An improved interwell connectivity model to obtain interwell connectivity information by using complex well data

Abstract

Information on interwell connectivity is important in reservoir field dynamic analysis. However, all conventional methods, such as the tracer test, interference well test, and numerical simulation, have disadvantages. These disadvantages include the length of time taken, high costs, and the effect on oilfield production. Thus, research focus has been directed toward the development of approaches that use production and injection data to obtain interwell connectivity data. Prevailing interwell connectivity models are sensitive to shut-ins, and their corresponding inversion methods are unreliable. The improved interwell connectivity model presented in this study exhibits enhanced robustness to shut-ins. The application of seepage theories and numerical simulation methods enables the main model parameters to absorb prior geological knowledge to characterize the reservoir and improve the initial estimate of connectivity. Corresponding inverse methods for model parameters are implemented based on Bayesian inverse theory and the projection gradient method and obtain greater robustness for the model parameter, compared with those in previous studies. Testing of a heterogeneous synthetic reservoir and a Z16 reservoir block demonstrates that the methodology can precisely determine the interwell connectivity and can be used in real oilfields.

Keywords

Interwell connectivity Bayesian theory projection gradient history matching

1. Introduction

Data on the interwell connectivity of any waterflooding development reservoirs are crucial for creating a waterflooding adjustment plan to modify flood patterns underground in order to enhance oil recovery. Owing to formation heterogeneity, water breakthrough, and field procedures, the interwell connectivity of the late waterflooding stage has become complicated. Finding the interwell connectivity profile and dynamic oil–water flows can show the remaining oil and help optimize procedures.

Tracer testing, well test analysis, and numerical simulation are the commonly used methods to test interwell connectivity. Interwell tracers^1,2 can provide reliable but ambiguous information about connectivity. They give the profile of the flow paths between injectors and producers under a fixed pressure pattern and help describe the continuity and direction characteristics in the reservoir. However, tracer monitoring entails a long wait for a breakthrough, which also involves pressure and well distance. Well test analysis^3,4 interprets transient pressure perturbations between the stimulus well and the response well to obtain the pressure transmission coefficient, permeability in different directions, and fracture direction. To ensure that the as much formation information as possible is obtained in the pressure test, the stimulus well tends to be a high-rate producer, which exhibits good permeability. Consequently, pressure testing affects the routine production, and the procedure cannot interpret long-distance wells because of signal attenuation. Conventional grid-based reservoir numerical simulation^5,6 can clearly depict the remaining oil after the cumbersome task of history matching, but cannot directly reveal the interwell connectivity and data uncertainty propagation with the grid increased. As such, these methods cannot be implemented on a large scale to provide an insight into the reservoir.

Production and injection rates are the most abundant data available in any injection projection.⁷ Numerous techniques have been proposed based on production and injection data to obtain information on interwell connectivity. In this category, Spearman analysis, multivariate linear regression (MLR),^8,9 the capacitance model (CM),^10–16 and systemic analysis methods¹⁷ are the prevailing methods. Spearman analysis and MLR operate under the assumption that production and injection rates exhibit a linear relationship and that the relevant coefficients reflect the interwell connectivity. Compared with Spearman analysis and MLR, the CM can characterize the connectivity, hysteresis, and attenuation between injection and production wells after integration with mass balance theory. On the basis of the features of first-order linear time-delay systems, systemic analysis treats the production and injection rates as the input and output signals of a reservoir; in addition, system analysis involves fewer model parameters than the CM. Previous studies have established several numerical models^7–17 and have been integrated with parameter estimation^17–22 to approach interwell connectivity. The inadequacies common to all of these methods are that the sensitivity of the model to shut-ins only works well within the stable period and that some model parameters lack geological characteristics, such as hydraulic fracturing and fractures,^23–26 because they are not rooted in reservoir theory. In addition, the inverse result obtained by parameter estimation without constraints in these studies is not reliable and even contradicts prior knowledge of geology.

With the aforementioned defects considered, an improved interwell dynamic connectivity model based on linear time-delay analysis and the inverse method (improved interwell connectivity model (IICM)) was established. This technique can be applied in the field with frequent shut-in events. The parameters of the IICM are well understood and characterize the reservoir properties. Based on Bayesian inverse theory,^27–29 an optimization algorithm was implemented on the model with constraints. The developed technique is applied in a synthetic reservoir model and a Z16 reservoir block. The results are reliable and can precisely estimate the interwell connectivity.

2. Improved interwell connectivity model

2.1. Modeling

Injection and production wells and an interwell porous medium comprise the system. The injection rate is considered the input signal (incentive) and the production rate (response) is the output signal of the reservoir. With constant producer bottom pressure, we can use numerical simulation to capture response signals when the injection rate is a unit step signal, as shown in Figure 1. Compared with the injection rate, the production rate shows reduction and time delay because of the dissipation of the signal intensity on the propagation process.

Figure 1.

Linear time-delay response signal of the injection and production system.

According to the response characteristics of the unit step signal shown in Figure 1, the injection and production system performance exhibits linear time delay. The transfer function of linear time-delay analysis can be stated as follows:

F (σ) = 1 / (β σ + 1)

(1)

where $β$ is the time constant for the linear time-delay method, indicating the hysteresis of the signal.

According to the transfer function of the injection and production system, the unit step response of the linear time-delay analysis at the zero state is as follows:

q (t) = F (t) = 1 - e^{- t / β} (t > 0)

(2)

Provided that the reservoir has I injectors and J producers, and for each producer j, the response production rate of all injection wells on j is $\sum_{i = 1}^{i = I} λ_{ij} q_{wi} (t)$ . $λ_{ij}$ is the weight factor of the impact of injector i imposed on producer j, and $\sum_{i = 1}^{i = I} λ_{ij} = 1$ . In accordance with the actual condition, the production rate can be discretized with a monthly interval (the injection rate is the mean of every month), as shown in Equation (3):

q_{j} (t) = {\begin{matrix} \sum_{i = 1}^{i = I} λ_{ij} q_{wi} (t) (1 - e^{- t / β_{ij}}) (n_{0} < t \leq n_{0} + 1) \\ \sum_{i = 1}^{i = I} λ_{ij} q_{wi} (1) (1 - e^{- 1 / β_{ij}}) e^{- (t - 1) / β_{ij}} (t \geq n_{0} + 1) \end{matrix}

(3)

Considering the effect of the initial production rate, adding up all injection impulses responds to the production well at each time step, whereas the injection rate varies continuously and the real response of the producer can be indicated by the following:

q_{j} (n) = q_{j} (n_{0}) e^{\frac{- (n - n_{0})}{β_{p}}} + \sum_{i = 1}^{i = I} λ_{ij} \sum_{m = 1}^{m = n} e^{\frac{(m - n)}{β_{ij}}} (1 - e^{- \frac{1}{β_{ij}}}) q_{wi} (m)

(4)

Considering the change in bottom pressure and well interference, the approximate production rate of producer j is as follows:

\overset{\land}{q_{j}} (n) = α_{j} + q_{j} (n_{0}) e^{\frac{- (n - n_{0})}{β_{p}}} + \sum_{i = 1}^{i = I} λ_{ij} \sum_{m = 1}^{m = n} e^{\frac{(m - n)}{β_{ij}}} (1 - e^{- \frac{1}{β_{ij}}}) q_{wi} (m)

(5)

where ${\hat{q}}_{j} (n)$ is the approximate production rate of producer j, m³•d⁻¹; $α_{j}$ is the unbalance constant, m³•d⁻¹; $q_{j} (n_{0})$ is the initial production rate of producer j, m³•d⁻¹; $q_{wi} (m)$ is the injection rate of injector i, m³•d⁻¹; and $β_{p}$ is the weight affecting the initial production rate of producer j and injector i. $λ_{ij}$ is the well distribution coefficient, which indicates the degree of connectivity of the dynamic oil and water. $β_{ij}$ is the hysteresis coefficient, which indicates the dissipation of signals between injector–producer pairs.

Equation (5) includes three parts: the first part, which indicates the injection–production imbalance constant; the second part, which denotes the effect of the initial production rate; the third part, which is the modified value after injected signal pretreatment. The effect of the initial production rate is generally so small that it can be neglected.

When some wells are shut in, according to the latter theoretical analysis of connectivity, the relevant distribution coefficient changes, as shown in Equation (12). If the distribution coefficient of Equation (5) was rectified by Equation (12), we can obtain an IICM that could concern a shut-in event, as follows:

\begin{matrix} \overset{\land}{q_{j}} (n) = α_{j} + q_{j} (n_{0}) e^{\frac{- (n - n_{0})}{β_{p}}} + \sum_{i = 1}^{i = I} \frac{λ_{ij} δ_{j}}{\sum_{k = 1}^{k = J} λ_{ik} δ_{k}} \sum_{m = 1}^{m = n} \\ e^{\frac{(m - n)}{β_{ij}}} (1 - e^{- \frac{1}{β_{ij}}}) q_{wi} (m) \end{matrix}

(6)

where $δ_{j}$ is the Dirichlet function. When producer j is shut in, $δ_{j} = 0$ ; when producer j is normal, $δ_{j} = 1$ .

For the steady-state flow, the transmissivity is $T_{ij} = \frac{{\bar{k}}_{ij} {\bar{A}}_{ij}}{μ L_{ij}}$ , according to Darcy‘s Law:

q_{ij} = T_{ij} (p_{wfi} - p_{wfj}) \approx λ_{ij} q_{wi}

(7)

where $p_{wfi}$ and $p_{wfj}$ denote the bottom pressure of injector i and that of producer j (MPa), respectively; ${\bar{k}}_{ij}$ , ${\bar{A}}_{ij}$ , $L_{ij}$ , $T_{ij}$ , and $Δ {p_{wf}}_{ij}$ indicate the average permeability (μm²), sectional area (m²), well spacing (m), transmissivity (m³•d⁻¹•MPa⁻¹), and differential pressure (MPa), respectively.

View injector i as the center well:

{q_{w}}_{i} = \sum_{j = 1}^{J} λ_{ij} q_{wi} = \sum_{j = 1}^{J} T_{ij} Δ {p_{wf}}_{ij}

(8)

Under the normal production state, if the production rate fluctuation is regarded as the main result of injection rate fluctuation and neglects the interference of producers:

λ_{ij} = \frac{T_{ij}}{\sum_{j = 1}^{N_{j}} T_{ij}} = \frac{{\bar{K}}_{ij} {\bar{A}}_{ij}}{L_{ij}} / \sum_{j = 1}^{N_{j}} \frac{{\bar{K}}_{ij} {\bar{A}}_{ij}}{L_{ij}}

(9)

$λ_{ij}$ is the ratio of the average transmissibility between injection i and producer j to the total transmissivity of well pairs. When some oil wells are shut in at time t, the other producers are denoted as $O_{p}$ , provided that the injection rate remains stable throughout the entire period. As shown in Equation (8), the pressure drop of injector i between the surrounding producers is denoted as $Δ p_{wfij}'$ :

Δ p_{wfij}' = \frac{q_{wi}}{\sum_{j \in O_{p}} T_{ij}}

(10)

According to Equations (8) and (9), when some wells are shut-in, the ratio of the production rate of a normal producer before and after the shut-in procedure is expressed as follows:

\frac{q_{ij}^{'}}{q_{ij}} = \frac{T_{ij} Δ p_{wfij}'}{T_{ij} Δ p_{wfij}} = \frac{{q_{w}}_{i} / \sum_{j \in O_{p}} T_{ij}}{{q_{w}}_{i} / \sum_{j = 1}^{J} T_{ij}} = \frac{\sum_{j = 1}^{J} T_{ij}}{\sum_{j \in O_{p}} T_{ij}} = \frac{1}{\sum_{j \in O_{p}} λ_{ij}}

(11)

The distribution coefficient from the injector to the normal producer is given by the following:

λ_{ij}' = \frac{q_{ij}^{'}}{q_{wi}} = \frac{1}{\sum_{j \in O_{p}} λ_{ij}} \frac{q_{ij}}{q_{wi}} = \frac{λ_{ij}}{\sum_{j \in O_{p}} λ_{ij}}

(12)

and the distribution coefficient between the shut-in producers to the surrounding injector is 0. To indicate the problem, the Dirichlet function is introduced as follows:

δ_{j} = {\begin{matrix} 1 \begin{matrix} j \in I_{p} \end{matrix} \\ 0 \begin{matrix} j \notin I_{p} \end{matrix} \end{matrix}

(13)

Thus, the distribution coefficient between injector i and producer j at time t can be presented as follows:

λ_{ij}' = \frac{λ_{ij} δ_{j}}{\sum_{k = 1}^{N_{j}} λ_{ik} δ_{k}}

(14)

2.2. Hysteresis coefficient analysis

The hysteresis coefficient has been introduced in numerous studies, but the geological data they contain has not been analyzed.^10–19 The difference in the hysteresis coefficient qualitatively reflects the heterogeneity of the reservoir. Implemented by factor analysis, the relationship between the hysteresis coefficient and the geological parameter is quantified.

A synthetic homogeneous two-dimensional (2D) model is constructed for the influence factor analysis, as shown in Figure 2. The grid of the 2D model is 7×16, and DX = DY = 20 m, DZ = 10 m. The porosity is 0.2, and the oil viscosity is 5 mPa•s. The model has one injector and one producer. The injector operates at a fixed rate of 25 m³/d, and one producer works in steady BHP. Total compressibility, permeability, height of formation, viscosity, and distance of well pairs are among the geological parameters. The response production rate under different sets of geological parameters is analyzed and history-matched based on the IICM. The relationship curves of total compressibility and permeability to the response production data are depicted in Figure 3. Combined with orthogonal experimental design and multivariate statistical analysis, the relationship between the hysteresis coefficient and geologic parameters is quantified.

Figure 2.

The two-dimensional model.

Figure 3.

Relationship between response production rate and geological characteristic.

On the basis of single-factor analysis, combined with orthogonal design and multivariate statistical methods, the relationship between the hysteresis coefficient and geological parameters can be quantified as follows:

β_{ij} = 0.021 (\frac{ϕ L_{ij}^{2}}{η})^{0.92}

(15)

where $L_{ij}$ is the well spacing (m); $ϕ$ is the porosity; $η$ is the pressure transmitting conductivity (cm²•s⁻¹); and $η = k_{ij} / μ C_{t}$ . As shown in Equation (15), $β_{ij}$ is increased with increasing well spacing and porosity and decreasing pressure transmitting conductivity. On the basis of the function and early geological knowledge, we can provide a preferable initial pressure transmitting conductivity estimation for the IICM.

3. Inverse method of the improved interwell connectivity model

3.1. Basic theory

The inverse problem refers to the determination of the parameters of current interwell connectivity models typically solved using an optimization algorithm.^12–14 In MLR and the CM, the transmissibility of a production–injection pair is independent of the other producers. Therefore, we can easily obtain the solution for a single well in the field individually.

However, this method has a disadvantage: the inversion result may not be realistic. The sum of transmissivity of the individual injector may be greater than 1, given that the model parameters cannot be completely constrained. Moreover, the model cannot be applied in fields in which wells are shut in frequently or over a long period of time. Only the injection and production data captured during the steady-production period can be used in the inverse problem. The data limitation renders the inversion result unreliable.

In the IICM, the distribution coefficient reflects the connectivity information of all producers in the reservoir. To diminish the ill-posed nature of the solution, the production data of all producers are history-matched using a constraint optimization algorithm.³⁰ In addition, we can use the production data during the entire period of reservoir development as the IICM is capable of considering the shut-in procedure. On the basis of a Bayesian framework,^27–29 we establish the objective function of the inverse problem as follows:

\begin{matrix} O (s) = \frac{1}{2} (s - s_{pr})^{T} C_{S}^{- 1} (s - s_{pr}) + \frac{1}{2} (q_{obs} - g (s))^{T} \\ C_{Q}^{- 1} (q_{obs} - g (s)) \end{matrix}

(16)

where s is an $N_{s}$ -dimensional vector consisting of the distribution coefficient, hysteresis coefficient, unbalanced initial constant, and so on; $C_{S}$ is the covariance of s; $s_{pr}$ is the transcendental estimation of s; $q_{obs}$ is an $N_{q}$ -dimensional vector of observed data whose element is the actual production rate; $C_{Q}$ is the covariance of the measurement error; $g (s)$ is an $N_{q}$ -dimensional vector of calculating data whose element is the production rate calculated by the IICM.

The practical solution usually neglects the first part of Equation (16)²⁸ because of the difficulty of obtaining $C_{s}^{- 1}$ and the strong uncertainty of s. For the purpose of physical meaning, the boundary constraint and the equilibrium constraint are added to the equation set. As stated in Equations (18) and (19), the sum of the distribution coefficient is 1, and the distribution and hysteresis coefficients are greater than 0. Thus, the inverse problem equation set is expressed as follows:

\min O (s) = \frac{1}{2} (q_{obs} - g (s))^{T} C_{Q}^{- 1} (q_{obs} - g (s))

(17)

subject to:

\sum_{j = 1}^{N_{m}} d_{ij} s_{j} - 1 = 0 \begin{matrix} i = 1, 2, \dots, I \end{matrix}

(18)

s_{j} \geq 0 j = 1, 2, \dots, J

(19)

Calculating the gradient of the target function $\nabla O (s)$ is the key to the optimization problem. $\nabla O (s)$ can be expressed as follows:

\nabla O (s) = G^{T} C_{Q}^{- 1} (g (s) - q_{obs})

(20)

where G is the sensitivity coefficient matrix of $g (s)$ to s, and the element at the ath row and bth column $G_{a, b}$ is the partial derivative of the ath calculated production data $g_{a}$ to $m_{b}$ . If the production rate of producer j at time n is denoted as $q_{j} (n)$ , and the distribution and hysteresis coefficients between injector i and producer j are denoted as $λ_{ij}$ and $β_{ij}$ , respectively, $G_{a, b}$ and $G_{a, b'}$ can be calculated as follows:

\begin{matrix} G_{a, b} = \frac{\partial g_{a}}{\partial s_{b}} = \frac{\partial q_{j} (n)}{\partial λ_{ij}} \\ = \frac{δ_{j}}{\sum_{k = 1}^{J} δ_{k} λ_{ik}} (1 - \frac{λ_{ij} δ_{j}}{\sum_{k = 1}^{J} δ_{k} λ_{ik}}) \sum_{m = 1}^{m = n} e^{\frac{(m - n)}{β_{ij}}} (1 - e^{- \frac{1}{β_{ij}}}) q_{wi} (m) \end{matrix}

(21)

\begin{array}{l} G_{a, b'} = \frac{\partial g_{a}}{\partial s_{b'}} = \frac{\partial q_{j} (n)}{\partial β_{i j}} \\ = \frac{δ_{j} λ_{i j}}{\sum_{k = 1}^{J} δ_{k} λ_{i k}} \sum_{m = 1}^{m = n} e^{\frac{(m - n)}{β_{i j}}} \frac{q_{w i} (m)}{{β_{i j}}^{2}} [(m - n - 1) e^{- \frac{1}{β_{i j}}} - m + n] \end{array}

(22)

Combined with the gradient projection³⁰ and after the gradient $\nabla O (s)$ is obtained, the target function $O (s)$ with constraints can be solved, and the iterative formula is shown as follows:

s^{r + 1} = s^{r} - γ (I - D {(D^{T} D)}^{- 1} D^{T}) \nabla O (s^{r})

(23)

where r is the number of iteration steps; I is the unit matrix; D is the coefficient matrix of the constraints whose element $d_{ij}$ can be seen in Equation (19); and $γ$ is the search step.

3.2. Calculation steps

Assign the actual production data $q_{obs}$ , covariance $C_{Q}$ , and initial model parameters $s^{0}$ , and set $r = 0$ .

Calculate the production rate by the IICM, obtain relevant $g (s^{r})$ , and compute for the target function $O (s^{r})$ .

On the basis of Equations (21) and (22), calculate the sensitivity coefficient matrix $G^{r}$ , and on account of Equation (20), obtain the gradient of target function $\nabla O (s^{r})$ .

Substitute $\nabla O (s^{r})$ into Equation (23) and solve the constraints by gradient projection.

If $O (s^{r + 1}) < O (s^{r})$ , the optimal solution is obtained, and go to Step (6); otherwise, half the iteration step $γ$ , and go back to Step (4).

When it satisfies the following conditions of convergence, the optimization process is terminated, and the optimal solution of inversion is obtained:

| O (s^{r + 1}) - O (s^{r}) | / O (s^{r}) \leq 0.0001

(24)

Otherwise, let $r = r + 1$ , and go back to Step (2) and repeat the iteration.

4. Model application

4.1. Synthetic model application

We examine the IICM by applying it in a high-permeability channel synthetic reservoir, as depicted in Figure 4, by numerical simulation. The field uses five-spot network waterflooding, and the injection rates are fixed. The four producers operate at a fixed bottom-hole pressure in the different corners of the square field. Some of the producers are shut in for a long time while all the injectors remain working. The WLPR (well liquid production rate) data history-matched by the IICM and CM is presented in Figure 5. The inverse distribution coefficients of the IICM and CM are shown in Table 1 and are converted to the arrow size depicted in Figure 6.

Figure 4.

Permeability distribution of the channel reservoir.

Figure 5.

WLPR matched data of P1. IICM: improved interwell connectivity model; CM: capacitance model; WLPR: well liquid production rate.

Table 1.

Inverse distribution coefficient of the improved interwell connectivity model (IICM) and capacitance model (CM).

Distribution coefficient	IICM					CM
Distribution coefficient	I1	I2	I3	I4	I5	I1	I2	I3	I4	I5
P1	0.264	0.68	0.323	0.222	0.153	0.375	0.448	0.293	0.375	0.345
P2	0.484	0.006	0.045	0.153	0.051	0.224	0.171	0.156	0.182	0.122
P3	0.018	0.11	0.038	0.003	0.157	0.076	0.206	0.123	0.034	0.241
P4	0.234	0.204	0.594	0.622	0.639	0.449	0.364	0.337	0.574	0.353
∑	1	1	1	1	1	1.124	1.189	0.909	1.165	1.061

Figure 6.

Inverse distribution coefficient of the improved interwell connectivity model (IICM) and capacitance model (CM).

According to the estimated parameters and matching data shown in the figures and table, the CM is sensitive to the shut-in procedure, and the estimated distribution coefficients conflict with the known geological characteristics. The IICM can handle the rate of change of producer caused by the shut-in, and the distribution coefficients are coincidental with the high-permeability channel. Moreover, the sum of the distribution coefficient of a single well is 1, which demonstrates that the parameter estimation method implemented in the IICM is reliable.

4.2. Real reservoir application

We implemented the methodology in Z16 reservoir blocks. The block has an oil-bearing area of 2.0 km² and has thin and low permeability layers. The average permeability and oil viscosity of the layers are 7.9 mD and 27.55 mPa•s, respectively. After long-term waterflooding, the pressure distribution is uneven, and water influx to the channel commonly occurs in the field.

To obtain the well connectivity under the current well pattern, we history-matched the injection and production data for the last six years. Some of the matched data are shown in Figure 7. The fitting correlation coefficient of the production data exceeds 85%. According to the well connectivity obtained by the IICM, shown in Figure 8, all injectors have a water channel formed in the well groups, except Z16X-4-3, which leads to low oil recovery. On the basis of the knowledge of well connectivity, the field can perform water shutoff procedures for producers and optimize the well pattern to improve waterflooding management.

Figure 7.

Part of the WLPR matched data. IICM: improved interwell connectivity mode; WLPR: well liquid production rate.

Figure 8.

Well connectivity of the Z16 reservoir block.

5. Conclusions

The IICM can consider the shut-in event in the reservoir, and the quantitative model parameters have explicit geological significance.

Combined with the projection gradient method, the methodology implemented can confine whole model parameters and ensure that the estimated parameters have definite physical meaning and are reliable.

Compared with the CM, the IICM can more precisely estimate the model parameter and more closely match the coefficient. The inverse well connectivity reveals the oil–water dynamics and can help adjust the waterflooding management to enhance oil recovery.

Footnotes

Funding

This work was supported by the PetroChina Innovation Foundation (No. 2016D-5007-0207), the Project of the Open Fund of the State Key Laboratory of Offshore Oil Exploitation (Grant No. CCL2015RCPS0223RNN), the National Natural Science Foundation of China (Grant No. 51674039 and No. 51604035), the China Important National Science & Technology Specific Projects (Grant No. 2016ZX05050-009), and the China Important National Science & Technology Specific Projects (Grant No. 2016ZX05014).

Author biographies

Shun Liu is an associate professor in the college of petroleum engineering at Xi’an Shiyou University, China and was a visiting scholar at the Colorado School of Mines. His main research interests are hydraulic fracturing and fracturing crack simulation. Liu holds a bachelor’s degree and a master’s degree in petroleum engineering from Xi’an Shiyou University. He holds PhD degree in oil and gas field development engineering from China University of Petroleum, East China.

Lin Cao is a lecturer in school of petroleum engineering at Yangtze University, China. Her main research interests are reservoir interwell connectivity modeling and numerical simulation. Cao holds a bachelor’s degree in petroleum engineering and a master’s degree in oil and gas well engineering from China University of Petroleum, East China.

Heng He is an engineer, working at Research Institute of Oil and Gas Technology, Changqing Oilfield Company, China. He is mainly engaged in the research and application of fracturing acidification technology. He holds a master’s degree in oil and gas field development engineering from Xi’an Shiyou University, China.

Tao Yang is a lecturer in school of safety engineering at North China Institute of Science & Technology, China. His main research interest is coal mine gas dynamic disaster prevention. Yang holds a PhD degree in safety technology and engineering from China University of Mining & Technology, Beijing.

De-sheng Zhou is a professor in the college of petroleum engineering at Xi’an Shiyou University, China. His main research interests are hydraulic fracturing technology and drainage and gas production technology. Zhou holds a PhD degree in petroleum engineering from Louisiana State University.

References

Sui

Wang

, et al. Advance on the tracer test technology among wells. SPE Asia Pacific Oil and Gas Conference and Exhibition, Melbourne, Australia, 2002; SPE 77874, 1–5, Richardson, TX: Society of Petroleum Engineers.

Guan

Interwell tracer tests: lessons learnted from past field studies. In: SPE Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, Indonesia, 2005; SPE 93140, 1–9. Richardson, TX: Society of Petroleum Engineers.

Whittle

T M

Lee

Gringarten

AC.

Will wireline formation tests replace well tests? In: SPE Annual Technical Conference and Exhibition, Denver, CO, 2003, pp. 1–12. Richardson, TX: Society of Petroleum Engineers.

Bybee

Well test analysis in gas/condensate reservoirs. Journal of Petroleum Technology 2000; 52: 68, 70.

Rwechungura

Dadashpour

Kleppe

Advanced history matching techniques reviewed. In: SPE Middle East Oil and Gas Show and Conference, Manama, 2011, SPE 142497, pp. 1–19. Richardson, TX: Society of Petroleum Engineers.

Emerick

Technology focus: history matching and forecasting. Journal of Petroleum Technology 2015; 67: 132.

Zhao

Yao

, et al. Reservoir interwell dynamic connectivity inversion based on injection and production data. J Chin Univ Petrol 2010; 34: 91–98.

Gentil

The use of multilinear regression models in patterned waterfloods: physical meaning of the regression coefficients. Dissertation, The University of Texas, Australian, 2005.

Anh

Djebbar

Inferring Interwell Connectivity from Well Bottom Hole Pressure Fluctuations in Waterfloods. In: Production and Operations Symposium, Oklahoma City, OK, 2007; SPE106881, pp. 1–11. Richardson, TX: Society of Petroleum Engineers.

10.

Yousef

Gentil

Jensen

, et al. A capacitance model to infer interwell connectivity from production and injection rate fluctuations. In: SPE Annual Technical Conference and Exhibition, Dallas, TX, 2005; SPE95322, pp. 1–19. Richardson, TX: Society of Petroleum Engineers.

11.

Yousef

Jensen

Lake

LW.

Integrated interpretation of interwell connectivity using injection and production fluctuations. Math Geosci 2009; 41: 81–102.

12.

Yousef

Lake

Jensen

. Analysis and interpretation of interwell connectivity from production and injection rate fluctuations using a capacitance model. In: SPE/DOE Symposium on Improved Oil Recovery, Tulsa, OK, 2006; SPE 99998, pp. 1–15, Richardson, TX: Society of Petroleum Engineers.

13.

Weber

Edgar

Lake

, et al. Improvements in capacitance-resistive modeling and optimization of large scale reservoirs. In: SPE Western Regional Meeting, San Jose, CA, 2009; SPE 121299, pp. 1–17, Richardson, TX: Society of Petroleum Engineers.

14.

Sayarpour

Kabir

Sepehrnoori

, et al. Probabilistic history matching with the capacitance-resistance model in waterfloods: A precursor to numerical modeling. Journal of Petroleum Science and Engineering 2011; 78: 96–108.

15.

Nguyen

Kim

Lake

, et al. Integrated capacitance resistive model for reservoir characterization in primary and secondary recovery. In: SPE Annual Technical Conference and Exhibition, Denver, CO, 2011; SPE 147344, pp. 1–20, Richardson, TX: Society of Petroleum Engineers.

16.

Salazar-Bustamante

Gonzalez-Gomez

Matringe

S F

, et al. Combining decline-curve analysis and capacitance/resistance models to understand and predict the behavior of a mature naturally fractured carbonate reservoir under gas injection. In: SPE Latin America and Caribbean Petroleum Engineering Conference, Mexico City, Mexico, 2012; SPE 153252, pp. 1–17, Richardson, TX: Society of Petroleum Engineers.

17.

Zhao

Gao

, et al. Study on reservoir interwell dynamic connectivity based on systemic analysis method. Acta Petrolei Sinica 2010; 31: 633–636.

18.

Lee

Yao

Okpani

, et al. Identifying injector-producer relationship in waterflood using hybrid constrained nonlinear optimization. In: SPE Western Regional Meeting, Anaheim, CA, 2010; SPE, 132359, pp. 1–11, Richardson, TX: Society of Petroleum Engineers.

19.

Zhang

Research on reservoir interwell dynamic connectivity inversion method. Petrol Geol Recover Efficiency 2011; 18: 70–73.

20.

Guo

Reynolds

Zhao

A physics-based data-driven model for history matching, prediction, and characterization of waterflooding performance. SPE J 2018; 23: 367–395.

21.

Guo

Reynolds

Zhao

Waterflooding optimization with the INSIM-FT data-driven model. ComputGeosci 2018; 22: 745–761.

22.

Zhao

Cui

, et al. History matching and production optimization of water flooding based on a data-driven interwell numerical simulation model. J Nat Gas SciEng 2016; 31: 48–66.

23.

Liu

Zhao

, et al. Staggered extension laws of hydraulic fracture and natural fracture. Acta Petrolei Sinica 2018; 39: 320–326, 334.

24.

Zhao

Cui

, et al. History matching and production optimization of water flooding based on a data-driven interwell numerical simulation model. J Nat Gas SciEng 2016; 31: 48–66.

25.

You

Wang

Zhang

, et al. Experimental study on spontaneous imbibition of recycled fracturing flow-back fluid to enhance oil recovery in low permeability sandstone reservoirs. J Petrol Sci Eng 2018; 166: 375–380.

26.

Liu

Wang

, et al. Simple model for natural fracture propagation through hydraulic fracturing. Fault Block Oil Gas Field 2016; 23: 488–491, 513.

27.

Deans

On conditional simulation to inaccurate data. Math Geol 1996; 28: 811–817.

28.

Oliver

Reynolds

Liu

Inverse theory for petroleum reservoir characterization and history matching. New York: Cambridge University Press, 2008.

29.

Zhao

Yao

, et al. Theoretical research on reservoir closed-loop production management. Sci China Tech Sci 2011; 54: 2815–2824.

30.

Yuan

Sun

Optimization theory and methods. Beijing: Science and Technology Press, 1999.