Average pore pressure in oil formation is an important parameter to measure energy in the formation and the capacity of injection–production. In past studies, average pore pressure mainly depends on pressure build-up test results, which have a high cost and need a long testing time, so it is not conducive to utilize. In this paper, the vertical deformation of the Earth’s surface was used to calculate changes in reservoir pore pressure. We set up the marker stake for measuring ground displacement and measured the vertical deformation at the in situ position. Furthermore, we provided an improved new convolutional neural network (CNN), which adopted image-to-image mode, removed pooling layers and full connection layers, and used a new loss function considering the boundary influence coefficient matrix. Then, the machine learning method was used to invert surface vertical deformation to change pore pressure in the oil reservoir. The method was tested in the block of the Sazhong X development zone, in the Daqing Oilfield of China. The average pore-pressure change values of 20 grids whose area was 900×900m each were determined by the inversion of the corresponding surface vertical deformation at 27 marker stakes. The pore pressure change accuracy reached 82.34%. This study provides a new method for calculating average pore pressure in terms of an oil block. At the same time, it also provides a new technical method for inversion calculations.

1. Introduction

An oil formation consists of porous media, and formation pore pressure refers to the pressure of the pore fluid. Furthermore, average formation pore pressure in the development block is an important parameter for adjusting injection–production intensity and measuring the injecting–producing capacity [1]. The phenomenon of continuous casing damage in a large area was clarified to have as its main reason the mud shale horizontal section slip induced by interregional pore pressure. This nonoil section, used as the marker bed for the correlation layer, is distributed in the whole Daqing oilfield. Thickness is about 10 m, containing a fragile horizontal fossil layer. The section slip of the standard horizontal fossil layer is caused by uneven formation deformation induced by the interregional pore pressure [24]. It is necessary to control the difference of average interregional formation pressure to prevent the expansion of casing damage area in the Daqing oilfield [5, 6].

The average pore pressure of the formation cannot be directly obtained. Additionally, the formation-pressure gradient near the water-injection and oil-production wells is so large that it cannot be used as the basis and parameter for calculating the average formation pressure. In past studies, average pore pressure mainly depended on pressure build-up test results, which have a high cost and need a long testing time, so it is not conducive to utilize. Pressure build-up test data of a single well can also only reflect the average pore pressure of the formation near the test well, not the formation pore pressure of the far area [7, 8]. Consequently, these problems make it difficult to evaluate the average formation pressure in many oilfields.

At present, in the Sazhong development area of the Daqing oilfield, the vertical deformation of the formation is measured by setting up a displacement-measuring marker stake on the ground and then measuring changes of formation pore pressure. The height of the displacement-measuring marker stake exposed to Earth’s surface is 1.5 m, the depth of inserting into the stratum is 4.0 m, and the center is made of steel, which was designed to shield the deformation of the shallow soil. The height difference between two adjacent marker stakes is measured by a precise level, and the vertical deformation of the stratum at each marker-stake position in the plane is calculated by using four benchmark stakes in the nondevelopment area. The accuracy of surface deformation measurement accuracy is 0.3 mm per kilometer. The marker stake is shown in Figure 1.

There is a correlation between surface deformation and formation pore pressure that is not simple linear one-to-one correspondence. Many studies have confirmed this problem. In 2009, Rutqvist et al. [9] carried out a series of rock-mechanics simulation analyses of a coupling reservoir. They showed that surface displacement was consistent with the volume expansion of the injection area and the adjacent strata caused by the pressure change, which was a result of the change of reservoir pore pressure. Surface displacement depended on the change of reservoir pressure, the volume of the injected fluid, and the elastic property of the reservoir cap. Mathieson et al. [10] pointed out that the four-year gas injection in the Salah carbon dioxide burial project caused a surface uplift of about 2 cm near the injection well. Morris et al. [11] simulated the mechanical response caused by carbon dioxide injection near well KB-502, simulated surface displacement at the millimeter scale, and compared it with surface displacement monitored by an interferometry synthetic aperture radar. In 2015, Zheng [12] carried out a series of theoretical studies on surface deformation caused by changes of reservoir pressure. By using analytical solution derivation, a boundary-element program solution, and FLAC-3D software simulation, a method to calculate surface deformation through changes of formation pore pressure was created. Xue et al. [13] studied an abrupt transition of coupled gas–stress behavior at the dilatancy boundary by the strain-based percolation model. On the basis of orthogonal triaxial-stress experiments with CH4 seepage, a complete stress–strain relationship and the corresponding evolution of volumetric strain and permeability were obtained. Wang et al. [14] investigated the effect of pore-fluid pressure on normal deformation by laboratory experiments. Experiment results indicated that pore-fluid pressure significantly affects the normal deformation of a jointed sample. The relative normal deformation of the host rock during fluid injection had a linear relationship with pore-fluid pressure.

However, past research mainly focused on changes of formation pore pressure to calculate surface deformation. At present, the method of predicting the underground parameters by the surface deformation has been applied [1517]. This is the first time of calculating pore pressure of underground oil reservoirs by surface deformation. Although positive evolution is possible, it does not mean that reverse evolution is easy to perform. Forward calculation is an explicit calculation equation, but inverse calculation can be interpreted as solving linear equations with constant coefficients. Although the equations can be listed, the accuracy of the solution is greatly affected. It can be proved by calculation that for a system of linear equations with constant coefficients and 20 unknowns, if the constant term of the system fluctuates by 0.5%, the accuracy of solving the equation will drop to about 90%. If the constant term of the equations fluctuates to 1%, the accuracy of solving the unknown equation will drop to about 80%. The more the unknowns, the more the accuracy of solving the equation will drop. According to the relationship between formation pore pressure and surface deformation, forward calculation is an explicit equation, while backward calculation is an implicit equation that cannot be directly solved.

In this paper, the convolution method was simplified to a convolution algorithm by using the convolution characteristic of the linear-deformation stage. Artificial intelligence can help us better discover the relationship between two things. Machine learning can play a good role in images, or two-dimensional matrix, and a convolutional neural network (CNN) has the most prominent advantage. An improved new CNN was established to retrieve the pore pressure variation from the surface deformation. Accuracy was verified by using marker-stake monitoring data and the average formation-pressure data of the pressure build-up test interpretation in the Sazhong development zone, Daqing oilfield, China.

2. Inversion Method of Reservoir Average Pore Pressure Changes

The relationship between average pore pressure and surface deformation is the key to the inversion of reservoir pore pressure. The change range of reservoir volume caused by the change of reservoir pore pressure is within the range of linear reservoir elastic deformation. The deformation of the reservoir caused by a change of pore pressure simultaneously causes the deformation of the surrounding strata, and the deformation is transferred to the surface. In this process, the deformation of the formation rock is still in the stage of linear elasticity. According to the superposition principle of stress and strain in the linear elastic stage, final surface deformation is the linear superposition of the change of pore pressure of each underground storage grid on the surface-deformation results generated by each grid. In order to simplify the calculation, the thickness of the oil formation in the vertical multi-layer system is added and assumed to be one oil formation. This assumption has little effect on the effect of oil formation on surface deformation. We assumed that there was no gradient in the plane, and that mechanical and seepage parameters were equal in the plane. In this way, if we knew that a single grid changed the surface-deformation distribution caused by unit pressure, surface deformation caused by any formation pressure can be linearly superposed by formation deformation generated by the unit pressure of a single grid. This method can be expressed by the convolution equation.

The actual measured or calculated formation-pore-pressure and surface-deformation parameters were a plane composed of limited data point planes. We needed to perform plane discretization in advance and discretize the stratum plane into a finite mesh with a length of M and a width of N. The two-dimensional discrete convolution equation is
where ux,y is the vertical deformation of the surface at point x,y (m); kx,y is the convolution kernel (m), that is, a storage grid changes the distribution of surface deformation caused by unit pressure (m/MPa); and px,y is formation-pore-pressure distribution (MPa). Convolution kernel kx,y is the vertical deformation of formation with a single grid changing unit pressure, which can be obtained by numerical simulation. Correspondingly, px,y is formation-pore-pressure distribution. Therefore, it can be seen that the solution of surface deformation by formation pore pressure is forward modeling and explicit calculation, that is, for vertical deformation ux,y at any surface location, there is no unknown number in the solution process.

In order to obtain the pore-pressure distribution of the px,y formation through the vertical displacement of the ux,y surface, it was necessary to solve a huge set of equations in which the number of unknowns was equal to the square of the convolution kernel. However, the position of the convolution core far away from the center was almost zero (the value of the far end of the convolution core caused by the change of unit displacement and unit pressure is almost zero). This means that the inversion of equations was impossible, as some unknown coefficients in some equations are close to 0, which greatly affects solution accuracy [18].

In the calculation of two-dimensional data, machine learning can use convolution to deal with this kind of problem. The surface deformation field is a group of plane data; it is similar to a blurred image. CNN is the most outstanding model in machine learning algorithm, which takes image as input layer. However, the calculation of formation pressure changes by surface deformation inversion is deduced from one field to another, which requires some changes to the existing convolution neural network.

The traditional CNN includes input layer, convolution layer, pooling layer, full connection layer, and output layer. Convolution is carried out as in Equation (1). The number of convolution kernels required after each convolution calculation is the product of the number of images before and after convolution. The convolution kernel size is generally 3×3. After convolution, a bias value is usually added. Pooling is to divide the image into some regions, which are usually 2×2. For example, the maximum pooling is to calculate the maximum value of each pooling area. After pooling, the size of the image will decrease rapidly. Through continuous convolution and pooling of the input image, the input image is abstracted into a number of low pixel two-dimensional parameters and then output the results through several fully connected neural networks. The output results are used for image recognition and other applications. The convolution layer is used to increase the thickness (number of images), the pooling layer is used to highlight features and reduce the intermediate parameters, and the full connection layer is used to convert two-dimensional data into one-dimensional. For this method, the above layers need to be changed. As the relationship between the surface vertical deformation field and the formation pressure change field comes from the deformation of the formation, the process is smooth and continuous, so there is no need for pooling layers to get the characteristics of image collection, nor need too much layer thickness. Since the output layer is also two-dimensional data, it is an image-to-image issue, and there is no need for the full connection layer to transform the dimensions of the data. The convolutional neural network for this word does not include a pooling layer and full connection layer. The traditional CNN model and the improved new model are shown in Figure 2.

CNN model needs millions of samples to train it. As the surface vertical deformation can be calculated by the average formation pressure, Equation (1), it is easy to obtain machine learning samples. Since the convolution kernel kx,y can be obtained by finite element method, when we assume a formation pressure, we get a result of surface vertical deformation by Equation (1), and the corresponding average formation pore pressure and surface vertical deformation is a created training sample. In this way, we can obtain infinite training samples by constantly assuming the formation pressure field and calculate the corresponding surface vertical deformation. By using the samples from this method, the CNN can be trained to meet the needs of practical problems.

In addition, the loss function needs to be adjusted. The surface vertical deformation affected by the change of pore pressure at a certain underground point includes not only the point with the same plane coordinate but also the adjacent area around it. On the contrary, the surface vertical deformation of a certain point on the surface is the result of the joint action of pore pressure changes in a certain range of underground. The influence range of the boundary area of the underground pore pressure is larger than that of the surface vertical deformation observation area, which makes the influence of the boundary pore pressure in the training sample cannot be fully reflected in the surface vertical deformation. In the loss function, we should reduce the influence of the adjacent area, which is not considered by the surface deformation, so that CNN training can focus on the relevant grid. In order to realize this consideration, we should establish a hypothetical surface pressure field to analyze the influence of the surrounding boundary areas.

Assuming two large area pressure fields, the formation pressure of the first pressure field pax,y in the study area is the maximum formation pressure, and the formation pressure outside the study area is the minimum formation pressure. The second pressure field pbx,y is the maximum pressure in all regions. The difference of surface deformation uax,y and ubx,y caused by the two assumed pressure fields is then calculated. The difference between the two surface deformation fields can reflect the influence degree of the unmeasured area outside the boundary. The loss function of the improved new CNN model should focus on the influence of the unmeasured area on the calculation results, so we propose a coefficient matrix considering the boundary influence. In CNN training, the final loss function is the square of the 2 times error of each point multiplied by the coefficient of the corresponding position of the coefficient matrix. The coefficient of each point in the coefficient matrix is the second power of the influence ratio outside the boundary as:
where cx,y is the coefficient matrix. When uax,y was multiplied by 2, it is because the formation pressure difference can be positive or negative. If the numerator in Equation (2) is less than 0 before the square, it means that position is mainly controlled by the deformation out of the grid. The loss function should not consider that grid, and the corresponding position value of the loss coefficient matrix will be 0. The loss function coefficient matrix obtained by this method can fully consider the influence of the formation outside the grid.

3. The Oilfield Test

3.1. Forward Evolutionary Computation

The test of the matching relationship between formation pore pressure change and surface vertical deformation was carried out in the Sazhong development zone in the Daqing oilfield, China. Using COMSOL finite-element software, a two-dimensional radial geomechanical model of 10,000 m radius and 2,000 m depth was established to calculate the surface-deformation function caused by unit grid pressure difference, that is, the convolution kernel for test simples. Reservoir thickness was calculated as the total thickness of the vertical sandstone of the reservoir, which was 300 m. The geometric model of the finite element is shown in Figure 3:

From top to bottom, the geological model is upper stratum (700 m), mud shale horizontal section (0 m, an interface with continuous vertical displacement and discontinuous horizontal displacement), reservoir top layer (50 m), reservoir (300 m), and reservoir bottom layer (3,950 m). Other parameters are shown in Table 1.

As the geological parameters of the study block are relatively stable, the change of the parameters used in Table 1 in the actual block is generally not more than 5%. Through finite-element calculation, the distribution of surface vertical deformation caused by the center unit grid pressure difference is shown in Figure 4:

The landform of the study area is flat without obvious elevation change. There is no underground energy exploitation other than petroleum exploitation in the block. The surface deformation comes from the exploitation of petroleum. The surface-marker-stake monitoring network included 27 survey marker stakes in the development zone and 4 benchmark stakes outside the development zone. The spacing of the measuring-marker-stake points was 700–1000 m, which was more evenly arranged in the monitoring area. The Dini03 precision electronic level produced by the Trimble company (United States) was used as the precision level, and the standard deviation of the round-trip measurement per kilometer was ±0.3 mm. The maximum distance of nearest marker stakes is 1.13 km, and the accuracy of surface deformation measurement accuracy is 0.3 mm. The average surface deformation caused by a 1 MPa formation pore pressure change of a grid is about 4 mm. That means that a 900×900m grid with an average formation pressure change of 0.08 MPa can be measured on a surface marker stake. The marker stakes were built in Block X of the Sazhong development zone in the Daqing oilfield. There are 2499 oil and water wells in this block covering 3 series of development well groups. The block area is 19.28 km2, and the original formation pressure of the block was 10.13 MPa. The top depth of the oil layer is about 800 m, and the bottom depth is about 1250 m (the total thickness of the interlayer between reservoirs is about 150 m). The total effective thickness of the oil layer is about 150 m. Figure 5 shows the location of the surface-deformation marker stakes.

From November 2017 to April 2018, vertical distance measurement of the marker stakes and well test analysis of the average formation pressure by a build-up test were conducted for Block X (Figure 5). Vertical deformation, measured twice, was surface vertical deformation at the marker-stake locations. It takes about 3 hours to measure the vertical deformation of the ground. In order to ensure the accuracy of the measurement, both measurements were made on the first sunny morning of the current month. The height survey is a closed loop, and there are four fixed points outside the 27 stakes as the position reference. Only when the accuracy of the closed loop meets the requirements, the whole measurement data can be regarded as effective. The surface vertical deformation field measured by vertical position change at two time points is shown in Figure 6.

From November to December 2017, 35 well pressure build-up tests in this block were conducted, and these formation pressure parameters were used as initial values. From March to April 2018, 42 well pressure build-up tests were carried out in this block, and these formation pressure parameters were different from the previous period as the pressure change values. Due to the fewer monitoring times of formation pressure in the pressure build-up test, grid size was expanded to 900×900m when using formation pressure in the pressure build-up test. The change value of the average pore pressure obtained from the well pressure build-up test in the period from November 2017 to April 2018 is shown in Figure 7.

The formation-pore-pressure distribution obtained by the well test was used to calculate vertical surface deformation by Equation (1) and compared with the vertical deformation of the surface monitoring formation. Results are shown in Figure 8.

Vertical deformation calculated by the forward evolution of formation pore pressure was in good agreement with the measured deformation on the marker-stake points, and the coincidence rate of the vertical deformation trend was 92.6%. In order to measure the accuracy of calculation results, the overall compliance degree was defined as follows:
where pk is the vertical deformation of the surface calculated by the positive evolution at each marker-stake point (mm), and pk is the measured vertical deformation of the surface at each marker-stake point (mm).

According to Equation (3), the overall compliance degree is 82.51%. The main reason for the error was that there were few recovered logging data and few data points to be calculated, and too large a grid size.

3.2. Backward Evolutionary Computation by Improved New CNN Model

To compare with the average pore-pressure variation (Figure 7) from the pressure build-up test, formation-pressure dispersion of a 900×900m block was carried out. There were 20900×900m blocks (5 in x direction and 4 in y direction) in the range of marker-stake measurement in the inversion of formation pore pressure through the vertical deformation of the surface and the formation pore pressure obtained by the pressure build-up test method. The blocks outside the range of the marker-stake measurement were not considered.

The input and output of the improved new CNN are a square matrix with the same length and width. Since the boundary grid is ignored in the training of the CNN model, the input and output matrices of the established CNN model should be larger than the actual grid. According to the actual situation of Block X, a 7×7 input and output layers model was established.

The improved new CNN model used 8 hidden layers, the number of channels of the hidden layers is 2. As the grid sizes of input layer and output layer are the same, the same padding convolutional method was used to calculate the middle results. In order to reduce the complexity of the model, we use a CNN structure with less variables. The convolution kernels in the CNN size are 3×3. In this way, the total number of weight variables is 288 and bias variables is 32, and the total training parameters are only 320. At the same time, the number of intermediate images is 16, so the intermediate variables are only 1264. This number is far less than the training variables in common machine vision models. For this model, the more the hidden layer and the number of images in each layer are, the more accurate the final training result is. However, the calculation accuracy of the model with only 320 training parameters has fully met the requirements. The activation function is not used since there is a physical causal relationship between formation pore pressure and surface vertical deformation. The improved new CNN model we built is shown in Figure 9.

The convolution kernel for forward evolutionary is the surface vertical deformation field of single grid formation pore pressure change unit pressure, and its value was obtained through the curve in Figure 4. In the convolution kernel, only the grids near the center have values, and the values of the 3 grids outside the center are all less than 0.01 MPa/mm. The convolution kernel size for forward evolutionary is 7×7. The test data was obtained by assuming a pressure field and forward calculating the surface vertical deformation. Each point of 7×7 formation pressure field sample randomly takes a pressure value from -1.0 MPa to +1.0 MPa. Since the activation function is not used in this method, it can be used to calculate the pressure difference between any regions in the linear deformation stage.

The surface vertical deformation field was obtained by convolution calculation with Equation (1) from an assumed pressure field (Figure 9). We load 32 samples of training samples at a time for training. In order to reduce the time cost of creating training samples, the training sample batch was changed per 20 trainings. The adaptive moment estimation method was used to minimize the loss value to train the CNN variables. We used the same method to build 100 samples for testing, the testing samples were not used for training, and they were only used to test the accuracy of the predicted pressure field. The learning efficiency was 0.001. The loss function is considered boundary influence coefficient matrix form Equation (2).

We use Python 3.6 with TensorFlow 1.10 to develop the improved new CNN program. The CNN has trained 100,000 times. The formation pressure changes and the calculated surface vertical deformation field of the first of the 100 test samples are shown in Figure 10. The average error of formation pressure change based on test samples with the times of training iterations is shown in Figure 11. After 100,000 times of training, the average error of formation pressure is 0.0198 MPa, and the formation pressure change values of each grid in the figure have nearly no difference with the real values. If the relationship between underground pore pressure change and the surface vertical deformation measurement is completely accurate, the accurate distribution of formation pore pressure can be obtained through the improved new CNN model.

Using the measured surface vertical deformation in November 2017 and April 2018 of the Block X, we calculated the average formation pressure change of each grid within the coverage of the maker stake. The average formation pressure of 20900×900m blocks is shown in Figure 12.

By comparing the average pore-pressure changes of Figures 7 and 12 in the same position, we found that the results were in good agreement with each other. By Equation (3), the overall coincidence degree of pore pressure change was 82.34%. The accuracy of the change of unit pressure change, or convolution kernel, is the most important factor affecting the final accuracy. The lithology distribution and mechanical parameters of actual strata are not the same in plane. The discontinuous interfaces such as faults in the formation also affect the convolution kernel. These effects make the convolution kernel varies with the plane position. However, this effect has not been considered in this study. This is the main reason for the error of the calculation. Furthermore, during this period, surface deformation was larger. The basic coverage of marker-stake measurement parameters was larger, and the measurement error was much smaller than the measurement value. In addition, the average pore-pressure difference used for correlation came from the pressure build-up test. In this case, the grid area was larger, and there were only 20 grids. The amount of data was also small. Nevertheless, such data density is enough to meet the needs of oilfield management.

The inversion method in this paper is a kind of inversion algorithm from two-dimensional matrix to two-dimensional matrix. The training sample generation method, coefficient matrix loss function, and convolution neural network model without pooling layer and full connection layer can provide reference for similar calculation. The method of the inversion of average formation pore pressure by surface deformation is faster, cheaper, and more accurate.

4. Conclusions

  • (1)

    There is a corresponding relationship between formation pore pressure and surface vertical deformation. The range of formation vertical deformation caused by the change of formation pore pressure is within the range of linear elastic deformation of the formation, and the deformation per unit grid pressure difference can be used as a unit convolution kernel. Taking the surface deformation field of unit pressure as convolution kernel, the surface deformation field can be obtained by convolution calculation of underground pore pressure field

  • (2)

    The formation pore pressure of the boundary grid is seriously affected by the boundary effect. The calculation error of the boundary grid in the loss function can be reduced by using the coefficient matrix. The loss function obtained by the coefficient matrix can fully consider the influence of the formation outside the grid

  • (3)

    Through the improved new CNN method, the changes of pore pressure can be inverted by surface vertical deformation. With only 320 training parameters, the new CNN model can get good calculation accuracy through self-training. A field test in the Daqing oilfield in China showed that the variation of formation pore pressure obtained by inversion was 82.34%, in accordance with the result of 20 groups of pressure build-up tests within the range of marker-stake measurement. The calculation method of average formation pore pressure by surface deformation has a high accuracy. The improved new CNN model has a better speed and lower cost than the present method

Data Availability

The data used to support the findings of this study are currently under embargo while the research findings are commercialized by Daqing Oilfield in China. Requests for data, 12 months after publication of this article, will be considered by the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This research was supported by the National Natural Science Foundation of China (Youth Project) (Grant no. 51804076), Heilongjiang Provincial Natural Science Foundation (Youth Project) (Grant no. QC2018047), Youth Science Foundation of Northeast Petroleum University (Grant no. 2018NL-19), and Research Initiation Foundation of Northeast Petroleum University (Grant no. 2019KQ14).

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