Water Invasion Prediction Method for Edge–Bottom Water Reservoirs: A Case Study in an Oilfield in Xinjiang, China

Abstract

Clarifying the water invasion rule of edge and bottom water reservoirs can adjust the reservoir development mode and improve the recovery factor of edge and bottom water reservoirs in a timely manner. Influenced by the size of a reservoir water body, energy intensity and reservoir seepage capacity, the change model of reservoir water influx basically belongs to the exponential growth model of the GM (1,1) model or the self-constraint growth model of the logistic model. The above two models are used to predict and analyze the water inflow of edge and bottom water reservoirs, respectively, and it is found that the change in water inflow of the reservoir with sufficient edge and bottom water energy is more consistent with the prediction results of the GM (1,1) model, but it has a large error compared to the prediction results of the logistic model. The change in water influx in the reservoir with insufficient edge and bottom water energy is consistent with the prediction results of the logistic model and GM (1,1) model. The research shows that the strength of edge and bottom water energy of the reservoir can be determined by analyzing the error of the logistic model in predicting water influx. If we focus on the change in reservoir water influx, the improved GM (1,1) model formed by a Newton parabola interpolation polynomial is used to optimize its background value, which can further improve the prediction accuracy and reduce the prediction error of water inflow of edge and bottom water reservoirs. The method in this paper has certain reference significance for studying the water invasion rule and energy intensity of edge and bottom water reservoirs.

1. Introduction

During the development of edge–bottom water reservoirs, with the production of crude oil, the formation pressure inside the reservoir drops, and the pressure drop will gradually propagate to the external natural water surface in an elastic manner, causing the formation water in the natural water body and storage Elastic Expansion of Cluster Rocks [1,2,3]. Under the pressure difference between the natural water body and the reservoir formation, the water of the natural water body invades the oil reservoir from the edge or bottom, thereby making up or slowing down the drop in formation pressure inside the oil reservoir [4]. Near the bottom of the well, the surface gas and oil will be changed, affecting the production efficiency [5,6]. In edge–bottom water reservoirs, if the water influx continues to increase, water influx will continue to destroy the continuity of oil flow and form oil droplets of different sizes and shapes, changing the single-phase flow of crude oil to a two-phase flow of oil and water, further increasing the oil flow [7,8,9,10] and causing seepage resistance of the flow. On the other hand, with the increase in water influx, water will become the continuous phase flowing in the formation, and the surface of the sand grains will become the shear surface of the flow. Therefore, if the flow rate of formation water increases, it will wash away the original fixed phase. Loose particles on the surface of sand grains reduce the cementation strength of the formation and easily lead to sand production in oil wells [11,12]. In addition, loose particles will accumulate in appropriate places, leading to the blockage of flow pores and reducing the permeability of the oil layer [13,14]. Therefore, reducing the water invasion rate can achieve the purpose of reservoir protection to a certain extent. In the area where bottom water is developed, due to the influence of bottom water coning, the water-flooding efficiency is low, and the remaining oil is difficult to recover [15,16,17,18,19]. Mastering the variation law of water influx in edge and bottom water reservoirs and predicting future water influx play a crucial role in rationally developing edge and bottom water reservoirs. Well logging data can not completely predict the volume of a water body [20]; therefore, a new method is needed to predict the size of natural energy water in the reservoir. The reliability of prediction results depends not only on the accuracy of information and data, but also on the accuracy of the selected prediction method and the established mathematical model. Among these factors, the most important factor is the established mathematical model [21]. Without proper prediction methods, it is impossible to have good prediction results.

At present, the material balance method, stable-state model, quasi-stable-state model and modified steady-state model [22,23,24,25] are often used to calculate the water influx of oil and gas fields. The above model methods require many parameters, and the process of calculating the water influx is quite complex [26,27]. It is found that the logistic model [28,29,30] and GM (1,1) model [31,32,33] focus on the prediction of oil and gas field production, and the analysis of the water influx rate needs to be deepened. The GM (1,1) forecasting model has the advantages of less sample information and high fitting accuracy. It has unique advantages for dealing with complex internal systems and is suitable for short-term forecasting and exponential growth forecasting [34]. The model fitting generally refers to the number of recent periods in history, so the forecast value tends to increase or decrease too fast when the accumulated data fluctuate greatly, and the forecast error increases continuously over time. Because the logistic model is a global fitting, the accumulated data will have a large error in the area where there is a small local fluctuation. Furthermore, the logistic model is an equation that describes resource constraints [35,36]. The regularity of description reflects the process of things happening, developing, maturing and reaching a certain limit. Therefore, if the water body of the edge and bottom water reservoir is huge and the energy is sufficient, the logistic model cannot predict the water influx well, while the GM (1,1) model can effectively predict the water influx. If the reservoir water body is small, the change in inflow is limited by the size of the water body, and the logistic model can well predict the size of a water body. In the process of water invasion of edge and bottom water reservoirs, sometimes the water body size of the reservoir is unknown, which makes the selection of a water invasion prediction model very important. Based on the size and energy of the reservoir water body and the change law of water inflow during the reservoir development process, a method for judging the energy of the edge and bottom water invasion is proposed by comprehensively considering the principles of the GM (1,1) model and logistic model. That is, by comparing the error of the results of the two prediction models, the strength of the edge and bottom water energy of the reservoir can be determined. If the error of the logistic model in predicting water influx is large, the prediction accuracy of the GM (1,1) model is high, and the error is small, it is inclined to judge that the water invasion energy of the edge and bottom water reservoir is strong. If the accuracy of the logistic model in predicting water influx is high, it tends to judge that the water influx energy of edge and bottom water reservoirs is weak. Thus, this method can not only improve the prediction accuracy of water influx, but also judge the strength of water influx energy, which can provide a powerful reference for the reasonable development of edge and bottom water reservoirs.

2. Model Building

2.1. Logistic Model

The traditional differential equation of the logistic model is [37,38]

$$\frac{dy}{ydt}=a\left(1-\frac{y}{b}\right)$$

(1)

Separate variables and integrate Equation (1):

$$\ln \frac{y}{y-b}-\ln \frac{y_0}{b-y_0}=at$$

(2)

After finishing Equation (2), we obtain the following:

$$y=\frac{b}{1+\frac{b-y_0}{y_0}{e}^{-at}}$$

(3)

Then, make

$$c=\frac{b-y_0}{y_0}$$

(4)

In addition,

$$y=\frac{b}{1+c{e}^{-at}}$$

(5)

Equation (5) is the original formula of the logistic model; as can be seen, when $t\to \infty$, y is approaching its extremum b. Since the time-dependent change in water influx in edge-bottom water reservoirs belongs to the type of growth curve, in order to realize the cumulative water influx prediction for edge-bottom water reservoirs, make y be $W_e$ and b be $W_{er}$, and Equation (5) can be rewritten as follows:

$$W_e=\frac{W_{er}}{1+c{e}^{-at}}$$

(6)

In the formula, $W_e$ is the water invasion in edge-bottom water reservoirs, 10⁴ m³, and $W_{er}$ is the final water invasion in edge-bottom water reservoirs, 10⁴ m³; and t—development period. When $t\to \infty$, $W_e$ is approaching its extremum $W_{er}$.

By derivation of Equation (6) with respect to time, the relational expression for predicting the water invasion velocity of edge-bottom water reservoirs can be obtained:

$$q_e=\frac{d{W}_e}{dt}=\frac{ac{W}_{er}{e}^{-at}}{{(1+c{e}^{-at})}^2}$$

(7)

In the formula, $q_e$ is the annual water invasion rate of edge-bottom water reservoirs, $m^3/a$.

In Equations (6) and (7), there are three unknown parameters, a, c and W_er, according to the actual water influx; only after these three constants are determined can the water influx prediction of the edge-bottom water reservoir be carried out.

$$\frac{q_e}{W_e}=\frac{ac{e}^{-at}}{1+c{e}^{-at}}$$

(8)

Equation (5) can be rewritten as

$$1+c{e}^{-at}=\frac{W_{er}}{W_e}$$

(9)

Then, Equation (8) is

$$\frac{q_e}{W_e}=\frac{ac{W}_e{e}^{-at}}{W_{er}}$$

(10)

Add and subtract aW_e to the numerator on the right side of Equation (10) at the same time, obtaining

$$\frac{q_e}{W_e}=\frac{ac{W}_e{e}^{-at}+a{W}_e-a{W}_e}{W_{er}}=\frac{a{W}_e(1+c{e}^{-at})-a{W}_e}{W_{er}}=a-b{W}_e$$

(11)

where

$$b=\frac{a}{W_{er}}$$

(12)

The water invasion amount and the water invasion rate can be regressed into a straight line according to the Equation (11), to obtain the intercept a and the slope b, and then obtain W_er. Rewrite Equation (6) as follow:

$$\frac{W_{er}-W_e}{W_e}=c{e}^{-at}$$

(13)

Taking the logarithm of both sides of Equation (11), obtain the following:

$$\lg \frac{W_{er}-W_e}{W_e}=A-Bt$$

(14)

where $A=\lg c$ and $B=\frac{a}{2.303}$. It can be seen from Equation (12) that this is a straight-line relationship in the semi-logarithmic coordinate system. The intercept A and the slope B are obtained by knowing W_er and W_e, and determine the value of c.

$$c={10}^A$$

(15)

According to the obtained three parameters, a, c, W_er, establish a prediction formula for water invasion or water invasion rate.

2.2. GM (1,1) Model

The main steps of the GM (1,1) grey model are as follows: first-order accumulation of non-negative raw time series; perform quasi-smoothness and quasi-exponential tests; determine data matrix and parameter columns; and build cumulative data series models and raw time series forecasting models [39,40].

The grey prediction model is mainly composed of a grey differential equation and a whitening differential equation, where the grey differential equation is

$$X_{}^{(0)}(k)+a{Z}_{}^{(1)}(k)=u$$

(16)

The corresponding whitening differential equation is

$$\frac{d{X}_{}^{(1)}(t)}{dt}+a{X}_{}^{(1)}(t)=u$$

(17)

Using the least squares method to estimate the parameters of grey differential equations, to find the development coefficient a and the grey action u, obtain

$$A=\left[\begin{array}{l}a\\ u\end{array}\right]={(B^{T}B)}^{-1}{B}^{T}Y$$

(18)

In the formula

$$Y={\left[X_{}^{(0)}(1),X_{}^{(0)}(2),X_{}^{(0)}(3)\dots \dots X_{}^{(0)}(n)\right]}^T$$

(19)

$$B=\left[\begin{array}{l}-Z^{(1)}(2) 1\\ -Z^{(1)}(3) 1\\ .\\ .\\ -Z^{(1)}(n) 1\end{array}\right]$$

(20)

Solve the Equation (18) to obtain

$${\hat{X}}^{(0)}(k+1)=\left[X^{(0)}(1)-\frac{u}{a}\right]e^{-at}+\frac{u}{a} (k=0,1,2,\dots ,n)$$

(21)

Equation (21) is the original prediction model of GM (1,1). From the above steps, we can see that the GM (1,1) grey prediction model is a special modeling method. Based on the gray exponential law generated by the cumulative generation operation, it has a good fitting prediction for the data series with homogeneous exponential characteristics [41]. Therefore, according to the model, we can predict and analyze the cumulative water influx.

2.3. Improved GM (1,1) Model

The traditional GM (1,1) model is used as the core and basis of grey prediction. It has been widely used in various fields. However, its background value coefficient is usually obtained by using the adjacent mean [42]. That is, to replace the integral value ${\displaystyle {\int }_k^{k+1}{X}^{(1)}}(t)dt$ by the area of the trapezoid, the coefficient value is fixed (a = 0.5), and it has low flexibility and poor anti-interference ability. In this regard, this paper optimizes the background value coefficient by using the Newton parabolic interpolation method [43,44,45]. To overcome the problem that the background value of the traditional GM (1,1) model gray prediction model adopts equal weights, the recursive gray prediction equation is established instead of the traditional whitening equation and the improved background values are substituted into the equation to obtain the improved GM (1,1) model.

Specific steps are as follows [46]:

Step 1. According to the definition of Lagrangian interpolation, if the number of data points increases, the interpolation basis function needs to be recalculated; therefore, the Lagrangian interpolation has large defects, so the Newton interpolation formula is used to introduce a different quotient for optimization.

The original formula for Lagrangian interpolation is

$$P_1(x)=f_0+\frac{f_1-f_0}{x_1-x_0}(x-x_0)$$

(22)

It is extended to the case with n + 1 interpolation points (x₀, f₀), (x₁, f₁), ……, (x_n, f_n). Then, the interpolation polynomial can be expressed as follows:

$$P_n(x)=a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\dots \dots +a_n(x-x_0)\dots \dots (x-x_{n-1})$$

(23)

of which a₀, a₁……, a_n is the undetermined coefficient, which can be determined by interpolation conditions $P_n(x_j)=f_j(j=0,1,\dots \dots ,n)$. When x = x₀, Pn(x₀) = a₀ = f₀, and when x = x₁, Pn(x₁) = a₀ + a₁(x − x₀) = f₁, and the following is obtained:

$$a_1=\frac{f_1-f_0}{x_1-x_0}$$

(24)

Similarly, when x = x₂, obtain

$$a_1=\frac{\frac{f_2-f_0}{x_2-x_0}-\frac{f_1-f_0}{x_1-x_0}}{x_2-x_1}$$

(25)

and so on; a₃, a₄……, a_n can obtained, in order to write a general expression for the coefficient a_k. Suppose $f\left[x_0,x_k\right]=\frac{f(x_k)-f(x_0)}{x_k-x_0}$ is the first-order different quotient of the function f(x) with respect to the points x₀, x_k;

$$f\left[x_0,x_1,x_k\right]=\frac{f\left[x_0,x_k\right]-f\left[x_0,x_1\right]}{x_k-x_1}$$

(26)

is the second-order different quotient of the function f(x) about the points x₀, x₁, x_k;

$$f\left[x_0,x_1,\dots ,x_k\right]=\frac{f\left[x_0,x_1,\dots ,x_{k-2},x_k\right]-f\left[x_0,x_1,\dots ,x_{k-1}\right]}{x_k-x_{k-1}}$$

(27)

is the k-order different quotient of the function f(x) with respect to the point.

Step 2. Defined according to the difference quotient, consider x as a point above [a, b], and we can obtain the following:

$$f(x)=f(x_0)+f\left[x,x_0\right](x-x_0)$$

(28)

$$f\left[x,x_0\right]=f\left[x_0,x_1\right]+f\left[x,x_0,x_1\right](x-x_1)$$

(29)

$$\mathrm{\dots }\mathrm{\dots }$$

$$f\left[x,x_0,\dots ,x_{n-1}\right]=f\left[x_0,x_1,\dots ,x_n\right]+f\left[x,x_0,\dots ,x_n\right](x-x_n)$$

(30)

Taking the latter form into the former form, we obtain

$$\begin{array}{l}f(x)=f(x_0)+f\left[x_0,x_1\right](x-x_0)+f\left[x,x_0,x_2\right](x-x_0)(x-x_1)+\\ \dots f\left[x_0,x_1,\dots ,x_n\right](x-x_0)(x-x_1)\dots (x-x_{n-1})\end{array}$$

(31)

The above Equation (31) is the Newton difference quotient interpolation polynomial, which is simpler to calculate than the Lagrangian polynomial.

Step 3. If the Equation f(x) = 0 has three approximate roots, x_k, x_k−1, x_k−2, construct the quadratic interpolation polynomial P₂(x) with these three points as nodes, and appropriately select a zero-point x_k+1 of P₂(x) as a new approximate root. In this way, the process of determining the iteration is the parabolic method [47].

According to the previously derived Newton interpolation polynomial, the calculation formula of the parabola method is

$$P_2(x)=f(x_k)+f\left[x_k,x_{k-1}\right](x-x_k)+f\left[x_k,x_{k-1},x_{k-2}\right](x-x_k)(x-x_{k-1})$$

(32)

Equation (32) has two zeros:

$${\overline{x}}_k=x_k-\frac{2f(x_k)}{\omega \pm \sqrt{{\omega }^2-4f(x_k)f\left[x_k,x_{k-1},x_{k-2}\right]}}$$

(33)

In addition,

$$\omega =f\left[x_k,x_{k-1}\right]+f\left[x_k,x_{k-1},x_{k-2}\right](x_k-x_{k-1})$$

(34)

In order to determine the value from Equation (36), it is necessary to discuss the choice of the sign before the root sign. Among the three approximate roots of x_k, x_k−1, x_k−2, suppose x_k is closer to the desired root x*. It is possible to choose a value closer to x_k in Equation (30) as the new approximate root ${\overline{x}}_k$ (i.e., the modified background value) to ensure accuracy, for this reason, if the sign before the root equation is the same as the sign of $\omega$. Under certain conditions, for parabolic interpolation, the iterative error has the following asymptotic relationship:

$$\frac{\left|e_{k+1}\right|}{{\left|e_k\right|}^{1.840}}\to {\left|\frac{f^{‴}(x^{\ast })}{6f^{\prime }(x^{\ast })}\right|}^{0.42}$$

(35)

Equation (35) shows that the parabolic method of interpolation is superlinearly convergent, with the order of convergence p = 1.840, and the convergence rate is closer to that of the Newton interpolation method.

Using the above steps, the ${\overline{x}}_k$ value with a small relative error in prediction is obtained and substituted into the traditional gray prediction model to obtain the new background value formula.

$$Z^{(1)}(k)={\overline{x}}_k=x_k-\frac{2f(x_k)}{\omega \pm \sqrt{{\omega }^2-4f(x_k)f\left[x_k,x_{k-1},x_{k-2}\right]}}$$

(36)

3. Case Analysis

In this case study, an oilfield in Xinjiang, northwest China, is taken as the study object. The oil reservoir is a thin layer, has low amplitude and high permeability and has an edge and bottom water reservoir with a basically uniform oil–water interface (as shown in Figure 1). The most important feature is that the thin (single) layer is not filled in the process of reservoir formation due to the influence of a low-amplitude structure, resulting in almost all wells except individual wells forming the distribution characteristics of upper oil and water in the same layer. At present, the formation energy at the edge of this area is relatively sufficient, and the injection–production ratio is basically kept between 0.9 and 1.0. Maintaining a high injection–production ratio has an obvious inhibiting effect on bottom water coning [48,49,50,51]. The formation pressure at the edge is high. Whether we can make full use of natural energy to achieve maximum development benefits requires us to evaluate the size of natural water bodies. In the process of reservoir development, the decline in formation pressure in the reservoir itself will cause natural water invasion in the external water supply area, thus making up or slowing down the decline in formation pressure in the reservoir. Under the condition of a certain layer pressure drop between the water supply area and the reservoir, the magnitude of natural water invasion mainly depends on the geometry and size of the water supply area, the permeability and porosity of the reservoir, the oil–water viscosity ratio and the elastic expansion coefficient of the formation of water and rock. The size of natural energy and whether it can be supplied adequately is the key to determine the development mode. Based on the study of static data and combined with reservoir dynamic data, various mathematical models are adopted to evaluate natural energy.

According to the development dynamics, the reservoir is characterized as a stratified reservoir with active side water outside the reservoir, and bottom water exists inside the reservoir, causing the wells to show more characteristics of water invasion in bottom water reservoirs. According to the actual production data of the oilfield, from 2010 to 2021, the amount of water invasion in this block is of the type that has increased. To show the prediction performance of the developed model and its improved model, the actual water invasion data of the site over the years were combined. Based on the logistic model, the GM (1,1) model and the modified GM (1,1) model developed in the previous paper, the historical data were fitted analytically. The modeling data for this example are shown in

Table 1. Cumulative water invasion data of an oilfield in Xinjiang, China, over the years.

Year	Cumulative Water Invasion (104 m3)	Year	Cumulative Water Invasion (104 m3)
2010	35.13	2016	101.52
2011	39.70	2017	117.06
2012	48.19	2018	141.07
2013	61.46	2019	146.72
2014	70.65	2020	147.76
2015	86.47	2021	165.95

.

With the above actual data, the corresponding prediction models were established, respectively.

Logistic model:

$$W_e=\frac{268}{1+8.64e^{-0.2144t}}$$

(37)

GM (1,1) model:

$$W_e=567.376e^{0.01849t}-532.242$$

(38)

Improved GM (1,1) model:

$$W_e=107.2514e^{0.07513t}-72.117$$

(39)

The predictions were carried out by the three models of Equation (37), Equation (38) and Equation (39), respectively, and the results are shown in

Table 2. Comparison of prediction results of different models.

Years	Logistic Model		GM (1,1)		Improve GM (1,1)
	Predicted (104 m3)	Error (%)	Predicted (104 m3)	Error (%)	Predicted (104 m3)	Error (%)
2010	33.60	4.36	35.13	0.00	35.13	0.00
2011	42.42	6.86	45.44	14.47	43.66	9.98
2012	50.35	4.47	56.05	16.30	52.81	9.58
2013	57.43	6.56	66.96	8.94	62.62	1.88
2014	67.69	4.19	78.17	10.65	73.14	3.52
2015	79.10	8.52	89.71	3.75	84.42	2.37
2016	91.56	9.81	101.58	0.06	96.53	4.92
2017	104.88	10.40	113.79	2.79	109.51	6.45
2018	118.84	15.76	126.35	10.44	123.43	12.51
2019	133.14	9.26	139.26	5.08	138.36	5.70
2020	150.46	1.83	152.55	3.24	154.38	4.48
2021	165.47	0.30	166.21	0.15	171.55	3.37

.

The prediction results show that the error of the logistic model is large, the error of the GM (1,1) model is relatively small and the change in water influx in the reservoir is more consistent with the GM (1,1) model. The improved GM (1.1) model has the least error.

Two typical oil wells are selected in the reservoir. Well A is located in the area of the bottom water reservoir with sufficient energy (as shown below in Figure 2). Well B is located in the reservoir area with insufficient bottom water energy (as shown below in Figure 3). Based on the water invasion data of the two wells, the logistic model and GM model are used for prediction and analysis to evaluate which natural water body energy is more suitable for different prediction methods.

It can be seen from

Table 3. Prediction effect of different methods.

Years	Well A (Sufficient Natural Energy)			Well B (Insufficient Natural Energy)
	Original	GM Model	Logistic Model	Original	GM Model	Logistic Model
2010	913.66	913.66	889.17	773.15	773.15	748.53
2011	1031.12	1107.64	1034.72	965.23	1006.87	942.36
2012	1263.30	1314.62	1200.35	1194.81	1224.24	1154.65
2013	1440.03	1535.47	1387.59	1450.91	1426.39	1377.57
2014	1691.81	1771.13	1597.69	1605.56	1614.40	1601.50
2015	1941.96	2022.59	1831.45	1780.79	1789.25	1816.67
2016	2228.65	2290.90	2089.11	1949.62	1951.87	2014.77
2017	2568.99	2577.20	2370.22	2120.46	2103.10	2190.11
2018	2907.24	2882.69	2673.48	2295.49	2243.76	2339.95
2019	3215.33	3208.65	2996.70	2419.32	2374.57	2464.24
2020	3667.26	3556.47	3336.77	2533.76	2496.23	2564.788
2021	3907.79	3927.60	3689.74	2604.19	2609.38	2644.50

that the GM (1.1) model has a good prediction effect on the water influx of the reservoir regarding whether the energy of the edge and bottom water is sufficient. Compared with the prediction results of water influx in edge and bottom water reservoirs with sufficient energy, the logistic model is more suitable for reservoirs with insufficient edge and bottom water energy.

4. Discussion

Evaluation of water invasion in marginal bottom water reservoirs is a complex task with important reservoir engineering applications. Al-Ghanim [52] used nonparametric optimal transformation models for the prediction of dimensionless water influx and dimensionless pressure drop for finite and infinite edge water drive reservoirs using Graphical Alternating Conditional Expectation (GRACE). However, all the terms involved in the models are used in dimensionless form. Yong et al. [53] identified water invasion in aquifers by establishing four diagnostic curves, which requires consideration of parameters such as well production, the pressure drop and material balance. Yan et al. [54] used the Buckley–Leverett equation to calculate the water saturation front in the bottom water invasion zone, considering the air–water velocity difference, the water invasion propulsion mode and the seepage pattern to establish the corresponding analytical equations. Understanding the reservoir water invasion rule is the premise to ensure the water control effect and must be supported by high-quality dynamic analysis. The main methods include production data analysis, numerical simulation, mathematical model prediction, etc.

Compared with other methods in the literature for predicting water invasion in marginal bottom water reservoirs, the established logistic, GM and improved GM models have the following advantages. The application of these methods to the prediction of water invasion in marginal bottom water reservoirs was not considered by previous authors, and it was found by analysis that all three methods have good results for the prediction of cumulative water invasion with less unknowns in the model, which can be easily obtained by multi-parameter fitting and the linear trial-and-error method. Because of the increasing trend of cumulative water invasion, the prediction of water invasion by these three models does not need to consider too many complex parameters. The predicted data of the three models established were compared with the actual data and the accuracy of the models was evaluated by the absolute value of the relative errors. The simulation results are shown in Figure 4, and the relative error evaluation results are shown in

Table 4. Average relative error of different model predictions.

	Logistic Model	GM (1,1)	Improved GM (1,1)
Average relative error (%)	6.86	6.32	5.40

.

As can be seen from Figure 4 and Table 4, the original data series are more volatile. However, the logistic model, the GM (1,1) model and the improved GM (1,1) model can achieve good predictions of water invasion in the marginal bottom water reservoir. As shown in Table 4, the average relative errors of the prediction results of the three models were 6.86%, 6.32% and 5.4%, respectively. The improved GM (1,1) model has a small prediction error, which is 0.92% lower than that of the conventional GM (1,1) model. In contrast, the logistic model has relatively large simulation error and the worst prediction effect. That is, the improved GM (1.1) model slightly outperforms the conventional GM (1.1) model in data smoothing; its processing of a series of actual data by Newton parabolic interpolation can effectively improve the simulation prediction accuracy and effect of the model to predict the growth direction of the original data more accurately. Therefore, it can be shown that the prediction of water influx of an oilfield in Xinjiang is more in line with the GM model, indicating that the volume of the water body is large.

As can be seen from Figure 5 and

Table 5. Average relative error of different model predictions on different wells.

Years	Well A (Average Relative Error (%))		Well B (Average Relative Error (%))
	GM Model	Logistic Model	GM Model	Logistic Model
2010	0.00	2.68	0.00	3.18
2011	7.42	0.35	4.31	2.37
2012	4.06	4.98	2.46	3.36
2013	6.63	3.64	1.69	5.05
2014	4.69	5.56	0.55	0.25
2015	4.15	5.69	0.47	2.02
2016	2.79	6.26	0.11	3.34
2017	0.32	7.74	0.82	3.28
2018	0.84	8.04	2.25	1.94
2019	0.21	6.80	1.85	1.86
2020	3.02	9.01	1.48	1.22
2021	0.51	5.58	0.20	1.55
Average	2.89	5.53	1.35	2.45

, in the prediction of different water invasion effects, the water invasion volume has been increasing in the area with sufficient natural energy, while the increase in water invasion volume in the area with insufficient natural energy has a certain limitation, and the two wells have an obvious separation trend after 2017. In the prediction result error, the GM model can better predict the water invasion law in the area with sufficient natural energy, with an average error of 2.89%. In areas with insufficient natural energy, both the logistic model and GM model can predict better. The average error is below 2.5%.

The comparative evaluation of the two methods shows that in the evaluation of the natural energy of the marginal and bottom water reservoirs, we can use the two water intrusion prediction models to evaluate whether the natural energy of the marginal and bottom water reservoirs is sufficient. Energy-sufficient bottom water reservoirs are more suitable for GM models. For marginal and bottom water reservoirs with insufficient energy, the logistic model and GM are more suitable. On the whole, the accuracy of the logistic model in predicting water intrusion is an important method to judge whether the natural energy of marginal and bottom water reservoirs is sufficient. This can help us better develop bottom water reservoirs and develop different development policies for different types.

5. Conclusions

(1) In this study, the GM (1,1) model and logistic model are applied to the prediction of water influx in edge and bottom water reservoirs. According to the accuracy of prediction results, it is found that the GM (1,1) model is more suitable for predicting the variation law of water influx in an oilfield in Xinjiang. Further combining the background of Newton parabola interpolation polynomial optimization, we can obtain the improved GM (1.1) model, which is superior to the traditional GM (1.1) model and the logistic model in data processing, has the smallest relative error and higher prediction accuracy, and can better reflect the growth rate of water invasion of edge and bottom water reservoirs.

(2) Combined with the characteristics of edge and bottom water reservoirs, the error size of water invasion volume predicted by the logistic model can be used to determine the strength of water invasion energy of edge and bottom water reservoirs. If the water invasion energy is large, the error of the logistic model in predicting the water invasion amount is large; if the water invasion energy of the reservoir is small, the accuracy of the logistic model is high.

(3) This study also has some limitations. During the development of the reservoir, the oil production rate, injection production ratio and formation pressure of the reservoir or the artificial profile control and water plugging will affect the invasion of edge and bottom water, which will lead to the fluctuation in water invasion data and affect the judgment results of the adaptive model. In view of this limitation, data mining methods can be considered in the future to shield out some interference factors or to reduce the impact of data fluctuations by increasing the length of fitting data. In a word, the change in water influx in edge and bottom water reservoirs can be divided into two types: exponential growth type and resource constrained type. By comparing the prediction results of the GM (1,1) model and logistic model, further improving the method of the model according to the accuracy of the prediction results of the two models can better grasp the rule of water influx in edge and bottom water reservoirs, which has certain practical significance and application value. It plays an important role in the prediction of water invasion in edge and bottom water reservoirs. When signs of water invasion are found, reducing production, controlling water and inhibiting water invasion speed in a timely manner can allow time for further research and control.