}{T_{\mathit{in},i}}}{\stackrel{•}{T_s}}-1\right),$$

   

   
   where Q_i is a flow rate of the ith pay zone,
   Ṫ_s is a rate of temperature recovery in the wellbore before perforation,
   Ṫ_{in, i} is a rate of temperature variation of the fluid inflowing into the wellbore from the ith pay zone at the initial stage of production,
   h_i is a thickness of the ith pay zone,
   a is a thermal diffusivity of the reservoir, $$\chi =\frac{c_f\cdot {\rho }_f}{{\rho }_r\cdot c_r},$$

   

   
   ρ _fc_f is a volumetric heat capacity of the fluid,
   ρ _rc_r = φ · ρ _fc_f + (1 - φ)·ρ _mC_m is a volumetric heat capacity of the rock saturated with the fluid,
   ρ_mc_m is a volumetric heat capacity of a rock matrix ,
   φ is a porosity of the reservoir.
3. The method of claim 1 wherein the wellbore-retum-to-thermal-equilibrium time is 5-10 days.
4. The method 1 of claim 1 wherein the temperature of the fluids inflowing into the wellbore from each pay zone at the initial stage of production is measured within 3-5 hours after start of production.
5. The method of claim 1 wherein the temperature of the fluids is determined by sensors installed on a tubing string being used for perforating, above each perforated interval, a specific flow rate of a lower pay zone is determined by the formula $$Q_1=\frac{4\pi }{\chi }\cdot a\cdot h_1\cdot \left(\frac{\stackrel{•}{T_1}}{\stackrel{•}{T_s}}-1\right),$$

   

   
   where Q₁ is a flow rate of the lower zone,
   Ṫ_s is a rate of temperature recovery in the wellbore before perforation,
   Ṫ ₁ is a rate of temperature change of the fluid inflowing into the wellbore from the pay zone at the initial stage of production as measured above the lower perforated interval,
   h ₁ is a thickness of the lower pay zone,
   a is a thermal diffusivity of the reservoir, $$\chi =\frac{c_f\cdot {\rho }_f}{{\rho }_r\cdot c_r},$$

   

   
   ρ _fc_f is a volumetric heat capacity of the fluid,
   ρ _rc_r = φ · ρ _fc_f + (1-φ)·ρ _mC_m is a volumetric heat capacity of the rock saturated by the fluid,
   ρ _mc_m is a volumetric heat capacity of the rock matrix,
   φ is a porosity of the reservoir,
   and specific flow rates of overlying pay zones are determined by temperatures measured by the sensors installed on the tubing string, using the flow rates determined for the underlying pay zones.
6. The method of claim 5 wherein the wellbore-retum-to-thermal-equilibrium time is 5-10 days.
7. The method of claim 5 wherein the temperature of the fluids inflowing into the wellbore from each pay zone at the initial stage of production is measured within 3-5 hours after start of production.

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