Figure 1 shows a physical model of a single parallel-plate fracture within one injection-production well. Fracture length was assumed to be l, fracture width was assumed to be b, and fracture height was assumed to be h.

The output concentration of a certain tracer slug can be obtained from the analytical solution of the 1D convection-diffusion equation with a definite tracer concentration boundary (Jing et al., 2016a): C ( t ) C 0 = Δ l 2 π σ 2 exp [ − ( l − l ¯ ) 2 2 σ 2 ] (1) where C(t) is the output concentration of the tracer at time t, mg/L; C ₀ is the concentration of the injected tracer slug, mg/L; ∆l is the tracer slug size (ratio of the injected volume of tracer slug V _d to flow-channel cross-sectional area A), m; l is the length of the flow channel where the tracer is transported, m; l ( — ) is the position of the tracer front at a concentration of 0.5 C ₀ (the product of average flow velocity v and time t), m; and σ ² is the variance of the tracer distribution curve (two times the product of tracer diffusion constant α, average flow velocity v, and time t), m².

Because the tracer follows the movement of the injected fluid, its average migration velocity can be obtained from the flow formula for a parallel-plate fracture: v = b 2 Δ P 12 μ l (2) where b is the fracture width, μm; μ is the hydrodynamic viscosity, mPas; and ∆P is the pressure difference at both ends of the fracture, MPa.

By substituting Eq. 2 into the relevant parameters of Eq. 1, the equation for tracer output concentration for an interwell single fracture can be obtained through dimensional analysis and unit conversion: C ( t ) C 0 = 10 6 V d μ l b 2 h 28.8 π α μ l Δ P t exp [ − ( μ l 2 − 7.2 b 2 Δ P t ) 2 28.8 α μ l b 2 Δ P t ] (3) where V _d is the injected volume of the tracer slug, m³; h is the fracture height, m; and α is the tracer diffusion constant, m.

In order to illustrate the characteristics of the tracer output concentration curve for an interwell single fracture, C ₀ was considered to be 0.5 mg/L, V _d was considered to be 5 m³, and α was considered to be 5 m. Eq. 3 was used to analyze the influence of four parameters—l, b, h, and ∆P—on the tracer-curve characteristics of an interwell single fracture (Figure 2).

As can be seen from Figure 2, the tracer output concentration curve for the interwell single fracture is a single-sharp-peak curve with basic symmetry between ascending and descending branches. Parameters l and b have effects on the peak value, peak time, and bandwidth of the tracer curve; h only affects the peak value; and ∆P affects the peak time and bandwidth, but has no effect on the peak value.

Figure 3 shows a physical model of a single pipe within one injection-production well. Pipe length was assumed to be l, and the equivalent diameter of the pipe was assumed to be D.

Similar to the single fracture, the tracer migration in a single pipe also follows Eq. 1, but the expression of its average velocity is different. The average velocity of the Hagen-Poiseuille equation under the condition of steady flow can be expressed as: v = D 2 Δ P 32 μ l (4)

By substituting Eq. 4 into the relevant parameters of Eq. 1, the equation of tracer output concentration for an interwell single pipe can be obtained through dimensional analysis and unit conversion: C ( t ) C 0 = V d μ l 15 π D 3 3 π α μ l Δ P t exp [ − ( μ l 2 − 2700 D 2 Δ P t ) 2 10800 α μ l D 2 Δ P t ] (5)

Similar to the single fracture, C ₀ was considered 0.5 mg/L, V _d was considered 5 m³, and α was considered 5 m. The influences of l, D, and ∆P on the output concentration curve of an interwell single pipe were analyzed (Figure 4).

As can be seen from Figure 4, the tracer output concentration curve of the interwell single pipe is a single-sharp-peak curve with basic symmetry between ascending and descending branches. Parameters l and D have effects on the peak value, peak time, and bandwidth of the tracer curve, while ∆P affects the peak time, but has no effect on the peak value and bandwidth.

Figure 5 shows a physical model of a single cavity within one injection-production well. Given the large scale of the cavity, the fluid reaches equilibrium quickly, so the concrete form of the fluid flow was not considered, but was regarded as an equipotential body. The tracer migration in the cavity is dominated by slow diffusion.

Assuming that the output concentration of the tracer is equal to the average concentration in the cavity, it can be deduced according to the tracer equilibrium relationship: d C ( t ) d t = C in ( t ) − C ( t ) t ¯ (6)

Eq. 6 can be solved to obtain the analytical solution of tracer output concentration of the cavity: C ( t ) = 1 t ¯ exp ( − t t ¯ ) ∫ 0 t C in ( t ) exp ( t t ¯ ) d t (7) where C _in(t) is the tracer concentration at the entrance of the cavity at time t, mg/L; and t ¯ is the average residence time of the fluid in the cavity (cavity volume V _c divided by cavity output flow Q), d.

Compared with the whole time of tracer monitoring, tracer slug injection can be regarded as instantaneous. The tracer concentration of the cavity entrance is C ₀, which is the constant concentration of the injected tracer slug. Residence time t ¯ is used to describe the delayed response of the tracer from the entrance to the outlet of the cavity, and the tracer output concentration equation of a single cavity can be obtained from Eq. 7: C ( t ) = { 0 t ≤ t ¯ C 0 V d V c exp ( − t − t ¯ t ¯ ) [ exp ( t − t ¯ t ¯ ) − 1 ] t ¯ > t > 2 t ¯ C 0 V d V c exp ( − t − t ¯ t ¯ ) ( e − 1 ) t ≥ 2 t ¯ (8)

In order to illustrate the characteristics of the tracer output concentration curve of an interwell single cavity, C ₀ was 500 mg/L and V _d was 0.05 m³. The influences of V _c and Q on the output concentration curve of the interwell single cavity were analyzed (Figure 6).

As can be seen from Figure 6, the tracer output concentration curve of the interwell single cavity is a single-peak curve with clearly asymmetrical wings. The ascending branch is steep, and the descending branch is slow, with an obvious trailing phenomenon. V _c has an effect on the peak, peak time, and bandwidth of the tracer curve. The greater V _c is, the much blunter the tracer curve is. Q affects the peak time and bandwidth, but has no effect on the peak value.

Noteworthily, the related parameters of the interwell fracture, pipe, and cavity have a great influence on the peak value, peak time, and bandwidth of the single-peak curve of the tracer, so it is not suitable to identify the interwell fracture-cavity combination structure by the peak value, peak time, or bandwidth.

It is worth noting that the tracer curve type of multiple series cavity is consistent with that of a single cavity, and multiple series cavity is equivalent to a large volume cavity. Therefore, the section only considers the tracer curve characteristics of single cavity in fracture/pipe series, and the morphological characteristics of tracer curve of single cavity in multi-cavity series can be referred to.

Figure 7 shows a physical model of a fracture series cavity within one injection-production well. The entrance and exit of the cavity are connected with a plate fracture with length l, width b, and height h.

Figure 9 shows a physical model of a pipe series cavity within one injection-production well. The entrance and exit of the cavity are connected with a pipe with length l and equivalent diameter D.

The derivation process for the tracer output equation of the pipe series cavity is the same as that for the fracture series cavity. Firstly, the Hagen-Poiseuille equation was used to represent ∆P in the tracer output concentration [Eq. 5] of the interwell single pipe as flow Q to get the tracer output equation of the pipe at the upstream of the cavity. Then, the tracer output equation of the cavity was deduced by applying Eq. 7. Finally, according to the superposition principle, the tracer output equation for the pipe series cavity was deduced the same as Eq. 10 and Eq. 12, except for the expressions of β, γ, η, and ω in the equation being different: β = C 0 V d π D α Q (13) γ = 16 π D 2 α Q (14) η = π D 2 l (15) ω = 4 Q (16)

Similarly, the tracer output concentration curve of the interwell pipe series cavity should be shifted to the right by l/v + t ¯ . In order to illustrate the characteristics of the tracer output concentration curve of the interwell pipe series cavity, C ₀ was 500 mg/L, V _d was 0.05 m³, and α was 5 m. The influences of D, l, V _c, and Q on the tracer curve were analyzed (Figure 10).

As can be seen from Figure 10, the tracer output concentration curve of the interwell pipe series cavity is a single-peak curve with clearly asymmetrical wings. The ascending branch is steep, and the descending branch is slow, with an obvious trailing phenomenon. Parameters D, l, and V _c all have an effect on the peak value, peak time, and bandwidth of the tracer curve, while Q affects the peak time and bandwidth, but not the peak value; these characteristics are similar to those of the tracer curve of the interwell fracture series cavity.

Figure 11 shows a physical model of a multifracture in parallel within one injection-production well (taking three fractures as an example). Assuming N branch fractures, branch fracture i has length l _i , width b _i , and height h _i .

After tracer slug injection, the slug is distributed to each branch fracture in a certain proportion. The flow and the volume of tracer slug distributed to each fracture are different, and the reciprocal of the flow resistance of each fracture can be used for splitting. Flow resistance R _i of fracture i can be expressed as (Jing et al., 2016b): R i = Δ p q i = 12 μ l i h i b i 3 (17) where q _i is the flow of fracture i, m³/d.

Then, the equation of tracer output concentration of fracture i is: C i ( t ) C 0 = V d T i 4 π α b i h i T i Q F t exp [ − ( b i h i l i − T i Q F t ) 2 4 α b i h i T i Q F t ] (18) where Q _F is the total flow of the multifracture in parallel, m³/d. Splitting coefficient T _i of fracture i is defined as: T i = h i b i 3 l i ∑ i = 1 N h i b i 3 / l i (19)

The tracer output concentration of an interwell multifracture in parallel should be the superposition of the output concentration of each fracture at the producing well: C ( t ) = ∑ i = 1 N q i C i ( t ) Q F (20)

Therefore, the output tracer concentration equation for the multifracture in parallel was obtained as follows: C ( t ) C 0 = 500 V d π α Q F ∑ i = 1 N T i 2 b i h i T i t exp [ − ( b i h i l i − 10 6 × T i Q F t ) 2 4 × 10 6 α b i h i T i Q F t ] (21)

In order to illustrate the characteristics of the tracer output concentration curve of the interwell multifracture in parallel, C ₀ was 500 mg/L, V _d was 0.05 m³, Q _F was 20 m³, h _i was 5 m, and α was 1 m. The characteristics of the tracer output concentration curve under different combinations of N (values of 2, 3, 4, and 5), l _i (values of 200, 300, 400, 500, and 600 m), and b _i (values of 5,000, 5,300, 5,600, 5,900, and 6,200 μm) were analyzed (Figure 12).

As can be seen from Figure 12, the tracer output concentration curve of the interwell multifracture in parallel can be divided into an independent multipeak curve and a continuous multipeak curve, showing a curve with multiple peaks and the two wings of each peak (ascending branch and descending branch) being basically symmetrical. The longer the fracture length, the larger the corresponding fracture width and the greater the difference in flow among the fractures, resulting in the obvious independent peak of each fracture; importantly, the curve shows a relatively independent multipeak shape (Figure 12A). The length of each fracture is the same, but the corresponding fracture width is slightly different. There is little difference in the flow and migration rate of the tracer in each fracture, resulting in the tracer peak of each fracture arriving successively, with the difference in arrival time being small. Therefore, the tracer curve shows a relatively continuous multipeak shape (Figure 12B).

Figure 13 shows a physical model of a multipipe in parallel within one injection-production well (taking three pipes as an example). Assuming N branch pipes, branch pipe i is described by length l _i and equivalent diameter D _i .

The derivation of the tracer output concentration equation for the multipipe in parallel is the same as that for the multifracture in parallel, and its concentration should be the superposition of the output concentration of each fracture at the producing well: C ( t ) C 0 = V d π α Q p ∑ i = 1 N T i 2 D i T i t exp [ − ( π D i 2 l i − 4 T i Q p t ) 2 16 π D i 2 α T i Q p t ] (22) where Q _p is the total flow of the multipipe in parallel, m³/d. Splitting coefficient T _i of pipe i is defined as: T i = D i 4 l i ∑ i = 1 N D i 4 / l i (23)

In order to illustrate the characteristics of the tracer output concentration curve of the interwell multipipe in parallel, C ₀ was 500 mg/L, V _d was 0.05 m³, Q _p was 20 m³, and α was 1 m. The characteristics of the tracer output concentration curve under different combinations of N (values of 2, 3, 4, and 5), l _i (values of 200, 300, 400, 500, and 600 m), and D _i (values of 0.80, 0.85, 0.90, 0.95, and 1.00 m) were analyzed (Figure 14).

As can be seen from Figure 14, the tracer output concentration curve of the interwell multipipe in parallel can be divided into a multipeak curve and a continuous multipeak curve, showing a curve with multiple peaks with the two wings of each peak (ascending branch and descending branch) being basically symmetrical. Similar to the multifracture in parallel, there is a large flow difference between the pipes, and the tracer concentration curve shows a relatively independent multipeak shape (Figure 14A). The flow difference between the pipes is small, and the tracer concentration curve shows a relatively continuous multipeak shape (Figure 14B).

Figure 15 shows a physical model of a multicavity in parallel within one injection-production well (taking three parallel pipes containing cavities as an example). Assuming N flow channels in parallel, the volume of the cavity on the i parallel branch is V _ci , the pipe length is l _i (the pipe length on both sides of the cavity is the same), and the equivalent diameter of the pipe is D _i (the pipe equivalent diameter on both sides of the cavity is the same).

According to Eq. 12, the tracer output concentration equation of the i parallel branch can be obtained as follows: C i ( t ) = 1 2 ∑ j = 1 n { [ C c i ( t j ) - C c i ( t j − 1 ) ] erfc [ η i − ω i ( t − t j − 1 ) γ i ( t − t j − 1 ) ] } (24)

C _ci can be expressed by Eq. 10, but the expressions of β, γ, η, and ω in the equation are different: β i = C 0 T i V d π D i α T i Q c (25) γ i = 16 π D i 2 α T i Q c (26) η i = π D i 2 l i (27) ω i = 4 T i Q c (28) 1 t ¯ i = T i Q c V ci (29) where Q _c is the total flow of the multicavity in parallel, m³/d. T _i is expressed by Eq. 23.

The tracer output concentration of the interwell multicavity in parallel should be the superposition of the output concentration of each parallel branch, which is the same as Eq. 20. Through further derivation and simplification, the tracer output concentration equation of the multicavity in parallel can be obtained as follows: C ( t ) = ∑ i = 1 N T i C i ( t ) (30)

As with the tracer output curve of the fracture/pipe series cavity, considering the residence times of tracer migration in different parallel branches, the tracer output concentration of the i parallel branch should be delayed l _i /v _i + t ¯ i , and then the curve after the superposition of each tracer concentration of parallel branch can be considered the true tracer curve.

In order to illustrate the characteristics of the tracer output concentration curve of the interwell multicavity in parallel, C ₀ was 500 mg/L, V _d was 0.05 m³, Q _c was 20 m³, α was 1 m, N was 3, and V _ci was 100 m³. Equivalent diameter D _i of the pipes at both ends of the cavity was 1 m, and the length of the pipes at both ends of the cavity was the same. Parameters of l _i = 80, 190, and 300 m and l _i = 100, 150, and 200 m were applied to analyze the curve characteristics of tracer output concentration under these two conditions (Figure 16).

As can be seen from Figure 16, similar to the tracer output concentration curve of the multifracture/multipipe in parallel, the tracer output concentration curve of the multicavity in parallel is a multipeak curve. There are two types of multipeak tracer curve: independent multipeak and continuous multipeak.

The peaks of the independent multipeak curve obviously show the curve shapes containing cavities. The two wings of the peaks are asymmetric with an obvious trailing phenomenon (the ascending branch is steep and the descending branch is slow). Each peak reflects a cavity, the flow difference between parallel branches is large, and there is no peak relative fusion.

The other type is the continuous multipeak curve. As can be seen from the continuous three-peak curve in Figure 16, there are only three parallel branches. The overall peak value lasted for about 100 days, reflecting the long fusion time of the peak values of the parallel branches, reflected in the long trailing time caused by cavities; and the flow difference between the parallel branches was small, resulting in a continuous multipeak curve shape.

Around injection well TK411, there are six production wells (TK408, TK428CH, TK476, TK467, T401, and S48), all of which are Ordovician producing zones. On 14 April 2007, 14 kg of BY-1 tracer was injected by pump truck according to the tracer injection design concentration for TK411 (100%). In tracer monitoring, 1,594 samples were sampled and 1,577 samples were tested for 200 days from the day following tracer injection to 31 October 2007. Meanwhile, tracer samples were also collected and tested from the second-line wells (TK457H and TK424CH).

Because of factors such as engineering, geology, and testing, tracer curves detected in the field often appear with errors, making the tracer curves rise and fall, and many real data points are masked. It is difficult to effectively judge the fracture-cavity combination structure according to the tracer-curve morphological characteristics of the different fracture-cavity combination structures. Therefore, a smoothing and filtering algorithm was used to denoise the tracer curve of eight wells corresponding to TK411, to identify the effective wave peak, to eliminate burrs, and to make the whole curve smooth and orderly. The tracer output concentration curve of each well and its denoising curve are shown in Figure 17.

According to the classification characteristics of the interwell tracer curve in the fracture-cavity reservoir from Table 1, the interwell fracture-cavity combination structure was qualitatively identified in terms of peak number, characteristics of two wings, and peak type of each curve in the TK411 well group. The results are shown in Table 2.

According to the tracer output concentration data and production performance data from a production well (cumulative water injection and cumulative water production during the tracer monitoring period), tracer recovery quality, tracer average residence time, average migration rate, distribution coefficient of injected water, and swept volume of injected water can be reliably calculated without using a theoretical model for inversion and fitting. These parameters provide an idea and means for verifying the reliability of interwell fracture-cavity structure identification with the morphological characteristics of tracer curve (Jing et al., 2016a; Dewaide et al., 2016).

Figure 18 displays the distribution coefficient and swept volume of injected water for the TK411 well group. It can be seen from the figure that the distribution coefficient of injected water from injection well TK411 to production well TK467 is the largest, indicating that there is a large connected channel. The swept volume of injected water is 1241.64 m³, and there is obviously a large cavity. According to its peak characteristics, it can be inferred that there is one cavity. This is basically consistent with the results obtained using the distribution coefficient and swept volume of injected water.

Figure 19 is the seismic multi-attribute plane superposition of 0–80 ms below plane T74 of TK411 block. In the figure, the darker the color is, the cavity reservoir development area (all the cavity development zones are in the area with strong amplitude change rate). It can be seen from the figure that TK411 has a strong amplitude change rate area in each path direction of the surrounding production wells, indicating that there is a high probability of cavity between each well. It also verifies the reliability of using tracer curve morphology to identify interwell fracture-cavity structure.

Meanwhile, by analyzing the results from karst tracer field testing and laboratory model testing in hydrologic exploration, it has been seen that the two wings of a karst pipe/fracture have the characteristics of a basically symmetrical tracer curve under relatively high velocity flow (Morales et al., 2006; Luhmann et al., 2012; Borghi et al., 2016; Dewaide et al., 2016; Zhang et al., 2016; Ji et al., 2017; Zhao et al., 2017), as well as that underground pools, water tanks, or underground rivers (caves) can cause obvious trailing of the tracer curve (Morales et al., 2006; Luhmann et al., 2012; Borghi et al., 2016; Dewaide et al., 2016; Zhang et al., 2016; Ji et al., 2017; Zhao et al., 2017). These conclusions are consistent with the conclusions obtained in this paper, and the curve characteristics obtained from the series and parallel combinations of pipe, fracture, and cavity are also consistent with them.